\input texinfo

@setfilename comms.info

@settitle Communications Package for Octave

@titlepage
@title  Communications Package for Octave
@subtitle November 2013
@author David Bateman
@author Paul Kienzle
@author Laurent Mazet
@author Mike Miller
@page
@vskip 0pt plus 1filll
Copyright @copyright{} 2003-2013

Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
are preserved on all copies.

Permission is granted to copy and distribute modified versions of this
manual under the conditions for verbatim copying, provided that the entire
resulting derived work is distributed under the terms of a permission
notice identical to this one.

Permission is granted to copy and distribute translations of this manual
into another language, under the same conditions as for modified versions.
@end titlepage

@contents

@ifnottex
@node Top, Introduction
@top
@end ifnottex

@menu
* Introduction::
* Random Signals::
* Source Coding::
* Block Coding::
* Convolutional Coding::
* Modulations::
* Special Filters::
* Galois Fields::
* Function Reference::
@end menu

@node Introduction, Random Signals, Top, Top
@chapter Introduction

This is the manual for the Communications Package for GNU Octave. All
functions provided by this package are described in this manual. In addition
many functions from Octave and other Octave packages are useful to or
required by this package, and so they may also be explained or shown in
examples in this manual.

This documentation is a work in progress, you are invited to help improve it
and submit patches.

@node Random Signals, Source Coding, Introduction, Top
@chapter Random Signals

The purpose of the functions described here is to create and add random
noise to a signal, to create random data and to analyze the eventually
errors in a received signal. The functions to perform these tasks can
be considered as either related to the creation or analysis of signals
and are treated separately below.

It should be noted that the examples below are based on the output of a
random number generator, and so the user can not expect to exactly recreate
the examples below.

@menu
* Signal Creation::
* Signal Analysis::
@end menu

@node Signal Creation, Signal Analysis, , Random Signals
@section Signal Creation

The signal creation functions here fall into to two classes. Those that
treat discrete data and those that treat continuous data. The basic
function to create discrete data is @code{randint}, that creates a
random matrix of equi-probable integers in a desired range. For example

@example
octave:1> a = randint (3, 3, [-1, 1])
a =

   0   1   0
  -1  -1   1
   0   1   1
@end example

@noindent
creates a 3-by-3 matrix of random integers in the range -1 to 1. To allow
for repeated analysis with the same random data, the function @code{randint}
allows the seed-value of the random number generator to be set. For instance

@example
octave:1> a = randint (3, 3, [-1, 1], 1)
a =

   0   1   1
   0  -1   0
   1  -1  -1
@end example

@noindent
will always produce the same set of random data. The range of the integers
to produce can either be a two element vector or an integer. In the case
of a two element vector all elements within the defined range can be produced.
In the case of an integer range @var{M}, @code{randint} returns the equi-probable
integers in the range
@tex
$[0:2^m-1]$.
@end tex
@ifnottex
[0:2^@var{m}-1].
@end ifnottex

The function @code{randsrc} differs from @code{randint} in that it allows
a random set of symbols to be created with a given probability. The symbols
can be real, complex or even characters. However characters and scalars
can not be mixed. For example

@example
octave:1> a = randsrc (2, 2, "ab");
octave:2> b = randsrc (4, 4, [1, 1i, -1, -1i]);
@end example

@noindent
are both legal, while

@example
octave:1> a = randsrc (2, 2, [1, "a"]);
@end example

@noindent
is not legal. The alphabet from which the symbols are chosen can be either
a row vector or two row matrix. In the case of a row vector, all of the
elements of the alphabet are chosen with an equal probability. In the case
of a two row matrix, the values in the second row define the probability
that each of the symbols are chosen. For example

@example
octave:1> a = randsrc (5, 5, [1, 1i, -1, -1i; 0.6 0.2 0.1 0.1])
a =

   1 + 0i   0 + 1i   0 + 1i   0 + 1i   1 + 0i
   1 + 0i   1 + 0i   0 + 1i   0 + 1i   1 + 0i
  -0 - 1i   1 + 0i  -1 + 0i   1 + 0i   0 + 1i
   1 + 0i   1 + 0i   1 + 0i   1 + 0i   1 + 0i
  -1 + 0i  -1 + 0i   1 + 0i   1 + 0i   1 + 0i
@end example

@noindent
defines that the symbol '1' has a 60% probability, the symbol '1i' has
a 20% probability and the remaining symbols have 10% probability each.
The sum of the probabilities must equal one. Like @code{randint},
@code{randsrc} accepts a fourth argument as the seed of the random
number generator allowing the same random set of data to be reproduced.

The function @code{randerr} allows a matrix of random bit errors to be
created, for binary encoded messages. By default, @code{randerr} creates
exactly one errors per row, flagged by a non-zero value in the returned
matrix. That is

@example
octave:1> a = randerr (5, 10)
a =

   0   1   0   0   0   0   0   0   0   0
   0   0   1   0   0   0   0   0   0   0
   0   0   1   0   0   0   0   0   0   0
   0   0   0   0   0   0   0   0   0   1
   0   0   0   0   0   0   0   0   0   1
@end example

The number of errors per row can be specified as the third argument to
@code{randerr}. This argument can be either a scalar, a row vector or
a two row matrix. In the case of a scalar value, exactly this number of
errors will be created per row in the returned matrix. In the case of
a row vector, each element of the row vector gives a possible number of
equi-probable bit errors. The second row of a two row matrix defines
the probability of each number of errors occurring. For example

@example
octave:1> n = 15; k = 11; nsym = 100;
octave:2> msg = randint (nsym, k);       ## Binary vector of message
octave:3> code = encode (msg, n, k, "bch");
octave:4> berrs = randerr (nsym, n, [0, 1; 0.7, 0.3]);
octave:5> noisy = mod (code + berrs, 2)  ## Add errors to coded message
@end example

@noindent
creates a vector @var{msg}, encodes it with a [15,11] BCH code, and then
add either none or one error per symbol with the chances of an error being
30%. As previously, @code{randerr} accepts a fourth argument as the seed of
the random number generator allowing the same random set of data to be
reproduced.

All of the above functions work on discrete random signals. The functions
@code{wgn} and @code{awgn} create and add white Gaussian noise to continuous
signals. The function @code{wgn} creates a matrix of white Gaussian noise
of a certain power. A typical call to @code{wgn} is then

@example
octave:1> nse = wgn (10, 10, 0);
@end example

@noindent
which creates a 10-by-10 matrix of noise with a root mean squared power
of 0dBW relative to an impedance of
@tex
$1\Omega$.
@end tex
@ifnottex
1 Ohm.
@end ifnottex

This effectively means that an equivalent result to the above can be
obtained with

@example
octave:1> nse = randn (10, 10);
@end example

The reference impedance and units of power to the function @code{wgn}
can however be modified, for example

@example
octave:1> nse_30dBm_50Ohm = wgn (10000, 1, 30, 50, "dBm");
octave:2> nse_0dBW_50Ohm = wgn (10000, 1, 0, 50, "dBW");
octave:3> nse_1W_50Ohm = wgn (10000, 1, 1, 50, "linear");
octave:4> [std(nse_30dBm_50Ohm), std(nse_0dBW_50Ohm), std(nse_1W_50Ohm)]
ans =

   7.0805   7.1061   7.0730
@end example

Each of these produces a 1W signal referenced to a
@tex
$50\Omega$
@end tex
@ifnottex
50 Ohm
@end ifnottex
impedance. @sc{matlab} uses the misnomer "dB" for "dBW", so "dB" is an
accepted type for @code{wgn} and is treated as a synonym for "dBW".

In all cases, the returned matrix @var{v}, will be related to the input
power @var{p} and the impedance @var{Z} as

@tex
$$p = {\sum_i \sum_j v(i,j)^2 \over Z} Watts$$
@end tex
@ifnottex
@var{p} = sum (@var{v}(:) .^ 2 ) / @var{imp} Watts
@end ifnottex

By default @code{wgn} produces real vectors of white noise. However, it can
produce both real and complex vectors like

@example
octave:1> rnse = wgn (10000, 1, 0, "dBm", "real");
octave:2> cnse = wgn (10000, 1, 0, "dBm", "complex");
octave:3> [std(rnse), std(real (cnse)), std(imag (cnse)), std(cnse)]
ans =

   0.031615   0.022042   0.022241   0.031313
@end example

@noindent
which shows that with a complex return value that the total power is the
same as a real vector, but that it is equally shared between the real and
imaginary parts. As previously, @code{wgn} accepts a fourth numerical argument
as the seed of the random number generator allowing the same random set of
data to be reproduced. That is

@example
octave:1> nse = wgn (10, 10, 0, 0);
@end example

@noindent
will always produce the same set of data.

The final function to deal with the creation of random signals is
@code{awgn}, that adds noise at a certain level relative to a desired
signal. This function adds noise at a certain level to a desired
signal. An example call to @code{awgn} is

@example
octave:1> x = [0:0.1:2*pi];
octave:2> y = sin (x);
octave:3> noisy = awgn (y, 10, "measured")
@end example

@ifnotinfo
@noindent
which produces a sine-wave with noise added as seen in Figure 1.

@center @image{awgn}

@center Figure 1: Sine-wave with 10dB signal-to-noise ratio
@end ifnotinfo

@noindent
which adds noise with a 10dB signal-to-noise ratio to the measured power
in the desired signal. By default @code{awgn} assumes that the desired
signal is at 0dBW, and the noise is added relative to this assumed
power. This behavior can be modified by the third argument to @code{awgn}.
If the third argument is a numerical value, it is assumed to define the
power in the input signal, otherwise if the third argument is the string
"measured", as above, the power in the signal is measured prior to the
addition of the noise.

The final argument to @code{awgn} defines the definition of the power and
signal-to-noise ratio in a similar manner to @code{wgn}. This final
argument can be either "dB" or "linear". In the first case the numerical
value of the input power is assumed to be in dBW and the signal-to-noise
ratio in dB. In the second case, the power is assumed to be in Watts
and the signal-to-noise ratio is expressed as a ratio.

The return value of @code{awgn} will be in the same form as the input
signal. In addition if the input signal is real, the additive noise will
be real. Otherwise the additive noise will also be complex and the noise
will be equally split between the real and imaginary parts.

As previously the seed to the random number generator can be specified
as the last argument to @code{awgn} to allow repetition of the same
scenario. That is

@example
octave:1> x = [0:0.1:2*pi];
octave:2> y = sin (x);
octave:3> noisy = awgn (y, 10, "dB", 0, "measured")
@end example

@noindent
which uses the seed-value of 0 for the random number generator.

@node Signal Analysis, , Signal Creation, Random Signals
@section Signal Analysis

It is important to be able to evaluate the performance of a
communications system in terms of its bit-error and symbol-error
rates. Two functions @code{biterr} and @code{symerr} exist within this
package to calculate these values, both taking as arguments the
expected and the actually received data. The data takes the form
of matrices or vectors, with each element representing a single
symbol. They are compared in the following manner

@table @asis
@item Both matrices
In this case both matrices must be the same size and then by default the
return values are the overall number of errors and the overall error rate.
@item One column vector
In this case the column vector is used for comparison column-wise
with the matrix. The return values are row vectors containing the number
of errors and the error rate for each column-wise comparison. The number
of rows in the matrix must be the same as the length of the column vector.
@item One row vector
In this case the row vector is used for comparison row-wise
with the matrix. The return values are column vectors containing the number
of errors and the error rate for each row-wise comparison. The number
of columns in the matrix must be the same as the length of the row vector.
@end table

For the bit-error comparison, the size of the symbol is assumed to be the
minimum number of bits needed to represent the largest element in the
two matrices supplied. However, the number of bits per symbol can (and
in the case of random data should) be specified. As an example of the
use of @code{biterr} and @code{symerr}, consider the example

@example
octave:1> m = 8;
octave:2> msg = randint (10, 10, 2^m);
octave:3> noisy = mod (msg + diag (1:10), 2^m);
octave:4> [berr, brate] = biterr (msg, noisy, m)
berr = 32
brate = 0.040000
octave:5> [serr, srate] = symerr (msg, noisy)
serr = 10
srate = 0.10000
@end example

@noindent
which creates a 10-by-10 matrix adds 10 symbols errors to the data and then
finds the bit and symbol error-rates.

Two other means of displaying the integrity of a signal are the
eye-diagram and the scatterplot. Although the functions
@code{eyediagram} and @code{scatterplot} have different appearance, the
information presented is similar and so are their inputs. The difference
between @code{eyediagram} and @code{scatterplot} is that @code{eyediagram}
segments the data into time intervals and plots the in-phase and
quadrature components of the signal against this time interval. While
@code{scatterplot} uses a parametric plot of quadrature versus in-phase
components.

Both functions can accept real or complex signals in the following
forms.

@table @asis
@item A real vector
In this case the signal is assumed to be real and represented by the vector
@var{x}.
@item A complex vector
In this case the in-phase and quadrature components of the signal are
assumed to be the real and imaginary parts of the signal.
@item A matrix with two columns
In this case the first column represents the in-phase and the second the
quadrature components of a complex signal.
@end table

An example of the use of the function @code{eyediagram} is

@example
octave:1> n = 50;
octave:2> ovsp = 50;
octave:3> x = 1:n;
octave:4> xi = 1:1/ovsp:n-0.1;
octave:5> y = randsrc (1, n, [1 + 1i, 1 - 1i, -1 - 1i, -1 + 1i]);
octave:6> yi = interp1 (x, y, xi);
octave:7> noisy = awgn (yi, 15, "measured");
octave:8> eyediagram (noisy, ovsp);
@end example

@ifnotinfo
@noindent
which produces a eye-diagram of a noisy signal as seen in Figure 2. Similarly
an example of the use of the function @code{scatterplot} is

@center @image{eyediagram}

@center Figure 2: Eye-diagram of a QPSK like signal with 15dB signal-to-noise ratio
@end ifnotinfo
@ifinfo
Similarly an example of the use of the function @code{scatterplot} is
@end ifinfo

@example
octave:1> n = 200;
octave:2> ovsp = 5;
octave:3> x = 1:n;
octave:4> xi = 1:1/ovsp:n-0.1;
octave:5> y = randsrc (1, n, [1 + 1i, 1 - 1i, -1 - 1i, -1 + 1i]);
octave:6> yi = interp1 (x, y, xi);
octave:7> noisy = awgn (yi, 15, "measured");
octave:8> f = scatterplot (noisy, 1, 0, "b");
octave:9> hold on;
octave:10> scatterplot (noisy, ovsp, 0, "r+", f);
@end example

@ifnotinfo
@noindent
which produces a scatterplot of a noisy signal as seen in Figure 3.

@center @image{scatterplot}

@center Figure 3: Scatterplot of a QPSK like signal with 15dB signal-to-noise ratio
@end ifnotinfo

@node Source Coding, Block Coding, Random Signals, Top
@chapter Source Coding

@menu
* Quantization::
* PCM Coding::
* Arithmetic Coding::
* Dynamic Range Compression::
@end menu

@node Quantization, PCM Coding, , Source Coding
@section Quantization

An important aspect of converting an analog signal to the digital domain
is quantization. This is the process of mapping a continuous signal to a
set of defined values. Octave contains two functions to perform quantization,
@code{lloyds} creates an optimal mapping of the continuous signal to a fixed
number of levels and @code{quantiz} performs the actual quantization.

The set of quantization points to use is represented by a partitioning
table (@var{table}) of the data and the signal levels (@var{codes} to
which they are mapped. The partitioning @var{table} is monotonically
increasing and if x falls within the range given by two points of this
table then it is mapped to the corresponding code as seen in Table 1.

@center Table 1: Table quantization partitioning and coding

@multitable @columnfractions 0.1 0.4 0.4 0.1
@item @tab x < table(1)               @tab codes(1)   @tab
@item @tab table(1) <= x < table(2)   @tab codes(2)   @tab
@item @tab @dots{}                    @tab @dots{}    @tab
@item @tab table(i-1) <= x < table(i) @tab codes(i)   @tab
@item @tab @dots{}                    @tab @dots{}    @tab
@item @tab table(n-1) <= x < table(n) @tab codes(n)   @tab
@item @tab table(n-1) <= x            @tab codes(n+1) @tab
@end multitable

These partition and coding tables can either be created by the user of
using the function @code{lloyds}. For instance the use of a linear
mapping can be seen in the following example.

@example
octave:1> m = 8;
octave:2> n = 1024;
octave:3> table = 2*[0:m-1]/m - 1 + 1/m;
octave:4> codes = 2*[0:m]/m - 1;
octave:5> x = 4*pi*[0:(n-1)]/(n-1);
octave:6> y = cos (x);
octave:7> [i, z] = quantiz (y, table, codes);
@end example

If a training signal is known that well represents the expected signals,
the quantization levels can be optimized using the @code{lloyds} function.
For example the above example can be continued

@example
octave:8> [table2, codes2] = lloyds (y, table, codes);
octave:9> [i, z2] = quantiz (y, table2, codes2);
@end example

@noindent
to use the mapping suggested by the function @code{lloyds}. It should be
noted that the mapping given by @code{lloyds} is highly dependent on the
training signal used. So if this signal does not represent a realistic
signal to be quantized, then the partitioning suggested by @code{lloyds}
will be sub-optimal.

@node PCM Coding, Arithmetic Coding, Quantization, Source Coding
@section PCM Coding

The DPCM function @code{dpcmenco}, @code{dpcmdeco} and @code{dpcmopt}
implement a form of predictive quantization, where the predictability
of the signal is used to further compress it. These functions are
not yet implemented.

@node Arithmetic Coding, Dynamic Range Compression, PCM Coding, Source Coding
@section Arithmetic Coding

The arithmetic coding functions @code{arithenco} and @code{arithdeco} are
not yet implemented.

@node Dynamic Range Compression, , Arithmetic Coding, Source Coding
@section Dynamic Range Compression

The final source coding function is @code{compand} which is used to
compress and expand the dynamic range of a signal. For instance
consider a logarithm quantized by a linear partitioning. Such a
partitioning is very poor for this large dynamic range. @code{compand}
can then be used to compress the signal prior to quantization, with
the signal being expanded afterwards. For example

@example
octave:1> mu = 1.95;
octave:2> x = [0.01:0.01:2];
octave:3> y = log (x);
octave:4> V = max (abs (y));
octave:5> [i, z, d] = quantiz (y, [-4.875:0.25:0.875], [-5:0.25:1]);
octave:6> c = compand (y, minmu, V, "mu/compressor");
octave:7> [i2, c2] = quantiz (c, [-4.875:0.25:0.875], [-5:0.25:1]);
octave:8> z2 = compand (c2, minmu, max (abs (c2)), "mu/expander");
octave:9> d2 = sumsq (y - z2) / length (y);
octave:10> [d, d2]
ans =

   0.0053885   0.0029935
@end example

@noindent
which demonstrates that the use of @code{compand} can significantly
reduce the distortion due to the quantization of signals with a large
dynamic range.

@node Block Coding, Convolutional Coding, Source Coding, Top
@chapter Block Coding

The error-correcting codes available in this package are discussed here.
These codes work with blocks of data, with no relation between one block
and the next. These codes create codewords based on the messages to
transmit that contain redundant information that allow the recovery of
the original message in the presence of errors.

@menu
* Data Formats::
* Binary Block Codes::
* BCH Codes::
* Reed-Solomon Codes::
@end menu

@node Data Formats, Binary Block Codes, , Block Coding
@section Data Formats

All of the codes described in this section are binary and share similar
data formats. The exception is the Reed-Solomon coder which has a
significantly longer codeword length in general and therefore uses a
different representation to efficiently pass data. The user should refer
to the section about the Reed-Solomon codes for the data format used for
Reed-Solomon codes.

In general @var{k} bits of data are considered to represent a single
message symbol. These @var{k} bits are coded into @var{n} bits of
data representing the codeword. The data can therefore be grouped
in one of three manners, to emphasis this grouping into bits, messages
and codewords

@table @asis
@item A binary vector
Each element of the vector is either one or zero. If the data represents
an uncoded message the vector length should be an integer number of @var{k}
in length.
@item A binary matrix
In this case the data is ones and zeros grouped into rows, with each
representing a single message or codeword. The number of columns in the
matrix should be equal to @var{k} in the case of a uncoded message or
@var{n} in the case of a coded message.
@item A non-binary vector
In this case each element of the vector represents a message or codeword
in an integer format. The bits of the message or codeword are represented
by the bits of the vector elements with the least-significant bit
representing the first element in the message or codeword.
@end table

An example demonstrating the relationship between the three data
formats can be seen below.

@example
octave:1> k = 4;
octave:2> bin_vec = randint (k*10, 1);         # Binary vector format
octave:3> bin_mat = reshape (bin_vec, k, 10)'; # Binary matrix format
octave:4> dec_vec = bi2de (bin_mat);           # Decimal vector format
@end example

The functions within this package will return data in the same format
to which it is given. It should be noted that internally the binary
matrix format is used, and thus if the message or codeword length is
large it is preferable to use the binary format to avoid internal
rounding errors.

@node Binary Block Codes, BCH Codes, Data Formats, Block Coding
@section Binary Block Codes

All of the codes presented here can be characterized by their

@table @asis
@item Generator Matrix
A @var{k}-by-@var{n} matrix @var{G} to generate the codewords @var{C} from
the messages @var{T} by the matrix multiplication
@tex
$ {\bf C} = {\bf T} {\bf G}$.
@end tex
@ifnottex
@var{C} = @var{T} * @var{G}.
@end ifnottex
@item Parity Check Matrix
A '@var{n}-@var{k}'-by-@var{n} matrix @var{H} to check the parity of the
received symbols. If
@tex
$ {\bf H} {\bf R} = {\bf S} \ne 0$,
@end tex
@ifnottex
@var{H} * @var{R} = @var{S} != 0,
@end ifnottex
then an error has been detected. @var{S} can be used with the syndrome
table to correct this error
@item Syndrome Table
A 2^@var{k}-by-@var{n} matrix @var{ST} with the relationship of the error
vectors to the non-zero parities of the received symbols. That is, if
the received symbol is represented as
@tex
$ {\bf R} = ( {\bf T} + {\bf E} )\ mod\ 2$,
@end tex
@ifnottex
@var{R} = mod (@var{T} + @var{E}, 2),
@end ifnottex
then the error vector @var{E} is
@tex
${\bf ST}({\bf S})$.
@end tex
@ifnottex
@var{ST}(@var{S}).
@end ifnottex
@end table

It is assumed for most of the functions in this package that the generator
matrix will be in a 'standard' form. That is the generator matrix can be
represented by

@tex
$$
{\bf G} =
\left[\matrix{g_{11} & g_{12}  & \ldots & g_{1k} & 1     & 0 & \ldots & 0 \cr
              g_{21} & g_{22}  &        & g_{2k} & 0     & 1 &        & 0 \cr
              \vdots &         &        & \vdots & \vdots&   &        & \vdots \cr
              g_{k1} & g_{k2}  & \ldots & g_{kk} & 0     & 0 & \ldots & 1}\right]
$$
@end tex
@ifnottex
@example
@group
 g(1,1)  g(1,2)  ...  g(1,k)  1  0  ...  0
 g(2,1)  g(2,2)       g(2,k)  0  1  ...  0
   .                    .     .          .
   .                    .     .          .
   .                    .     .          .
 g(k,1)  g(k,2)  ...  g(k,k)  0  0  ...  1
@end group
@end example
@end ifnottex

@noindent
or

@tex
$$
{\bf G} =
\left[\matrix{1     & 0 & \ldots & 0 & g_{11} & g_{12}  & \ldots & g_{1k} \cr
              0     & 1 &        & 0 & g_{21} & g_{22}  &        & g_{2k} \cr
             \vdots &   &   & \vdots & \vdots &         &        & \vdots \cr
              0     & 0 & \ldots & 1 & g_{k1} & g_{k2}  & \ldots & g_{kk}}\right]
$$
@end tex
@ifnottex
@example
@group
  1  0  ...  0  g(1,1)  g(1,2)  ...  g(1,k)
  0  1  ...  0  g(2,1)  g(2,2)       g(2,k)
  .          .     .                   .
  .          .     .                   .
  .          .     .                   .
  0  0  ...  1  g(k,1)  g(k,2)  ...  g(k,k)
@end group
@end example
@end ifnottex

@noindent
and similarly the parity check matrix can be represented by a combination
of an identity matrix and a square matrix.

Some of the codes can also have their representation in terms of a
generator polynomial that can be used to create the generator and parity
check matrices. In the case of BCH codes, this generator polynomial is
used directly in the encoding and decoding without ever explicitly forming
the generator or parity check matrix.

The user can create their own generator and parity check matrices, or
they can rely on the functions @code{hammgen}, @code{cyclgen} and
@code{cyclpoly}. The function @code{hammgen} creates parity check and
generator matrices for Hamming codes, while @code{cyclpoly} and
@code{cyclgen} create generator polynomials and matrices for generic
cyclic codes. An example of their use is

@example
octave:1> m = 3;
octave:2> n = 2^m - 1;
octave:2> k = 4;
octave:3> [par, gen] = hammgen (m);
octave:4> [par2, gen2] = cyclgen (n, cyclpoly (n, k));
@end example

@noindent
which create identical parity check and generator matrices for the
[7,4] Hamming code.

The syndrome table of the codes can be created with the function
@code{syndtable}, in the following manner

@example
octave:1> [par, gen] = hammgen (3);
octave:2> st = syndtable (par);
@end example

There exists two auxiliary functions @code{gen2par} and @code{gfweight},
that convert between generator and parity check matrices and calculate
the Hamming distance of the codes. For instance

@example
octave:1> par = hammgen (3);
octave:2> gen = gen2par (par);
octave:3> gfweight (gen)
ans = 3
@end example

It should be noted that for large values of @var{n}, the generator,
parity check and syndrome table matrices are very large. There is
therefore an internal limitation on the size of the block codes that
can be created that limits the codeword length @var{n} to less than 64.
This is still excessively large for the syndrome table, so use caution
with these codes. These limitations do not apply to the Reed-Solomon
or BCH codes.

The top-level encode and decode functions are @code{encode} and
@code{decode}, which can be used with all codes, except the Reed-Solomon
code. The basic call to both of these functions passes the message
to code/decode, the codeword length, the message length and the type
of coding to use. There are four basic types that are available with
these functions

@table @asis
@item "linear"
Generic linear block codes
@item "cyclic"
Cyclic linear block codes
@item "hamming"
Hamming codes
@item "bch"
Bose Chaudhuri Hocquenghem (BCH) block codes
@end table

It is not possible to distinguish between a binary vector and a decimal
vector coding of the messages that just happens to only have ones and
zeros. Therefore the functions @code{encode} and @code{decode} must be
told the format of the messages in the following manner.

@example
octave:1> m = 3;
octave:2> n = 7;
octave:3> k = 4;
octave:4> msg_bin = randint (10, k);
octave:5> cbin = encode (msg_bin, n, k, "hamming/binary");
octave:5> cdec = encode (bi2de (msg), n, k, "hamming/decimal");
@end example

@noindent
which codes a binary matrix and a non-binary vector representation of a
message, returning the coded message in the same format. The functions
@code{encode} and @code{decode} by default accept binary coded
messages. Therefore "hamming" is equivalent to "hamming/binary".

Except for the BCH codes, the function @code{encode} and @code{decode}
internally create the generator, parity check and syndrome table
matrices. Therefore if repeated calls to @code{encode} and @code{decode}
are made, it will often be faster to create these matrices externally
and pass them as an argument. For example

@example
n = 15;
k = 11;
[par, gen] = hammgen (4);
code1 = code2 = zeros (100, 15)
for i = 1:100
  msg = get_msg (i);
  code1(i,:) = encode (msg, n, k, "linear", gen);  # This is faster
  code2(i,:) = encode (msg, n, k, "hamming");      # than this !!!
endfor
@end example

In the case of the BCH codes the low-level functions described in the
next section are used directly by the @code{encode} and @code{decode}
functions.


@node BCH Codes, Reed-Solomon Codes, Binary Block Codes, Block Coding
@section BCH Codes

The BCH coder used here is based on code written by Robert Morelos-Zaragoza
(r.morelos-zaragoza@@ieee.org). This code was originally written in C
and has been converted for use as an Octave oct-file.

@iftex
Called without arguments, @code{bchpoly} returns a table of valid BCH
error correcting codes and their error-correction capability
as seen in Table 1.

@center Table 2: Table of valid BCH codes with codeword length less than 511.

@multitable @columnfractions .083 .083 .083 .083 .083 .083 .083 .083 .083 .083 .083 .083
@item  N  @tab  K  @tab  T  @tab  N  @tab  K  @tab  T  @tab  N  @tab  K  @tab  T  @tab  N  @tab  K  @tab  T
@item   7 @tab   4 @tab   1 @tab 127 @tab  36 @tab  15 @tab 255 @tab  45 @tab  43 @tab 511 @tab 268 @tab  29
@item  15 @tab  11 @tab   1 @tab 127 @tab  29 @tab  21 @tab 255 @tab  37 @tab  45 @tab 511 @tab 259 @tab  30
@item  15 @tab   7 @tab   2 @tab 127 @tab  22 @tab  23 @tab 255 @tab  29 @tab  47 @tab 511 @tab 250 @tab  31
@item  15 @tab   5 @tab   3 @tab 127 @tab  15 @tab  27 @tab 255 @tab  21 @tab  55 @tab 511 @tab 241 @tab  36
@item  31 @tab  26 @tab   1 @tab 127 @tab   8 @tab  31 @tab 255 @tab  13 @tab  59 @tab 511 @tab 238 @tab  37
@item  31 @tab  21 @tab   2 @tab 255 @tab 247 @tab   1 @tab 255 @tab   9 @tab  63 @tab 511 @tab 229 @tab  38
@item  31 @tab  16 @tab   3 @tab 255 @tab 239 @tab   2 @tab 511 @tab 502 @tab   1 @tab 511 @tab 220 @tab  39
@item  31 @tab  11 @tab   5 @tab 255 @tab 231 @tab   3 @tab 511 @tab 493 @tab   2 @tab 511 @tab 211 @tab  41
@item  31 @tab   6 @tab   7 @tab 255 @tab 223 @tab   4 @tab 511 @tab 484 @tab   3 @tab 511 @tab 202 @tab  42
@item  63 @tab  57 @tab   1 @tab 255 @tab 215 @tab   5 @tab 511 @tab 475 @tab   4 @tab 511 @tab 193 @tab  43
@item  63 @tab  51 @tab   2 @tab 255 @tab 207 @tab   6 @tab 511 @tab 466 @tab   5 @tab 511 @tab 184 @tab  45
@item  63 @tab  45 @tab   3 @tab 255 @tab 199 @tab   7 @tab 511 @tab 457 @tab   6 @tab 511 @tab 175 @tab  46
@item  63 @tab  39 @tab   4 @tab 255 @tab 191 @tab   8 @tab 511 @tab 448 @tab   7 @tab 511 @tab 166 @tab  47
@item  63 @tab  36 @tab   5 @tab 255 @tab 187 @tab   9 @tab 511 @tab 439 @tab   8 @tab 511 @tab 157 @tab  51
@item  63 @tab  30 @tab   6 @tab 255 @tab 179 @tab  10 @tab 511 @tab 430 @tab   9 @tab 511 @tab 148 @tab  53
@item  63 @tab  24 @tab   7 @tab 255 @tab 171 @tab  11 @tab 511 @tab 421 @tab  10 @tab 511 @tab 139 @tab  54
@item  63 @tab  18 @tab  10 @tab 255 @tab 163 @tab  12 @tab 511 @tab 412 @tab  11 @tab 511 @tab 130 @tab  55
@item  63 @tab  16 @tab  11 @tab 255 @tab 155 @tab  13 @tab 511 @tab 403 @tab  12 @tab 511 @tab 121 @tab  58
@item  63 @tab  10 @tab  13 @tab 255 @tab 147 @tab  14 @tab 511 @tab 394 @tab  13 @tab 511 @tab 112 @tab  59
@item  63 @tab   7 @tab  15 @tab 255 @tab 139 @tab  15 @tab 511 @tab 385 @tab  14 @tab 511 @tab 103 @tab  61
@item 127 @tab 120 @tab   1 @tab 255 @tab 131 @tab  18 @tab 511 @tab 376 @tab  15 @tab 511 @tab  94 @tab  62
@item 127 @tab 113 @tab   2 @tab 255 @tab 123 @tab  19 @tab 511 @tab 367 @tab  17 @tab 511 @tab  85 @tab  63
@item 127 @tab 106 @tab   3 @tab 255 @tab 115 @tab  21 @tab 511 @tab 358 @tab  18 @tab 511 @tab  76 @tab  85
@item 127 @tab  99 @tab   4 @tab 255 @tab 107 @tab  22 @tab 511 @tab 349 @tab  19 @tab 511 @tab  67 @tab  87
@item 127 @tab  92 @tab   5 @tab 255 @tab  99 @tab  23 @tab 511 @tab 340 @tab  20 @tab 511 @tab  58 @tab  91
@item 127 @tab  85 @tab   6 @tab 255 @tab  91 @tab  25 @tab 511 @tab 331 @tab  21 @tab 511 @tab  49 @tab  93
@item 127 @tab  78 @tab   7 @tab 255 @tab  87 @tab  26 @tab 511 @tab 322 @tab  22 @tab 511 @tab  40 @tab  95
@item 127 @tab  71 @tab   9 @tab 255 @tab  79 @tab  27 @tab 511 @tab 313 @tab  23 @tab 511 @tab  31 @tab 109
@item 127 @tab  64 @tab  10 @tab 255 @tab  71 @tab  29 @tab 511 @tab 304 @tab  25 @tab 511 @tab  28 @tab 111
@item 127 @tab  57 @tab  11 @tab 255 @tab  63 @tab  30 @tab 511 @tab 295 @tab  26 @tab 511 @tab  19 @tab 119
@item 127 @tab  50 @tab  13 @tab 255 @tab  55 @tab  31 @tab 511 @tab 286 @tab  27 @tab 511 @tab  10 @tab 127
@item 127 @tab  43 @tab  14 @tab 255 @tab  47 @tab  42 @tab 511 @tab 277 @tab  28 @tab     @tab     @tab
@end multitable

@end iftex
@ifnottex
Called without arguments, @code{bchpoly} returns a table of valid BCH
error correcting codes and their error-correction capability.
@end ifnottex
The first returned column of @code{bchpoly} is the codeword length,
the second the message length and the third the error correction capability
of the code. Called with one argument, @code{bchpoly} returns similar
output, but only for the specified codeword length. In this manner codes
with codeword length greater than 511 can be found.

In general the codeword length is of the form @code{2^@var{m} - 1}, where
@var{m} is an integer. However if [@var{n},@var{k}] is a valid BCH
code, then it is also possible to use a shortened BCH form of the form
@code{[@var{n}-@var{x},@var{k}-@var{x}]}.

With two or more arguments, @code{bchpoly} is used to find the generator
polynomial of a valid BCH code. For instance

@example
octave:1> bchpoly (15, 7)
ans =

   1   0   0   0   1   0   1   1   1

octave:2> bchpoly (14, 6)
ans =

   1   0   0   0   1   0   1   1   1
@end example

@noindent
show that the generator polynomial of a [15,7] BCH code with the default
primitive polynomial is

@tex
$$ 1 + x^4 + x^6 + x^7 + x^8 $$
@end tex
@ifnottex
1 + @var{x} ^ 4 + @var{x} ^ 6 + @var{x} ^ 7 + @var{x} ^ 8
@end ifnottex

Using a different primitive polynomial to define the Galois Field over
which the BCH code is defined results in a different generator polynomial
as can be seen in the example.

@example
octave:1> bchpoly ([1 1 0 0 1], 7)
ans =

   1   0   0   0   1   0   1   1   1

octave:2> bchpoly ([1 0 0 1 1], 7)
ans =

   1   1   1   0   1   0   0   0   1
@end example

It is recommend not to convert the generator polynomials created by
@code{bchpoly} into generator and parity check matrices with the
BCH codes, as the underlying BCH software is faster than the generic
coding software and can treat significantly longer codes.

As well as using the @code{encode} and @code{decode} functions previously
discussed, the user can directly use the low-level BCH functions
@code{bchenco} and @code{bchdeco}. In this case the messages must be
in the format of a binary matrix with @var{k} columns

@example
octave:1> n = 31;
octave:2> pgs = bchpoly (n);
octave:3> pg = pgs(floor (rand () * (rows (pgs) + 1)),:);  # Pick a poly
octave:4> k = pg(2);
octave:5> t = pg(3);
octave:6> msg = randint (10, k);
octave:7> code = bchenco (msg, n, k);
octave:8> noisy = code + [ones(10, 1), zeros(10, n-1)];
octave:9> dec = bchdeco (code, k, t);
@end example

@node Reed-Solomon Codes, , BCH Codes, Block Coding
@section Reed-Solomon Codes

@menu
* Representation of Reed-Solomon Messages::
* Creating and Decoding Messages::
* Shortened Reed-Solomon Codes::
@end menu

@node Representation of Reed-Solomon Messages, Creating and Decoding Messages, , Reed-Solomon Codes
@subsection Representation of Reed-Solomon Messages

The Reed-Solomon coder used in this package is based on code written by
Phil Karn (http://www.ka9q.net/code/fec). This code was originally written
in C and has been converted for use as an Octave oct-file.

Reed-Solomon codes are based on Galois Fields of even characteristics
GF(2^M). Many of the properties of Galois Fields are therefore important
when considering Reed-Solomon coders.

The representation of the symbols of the Reed-Solomon code differs from
the other block codes, in that the other block codes use a binary
representation, while the Reed-Solomon code represents each m-bit symbol
by an integer. The elements of the message and codeword must be elements
of the Galois Field corresponding to the Reed-Solomon code. Thus to
code a message with a [7,5] Reed-Solomon code an example is

@example
octave:1> m = 3;
octave:2> n = 7;
octave:3> k = 5;
octave:4> msg = gf (floor (2^m * rand (2, k)), m)
msg =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   5   0   6   3   2
   4   1   3   1   2

octave:5> code = rsenc (msg, n, k)
code =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   5   0   6   3   2   3   5
   4   1   3   1   2   6   3
@end example

The variable @var{n} is the codeword length of the Reed-Solomon coder,
while @var{k} is the message length. It should be noted that @var{k}
should be less than @var{n} and that @code{@var{n} - @var{k}} should be
even. The error correcting capability of the Reed-Solomon code is then
@code{(@var{n} - @var{k})/2} symbols. @var{m} is the number of bits per
symbol, and is related to @var{n} by @code{@var{n} = 2^@var{m} - 1}.
For a valid Reed-Solomon coder, @var{m} should be between 3 and 16.

@node Creating and Decoding Messages, Shortened Reed-Solomon Codes, Representation of Reed-Solomon Messages, Reed-Solomon Codes
@subsection Creating and Decoding Messages

The Reed-Solomon encoding function requires at least three arguments. The
first @var{msg} is the message in encodes, the second is @var{n} the codeword
length and @var{k} is the message length. Therefore @var{msg} must have
@var{k} columns and the output will have @var{n} columns of symbols.

The message itself is many up of elements of a Galois Field
GF(2^M). Normally, The order of the Galois Field (M), is related to the
codeword length by @code{@var{n} = 2^@var{m} - 1}. Another important
parameter when determining the behavior of the Reed-Solomon coder is
the primitive polynomial of the Galois Field (see @code{gf}). Thus
the messages

@example
octave:1> msg0 = gf ([0, 1, 2, 3], 3);
octave:2> msg1 = gf ([0, 1, 2, 3], 3, 13);
@end example

@noindent
will not result in the same Reed-Solomon coding. Finally, the parity of
the Reed-Solomon code are generated with the use of a generator
polynomial. The parity symbols are then generated by treating the message
to encode as a polynomial and finding the remainder of the division of
this polynomial by the generator polynomial. Therefore the generator
polynomial must have as many roots as @code{@var{n} - @var{k}}. Whether
the parity symbols are placed before or afterwards the message will then
determine which end of the message is the most-significant term of the
polynomial representing the message. The parity symbols are therefore
different in these two cases. The position of the parity symbols can be
chosen by specifying "beginning" or "end" to @code{rsenc} and @code{rsdec}.
By default the parity symbols are placed after the message.

Valid generator polynomials can be constructed with the @code{rsgenpoly}
function. The roots of the generator polynomial are then defined by

@tex
$$
g = (x - A^{bs}) (x - A^{(b+1)s})  \cdots (x - A ^{(b+2t-1)s}).
$$
@end tex
@ifnottex

@example
@var{g} = (@var{x} - A^(@var{b}*@var{s})) * (@var{x} - A^((@var{b}+1)*@var{s})) * @dots{} * (@var{x} - A^((@var{b}+2*@var{t}-1)*@var{s})).
@end example
@end ifnottex

@noindent
where @var{t} is @code{(@var{n} - @var{k})/2}, A is the primitive element
of the Galois Field, @var{b} is the first consecutive root, and @var{s}
is the step between roots. Generator polynomial of this form are constructed
by @code{rsgenpoly} and can be passed to both @code{rsenc} and @code{rsdec}.
It is also possible to pass the @var{b} and @var{s} values directly to
@code{rsenc} and @code{rsdec}. In the case of @code{rsdec} passing @var{b}
and @var{s} can make the decoding faster.

Consider the example below.

@example
octave:1> m = 8;
octave:2> n = 2^m - 1;
octave:3> k = 223;
octave:4> prim = 391;
octave:5> b = 112;
octave:6> s = 11;
octave:7> gg = rsgenpoly (n, k, prim, b, s);
octave:8> msg = gf (floor (2^m * rand (17, k)), m, prim);
octave:9> code = rsenc (msg, n, k, gg);
octave:10> noisy = code + [toeplitz([ones(1,17)], zeros(1,17)), zeros(17,238)];
octave:11> [dec, nerr] = rsdec (msg, n, k, b, s);
octave:12> nerr'
ans =

    1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   -1

octave:13> any (msg' != dec')
ans =

   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   1
@end example

This is an interesting example in that it demonstrates many of the
additional arguments of the Reed-Solomon functions. In particular
this example approximates the CCSDS standard Reed-Solomon coder,
lacking only the dual-basis lookup tables used in this standard.
The CCSDS uses non-default values to all of the basic functions
involved in the Reed-Solomon encoding, since it has a non-default
primitive polynomial, generator polynomial, etc.

The example creates 17 message blocks and adds between 1 and 17 error
symbols to these block. As can be seen @var{nerr} gives the number of
errors corrected. In the case of 17 introduced errors @var{nerr}
equals -1, indicating a decoding failure. This is normal as the
correction ability of this code is up to 16 error symbols. Comparing
the input message and the decoding it can be seen that as expected,
only the case of 17 errors has not been correctly decoded.

@node Shortened Reed-Solomon Codes, , Creating and Decoding Messages, Reed-Solomon Codes
@subsection Shortened Reed-Solomon Codes

In general the codeword length of the Reed-Solomon coder is chosen so
that it is related directly to the order of the Galois Field by the
formula @code{@var{n} = 2^@var{m} - 1}. Although, the underlying
Reed-Solomon coding must operate over valid codeword length, there
are sometimes reasons to assume that the codeword length will be shorter.
In this case the message is padded with zeros before coding, and the
zeros are stripped from the returned block. For example consider the
shortened [6,4] Reed-Solomon below

@example
octave:1> m = 3;
octave:2> n = 6;
octave:3> k = 4;
octave:4> msg = gf (floor (2^m * rand (2, k)), m)
msg =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   7   0   2   5
   1   5   7   1

octave:5> code = rsenc (msg, n, k)
code =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   7   0   2   5   2   3
   1   5   7   1   0   2
@end example


@node Convolutional Coding, Modulations, Block Coding, Top
@chapter Convolutional Coding

Some initial support for convolutional codes is provided by the
functions described in this chapter. Convolutional codes are different
from block codes in that the sequence of preceding symbols is taken
into account when computing the output symbol of the coder.

@menu
* Trellis Structure::
* Convolutional Encoding::
@end menu

@node Trellis Structure, Convolutional Encoding, , Convolutional Coding
@section Trellis Structure

Like block codes, convolutional codes can be described by a set of
generator polynomials. Each polynomial describes the combination of
current and previous input symbols used to compute one output bit of
the encoder.

The state transitions and outputs of a convolutional encoder can also be
described by a trellis diagram. This diagram describes the transitions
between states and the outputs of the encoder as a function of the
current state and the current input symbol. A trellis structure can be
created from a set of generator polynomials, specified as octal numbers
by convention,

@example
octave:1> g0 = 13;
octave:2> g1 = 17;
octave:3> trellis = poly2trellis (4, [g0, g1]);
@end example

@noindent
where @var{g0} and @var{g1} are the two polynomials of a rate 1/2
encoder with a constraint length of 4. The returned trellis structure
contains the following fields

@table @samp
@item numInputSymbols
The number of possible input symbols in the input sequence.
@item numOutputSymbols
The number of possible output symbols in the encoded sequence.
@item numStates
The number of possible states that the encoder can take.
@item nextStates
The state transition table for the encoder. Each row contains the
(zero-based) indices of the states reachable from the state represented
by that row for each possible input symbol.
@item outputs
The output table for the encoder. Each row contains the (octal-encoded)
output symbols produced by the encoder in the state represented by that
row for each possible input symbol.
@end table

To check if a variable references a structure that is a valid trellis
describing a convolutional encoder, use the @code{istrellis} function.

@node Convolutional Encoding, , Trellis Structure, Convolutional Coding
@section Convolutional Encoding

The convolutional encoding function takes the message to be encoded and
a trellis describing the encoder. The message must be a binary vector
containing an even number of symbols. For example, using the encoder
from the previous section,

@example
octave:1> trellis = poly2trellis (4, [13, 17]);
octave:2> msg = [1 1 0 1 1 0 0 0];
octave:3> out = convenc (msg, trellis)
out =

   1   1   1   0   1   0   1   1   0   1   1   0   0   0   1   1
@end example

The initial state of the encoder can also be passed in to @code{convenc},
and the ending state can be read with an optional output argument.
Encoding a different vector with a different initial state using the
same encoder,

@example
octave:4> msg = [0 1 1 0 1 0 1 1];
octave:5> [out, state] = convenc (msg, trellis, [], 4)
out =

   0   1   0   0   0   1   1   0   1   1   1   0   0   0   0   1

state =  6
@end example

@noindent
returns both the encoded array and the final state of the convolutional
encoder. This can be used to encode data in blocks, for example, saving
and restoring the internal state of the encoder for each subsequent
input block.

@node Modulations, Special Filters, Convolutional Coding, Top
@chapter Modulations

To be written.

Currently have functions amodce, ademodce, apkconst, demodmap, modmap,
qaskdeco, qaskenco, genqammod, pamdemod, pammod, pskdemod and pskmod.

@node Special Filters, Galois Fields, Modulations, Top
@chapter Special Filters

To be written.

@node Galois Fields, Function Reference, Special Filters, Top
@chapter Galois Fields

@menu
* Galois Field Basics::
* Manipulating Galois Fields::
@end menu

@node Galois Field Basics, Manipulating Galois Fields, , Galois Fields
@section Galois Field Basics

A Galois Field is a finite algebraic field. This package implements a
Galois Field type in Octave having 2^M members where M is an integer
between 1 and 16. Such fields are denoted as GF(2^M) and are used in
error correcting codes in communications systems. Galois Fields having
odd numbers of elements are not implemented.

The @emph{primitive element} of a Galois Field has the property that all
elements of the Galois Field can be represented as a power of this element.
The @emph{primitive polynomial} is the minimum polynomial of some primitive
element in GF(2^M) and is irreducible and of order M. This means that the
primitive element is a root of the primitive polynomial.

The elements of the Galois Field GF(2^M) are represented as the values
0 to 2^M -1 by Octave. The first two elements represent the zero and unity
values of the Galois Field and are unique in all fields. The element
represented by 2 is the primitive element of the field and all elements can
be represented as combinations of the primitive element and unity as follows

@multitable @columnfractions .33 .33 .33
@item Integer    @tab Binary     @tab Element of GF(2^M)
@item  0         @tab  000       @tab @code{0}
@item  1         @tab  001       @tab @code{1}
@item  2         @tab  010       @tab @code{A}
@item  3         @tab  011       @tab @code{A + 1}
@item  4         @tab  100       @tab @code{A^2}
@item  5         @tab  101       @tab @code{A^2 + 1}
@item  6         @tab  110       @tab @code{A^2 + A}
@item  7         @tab  111       @tab @code{A^2 + A + 1}
@end multitable

It should be noted that there is often more than a single primitive
polynomial of GF(2^M). Each Galois Field over a different primitive
polynomial represents a different realization of the Field. The
representations above however rest valid.

@menu
* Creating Galois Fields::
* Primitive Polynomials::
* Accessing Internal Fields::
* Function Overloading::
* Known Problems::
@end menu

@node Creating Galois Fields, Primitive Polynomials, , Galois Field Basics
@subsection Creating Galois Fields

To work with a Galois Field GF(2^M) in Octave, you must first create a variable
that Octave recognizes as a Galois Field. This is done with the function
@code{gf (@var{a}, @var{m})} as follows.

@example
octave:1> a = [0:7];
octave:2> b = gf (a, 4)
b =
GF(2^4) array. Primitive Polynomial = D^4+D+1 (decimal 19)

Array elements =

   0   1   2   3   4   5   6   7
@end example

This creates an array @var{b} with 8 elements that Octave recognizes as a
Galois Field. The field is created with the default primitive polynomial for
the field GF(2^4). It can be verified that a variable is in fact a Galois
Field with the functions @code{isgalois} or @code{whos}.

@example
octave:3> isgalois (a)
ans = 0
octave:4> isgalois (b)
ans = 1
octave:5> whos
Variables in the current scope:

   Attr Name        Size                     Bytes  Class
   ==== ====        ====                     =====  ===== 
        a           1x8                         24  double
        b           1x8                         32  galois

Total is 16 elements using 56 bytes
@end example

It is also possible to create a Galois Field with an arbitrary primitive
polynomial. However, if the polynomial is not a primitive polynomial of
the field, and error message is returned. For instance.

@example
octave:1> a = [0:7];
octave:2> b = gf (a, 4, 25)
b =
GF(2^4) array. Primitive Polynomial = D^4+D^3+1 (decimal 25)

Array elements =

   0   1   2   3   4   5   6   7

octave:3> c = gf (a, 4, 21)
error: gf: primitive polynomial (21) of Galois Field must be irreducible
@end example

The function @code{gftable} is included for compatibility with @sc{matlab}. In
@sc{matlab} this function is used to create the lookup tables used to accelerate
the computations over the Galois Field and store them to a file. However
Octave stores these parameters for all of the fields currently in use and
so this function is not required, although it is silently accepted.

@node Primitive Polynomials, Accessing Internal Fields, Creating Galois Fields, Galois Field Basics
@subsection Primitive Polynomials

The function @code{gf (@var{a}, @var{m})} creates a Galois Field using the default primitive
polynomial. However there exists many possible primitive polynomials for most
Galois Fields. Two functions exist for identifying primitive polynomials,
@code{isprimitive} and @code{primpoly}. @code{primpoly (@var{m}, @var{opt})} is
used to identify the primitive polynomials of the fields GF(2^M). For example

@example
octave:1> primpoly (4)

Primitive polynomial(s) =

D^4+D+1

ans = 19
@end example

@noindent
identifies the default primitive polynomials of the field GF(2^M), which
is the same as @code{primpoly (4, "min")}. All of the primitive polynomials
of a field can be identified with the function @code{primpoly (@var{m}, "all")}.
For example

@example
octave:1> primpoly (4, "all")

Primitive polynomial(s) =

D^4+D+1
D^4+D^3+1

ans =

   19   25
@end example

@noindent
while @code{primpoly (@var{m}, "max")} returns the maximum primitive polynomial
of the field, which for the case above is 25. The function @code{primpoly}
can also be used to identify the primitive polynomials having only a
certain number of non-zero terms. For instance

@example
octave:1> primpoly (5, 3)

Primitive polynomial(s) =

D^5+D^2+1
D^5+D^3+1

ans =

   37   41
@end example

@noindent
identifies the polynomials with only three terms that can be used as
primitive polynomials of GF(2^5). If no primitive polynomials existing
having the requested number of terms then @code{primpoly} returns an
empty vector. That is

@example
octave:1> primpoly (5, 2)
warning: primpoly: No primitive polynomial satisfies the given constraints

ans = [](1x0)
@end example

As can be seen above, @code{primpoly} displays the polynomial forms the
the polynomials that it finds. This output can be suppressed with the
"nodisplay" option, while the returned value is left unchanged.

@example
octave:1> primpoly (4, "all", "nodisplay")
ans =

   19   25
@end example

@code{isprimitive (@var{a})} identifies whether the elements of @var{a} can
be used as primitive polynomials of the Galois Fields GF(2^M). Consider
as an example the fields GF(2^4). The primitive polynomials of these fields
must have an order m and so their integer representation must be between
16 and 31. Therefore @code{isprimitive} can be used in a similar manner to
@code{primpoly} as follows

@example
octave:1> find (isprimitive (16:31)) + 15
ans =

   19   25
@end example

@noindent
which finds all of the primitive polynomials of GF(2^4).

@node Accessing Internal Fields, Function Overloading, Primitive Polynomials, Galois Field Basics
@subsection Accessing Internal Fields

Once a variable has been defined as a Galois Field, the parameters of the
field of this structure can be obtained by adding a suffix to the variable.
Valid suffixes are '.m', '.prim_poly' and '.x', which return the order of the
Galois Field, its primitive polynomial and the data the variable contains
respectively. For instance

@example
octave:1> a = [0:7];
octave:2> b = gf (a, 4);
octave:3> b.m
ans = 4
octave:4> b.prim_poly
ans = 19
octave:5> c = b.x;
octave:6> whos
Variables in the current scope:

   Attr Name        Size                     Bytes  Class
   ==== ====        ====                     =====  ===== 
        a           1x8                         24  double
        b           1x8                         32  galois
        c           1x8                         64  double

Total is 24 elements using 120 bytes
@end example

@c Note that if code compiled with GALOIS_DISP_PRIVATES then '.n', '.alpha_to'
@c and '.index_of' are also available. These give 2^m-1, the lookup table
@c and its inverse respectively.

Please note that it is explicitly forbidden to modify the Galois field by
accessing these variables. For instance

@example
octave:1> a = gf ([0:7], 3);
octave:2> a.prim_poly = 13;
@end example

@noindent
is explicitly forbidden. The result of this will be to replace the
Galois array @var{a} with a structure @var{a} with a single element
called '.prim_poly'. To modify the order or primitive polynomial of a
field, a new field must be created and the data copied. That is

@example
octave:1> a = gf ([0:7], 3);
octave:2> a = gf (a.x, a.m, 13);
@end example

@node Function Overloading, Known Problems, Accessing Internal Fields, Galois Field Basics
@subsection Function Overloading

An important consideration in the use of the Galois Field package is
that many of the internal functions of Octave, such as @code{roots}, can
not accept Galois Fields as an input. This package therefore uses Octave
classes to @emph{overload} the internal Octave functions with equivalent
functions that work with Galois Fields, so that the standard function names
can be used.

The version of the function that is chosen is determined by the first
argument of the function. This is a temporary situation until the
Galois Field class constructor can be rewritten to allow the use of the
@code{superiorto} function to define the galois class with a higher
precedence. So, considering the @code{filter} function,
if the first argument is a @emph{Matrix}, then the normal version of
the function is called regardless of whether the other arguments of the
function are Galois vectors or not.

Other Octave functions work correctly with Galois Fields and so overloaded
versions are not necessary. This include such functions as @code{size} and
@code{polyval}.

It is also useful to use the '.x' option discussed in the previous section,
to extract the raw data of the Galois field for use with some functions. An
example is

@example
octave:1> a = minpol (gf (14, 5));
octave:2> b = de2bi (a.x, [], "left-msb");
@end example

@noindent
converts the polynomial form of the minimum polynomial of 14 in GF(2^5) into
an integer. Finally help for the Galois specific versions of the functions
must explicitly call the correct function as

@example
octave:1> help @@galois/conv
@end example

@node Known Problems, , Function Overloading, Galois Field Basics
@subsection Known Problems

Please review the following list of known problems with the Galois type
before reporting a bug against this package.

@table @asis
@item Saving and loading Galois variables

Saving a Galois variable to a file is as simple as

@example
octave:1> a = gf (@dots{});
octave:2> save a.mat a
@end example

@noindent
where @var{a} is any Galois variable. Galois variables can be saved in the
Octave binary and ASCII formats, as well as the HDF5 format. To load a
Galois variable from a file, the Galois type must already be registered to
the Octave interpreter prior to the call to @code{load}. If no Galois
variables have been created yet, you will have to do something like

@example
octave:1> dummy = gf (1);
octave:2> load a.mat
@end example

@item Logarithm of zero does not return NaN
The logarithm of zero in a Galois field is not defined. However, to avoid
segmentation faults in later calculations the logarithm of zero is defined
as @code{2^@var{m} - 1}, whose value is not the logarithm of any other value
in the Galois field. A warning is also shown to tell the user about the
problem. For example

@example
octave:1> m = 3;
octave:2> a = log (gf ([0:2^m-1], m))
warning: log of zero undefined in Galois field
a =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   7   0   1   3   2   6   4   5
@end example

To fix this problem would require a major rewrite of all code, adding
an exception for the case of NaN to all basic operators. These
exceptions will certainly slow the code down.

@item Speed
The code was written piecemeal with no attention to optimization. Some
operations may be slower than they could be. Contributions are welcome.

@end table

@node Manipulating Galois Fields, , Galois Field Basics, Galois Fields
@section Manipulating Galois Fields

@menu
* Expressions manipulation and assignment::
* Unary operations::
* Arithmetic operations::
* Comparison operations::
* Polynomial manipulations::
* Linear Algebra::
* Signal Processing::
@end menu

@node Expressions manipulation and assignment, Unary operations, , Manipulating Galois Fields
@subsection Expressions, manipulation and assignment

Galois variables can be treated in similar manner to other variables within
Octave. For instance Galois fields can be accessed using index expressions
in a similar manner to all other Octave matrices. For example

@example
octave:1> a = gf ([[0:7]; [7:-1:0]], 3)
a =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   0   1   2   3   4   5   6   7
   7   6   5   4   3   2   1   0

octave:2> b = a(1,:)
b =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   0   1   2   3   4   5   6   7
@end example

Galois arrays can equally use indexed assignments. That is, the data
in the array can be partially replaced, on the condition that the two
fields are identical. An example is

@example
octave:1> a = gf (ones (2, 8), 3);
octave:2> b = gf (zeros (1, 8), 3);
octave:3> a(1,:) = b
a =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   0   0   0   0   0   0   0   0
   1   1   1   1   1   1   1   1
@end example

Implicit conversions between normal matrices and Galois arrays are possible.
For instance data can be directly copied from a Galois array to a real matrix
as follows.

@example
octave:1> a = gf (ones (2, 8), 3);
octave:2> b = zeros (2, 8);
octave:3> b(2,:) = a(2,:)
b =

   0   0   0   0   0   0   0   0
   1   1   1   1   1   1   1   1
@end example

The inverse is equally possible, with the proviso that the data in the matrix
is valid in the Galois field. For instance

@example
octave:1> a = gf ([0:7], 3);
octave:2> a(1) = 1;
@end example

@noindent
is valid, while

@example
octave:1> a = gf ([0:7], 3);
octave:2> a(1) = 8;
@end example

@noindent
is not, since 8 is not an element of GF(2^3). This is a basic rule of
manipulating Galois arrays. That is matrices and scalars can be used in
conjunction with a Galois array as long as they contain valid data
within the Galois field. In this case they will be assumed to be of the
same field.

Galois arrays can also be concatenated with real matrices or with other
Galois arrays in the same field. For example

@example
octave:1> a = [gf([0:7], 3); gf([7:-1:0], 3)];
octave:2> b = [a, a];
octave:3> c = [a, eye(2)];
octave:3> whos
Variables in the current scope:

   Attr Name        Size                     Bytes  Class
   ==== ====        ====                     =====  ===== 
        a           2x8                         64  galois
        b           2x16                       128  galois
        c           2x10                        80  galois

Total is 68 elements using 272 bytes
@end example

Other basic manipulations of Galois arrays are

@table @code
@item isempty
Returns true if the Galois array is empty.

@item size
Returns the number of rows and columns in the Galois array.

@item length
Returns the length of a Galois vector, or the maximum of rows or columns
of Galois arrays.

@item find
Find the indexes of the non-zero elements of a Galois array.

@item diag
Create a diagonal Galois array from a Galois vector, or extract a diagonal
from a Galois array.

@item reshape
Change the shape of the Galois array.

@end table

@node Unary operations, Arithmetic operations, Expressions manipulation and assignment, Manipulating Galois Fields
@subsection Unary operations

The same unary operators that are available for normal Octave matrices are
also available for Galois arrays. These operations are

@table @code
@item +@var{x}
Unary plus. This operator has no effect on the operand.

@item -@var{x}
Unary minus. Note that in a Galois Field this operator also has no effect
on the operand.

@item !@var{x}
Returns true for zero elements of Galois Array.

@item @var{x}'
Complex conjugate transpose. As the Galois Field only contains integer
values, this is equivalent to the transpose operator.

@item @var{x}.'
Transpose of the Galois array.
@end table

@node Arithmetic operations, Comparison operations, Unary operations, Manipulating Galois Fields
@subsection Arithmetic operations

The available arithmetic operations on Galois arrays are the same as on
other Octave matrices. It should be noted that both operands must be in
the same Galois Field. If one operand is a Galois array and the second is
a matrix or scalar, then the second operand is silently converted to the
same Galois Field. The element(s) of these matrix or scalar must however
be valid members of the Galois field. Thus

@example
octave:1> a = gf ([0:7], 3);
octave:2> b = a + [0:7];
@end example

@noindent
is valid, while

@example
octave:1> a = gf ([0:7], 3);
octave:2> b = a + [1:8];
@end example

@noindent
is not, since 8 is not a valid element of GF(2^3). The available arithmetic
operators are

@table @code
@item @var{x} + @var{y}
Addition. If both operands are Galois arrays or matrices, the number of rows
and columns must both agree. If one operand is a is a Galois array with a
single element or a scalar, its value is added to all the elements of the
other operand. The @code{+} operator on a Galois Field is equivalent to an
exclusive-or on normal integers.

@item @var{x} .+ @var{y}
Element by element addition. This operator is equivalent to @code{+}.

@item @var{x} - @var{y}
As both @code{+} and @code{-} in a Galois Field are equivalent to an
exclusive-or for normal integers, @code{-} is equivalent to the @code{+}
operator

@item @var{x} .- @var{y}
Element by element subtraction. This operator is equivalent to @code{-}.

@item @var{x} * @var{y}
Matrix multiplication. The number of columns of @var{x} must agree
with the number of rows of @var{y}.

@item @var{x} .* @var{y}
Element by element multiplication. If both operands are matrices, the
number of rows and columns must both agree.

@item @var{x} / @var{y}
Right division. This is conceptually equivalent to the expression

@example
(inverse (y') * x')'
@end example

@noindent
but it is computed without forming the inverse of @var{y'}.

If the matrix is singular then an error occurs. If the matrix is
under-determined, then a particular solution is found (but not minimum
norm). If the solution is over-determined, then an attempt is made
to find a solution, but this is not guaranteed to work.

@item @var{x} ./ @var{y}
Element by element right division.

@item @var{x} \ @var{y}
Left division. This is conceptually equivalent to the expression

@example
inverse (x) * y
@end example

@noindent
but it is computed without forming the inverse of @var{x}.

If the matrix is singular then an error occurs. If the matrix is
under-determined, then a particular solution is found (but not minimum
norm). If the solution is over-determined, then an attempt is made
to find a solution, but this is not guaranteed to work.

@item @var{x} .\ @var{y}
Element by element left division. Each element of @var{y} is divided
by each corresponding element of @var{x}.

@item @var{x} ^ @var{y}
@itemx @var{x} ** @var{y}
Power operator. If @var{x} and @var{y} are both scalars, this operator
returns @var{x} raised to the power @var{y}. Otherwise @var{x} must
be a square matrix raised to an integer power.

@item @var{x} .^ @var{y}
@item @var{x} .** @var{y}
Element by element power operator. If both operands are matrices, the
number of rows and columns must both agree.

@end table

@node Comparison operations, Polynomial manipulations, Arithmetic operations, Manipulating Galois Fields
@subsection Comparison operations

Galois variables can be tested for equality in the usual manner. That is

@example
octave:1> a = gf ([0:7], 3);
octave:2> a == ones (1, 8)
ans =

   0   1   0   0   0   0   0   0

octave:3> a != zeros (1, 8)
ans =

   0   1   1   1   1   1   1   1
@end example

Likewise, Galois vectors can be tested against scalar values (whether they are
Galois or not). For instance

@example
octave:4> a == 1
ans =

   0   1   0   0   0   0   0   0
@end example

To test if any or all of the values in a Galois array are non-zero, the
functions @code{any} and @code{all} can be used as normally.

In addition the comparison operators @code{>}, @code{>=}, @code{<} and
@code{<=} are available. As elements of the Galois Field are modulus
2^@var{m}, all elements of the field are both greater than and less than
all others at the same time. Thus these comparison operators don't make
that much sense and are only included for completeness. The comparison is
done relative to the integer value of the Galois Field elements.

@node Polynomial manipulations, Linear Algebra, Comparison operations, Manipulating Galois Fields
@subsection Polynomial manipulations

A polynomial in GF(2^M) can be expressed as a vector in GF(2^M). For instance
if @var{a} is the @emph{primitive element}, then the example

@example
octave:1> poly = gf ([2, 4, 5, 1], 3);
@end example

@noindent
represents the polynomial

@tex
$$
poly = a x^3 + a^2 x^2 + (a^2 + 1) x + 1
$$
@end tex
@ifnottex
@example
poly = @var{a} * x^3 + @var{a}^2 * x^2 + (@var{a}^2 + 1) * x + 1
@end example
@end ifnottex

Arithmetic can then be performed on these vectors. For instance to add
to polynomials an example is

@example
octave:1> poly1 = gf ([2, 4, 5, 1], 3);
octave:2> poly2 = gf ([1, 2], 3);
octave:3> sumpoly = poly1 + [0, 0, poly2]
sumpoly =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   2   4   4   3
@end example

Note that @var{poly2} must be zero padded to the same length as poly1 to
allow the addition to take place.

Multiplication and division of Galois polynomials is equivalent to convolution
and de-convolution of vectors of Galois elements. Thus to multiply two
polynomials in GF(2^3).

@example
octave:4> mulpoly = conv (poly1, poly2)
mulpoly =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   2   0   6   0   2
@end example

Likewise the division of two polynomials uses the de-convolution function
as follows

@example
octave:5> [poly, remd] = deconv (mulpoly, poly2)
poly =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   2   4   5   1

remd =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   0   0   0   0   0
@end example

Note that the remainder of this division is zero, as we performed the inverse
operation to the multiplication.

To evaluate a polynomial for a certain value in GF(2^M), use the Octave
function @code{polyval}.

@example
octave:1> poly1 = gf ([2, 4, 5, 1], 3);  ## a*x^3+a^2*x^2+(a^2+1)*x+1
octave:2> x0 = gf ([0, 1, 2], 3);
octave:3> y0 = polyval (poly1, x0);
y0 =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   1   2   0

octave:4> a = gf (2, 3);               ## The primitive element
octave:5> y1 = a .* x0.^3 + a.^2 .* x0.^2 + (a.^2 + 1) .* x0 + 1
y1 =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   1   2   0
@end example

It is equally possible to find the roots of Galois polynomials with the
@code{roots} function. Using the polynomial above over GF(2^3), we can
find its roots in the following manner

@example
octave:1> poly1 = gf ([2, 4, 5, 1], 3);
octave:2> root1 = roots (poly1)
root1 =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   2
   5
   5
@end example

Thus the example polynomial has 3 roots in GF(2^3) with one root of
multiplicity 2. We can check this answer with the @code{polyval} function
as follows

@example
octave:3> check1 = polyval (poly1, root1)
check1 =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   0
   0
   0
@end example

@noindent
which as expected gives a zero vector. It should be noted that both the
number of roots and their value, will depend on the chosen field. Thus
for instance

@example
octave:1> poly3 = gf ([2, 4, 5, 1], 3, 13);
octave:2> root3 = roots (poly3)
root3 =
GF(2^3) array. Primitive Polynomial = D^3+D^2+1 (decimal 13)

Array elements =

   5
@end example

@noindent
shows that in the field GF(2^3) with a different primitive polynomial,
has only one root exists.

The minimum polynomial of an element of GF(2^M) is the minimum degree
polynomial in GF(2), excluding the trivial zero polynomial, that has
that element as a root. The fact that the minimum polynomial is in GF(2)
means that its coefficients are one or zero only. The @code{minpol}
function can be used to find the minimum polynomial as follows

@example
octave:1> a = gf (2, 3);               ## The primitive element
octave:2> b = minpol (a)
b =
GF(2) array.

Array elements =

   1   0   1   1
@end example

Note that the minimum polynomial of the primitive element is the primitive
polynomial. Elements of GF(2^M) sharing the same minimum polynomial form a
partitioning of the field. This partitioning can be found with the
@code{cosets} function as follows

@example
octave:1> c = cosets (3)
c =
@{
  [1,1] =
  GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

  Array elements =

     1

  [1,2] =
  GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

  Array elements =

     2   4   6

  [1,3] =
  GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

  Array elements =

     3   5   7

@}
@end example

@noindent
which returns a cell array containing all of the elements of the GF(2^3),
partitioned into groups sharing the same minimum polynomial. The function
@code{cosets} can equally accept a second argument defining the primitive
polynomial to use in its calculations (i.e. @code{cosets (@var{a}, @var{p})}).

@node Linear Algebra, Signal Processing, Polynomial manipulations, Manipulating Galois Fields
@subsection Linear Algebra

The basic linear algebra operation of this package is the LU factorization
of a Galois array. That is the Galois array @var{a} is factorized in the
following way

@example
octave:2> [l, u, p] = lu (a)
@end example

@noindent
such that @code{@var{p} * @var{a} = @var{l} * @var{u}}. The matrix @var{p}
contains row permutations of @var{a}, such that @var{l} and @var{u} are
strictly upper and low triangular. The Galois array @var{a} can be
rectangular.

All other linear algebra operations within this package are based on this
LU factorization of a Galois array. An important consequence of this is that
no solution can be found for singular matrices, only a particular solution
will be found for under-determined systems of equation and the solution found
for over-determined systems is not always correct. This is identical to the
way @sc{matlab} performs linear algebra on Galois arrays.

For instance consider the under-determined linear equation

@example
octave:1> A = gf ([2, 0, 3, 3; 3, 1, 3, 1; 3, 1, 1, 0], 2);
octave:2> b = [0:2]';
octave:3> x = A \ b;
@end example

@noindent
gives the solution @code{@var{x} = [2, 0, 3, 2]}. There are in fact 4
possible solutions to this linear system; @code{@var{x} = [3, 2, 2, 0]},
@code{@var{x} = [0, 3, 1, 1]}, @code{@var{x} = [2, 0, 3, 2]} and
@code{@var{x} = [1, 1, 0, 3]}. No particular selection criteria are
applied to the chosen solution.

In addition, because singular matrices cannot be solved, unless you
know the matrix is not singular, you should test the determinant of the
matrix prior to solving the linear system. For instance

@example
octave:1> A = gf (floor (2^m * rand (3)), 2);
octave:2> b = [0:2]';
octave:3> if (det (A) != 0); x = A \ b; y = b' / A; endif;
octave:4> r = rank (A);
@end example

@noindent
solves the linear systems @code{@var{A} * @var{x} = @var{b}} and
@code{@var{y} * @var{A} = @var{b}}. Note that you do not need to take
into account rounding errors in the determinant, as the determinant can
only take values within the Galois Field. So if the determinant equals
zero, the array is singular.

@node Signal Processing, , Linear Algebra, Manipulating Galois Fields
@subsection Signal Processing with Galois Fields

Signal processing functions such as filtering, convolution, de-convolution
and Fourier transforms can be performed over Galois Fields. For instance
the @code{filter} function can be used with Galois vectors in the same
manner as usual. For instance

@example
octave:1> b = gf ([2, 0, 0, 1, 0, 2, 0, 1], 2);
octave:2> a = gf ([2, 0, 1, 1], 2);
octave:3> x = gf ([1, zeros(1, 20)], 2);
octave:4> y = filter (b, a, x)
y =
GF(2^2) array. Primitive Polynomial = D^2+D+1 (decimal 7)

Array elements =

   1   0   3   0   2   3   1   0   1   3   3   1   0   1   3   3   1   0   1   3   3
@end example

@noindent
gives the impulse response of the filter defined by @var{a} and @var{b}.

Two equivalent ways are given to perform the convolution of two Galois
vectors. Firstly the function @code{conv} can be used, or alternatively
the function @code{convmtx} can be used. The first of these function is
identical to the convolution function over real vectors, and has been
described in the section about multiplying two Galois polynomials.

In the case where many Galois vectors will be convolved with the same
vector, the second function @code{convmtx} offers an alternative method
to calculate the convolution. If @var{a} is a column vector and @var{x}
is a column vector of length @var{n}, then

@example
octave:1> m = 3;
octave:2> a = gf (floor (2^m * rand (4, 1)), m);
octave:3> b = gf (floor (2^m * rand (4, 1)), m);
octave:4> c0 = conv (a, b)';
octave:5> c1 = convmtx (a, length (b)) * b;
octave:6> check = all (c0 == c1)
check = 1
@end example

@noindent
shows the equivalence of the two functions. The de-convolution function has
been previously described above.

The final signal processing function available in this package are the
functions to perform Fourier transforms over a Galois field. Three
functions are available, @code{fft}, @code{ifft} and @code{dftmtx}. The
first two functions use the third to perform their work. Given an element
@var{a} of the Galois field GF(2^M), @code{dftmtx} returns the @code{2^M - 1}
square matrix used in the Fourier transforms with respect to @var{a}. The
minimum polynomial of @var{a} must be primitive in GF(2^M). In the case of
the @code{fft} function @code{dftmtx} is called with the primitive element of
the Galois Field as an argument. As an example

@example
octave:1> m = 4;
octave:2> n = 2^m - 1;
octave:2> alph = gf (2, m);
octave:3> x = gf (floor (2^m * rand (n, 1)), m);
octave:4> y0 = fft (x);
octave:5> y1 = dftmtx (alph) * x;
octave:6> z0 = ifft (y0);
octave:7> z1 = dftmtx (1/alph) * y1;
octave:8> check = all (y0 == y1) & all (z0 == x) & all (z1 == x)
check = 1
@end example

In all cases, the length of the vector to be transformed must be
@code{2^M -1}. As the @code{dftmtx} creates a matrix representing the
Fourier transform, to limit the computational task only Fourier
transforms in GF(2^M), where M is less than or equal to 8, are supported.

@node Function Reference, , Galois Fields, Top
@chapter Function Reference

@iftex
@section Functions by Category
@subsection Random Signals
@table @asis
@item awgn
Add white Gaussian noise to a voltage signal
@item biterr
Compares two matrices and returns the number of bit errors and the bit error rate.
@item eyediagram
Plot the eye-diagram of a signal.
@item randerr
Generate a matrix of random bit errors.
@item randint
Generate a matrix of random binary numbers.
@item randsrc
Generate a matrix of random symbols.
@item scatterplot
Display the scatter plot of a signal.
@item symerr
Compares two matrices and returns the number of symbol errors and the symbol error rate.
@item wgn
Returns a M-by-N matrix Y of white Gaussian noise.
@item bsc
Send DATA into a binary symmetric channel with probability P of error one each symbol
@end table
@subsection Source Coding
@table @asis
@item compand
Compresses and expanding the dynamic range of a signal using a mu-law or or A-law algorithm
@item dpcmdeco
Decode using differential pulse code modulation (DPCM)
@item dpcmenco
Encode using differential pulse code modulation (DPCM)
@item dpcmopt
Optimize the DPCM parameters and codebook
@item huffmandeco
Decode signal encoded by 'huffmanenco'
@item huffmandict
Builds a Huffman code, given a probability list.
@item huffmanenco
Returns the Huffman encoded signal using DICT.
@item lloyds
Optimize the quantization table and codes to reduce distortion.
@item lz77deco
Lempel-Ziv 77 source algorithm decoding implementation.
@item lz77enco
Lempel-Ziv 77 source algorithm implementation.
@item quantiz
Quantization of an arbitrary signal relative to a partitioning
@item shannonfanodict
Returns the code dictionary for source using Shannon-Fano algorithm Dictionary is built from SYMBOL_PROBABILITIES using the Shannon-Fano scheme.
@item shannonfanoenco
Returns the Shannon-Fano encoded signal using DICT This function uses a DICT built from the 'shannonfanodict' and uses it to encode a signal list into a Shannon-Fano code Restrictions include a signal set that strictly belongs in the 'range [1,N]' with 'N = length (dict)'.
@item shannonfanodeco
Returns the original signal that was Shannon-Fano encoded.
@item rleenco
Returns run-length encoded MESSAGE.
@item rledeco
Returns decoded run-length MESSAGE.
@item riceenco
Returns the Rice encoded signal using K or optimal K Default optimal K is chosen between 0-7.
@item ricedeco
Returns the Rice decoded signal vector using CODE and K Compulsory K is need to be specified A restrictions is that a signal set must strictly be non-negative The value of code is a cell array of row-vectors which have the encoded rice value for a single sample.
@item fiboenco
Returns the cell-array of encoded Fibonacci value from the column vectors NUM Universal codes like Fibonacci codes have a useful synchronization property, only for 255 maximum value we have designed these routines.
@item fibodeco
Returns the decoded Fibonacci value from the binary vectors CODE Universal codes like Fibonacci codes have a useful synchronization property, only for 255 maximum value we have designed these routines.
@item fibosplitstream
Returns the split data stream at the word boundaries Assuming the stream was originally encoded using 'fiboenco' and this routine splits the stream at the points where "11" occur together & gives us the code-words which can later be decoded from the 'fibodeco' This however doesn't mean that we intend to verify if all the codewords are correct, and in fact the last symbol in the return list can or can not be a valid codeword
@item golombenco
Returns the Golomb coded signal as cell array Also total length of output code in bits can be obtained This function uses a M need to be supplied for encoding signal vector into a Golomb coded vector.
@item golombdeco
Returns the Golomb decoded signal vector using CODE and M Compulsory m is need to be specified.
@end table
@subsection Block Interleavers
@table @asis
@item intrlv
Interleaved elements of DATA according to ELEMENTS See also: deintrlv
@item helscanintrlv
NROWS-by-NCOLS See also: helscandeintrlv
@item matintrlv
Interleaved elements of DATA with a temporary matrix of size NROWS-by-NCOLS See also: matdeintrlv
@item randintrlv
Interleaves elements of DATA with a random permutation See also: intrlv, deintrlv
@item deintrlv
Restore elements of DATA according to ELEMENTS See also: intrlv
@item matdeintrlv
Restore elements of DATA with a temporary matrix of size NROWS-by-NCOLS See also: matintrlv
@item randdeintrlv
Restore elements of DATA with a random permutation See also: randintrlv, intrlv, deintrlv
@end table
@subsection Block Coding
@table @asis
@item bchdeco
Decodes the coded message CODE using a BCH coder.
@item bchenco
Encodes the message MSG using a [N,K] BCH coding.
@item bchpoly
Calculates the generator polynomials for a BCH coder.
@item convenc
Encode the binary vector MSG with the convolutional encoder described by the trellis structure T
@item cyclgen
Produce the parity check and generator matrix of a cyclic code.
@item cyclpoly
This function returns the cyclic generator polynomials of the code [N,K].
@item decode
Top level block decoder.
@item encode
Top level block encoder.
@item egolaydec
Decode Extended Golay code
@item egolayenc
Encode with Extended Golay code
@item egolaygen
Extended Golay code generator matrix
@item gen2par
Converts binary generator matrix GEN to the parity check matrix PAR and visa-versa.
@item hammgen
Produce the parity check and generator matrices of a Hamming code.
@item reedmullerdec
Decode the received code word VV using the RM-generator matrix G, of order R, M, returning the code-word C.
@item reedmullerenc
Definition type construction of Reed-Muller code, of order R, length 2^M.
@item reedmullergen
Definition type construction of Reed-Muller code, of order R, length 2^M.
@item rsgenpoly
Creates a generator polynomial for a Reed-Solomon coding with message length of K and codelength of N.
@item rsdec
Decodes the message contained in CODE using a [N,K] Reed-Solomon code.
@item rsdecof
Decodes an ASCII file using a Reed-Solomon coder.
@item rsenc
Encodes the message MSG using a [N,K] Reed-Solomon coding.
@item rsencof
Encodes an ASCII file using a Reed-Solomon coder.
@item systematize
Given G, extract P parity check matrix.
@item syndtable
Create the syndrome decoding table from the parity check matrix H.
@end table
@subsection Modulations
@table @asis
@item ademodce
Baseband demodulator for analog signals.
@item amodce
Baseband modulator for analog signals.
@item ammod
Create the AM modulation of the signal x with carrier frequency fs.
@item amdemod
Compute the amplitude demodulation of the signal S with a carrier frequency of FC and a sample frequency of FS See also: ammod
@item apkconst
Plots a ASK/PSK signal constellation.
@item demodmap
Demapping of an analog signal to a digital signal.
@item fmmod
Create the FM modulation of the signal x with carrier frequency fs.
@item fmdemod
Create the FM demodulation of the signal x with carrier frequency fs Where x is sample at frequency fs See also: ammod, amdemod, fmmod
@item genqammod
Modulates an information sequence of integers X in the range '[0 ... M-1]' onto a quadrature amplitude modulated signal Y, where 'M = length (c) - 1' and C is a 1D vector specifying the signal constellation mapping to be used.
@item genqamdemod
General quadrature amplitude demodulation.
@item modmap
Mapping of a digital signal to an analog signal.
@item pamdemod
Demodulates a pulse amplitude modulated signal X into an information sequence of integers in the range '[0 ... M-1]' PHI controls the initial phase and TYPE controls the constellation mapping.
@item pammod
Modulates an information sequence of integers X in the range '[0 ... M-1]' onto a pulse amplitude modulated signal Y PHI controls the initial phase and TYPE controls the constellation mapping.
@item pskdemod
Demodulates a complex-baseband phase shift keying modulated signal into an information sequence of integers in the range '[0 ... M-1]'.
@item pskmod
Modulates an information sequence of integers X in the range '[0 ... M-1]' onto a complex baseband phase shift keying modulated signal Y.
@item qaskdeco
Demaps an analog signal using a square QASK constellation.
@item qaskenco
Map a digital signal using a square QASK constellation.
@item qammod
Create the QAM modulation of x with a size of alphabet m See also: qamdemod, pskmod, pskdemod
@item qamdemod
Create the QAM demodulation of x with a size of alphabet m See also: qammod, pskmod, pskdemod
@end table
@subsection Galois Fields of Even Characteristic
@table @asis
@item + -
Addition and subtraction in a Galois Field.

@item * / \
Matrix multiplication and division of Galois arrays.

@item .* ./ .\
Element by element multiplication and division of Galois arrays.

@item ** ^
Matrix exponentiation of Galois arrays.

@item .** .^
Element by element matrix exponentiation of Galois arrays.

@item ' .'
Matrix transpose of Galois arrays.

@item == ~= != > >= < <=
Logical operators on Galois arrays.

@item all
@emph{Not implemented}
@item any
@emph{Not implemented}
@item cosets
Finds the elements of GF(2^M) with primitive polynomial PRIM, that share the same minimum polynomial.
@item conv
Convolve two Galois vectors
@item convmtx
Create matrix to perform repeated convolutions with the same vector in a Galois Field.
@item deconv
Deconvolve two Galois vectors
@item det
Compute the determinant of the Galois array A
@item dftmtx
Form a matrix, that can be used to perform Fourier transforms in a Galois Field
@item diag
Return a diagonal matrix with Galois vector V on diagonal K The second argument is optional.
@item exp
Compute the anti-logarithm for each element of X for a Galois array
@item gf
#endif
@item fft
If X is a column vector, finds the FFT over the primitive element of the Galois Field of X.
@item filter
Digital filtering of vectors in a Galois Field.
@item gftable
This function exists for compatibility with matlab.
@item gfweight
Calculate the minimum weight or distance of a linear block code.
@item ifft
If X is a column vector, finds the IFFT over the primitive element of the Galois Field of X.
@item inv
Compute the inverse of the square matrix A.
@item inverse
See inv
@item isequal
Return true if all of X1, X2, ... are equal See also: isequalwithequalnans
@item log
Compute the natural logarithm for each element of X for a Galois array
@item lu
Compute the LU decomposition of A in a Galois Field.
@item prod
Product of elements along dimension DIM of Galois array.
@item sqrt
Compute the square root of X, element by element, in a Galois Field See also: exp
@item rank
Compute the rank of the Galois array A by counting the independent rows and columns
@item reshape
Return a matrix with M rows and N columns whose elements are taken from the Galois array A.
@item roots
For a vector V with N components, return the roots of the polynomial over a Galois Field
@item sum
Sum of elements along dimension DIM of Galois array.
@item sumsq
Sum of squares of elements along dimension DIM of Galois array If DIM is omitted, it defaults to 1 (column-wise sum of squares)
@item isempty
@emph{Not implemented}
@item isgalois
Return 1 if the value of the expression EXPR is a Galois Field.
@item isprimitive
Returns 1 is the polynomial represented by A is a primitive polynomial of GF(2).
@item length
@emph{Not implemented}
@item minpol
Finds the minimum polynomial for elements of a Galois Field.
@item polyval
@emph{Not implemented}
@item primpoly
Finds the primitive polynomials in GF(2^M).
@item size
@emph{Not implemented}
@end table
@subsection Utility Functions
@table @asis
@item comms
Manual and test code for the Octave Communications toolbox.
@item bi2de
Convert bit matrix to a vector of integers
@item de2bi
Convert a non-negative integer to bit vector
@item oct2dec
Convert octal to decimal values
@item istrellis
Return true if T is a valid trellis structure
@item poly2trellis
Convert convolutional code generator polynomials into trellis form
@item vec2mat
Converts the vector V into a C column matrix with row priority arrangement and with the final column padded with the value D to the correct length.
@item qfunc
Compute the Q function See also: erfc, erf
@item qfuncinv
Compute the inverse Q function See also: erfc, erf
@end table
@end iftex


@section Functions Alphabetically
@menu
* ademodce::	Baseband demodulator for analog signals.
* amdemod::	Compute the amplitude demodulation of the signal S with a
		carrier frequency of FC and a sample frequency of FS See
		also: ammod
* ammod::	Create the AM modulation of the signal x with carrier
		frequency fs.
* amodce::	Baseband modulator for analog signals.
* apkconst::	Plots a ASK/PSK signal constellation.
* awgn::	Add white Gaussian noise to a voltage signal
* bchdeco::	Decodes the coded message CODE using a BCH coder.
* bchenco::	Encodes the message MSG using a [N,K] BCH coding.
* bchpoly::	Calculates the generator polynomials for a BCH coder.
* bi2de::	Convert bit matrix to a vector of integers
* biterr::	Compares two matrices and returns the number of bit errors
		and the bit error rate.
* bsc:: 	Send DATA into a binary symmetric channel with probability
		P of error one each symbol
* comms::	Manual and test code for the Octave Communications toolbox.
* compand::	Compresses and expanding the dynamic range of a signal
		using a mu-law or or A-law algorithm
* conv::	Convolve two Galois vectors
* convenc::	Encode the binary vector MSG with the convolutional encoder
		described by the trellis structure T
* convmtx::	Create matrix to perform repeated convolutions with the
		same vector in a Galois Field.
* cosets::	Finds the elements of GF(2^M) with primitive polynomial
		PRIM, that share the same minimum polynomial.
* cyclgen::	Produce the parity check and generator matrix of a cyclic
		code.
* cyclpoly::	This function returns the cyclic generator polynomials of
		the code [N,K].
* de2bi::	Convert a non-negative integer to bit vector
* decode::	Top level block decoder.
* deconv::	Deconvolve two Galois vectors
* deintrlv::	Restore elements of DATA according to ELEMENTS See also:
		intrlv
* demodmap::	Demapping of an analog signal to a digital signal.
* det:: 	Compute the determinant of the Galois array A
* dftmtx::	Form a matrix, that can be used to perform Fourier
		transforms in a Galois Field
* diag::	Return a diagonal matrix with Galois vector V on diagonal K
		The second argument is optional.
* dpcmdeco::	Decode using differential pulse code modulation (DPCM)
* dpcmenco::	Encode using differential pulse code modulation (DPCM)
* dpcmopt::	Optimize the DPCM parameters and codebook
* egolaydec::	Decode Extended Golay code
* egolayenc::	Encode with Extended Golay code
* egolaygen::	Extended Golay code generator matrix
* encode::	Top level block encoder.
* exp:: 	Compute the anti-logarithm for each element of X for a
		Galois array
* eyediagram::	Plot the eye-diagram of a signal.
* fft:: 	If X is a column vector, finds the FFT over the primitive
		element of the Galois Field of X.
* fibodeco::	Returns the decoded Fibonacci value from the binary vectors
		CODE Universal codes like Fibonacci codes have a useful
		synchronization property, only for 255 maximum value we
		have designed these routines.
* fiboenco::	Returns the cell-array of encoded Fibonacci value from the
		column vectors NUM Universal codes like Fibonacci codes
		have a useful synchronization property, only for 255
		maximum value we have designed these routines.
* fibosplitstream:: Returns the split data stream at the word boundaries
		Assuming the stream was originally encoded using 'fiboenco'
		and this routine splits the stream at the points where "11"
		occur together & gives us the code-words which can later be
		decoded from the 'fibodeco' This however doesn't mean that
		we intend to verify if all the codewords are correct, and
		in fact the last symbol in the return list can or can not
		be a valid codeword
* filter::	Digital filtering of vectors in a Galois Field.
* fmdemod::	Create the FM demodulation of the signal x with carrier
		frequency fs Where x is sample at frequency fs See also:
		ammod, amdemod, fmmod
* fmmod::	Create the FM modulation of the signal x with carrier
		frequency fs.
* gen2par::	Converts binary generator matrix GEN to the parity check
		matrix PAR and visa-versa.
* genqamdemod:: General quadrature amplitude demodulation.
* genqammod::	Modulates an information sequence of integers X in the
		range '[0 ... M-1]' onto a quadrature amplitude modulated
		signal Y, where 'M = length (c) - 1' and C is a 1D vector
		specifying the signal constellation mapping to be used.
* gf::		#endif
* gftable::	This function exists for compatibility with matlab.
* gfweight::	Calculate the minimum weight or distance of a linear block
		code.
* golombdeco::	Returns the Golomb decoded signal vector using CODE and M
		Compulsory m is need to be specified.
* golombenco::	Returns the Golomb coded signal as cell array Also total
		length of output code in bits can be obtained This function
		uses a M need to be supplied for encoding signal vector
		into a Golomb coded vector.
* hammgen::	Produce the parity check and generator matrices of a
		Hamming code.
* helscanintrlv:: NROWS-by-NCOLS See also: helscandeintrlv
* huffmandeco:: Decode signal encoded by 'huffmanenco'
* huffmandict:: Builds a Huffman code, given a probability list.
* huffmanenco:: Returns the Huffman encoded signal using DICT.
* ifft::	If X is a column vector, finds the IFFT over the primitive
		element of the Galois Field of X.
* intrlv::	Interleaved elements of DATA according to ELEMENTS See
		also: deintrlv
* inv:: 	Compute the inverse of the square matrix A.
* inverse::	See inv
* isequal::	Return true if all of X1, X2, ... are equal See also:
		isequalwithequalnans
* isgalois::	Return 1 if the value of the expression EXPR is a Galois
		Field.
* isprimitive:: Returns 1 is the polynomial represented by A is a primitive
		polynomial of GF(2).
* istrellis::	Return true if T is a valid trellis structure
* lloyds::	Optimize the quantization table and codes to reduce
		distortion.
* log:: 	Compute the natural logarithm for each element of X for a
		Galois array
* lu::		Compute the LU decomposition of A in a Galois Field.
* lz77deco::	Lempel-Ziv 77 source algorithm decoding implementation.
* lz77enco::	Lempel-Ziv 77 source algorithm implementation.
* matdeintrlv:: Restore elements of DATA with a temporary matrix of size
		NROWS-by-NCOLS See also: matintrlv
* matintrlv::	Interleaved elements of DATA with a temporary matrix of
		size NROWS-by-NCOLS See also: matdeintrlv
* minpol::	Finds the minimum polynomial for elements of a Galois
		Field.
* modmap::	Mapping of a digital signal to an analog signal.
* oct2dec::	Convert octal to decimal values
* pamdemod::	Demodulates a pulse amplitude modulated signal X into an
		information sequence of integers in the range '[0 ... M-1]'
		PHI controls the initial phase and TYPE controls the
		constellation mapping.
* pammod::	Modulates an information sequence of integers X in the
		range '[0 ... M-1]' onto a pulse amplitude modulated signal
		Y PHI controls the initial phase and TYPE controls the
		constellation mapping.
* poly2trellis:: Convert convolutional code generator polynomials into
		trellis form
* primpoly::	Finds the primitive polynomials in GF(2^M).
* prod::	Product of elements along dimension DIM of Galois array.
* pskdemod::	Demodulates a complex-baseband phase shift keying modulated
		signal into an information sequence of integers in the
		range '[0 ... M-1]'.
* pskmod::	Modulates an information sequence of integers X in the
		range '[0 ... M-1]' onto a complex baseband phase shift
		keying modulated signal Y.
* qamdemod::	Create the QAM demodulation of x with a size of alphabet m
		See also: qammod, pskmod, pskdemod
* qammod::	Create the QAM modulation of x with a size of alphabet m
		See also: qamdemod, pskmod, pskdemod
* qaskdeco::	Demaps an analog signal using a square QASK constellation.
* qaskenco::	Map a digital signal using a square QASK constellation.
* qfunc::	Compute the Q function See also: erfc, erf
* qfuncinv::	Compute the inverse Q function See also: erfc, erf
* quantiz::	Quantization of an arbitrary signal relative to a
		partitioning
* randdeintrlv:: Restore elements of DATA with a random permutation See
		also: randintrlv, intrlv, deintrlv
* randerr::	Generate a matrix of random bit errors.
* randint::	Generate a matrix of random binary numbers.
* randintrlv::	Interleaves elements of DATA with a random permutation See
		also: intrlv, deintrlv
* randsrc::	Generate a matrix of random symbols.
* rank::	Compute the rank of the Galois array A by counting the
		independent rows and columns
* reedmullerdec:: Decode the received code word VV using the RM-generator
		matrix G, of order R, M, returning the code-word C.
* reedmullerenc:: Definition type construction of Reed-Muller code, of
		order R, length 2^M.
* reedmullergen:: Definition type construction of Reed-Muller code, of
		order R, length 2^M.
* reshape::	Return a matrix with M rows and N columns whose elements
		are taken from the Galois array A.
* ricedeco::	Returns the Rice decoded signal vector using CODE and K
		Compulsory K is need to be specified A restrictions is that
		a signal set must strictly be non-negative The value of
		code is a cell array of row-vectors which have the encoded
		rice value for a single sample.
* riceenco::	Returns the Rice encoded signal using K or optimal K
		Default optimal K is chosen between 0-7.
* rledeco::	Returns decoded run-length MESSAGE.
* rleenco::	Returns run-length encoded MESSAGE.
* roots::	For a vector V with N components, return the roots of the
		polynomial over a Galois Field
* rsdec::	Decodes the message contained in CODE using a [N,K]
		Reed-Solomon code.
* rsdecof::	Decodes an ASCII file using a Reed-Solomon coder.
* rsenc::	Encodes the message MSG using a [N,K] Reed-Solomon coding.
* rsencof::	Encodes an ASCII file using a Reed-Solomon coder.
* rsgenpoly::	Creates a generator polynomial for a Reed-Solomon coding
		with message length of K and codelength of N.
* scatterplot:: Display the scatter plot of a signal.
* shannonfanodeco:: Returns the original signal that was Shannon-Fano
		encoded.
* shannonfanodict:: Returns the code dictionary for source using
		Shannon-Fano algorithm Dictionary is built from
		SYMBOL_PROBABILITIES using the Shannon-Fano scheme.
* shannonfanoenco:: Returns the Shannon-Fano encoded signal using DICT This
		function uses a DICT built from the 'shannonfanodict' and
		uses it to encode a signal list into a Shannon-Fano code
		Restrictions include a signal set that strictly belongs in
		the 'range [1,N]' with 'N = length (dict)'.
* sqrt::	Compute the square root of X, element by element, in a
		Galois Field See also: exp
* sum:: 	Sum of elements along dimension DIM of Galois array.
* sumsq::	Sum of squares of elements along dimension DIM of Galois
		array If DIM is omitted, it defaults to 1 (column-wise sum
		of squares)
* symerr::	Compares two matrices and returns the number of symbol
		errors and the symbol error rate.
* syndtable::	Create the syndrome decoding table from the parity check
		matrix H.
* systematize:: Given G, extract P parity check matrix.
* vec2mat::	Converts the vector V into a C column matrix with row
		priority arrangement and with the final column padded with
		the value D to the correct length.
* wgn:: 	Returns a M-by-N matrix Y of white Gaussian noise.
@end menu

@node ademodce, amdemod, , Function Reference
@subsection ademodce

@deftypefn  {Function File} {@var{y} =} ademodce (@var{x}, @var{Fs}, "amdsb-tc", offset)
@deftypefnx {Function File} {@var{y} =} ademodce (@var{x}, @var{Fs}, "amdsb-tc/costas", offset)
@deftypefnx {Function File} {@var{y} =} ademodce (@var{x}, @var{Fs}, "amdsb-sc")
@deftypefnx {Function File} {@var{y} =} ademodce (@var{x}, @var{Fs}, "amdsb-sc/costas")
@deftypefnx {Function File} {@var{y} =} ademodce (@var{x}, @var{Fs}, "amssb")
@deftypefnx {Function File} {@var{y} =} ademodce (@var{x}, @var{Fs}, "qam")
@deftypefnx {Function File} {@var{y} =} ademodce (@var{x}, @var{Fs}, "qam/cmplx")
@deftypefnx {Function File} {@var{y} =} ademodce (@var{x}, @var{Fs}, "fm", @var{dev})
@deftypefnx {Function File} {@var{y} =} ademodce (@var{x}, @var{Fs}, "pm", @var{dev})
@deftypefnx {Function File} {@var{y} =} ademodce (@var{x}, [@var{Fs}, @var{iphs}], @dots{})
@deftypefnx {Function File} {@var{y} =} ademodce (@dots{}, @var{num}, @var{den})

Baseband demodulator for analog signals. The input signal is specified by
@var{x}, its sampling frequency by @var{Fs} and the type of modulation
by the third argument, @var{typ}. The default values of @var{Fs} is 1 and
@var{typ} is "amdsb-tc"

If the argument @var{Fs} is a two element vector, the first element
represents the sampling rate and the second the initial phase

The different types of demodulations that are available are

@table @asis
@item  "am"
@itemx "amdsb-tc"
Double-sideband with carrier
@item "amdsb-tc/costas"
Double-sideband with carrier and Costas phase locked loop
@item "amdsb-sc"
Double-sideband with suppressed carrier
@item "amssb"
Single-sideband with frequency domain Hilbert filtering
@item "qam"
Quadrature amplitude demodulation. In-phase in odd-columns and quadrature
in even-columns
@item "qam/cmplx"
Quadrature amplitude demodulation with complex return value
@item "fm"
Frequency demodulation
@item "pm"
Phase demodulation
@end table

Additional arguments are available for the demodulations "amdsb-tc", "fm",
"pm". These arguments are

@table @code
@item offset
The offset in the input signal for the transmitted carrier
@item dev
The deviation of the phase and frequency modulation
@end table

It is possible to specify a low-pass filter, by the numerator @var{num}
and denominator @var{den} that will be applied to the returned vector

See also: ademodce, dmodce
@end deftypefn



@node amdemod, ammod, ademodce, Function Reference
@subsection amdemod

@deftypefn {Function File} {@var{m} =} amdemod (@var{s}, @var{fc}, @var{fs})
Compute the amplitude demodulation of the signal @var{s} with a carrier
frequency of @var{fc} and a sample frequency of @var{fs}
See also: ammod
@end deftypefn



@node ammod, amodce, amdemod, Function Reference
@subsection ammod

@deftypefn {Function File} {} ammod (@var{x}, @var{fc}, @var{fs})
Create the AM modulation of the signal x with carrier frequency fs. Where
x is sample at frequency fs
See also: amdemod, fmmod, fmdemod
@end deftypefn



@node amodce, apkconst, ammod, Function Reference
@subsection amodce

@deftypefn  {Function File} {@var{y} =} amodce (@var{x}, @var{Fs}, "amdsb-tc", offset)
@deftypefnx {Function File} {@var{y} =} amodce (@var{x}, @var{Fs}, "amdsb-sc")
@deftypefnx {Function File} {@var{y} =} amodce (@var{x}, @var{Fs}, "amssb")
@deftypefnx {Function File} {@var{y} =} amodce (@var{x}, @var{Fs}, "amssb/time", @var{num}, @var{den})
@deftypefnx {Function File} {@var{y} =} amodce (@var{x}, @var{Fs}, "qam")
@deftypefnx {Function File} {@var{y} =} amodce (@var{x}, @var{Fs}, "fm", @var{dev})
@deftypefnx {Function File} {@var{y} =} amodce (@var{x}, @var{Fs}, "pm", @var{dev})
@deftypefnx {Function File} {@var{y} =} amodce (@var{x}, [@var{Fs}, @var{iphs}], @dots{})

Baseband modulator for analog signals. The input signal is specified by
@var{x}, its sampling frequency by @var{Fs} and the type of modulation
by the third argument, @var{typ}. The default values of @var{Fs} is 1 and
@var{typ} is "amdsb-tc"

If the argument @var{Fs} is a two element vector, the first element
represents the sampling rate and the second the initial phase

The different types of modulations that are available are

@table @asis
@item  "am"
@itemx "amdsb-tc"
Double-sideband with carrier
@item "amdsb-sc"
Double-sideband with suppressed carrier
@item "amssb"
Single-sideband with frequency domain Hilbert filtering
@item "amssb/time"
Single-sideband with time domain filtering. Hilbert filter is used by
default, but the filter can be specified
@item "qam"
Quadrature amplitude modulation
@item "fm"
Frequency modulation
@item "pm"
Phase modulation
@end table

Additional arguments are available for the modulations "amdsb-tc", "fm",
"pm" and "amssb/time". These arguments are

@table @code
@item offset
The offset in the input signal for the transmitted carrier
@item dev
The deviation of the phase and frequency modulation
@item  num
@itemx den
The numerator and denominator of the filter transfer function for the
time domain filtering of the SSB modulation
@end table

See also: ademodce, dmodce
@end deftypefn



@node apkconst, awgn, amodce, Function Reference
@subsection apkconst

@deftypefn  {Function File} {} apkconst (@var{nsig})
@deftypefnx {Function File} {} apkconst (@var{nsig}, @var{amp})
@deftypefnx {Function File} {} apkconst (@var{nsig}, @var{amp}, @var{phs})
@deftypefnx {Function File} {} apkconst (@dots{}, "n")
@deftypefnx {Function File} {} apkconst (@dots{}, @var{str})
@deftypefnx {Function File} {@var{y} =} apkconst (@dots{})

Plots a ASK/PSK signal constellation. Argument @var{nsig} is a real vector
whose length determines the number of ASK radii in the constellation
The values of vector @var{nsig} determine the number of points in each
ASK radii

By default the radii of each ASK modulated level is given by the index of
@var{nsig}. The amplitudes can be defined explicitly in the variable
@var{amp}, which is a vector of the same length as @var{nsig}

By default the first point in each ASK radii has zero phase, and following
points are coding in an anti-clockwise manner. If @var{phs} is defined then
it is a vector of the same length as @var{nsig} defining the initial phase
in each ASK radii

In addition @code{apkconst} takes two string arguments "n" and @var{str}
If the string "n" is included in the arguments, then a number is printed
next to each constellation point giving the symbol value that would be
mapped to this point by the @code{modmap} function. The argument @var{str}
is a plot style string (example "r+") and determines the default gnuplot
point style to use for plot points in the constellation

If @code{apkconst} is called with a return argument, then no plot is
created. However the return value is a vector giving the in-phase and
quadrature values of the symbols in the constellation
See also: dmod, ddemod, modmap, demodmap
@end deftypefn



@node awgn, bchdeco, apkconst, Function Reference
@subsection awgn

@deftypefn  {Function File} {@var{y} =} awgn (@var{x}, @var{snr})
@deftypefnx {Function File} {@var{y} =} awgn (@var{x}, @var{snr}, @var{pwr})
@deftypefnx {Function File} {@var{y} =} awgn (@var{x}, @var{snr}, @var{pwr}, @var{seed})
@deftypefnx {Function File} {@var{y} =} awgn (@dots{}, @var{type})

Add white Gaussian noise to a voltage signal

The input @var{x} is assumed to be a real or complex voltage signal. The
returned value @var{y} will be the same form and size as @var{x} but with
Gaussian noise added. Unless the power is specified in @var{pwr}, the
signal power is assumed to be 0dBW, and the noise of @var{snr} dB will be
added with respect to this. If @var{pwr} is a numeric value then the signal
@var{x} is assumed to be @var{pwr} dBW, otherwise if @var{pwr} is
"measured", then the power in the signal will be measured and the noise
added relative to this measured power

If @var{seed} is specified, then the random number generator seed is
initialized with this value

By default the @var{snr} and @var{pwr} are assumed to be in dB and dBW
respectively. This default behavior can be chosen with @var{type}
set to "dB". In the case where @var{type} is set to "linear", @var{pwr}
is assumed to be in Watts and @var{snr} is a ratio
See also: randn, wgn
@end deftypefn



@node bchdeco, bchenco, awgn, Function Reference
@subsection bchdeco

@deftypefn  {Loadable Function} {@var{msg} =} bchdeco (@var{code}, @var{k}, @var{t})
@deftypefnx {Loadable Function} {@var{msg} =} bchdeco (@var{code}, @var{k}, @var{t}, @var{prim})
@deftypefnx {Loadable Function} {@var{msg} =} bchdeco (@dots{}, @var{parpos})
@deftypefnx {Loadable Function} {[@var{msg}, @var{err}] =} bchdeco (@dots{})
@deftypefnx {Loadable Function} {[@var{msg}, @var{err}, @var{ccode}] =} bchdeco (@dots{})
Decodes the coded message @var{code} using a BCH coder. The message length
of the coder is defined in variable @var{k}, and the error correction
capability of the code is defined in @var{t}.

The variable @var{code} is a binary array with @var{n} columns and an
arbitrary number of rows. Each row of @var{code} represents a single symbol
to be decoded by the BCH coder. The decoded message is returned in the
binary array @var{msg} containing @var{k} columns and the same number of
rows as @var{code}.

The use of @code{bchdeco} can be seen in the following short example.

@example
m = 3; n = 2^m -1; k = 4; t = 1;
msg = randint (10, k);
code = bchenco (msg, n, k);
noisy = mod (randerr (10,n) + code, 2);
[dec, err] = bchdeco (msg, k, t);
@end example

Valid codes can be found using @code{bchpoly}. In general the codeword
length @var{n} should be of the form @code{2^@var{m}-1}, where m is an
integer. However, shortened BCH codes can be used such that if
@code{[2^@var{m}-1,@var{k}]} is a valid code
@code{[2^@var{m}-1-@var{x},@var{k}-@var{x}]}
 is also a valid code using
the same generator polynomial.

By default the BCH coding is based on the properties of the Galois
Field GF(2^@var{m}). The primitive polynomial used in the Galois
can be overridden by a primitive polynomial in @var{prim}. Suitable
primitive polynomials can be constructed with @code{primpoly}. The form
of @var{prim} maybe be either a integer representation of the primitive
polynomial as given by @code{primpoly}, or a binary representation that
might be constructed like

@example
m = 3;
prim = de2bi (primpoly (m));
@end example

By default the parity symbols are assumed to be placed at the beginning of
the coded message. The variable @var{parpos} controls this positioning and
can take the values @code{"beginning\"} or @code{\"end\"}.
See also: bchpoly, bchenco, decode, primpoly
@end deftypefn



@node bchenco, bchpoly, bchdeco, Function Reference
@subsection bchenco

@deftypefn  {Loadable Function} {@var{code} =} bchenco (@var{msg}, @var{n}, @var{k})
@deftypefnx {Loadable Function} {@var{code} =} bchenco (@var{msg}, @var{n}, @var{k}, @var{g})
@deftypefnx {Loadable Function} {@var{code} =} bchenco (@dots{}, @var{parpos})
Encodes the message @var{msg} using a [@var{n},@var{k}] BCH coding.
The variable @var{msg} is a binary array with @var{k} columns and an
arbitrary number of rows. Each row of @var{msg} represents a single symbol
to be coded by the BCH coder. The coded message is returned in the binary
array @var{code} containing @var{n} columns and the same number of rows as
@var{msg}.

The use of @code{bchenco} can be seen in the following short example.

@example
m = 3; n = 2^m -1; k = 4;
msg = randint (10,k);
code = bchenco (msg, n, k);
@end example

Valid codes can be found using @code{bchpoly}. In general the codeword
length @var{n} should be of the form @code{2^@var{m}-1}, where m is an
integer. However, shortened BCH codes can be used such that if
@code{[2^@var{m}-1,@var{k}]} is a valid code
@code{[2^@var{m}-1-@var{x},@var{k}-@var{x}]}
 is also a valid code using
the same generator polynomial.

By default the generator polynomial used in the BCH coding is
based on the properties of the Galois Field GF(2^@var{m}). This
default generator polynomial can be overridden by a polynomial in @var{g}.
Suitable generator polynomials can be constructed with @code{bchpoly}.

By default the parity symbols are placed at the beginning of the coded
message. The variable @var{parpos} controls this positioning and can take
the values @code{"beginning\"} or @code{\"end\"}.
See also: bchpoly, bchdeco, encode
@end deftypefn



@node bchpoly, bi2de, bchenco, Function Reference
@subsection bchpoly

@deftypefn  {Function File} {@var{p} =} bchpoly ()
@deftypefnx {Function File} {@var{p} =} bchpoly (@var{n})
@deftypefnx {Function File} {@var{p} =} bchpoly (@var{n}, @var{k})
@deftypefnx {Function File} {@var{p} =} bchpoly (@var{prim}, @var{k})
@deftypefnx {Function File} {@var{p} =} bchpoly (@var{n}, @var{k}, @var{prim})
@deftypefnx {Function File} {@var{p} =} bchpoly (@dots{}, @var{probe})
@deftypefnx {Function File} {[@var{p}, @var{f}] =} bchpoly (@dots{})
@deftypefnx {Function File} {[@var{p}, @var{f}, @var{c}] =} bchpoly (@dots{})
@deftypefnx {Function File} {[@var{p}, @var{f}, @var{c}, @var{par}] =} bchpoly (@dots{})
@deftypefnx {Function File} {[@var{p}, @var{f}, @var{c}, @var{par}, @var{t}] =} bchpoly (@dots{})

Calculates the generator polynomials for a BCH coder. Called with no input
arguments @code{bchpoly} returns a list of all of the valid BCH codes for
the codeword length 7, 15, 31, 63, 127, 255 and 511. A three column matrix
is returned with each row representing a separate valid BCH code. The first
column is the codeword length, the second the message length and the third
the error correction capability of the code

Called with a single input argument, @code{bchpoly} returns the valid BCH
codes for the specified codeword length @var{n}. The output format is the
same as above

When called with two or more arguments, @code{bchpoly} calculates the
generator polynomial of a particular BCH code. The generator polynomial
is returned in @var{p} as a vector representation of a polynomial in
GF(2). The terms of the polynomial are listed least-significant term
first

The desired BCH code can be specified by its codeword length @var{n}
and its message length @var{k}. Alternatively, the primitive polynomial
over which to calculate the polynomial can be specified as @var{prim}
If a vector representation of the primitive polynomial is given, then
@var{prim} can be specified as the first argument of two arguments,
or as the third argument. However, if an integer representation of the
primitive polynomial is used, then the primitive polynomial must be
specified as the third argument

When called with two or more arguments, @code{bchpoly} can also return the
factors @var{f} of the generator polynomial @var{p}, the cyclotomic coset
for the Galois field over which the BCH code is calculated, the parity
check matrix @var{par} and the error correction capability @var{t}. It
should be noted that the parity check matrix is calculated with
@code{cyclgen} and limitations in this function means that the parity
check matrix is only available for codeword length up to 63. For
codeword length longer than this @var{par} returns an empty matrix

With a string argument @var{probe} defined, the action of @code{bchpoly}
is to calculate the error correcting capability of the BCH code defined
by @var{n}, @var{k} and @var{prim} and return it in @var{p}. This is
similar to a call to @code{bchpoly} with zero or one argument, except that
only a single code is checked. Any string value for @var{probe} will
force this action

In general the codeword length @var{n} can be expressed as
@code{2^@var{m}-1}, where @var{m} is an integer. However, if
[@var{n},@var{k}] is a valid BCH code, then a shortened BCH code of
the form [@var{n}-@var{x},@var{k}-@var{x}] can be created with the
same generator polynomial

See also: cyclpoly, encode, decode, cosets
@end deftypefn



@node bi2de, biterr, bchpoly, Function Reference
@subsection bi2de

@deftypefn  {Function File} {@var{d} =} bi2de (@var{b})
@deftypefnx {Function File} {@var{d} =} bi2de (@var{b}, @var{f})
@deftypefnx {Function File} {@var{d} =} bi2de (@var{b}, @var{p})
@deftypefnx {Function File} {@var{d} =} bi2de (@var{b}, @var{p}, @var{f})

Convert bit matrix to a vector of integers

Each row of the matrix @var{b} is treated as a single integer represented
in binary form. The elements of @var{b}, must therefore be '0' or '1'

If @var{p} is defined then it is treated as the base of the decomposition
and the elements of @var{b} must then lie between '0' and 'p-1'

The variable @var{f} defines whether the first or last element of @var{b}
is considered to be the most-significant. Valid values of @var{f} are
"right-msb" or "left-msb". By default @var{f} is "right-msb"

See also: de2bi
@end deftypefn



@node biterr, bsc, bi2de, Function Reference
@subsection biterr

@deftypefn  {Function File} {[@var{num}, @var{rate}] =} biterr (@var{a}, @var{b})
@deftypefnx {Function File} {[@var{num}, @var{rate}] =} biterr (@dots{}, @var{k})
@deftypefnx {Function File} {[@var{num}, @var{rate}] =} biterr (@dots{}, @var{flag})
@deftypefnx {Function File} {[@var{num}, @var{rate} @var{ind}] =} biterr (@dots{})

Compares two matrices and returns the number of bit errors and the bit
error rate. The binary representations of the variables @var{a} and
@var{b} are treated and @var{a} and @var{b} can be either:

@table @asis
@item Both matrices
In this case both matrices must be the same size and then by default the
return values @var{num} and @var{rate} are the overall number of bit
errors and the overall bit error rate
@item One column vector
In this case the column vector is used for bit error comparison column-wise
with the matrix. The returned values @var{num} and @var{rate} are then
row vectors containing the number of bit errors and the bit error rate for
each of the column-wise comparisons. The number of rows in the matrix
must be the same as the length of the column vector
@item One row vector
In this case the row vector is used for bit error comparison row-wise
with the matrix. The returned values @var{num} and @var{rate} are then
column vectors containing the number of bit errors and the bit error rate
for each of the row-wise comparisons. The number of columns in the matrix
must be the same as the length of the row vector
@end table

This behavior can be overridden with the variable @var{flag}. @var{flag}
can take the value "column-wise", "row-wise" or "overall". A column-wise
comparison is not possible with a row vector and visa-versa

By default the number of bits in each symbol is assumed to be give by the
number required to represent the maximum value of @var{a} and @var{b}
The number of bits to represent a symbol can be overridden by the variable
@var{k}
@end deftypefn



@node bsc, comms, biterr, Function Reference
@subsection bsc

@deftypefn {Function File} {@var{y} =} bsc (@var{data}, @var{p})
Send @var{data} into a binary symmetric channel with probability
@var{p} of error one each symbol
@end deftypefn



@node comms, compand, bsc, Function Reference
@subsection comms

@deftypefn  {Function File} {} comms ("help")
@deftypefnx {Function File} {} comms ("info")
@deftypefnx {Function File} {} comms ("info", @var{mod})
@deftypefnx {Function File} {} comms ("test")
@deftypefnx {Function File} {} comms ("test", @var{mod})

Manual and test code for the Octave Communications toolbox. There are
5 possible ways to call this function

@table @code
@item comms ("help")
Display this help message. Called with no arguments, this function also
displays this help message
@item comms ("info")
Open the Communications toolbox manual
@item comms ("info", @var{mod})
Open the Communications toolbox manual at the section specified by
@var{mod}
@item comms ("test")
Run all of the test code for the Communications toolbox
@item comms ("test", @var{mod})
Run only the test code for the Communications toolbox in the module
@var{mod}
@end table

Valid values for the variable @var{mod} are

@table @asis
@item "all"
All of the toolbox
@item "random"
The random signal generation and analysis package
@item "source"
The source coding functions of the package
@item "block"
The block coding functions
@item "convol"
The convolution coding package
@item "modulation"
The modulation package
@item "special"
The special filter functions
@item "galois"
The Galois fields package
@end table

Please note that this function file should be used as an example of the
use of this toolbox
@end deftypefn



@node compand, conv, comms, Function Reference
@subsection compand

@deftypefn  {Function File} {@var{y} =} compand (@var{x}, @var{mu}, @var{V}, "mu/compressor")
@deftypefnx {Function File} {@var{y} =} compand (@var{x}, @var{mu}, @var{V}, "mu/expander")
@deftypefnx {Function File} {@var{y} =} compand (@var{x}, @var{mu}, @var{V}, "A/compressor")
@deftypefnx {Function File} {@var{y} =} compand (@var{x}, @var{mu}, @var{V}, "A/expander")

Compresses and expanding the dynamic range of a signal using a mu-law or
or A-law algorithm

The mu-law compressor/expander for reducing the dynamic range, is used
if the fourth argument of @code{compand} starts with "mu/". Whereas the
A-law compressor/expander is used if @code{compand} starts with "A/"
The mu-law algorithm uses the formulation

@tex
$$
y = {V log (1 + \mu / V \|x\|) \over log (1 + \mu)} sgn(x)
$$
@end tex
@ifnottex
@example
@group

        V log (1 + \mu/V |x|)
    y = -------------------- sgn(x)
            log (1 + \mu)

@end group
@end example
@end ifnottex

while the A-law algorithm used the formulation

@tex
$$
y = { \left\{  \matrix{ {A / (1 + log A) x}, & 0 <= \|x\| <= V/A \cr
                 & \cr
                 {V log (1 + log(A/V \|x\|) ) \over 1 + logA}, &
                 V/A < \|x\| <= V} \right. }
$$
@end tex
@ifnottex
@example
@group

        /    A / (1 + log A) x,               0 <= |x| <= V/A
        |
    y = <    V ( 1 + log (A/V |x|) )
        |    ----------------------- sgn(x),  V/A < |x| <= V
        \        1 + log A
@end group
@end example
@end ifnottex

Neither converts from or to audio file ulaw format. Use mu2lin or lin2mu
instead

See also: m2ulin, lin2mu
@end deftypefn



@node conv, convenc, compand, Function Reference
@subsection conv

@deftypefn {Function File} {} conv (@var{a}, @var{b})
Convolve two Galois vectors

@code{y = conv (a, b)} returns a vector of length equal to
@code{length (a) + length (b) - 1}
If @var{a} and @var{b} are polynomial coefficient vectors, @code{conv}
returns the coefficients of the product polynomial
See also: deconv
@end deftypefn



@node convenc, convmtx, conv, Function Reference
@subsection convenc

@deftypefn  {Function File} {@var{y} =} convenc (@var{msg}, @var{t})
@deftypefnx {Function File} {@var{y} =} convenc (@var{msg}, @var{t}, @var{punct})
@deftypefnx {Function File} {@var{y} =} convenc (@var{msg}, @var{t}, @var{punct}, @var{s0})
@deftypefnx {Function File} {[@var{y}, @var{state_end}] =} convenc (@dots{})
Encode the binary vector @var{msg} with the convolutional encoder
described by the trellis structure @var{t}

The rate @math{k/n} convolutional encoder encodes @math{k} bits at a
time from the input vector and produces @math{n} bits at a time into the
output vector.  The input @var{msg} must have a length that is a multiple
of @math{k}

If the initial state @var{s0} is specified, it indicates the internal
state of the encoder when the first @math{k} input bits are fed in.  The
default value of @var{s0} is 0

The optional output argument @var{state_end} indicates the internal state
of the encoder after the last bits are encoded.  This allows the state of
the encoder to be saved and applied to the next call to @code{convenc} to
process data in blocks

See also: poly2trellis
@end deftypefn



@node convmtx, cosets, convenc, Function Reference
@subsection convmtx

@deftypefn {Function File} {} convmtx (@var{a}, @var{n})

Create matrix to perform repeated convolutions with the same vector
in a Galois Field. If @var{a} is a column vector and @var{x} is a
column vector of length @var{n}, in a Galois Field then

@code{convmtx (@var{a}, @var{n}) * @var{x}}

gives the convolution of of @var{a} and @var{x} and is the
same as @code{conv (@var{a}, @var{x})}. The difference is if
many vectors are to be convolved with the same vector, then
this technique is possibly faster

Similarly, if @var{a} is a row vector and @var{x} is a row
vector of length @var{n}, then

@code{@var{x} * convmtx (@var{a}, @var{n})}

is the same as @code{conv (@var{x}, @var{a})}
See also: conv
@end deftypefn



@node cosets, cyclgen, convmtx, Function Reference
@subsection cosets

@deftypefn {Function File} {} cosets (@var{m}, @var{prim})

Finds the elements of GF(2^@var{m}) with primitive polynomial @var{prim},
that share the same minimum polynomial. Returns a cell array of the
partitioning of GF(2^@var{m})
@end deftypefn



@node cyclgen, cyclpoly, cosets, Function Reference
@subsection cyclgen

@deftypefn  {Loadable Function} {@var{h} =} cyclgen (@var{n}, @var{p})
@deftypefnx {Loadable Function} {@var{h} =} cyclgen (@var{n}, @var{p}, @var{typ})
@deftypefnx {Loadable Function} {[@var{h}, @var{g}] =} cyclgen (@dots{})
@deftypefnx {Loadable Function} {[@var{h}, @var{g}, @var{k}] =} cyclgen (@dots{})
Produce the parity check and generator matrix of a cyclic code. The parity
check matrix is returned as a @var{m} by @var{n} matrix, representing the
[@var{n},@var{k}] cyclic code. @var{m} is the order of the generator
polynomial @var{p} and the message length @var{k} is given by
@code{@var{n} - @var{m}}.

The generator polynomial can either be a vector of ones and zeros,
and length @var{m} representing,
@tex
$$ p_0 + p_1 x + p_2 x^2 + \cdots + p_m x^{m-1} $$
@end tex
@ifnottex

@example
@var{p}(1) + @var{p}(2) * x + @var{p}(3) * x^2 + ... + @var{p}(@var{m}) * x^(m-1)
@end example
@end ifnottex

The terms of the polynomial are stored least-significant term first.
Alternatively, @var{p} can be an integer representation of the same
polynomial.

The form of the parity check matrix is determined by @var{typ}. If
@var{typ} is 'system', a systematic parity check matrix is produced. If
@var{typ} is 'nosys' and non-systematic parity check matrix is produced.

If requested @code{cyclgen} also returns the @var{k} by @var{n} generator
matrix @var{g}.
See also: hammgen, gen2par, cyclpoly
@end deftypefn



@node cyclpoly, de2bi, cyclgen, Function Reference
@subsection cyclpoly

@deftypefn  {Loadable Function} {@var{y} =} cyclpoly (@var{n}, @var{k})
@deftypefnx {Loadable Function} {@var{y} =} cyclpoly (@var{n}, @var{k}, @var{opt})
@deftypefnx {Loadable Function} {@var{y} =} cyclpoly (@var{n}, @var{k}, @var{opt}, @var{rep})
This function returns the cyclic generator polynomials of the code
[@var{n},@var{k}]. By default the polynomial with the smallest weight
is returned. However this behavior can be overridden with the @var{opt}
flag. Valid values of @var{opt} are:

@table @asis
@item @code{"all\"}
Returns all of the polynomials of the code [@var{n},@var{k}]
@item @code{\"min\"}
Returns the polynomial of minimum weight of the code [@var{n},@var{k}]
@item @code{\"max\"}
Returns the polynomial of the maximum weight of the code [@var{n},@var{k}]
@item @var{l}
Returns the polynomials having exactly the weight @var{l}
@end table

The polynomials are returns as row-vectors in the variable @var{y}. Each
row of @var{y} represents a polynomial with the least-significant term
first. The polynomials can be returned with an integer representation
if @var{rep} is @code{\"integer\"}. The default behavior is given if @var{rep}
is @code{\"polynomial\"}.
See also: gf, isprimitive
@end deftypefn



@node de2bi, decode, cyclpoly, Function Reference
@subsection de2bi

@deftypefn  {Function File} {@var{b} =} de2bi (@var{d})
@deftypefnx {Function File} {@var{b} =} de2bi (@var{d}, @var{n})
@deftypefnx {Function File} {@var{b} =} de2bi (@var{d}, @var{n}, @var{p})
@deftypefnx {Function File} {@var{b} =} de2bi (@var{d}, @var{n}, @var{p}, @var{f})

Convert a non-negative integer to bit vector

The variable @var{d} must be a vector of non-negative integers. @code{de2bi}
then returns a matrix where each row represents the binary representation
of elements of @var{d}. If @var{n} is defined then the returned matrix
will have @var{n} columns. This number of columns can be either larger
than the minimum needed and zeros will be added to the msb of the
binary representation or smaller than the minimum in which case the
least-significant part of the element is returned

If @var{p} is defined then it is used as the base for the decomposition
of the returned values. That is the elements of the returned value are
between '0' and 'p-1'

The variable @var{f} defines whether the first or last element of @var{b}
is considered to be the most-significant. Valid values of @var{f} are
"right-msb" or "left-msb". By default @var{f} is "right-msb"

See also: bi2de
@end deftypefn



@node decode, deconv, de2bi, Function Reference
@subsection decode

@deftypefn  {Function File} {@var{msg} =} decode (@var{code}, @var{n}, @var{k})
@deftypefnx {Function File} {@var{msg} =} decode (@var{code}, @var{n}, @var{k}, @var{typ})
@deftypefnx {Function File} {@var{msg} =} decode (@var{code}, @var{n}, @var{k}, @var{typ}, @var{opt1})
@deftypefnx {Function File} {@var{msg} =} decode (@var{code}, @var{n}, @var{k}, @var{typ}, @var{opt1}, @var{opt2})
@deftypefnx {Function File} {[@var{msg}, @var{err}] =} decode (@dots{})
@deftypefnx {Function File} {[@var{msg}, @var{err}, @var{ccode}] =} decode (@dots{})
@deftypefnx {Function File} {[@var{msg}, @var{err}, @var{ccode}, @var{cerr}] =} decode (@dots{})

Top level block decoder. This function makes use of the lower level
functions such as @code{cyclpoly}, @code{cyclgen}, @code{hammgen}, and
@code{bchenco}. The coded message to decode is pass in @var{code}, the
codeword length is @var{n} and the message length is @var{k}. This
function is used to decode messages using either:

@table @asis
@item A [n,k] linear block code defined by a generator matrix
@item A [n,k] cyclic code defined by a generator polynomial
@item A [n,k] Hamming code defined by a primitive polynomial
@item A [n,k] BCH code code defined by a generator polynomial
@end table

The type of coding to use is defined by the variable @var{typ}. This
variable is a string taking one of the values

@table @code
@item  "linear"
@itemx "linear/binary"
A linear block code is assumed with the message @var{msg} being in a
binary format. In this case the argument @var{opt1} is the generator
matrix, and is required. Additionally, @var{opt2} containing the
syndrome lookup table (see @code{syndtable}) can also be passed
@item  "cyclic"
@itemx "cyclic/binary"
A cyclic code is assumed with the message @var{msg} being in a binary
format. The generator polynomial to use can be defined in @var{opt1}
The default generator polynomial to use will be
@code{cyclpoly (@var{n}, @var{k})}. Additionally, @var{opt2} containing the
syndrome lookup table (see @code{syndtable}) can also be passed
@item  "hamming"
@itemx "hamming/binary"
A Hamming code is assumed with the message @var{msg} being in a binary
format. In this case @var{n} must be of an integer of the form
@code{2^@var{m}-1}, where @var{m} is an integer. In addition @var{k}
must be @code{@var{n}-@var{m}}. The primitive polynomial to use can
be defined in @var{opt1}. The default primitive polynomial to use is
the same as defined by @code{hammgen}. The variable @var{opt2} should
not be defined
@item  "bch"
@itemx "bch/binary"
A BCH code is assumed with the message @var{msg} being in a binary
format. The primitive polynomial to use can be defined in @var{opt2}
The error correction capability of the code can also be defined in
@var{opt1}. Use the empty matrix [] to let the error correction
capability take the default value
@end table

In addition the argument "binary" above can be replaced with "decimal",
in which case the message is assumed to be a decimal vector, with each
value representing a symbol to be coded. The binary format can be in two
forms

@table @code
@item An @var{x}-by-@var{n} matrix
Each row of this matrix represents a symbol to be decoded
@item A vector with length divisible by @var{n}
The coded symbols are created from groups of @var{n} elements of this vector
@end table

The decoded message is return in @var{msg}. The number of errors encountered
is returned in @var{err}. If the coded message format is "decimal" or a
"binary" matrix, then @var{err} is a column vector having a length equal
to the number of decoded symbols. If @var{code} is a "binary" vector, then
@var{err} is the same length as @var{msg} and indicated the number of
errors in each symbol. If the value @var{err} is positive it indicates the
number of errors corrected in the corresponding symbol. A negative value
indicates an uncorrectable error. The corrected code is returned in
@var{ccode} in a similar format to the coded message @var{msg}. The
variable @var{cerr} contains similar data to @var{err} for @var{ccode}

It should be noted that all internal calculations are performed in the
binary format. Therefore for large values of @var{n}, it is preferable
to use the binary format to pass the messages to avoid possible rounding
errors. Additionally, if repeated calls to @code{decode} will be performed,
it is often faster to create a generator matrix externally with the
functions @code{hammgen} or @code{cyclgen}, rather than let @code{decode}
recalculate this matrix at each iteration. In this case @var{typ} should
be "linear". The exception to this case is BCH codes, where the required
syndrome table is too large. The BCH decoder, decodes directly from the
polynomial never explicitly forming the syndrome table

See also: encode, cyclgen, cyclpoly, hammgen, bchdeco, bchpoly, syndtable
@end deftypefn



@node deconv, deintrlv, decode, Function Reference
@subsection deconv

@deftypefn {Function File} {} deconv (@var{y}, @var{a})
Deconvolve two Galois vectors

@code{[b, r] = deconv (y, a)} solves for @var{b} and @var{r} such that
@code{y = conv (a, b) + r}

If @var{y} and @var{a} are polynomial coefficient vectors, @var{b} will
contain the coefficients of the polynomial quotient and @var{r} will be
a remainder polynomial of lowest order
See also: conv
@end deftypefn



@node deintrlv, demodmap, deconv, Function Reference
@subsection deintrlv

@deftypefn {Function File} {@var{deintrlvd} =} deintrlv (@var{data}, @var{elements})
Restore elements of @var{data} according to @var{elements}
See also: intrlv
@end deftypefn



@node demodmap, det, deintrlv, Function Reference
@subsection demodmap

@deftypefn  {Function File} {z =} demodmap (@var{y}, @var{fd}, @var{fs}, "ask", @var{m})
@deftypefnx {Function File} {z =} demodmap (@var{y}, @var{fd}, @var{fs}, "fsk", @var{m}, @var{tone})
@deftypefnx {Function File} {z =} demodmap (@var{y}, @var{fd}, @var{fs}, "msk")
@deftypefnx {Function File} {z =} demodmap (@var{y}, @var{fd}, @var{fs}, "psk", @var{m})
@deftypefnx {Function File} {z =} demodmap (@var{y}, @var{fd}, @var{fs}, "qask", @var{m})
@deftypefnx {Function File} {z =} demodmap (@var{y}, @var{fd}, @var{fs}, "qask/cir", @var{nsig}, @var{amp}, @var{phs})
@deftypefnx {Function File} {z =} demodmap (@var{y}, @var{fd}, @var{fs}, "qask/arb", @var{inphase}, @var{quadr})
@deftypefnx {Function File} {z =} demodmap (@var{y}, @var{fd}, @var{fs}, "qask/arb", @var{map})
@deftypefnx {Function File} {z =} demodmap (@var{y}, [@var{fd}, @var{off}], @dots{})

Demapping of an analog signal to a digital signal. The function
@code{demodmap} must have at least three input arguments and one output
argument. Argument @var{y} is a complex variable representing the analog
signal to be demapped. The variables @var{fd} and @var{fs} are the
sampling rate of the of digital signal and the sampling rate of the
analog signal respectively. It is required that @code{@var{fs}/@var{fd}}
is an integer

The available mapping of the digital signal are

@table @asis
@item "ask"
Amplitude shift keying
@item "fsk"
Frequency shift keying
@item "msk"
Minimum shift keying
@item "psk"
Phase shift keying
@item  "qask"
@itemx "qsk"
@itemx "qam"
Quadrature amplitude shift keying
@end table

In addition the "qask", "qsk" and "qam" method can be modified with the
flags "/cir" or "/arb". That is "qask/cir" and "qask/arb", etc are valid
methods and give circular- and arbitrary-qask mappings respectively. Also
the method "fsk" and "msk" can be modified with the flag "/max", in which
case @var{y} is assumed to be a matrix with @var{m} columns, representing
the symbol correlations

The variable @var{m} is the order of the modulation to use. By default
this is 2, and in general should be specified

For "qask/cir", the additional arguments are the same as for
@code{apkconst}, and you are referred to @code{apkconst} for the definitions
of the additional variables

For "qask/arb", the additional arguments @var{inphase} and @var{quadr} give
the in-phase and quadrature components of the mapping, in a similar mapping
to the outputs of @code{qaskenco} with one argument. Similar @var{map}
represents the in-phase and quadrature components of the mapping as
the real and imaginary parts of the variable @var{map}
See also: modmap, ddemodce, ademodce, apkconst, qaskenco
@end deftypefn



@node det, dftmtx, demodmap, Function Reference
@subsection det

@deftypefn {Loadable Function} {@var{d} =} det (@var{a})
Compute the determinant of the Galois array @var{a}
@end deftypefn



@node dftmtx, diag, det, Function Reference
@subsection dftmtx

@deftypefn {Function File} {@var{d} =} dftmtx (@var{a})

Form a matrix, that can be used to perform Fourier transforms in
a Galois Field

Given that @var{a} is an element of the Galois Field GF(2^m), and
that the minimum value for @var{k} for which @code{@var{a} ^ @var{k}}
is equal to one is @code{2^m - 1}, then this function produces a
@var{k}-by-@var{k} matrix representing the discrete Fourier transform
over a Galois Field with respect to @var{a}. The Fourier transform of
a column vector is then given by @code{dftmtx (@var{a}) * @var{x}}

The inverse Fourier transform is given by @code{dftmtx (1 / @var{a})}
@end deftypefn



@node diag, dpcmdeco, dftmtx, Function Reference
@subsection diag

@deftypefn {Loadable Function} {} diag (@var{v}, @var{k})
Return a diagonal matrix with Galois vector @var{v} on diagonal @var{k}
The second argument is optional.  If it is positive, the vector is placed on
the @var{k}-th super-diagonal.  If it is negative, it is placed on the
@var{-k}-th sub-diagonal.  The default value of @var{k} is 0, and the
vector is placed on the main diagonal.  For example,

@example
diag (gf ([1, 2, 3], 2), 1)
ans =
GF(2^2) array. Primitive Polynomial = D^2+D+1 (decimal 7)

Array elements =

   0   1   0   0
   0   0   2   0
   0   0   0   3
   0   0   0   0

@end example
@end deftypefn



@node dpcmdeco, dpcmenco, diag, Function Reference
@subsection dpcmdeco

@deftypefn {Function File} {@var{sig} =} dpcmdeco (@var{indx}, @var{codebook}, @var{predictor})
Decode using differential pulse code modulation (DPCM)

@table @code
@item sig = dpcmdeco (indx, codebook, predictor)
Decode the signal coded by DPCM
Use the prediction model and the coded prediction error given by a codebook and
the index of each sample in this codebook

@end table
See also: dpcmenco, dpcmopt
@end deftypefn



@node dpcmenco, dpcmopt, dpcmdeco, Function Reference
@subsection dpcmenco

@deftypefn  {Function File} {@var{qidx} =} dpcmenco (@var{sig}, @var{codebook}, @var{partition}, @var{predictor})
@deftypefnx {Function File} {[@var{qidx}, @var{q}] =} dpcmenco (@var{sig}, @var{codebook}, @var{partition}, @var{predictor})
@deftypefnx {Function File} {[@var{qidx}, @var{q}, @var{d}] =} dpcmenco (@dots{})
Encode using differential pulse code modulation (DPCM)

@table @code
@item qidx = dpcmenco (sig, codebook, partition, predictor)
Determine position of the prediction error in a strictly monotonic table (partition)
The predictor vector describes a m-th order prediction for the
output according to the following equation
y(k) = p(1)sig(k-1) + p(2)sig(k-2) + ... + p(m-1)sig(k-m+1) + p(m)sig(k-m) ,
where the predictor vector is given by
predictor = [0, p(1), p(2), p(3),..., p(m-1), p(m)]

@item [qidx, q] = dpcmenco (sig, codebook, partition, predictor)
Also return the quantized values

@item [qidx, q, d] = dpcmenco (...)
Also compute distortion: mean squared distance of original sig from the
corresponding quantized values

@end table
See also: dpcmdeco, dpcmopt, quantiz
@end deftypefn



@node dpcmopt, egolaydec, dpcmenco, Function Reference
@subsection dpcmopt

@deftypefn  {Function File} {@var{predictor} =} dpcmopt (@var{training_set}, @var{ord})
@deftypefnx {Function File} {[@var{predictor}, @var{partition}, @var{codebook}] =} dpcmopt (@var{training_set}, @var{ord}, @var{cb})
Optimize the DPCM parameters and codebook

It uses the Levinson-Durbin algorithm to find the all-pole IIR filter
using the autocorrelation sequence. After the best predictor is found,
it uses the Lloyds algorithm to find the best codebook and partition
for the interval

@table @code
@item predictor = dpcmopt (training_set, ord)
Optimize the DPCM parameters using the Levinson-Durbin algorithm
The predictor vector describes a m-th order prediction for the
output according to the following equation
y(k) = p(1)sig(k-1) + p(2)sig(k-2) + ... + p(m-1)sig(k-m+1) + p(m)sig(k-m)
where the predictor vector is given by
predictor = [0, p(1), p(2), p(3),..., p(m-1), p(m)]

training_set is the training data used to find the best predictor

ord is the order of the desired prediction model

@item [predictor, partition, codebook] = dpcmopt (training_set,ord,cb)
Optimize the DPCM parameters and also uses the Lloyds algorithm to find
the best codebook and partition for the given training signal

cb might be the initial codebook used by Lloyds algorithm or
the length of the desired codebook

@end table
See also: dpcmenco, dpcmdeco, levinson, lloyds
@end deftypefn



@node egolaydec, egolayenc, dpcmopt, Function Reference
@subsection egolaydec

@deftypefn {Function File} {[@var{C}, @var{err}] =} egolaydec (@var{R})
Decode Extended Golay code

Given @var{R}, the received Extended Golay code, this function tries to
decode it using the Extended Golay code parity check matrix
Extended Golay code (24,12) which can correct up to 3 errors

The received code @var{R}, needs to be of length Nx24, for encoding. We can
decode several codes at once, if they are stacked as a matrix of 24 columns,
each code in a separate row

The generator used in here is same as obtained from the function
@code{egolaygen}

The function returns @var{C}, the error-corrected code word from the received
word. If decoding failed, @var{err} value is 1, otherwise it is 0

Extended Golay code (24,12) which can correct up to 3
errors. Decoding algorithm follows from Lin & Costello

Ref: Lin & Costello, pg 128, Ch4, "Error Control Coding", 2nd ed, Pearson

@example
@group
msg = rand (10, 12) > 0.5;
c1 = egolayenc (msg);
c1(:,1) = mod (c1(:,1) + 1, 2)
c2 = egolaydec (c1)
@end group
@end example

See also: egolaygen, egolayenc
@end deftypefn



@node egolayenc, egolaygen, egolaydec, Function Reference
@subsection egolayenc

@deftypefn {Function File} {@var{C} =} egolayenc (@var{M})
Encode with Extended Golay code

The message @var{M}, needs to be of size Nx12, for encoding
We can encode several messages, into codes at once, if they
are stacked in the order suggested

The generator used in here is same as obtained from the
function @code{egolaygen}. Extended Golay code (24,12) which can correct
up to 3 errors

@example
@group
msg = rand (10, 12) > 0.5;
c = egolayenc (msg)
@end group
@end example

See also: egolaygen, egolaydec
@end deftypefn



@node egolaygen, encode, egolayenc, Function Reference
@subsection egolaygen

@deftypefn {Function File} {[@var{G}, @var{P}] =} egolaygen ()
Extended Golay code generator matrix

Returns @var{G}, the Extended Golay code (24,12) generator matrix,
which can correct up to 3 errors. @var{P} is the parity
check matrix, for this code

See also: egolaydec, egolayenc
@end deftypefn



@node encode, exp, egolaygen, Function Reference
@subsection encode

@deftypefn  {Function File} {@var{code} =} encode (@var{msg}, @var{n}, @var{k})
@deftypefnx {Function File} {@var{code} =} encode (@var{msg}, @var{n}, @var{k}, @var{typ})
@deftypefnx {Function File} {@var{code} =} encode (@var{msg}, @var{n}, @var{k}, @var{typ}, @var{opt})
@deftypefnx {Function File} {[@var{code}, @var{added}] =} encode (@dots{})

Top level block encoder. This function makes use of the lower level
functions such as @code{cyclpoly}, @code{cyclgen}, @code{hammgen}, and
@code{bchenco}. The message to code is pass in @var{msg}, the
codeword length is @var{n} and the message length is @var{k}. This
function is used to encode messages using either:

@table @asis
@item A [n,k] linear block code defined by a generator matrix
@item A [n,k] cyclic code defined by a generator polynomial
@item A [n,k] Hamming code defined by a primitive polynomial
@item A [n,k] BCH code code defined by a generator polynomial
@end table

The type of coding to use is defined by the variable @var{typ}. This
variable is a string taking one of the values

@table @code
@item  "linear"
@itemx "linear/binary"
A linear block code is assumed with the coded message @var{code} being in
a binary format. In this case the argument @var{opt} is the generator
matrix, and is required
@item  "cyclic"
@itemx "cyclic/binary"
A cyclic code is assumed with the coded message @var{code} being in a
binary format. The generator polynomial to use can be defined in @var{opt}
The default generator polynomial to use will be
@code{cyclpoly (@var{n}, @var{k})}
@item  "hamming"
@itemx "hamming/binary"
A Hamming code is assumed with the coded message @var{code} being in a
binary format. In this case @var{n} must be of an integer of the form
@code{2^@var{m}-1}, where @var{m} is an integer. In addition @var{k}
must be @code{@var{n}-@var{m}}. The primitive polynomial to use can
be defined in @var{opt}. The default primitive polynomial to use is
the same as defined by @code{hammgen}
@item  "bch"
@itemx "bch/binary"
A BCH code is assumed with the coded message @var{code} being in a binary
format. The generator polynomial to use can be defined in @var{opt}
The default generator polynomial to use will be
@code{bchpoly (@var{n}, @var{k})}
@end table

In addition the argument "binary" above can be replaced with "decimal",
in which case the message is assumed to be a decimal vector, with each
value representing a symbol to be coded. The binary format can be in two
forms

@table @code
@item An @var{x}-by-@var{k} matrix
Each row of this matrix represents a symbol to be coded
@item A vector
The symbols are created from groups of @var{k} elements of this vector
If the vector length is not divisible by @var{k}, then zeros are added
and the number of zeros added is returned in @var{added}
@end table

It should be noted that all internal calculations are performed in the
binary format. Therefore for large values of @var{n}, it is preferable
to use the binary format to pass the messages to avoid possible rounding
errors. Additionally, if repeated calls to @code{encode} will be performed,
it is often faster to create a generator matrix externally with the
functions @code{hammgen} or @code{cyclgen}, rather than let @code{encode}
recalculate this matrix at each iteration. In this case @var{typ} should
be "linear". The exception to this case is BCH codes, whose encoder
is implemented directly from the polynomial and is significantly faster

See also: decode, cyclgen, cyclpoly, hammgen, bchenco, bchpoly
@end deftypefn



@node exp, eyediagram, encode, Function Reference
@subsection exp

@deftypefn {Loadable Function} {} exp (@var{x})
Compute the anti-logarithm for each element of @var{x} for a Galois
array
@end deftypefn



@node eyediagram, fft, exp, Function Reference
@subsection eyediagram

@deftypefn  {Function File} {} eyediagram (@var{x}, @var{n})
@deftypefnx {Function File} {} eyediagram (@var{x}, @var{n}, @var{per})
@deftypefnx {Function File} {} eyediagram (@var{x}, @var{n}, @var{per}, @var{off})
@deftypefnx {Function File} {} eyediagram (@var{x}, @var{n}, @var{per}, @var{off}, @var{str})
@deftypefnx {Function File} {} eyediagram (@var{x}, @var{n}, @var{per}, @var{off}, @var{str}, @var{h})
@deftypefnx {Function File} {@var{h} =} eyediagram (@dots{})

Plot the eye-diagram of a signal. The signal @var{x} can be either in one
of three forms

@table @asis
@item A real vector
In this case the signal is assumed to be real and represented by the vector
@var{x}. A single eye-diagram representing this signal is plotted
@item A complex vector
In this case the in-phase and quadrature components of the signal are
plotted separately
@item A matrix with two columns
In this case the first column represents the in-phase and the second the
quadrature components of a complex signal
@end table

Each line of the eye-diagram has @var{n} elements and the period is assumed
to be given by @var{per}. The time axis is then [-@var{per}/2 @var{per}/2]
By default @var{per} is 1

By default the signal is assumed to start at -@var{per}/2. This can be
overridden by the @var{off} variable, which gives the number of samples
to delay the signal

The string @var{str} is a plot style string (example "r+"),
and by default is the default gnuplot line style

The figure handle to use can be defined by @var{h}. If @var{h} is not
given, then the next available figure handle is used. The figure handle
used in returned on @var{hout}
See also: scatterplot
@end deftypefn



@node fft, fibodeco, eyediagram, Function Reference
@subsection fft

@deftypefn {Function File} {} fft (@var{x})

If @var{x} is a column vector, finds the FFT over the primitive element
of the Galois Field of @var{x}. If @var{x} is in the Galois Field
GF(2^@var{m}), then @var{x} must have @code{2^@var{m} - 1} elements
@end deftypefn



@node fibodeco, fiboenco, fft, Function Reference
@subsection fibodeco

@deftypefn {Function File} {} fibodeco (@var{code})

Returns the decoded Fibonacci value from the binary vectors @var{code}
Universal codes like Fibonacci codes have a useful synchronization property,
only for 255 maximum value we have designed these routines. We assume
user has partitioned the code into several unique segments based on
the suffix property of unique strings "11" and we just decode the
parts. Partitioning the stream is as simple as identifying the
"11" pairs that occur, at the terminating ends. This system implements
the standard binary Fibonacci codes, which means that row vectors
can only contain 0 or 1. Ref: @url{http://en.wikipedia.org/wiki/Fibonacci_coding}

@example
@group
fibodeco (@{[0 1 0 0 1 1]@})
    @result{} 10
fibodeco (@{[1 1], [0 1 1], [0 0 1 1], [1 0 1 1]@})
    @result{} [1, 2, 3, 4]
@end group
@end example
See also: fiboenco
@end deftypefn



@node fiboenco, fibosplitstream, fibodeco, Function Reference
@subsection fiboenco

@deftypefn {Function File} {} fiboenco (@var{num})

Returns the cell-array of encoded Fibonacci value from the column vectors @var{num}
Universal codes like Fibonacci codes have a useful synchronization
property, only for 255 maximum value we have designed these routines. We assume
user has partitioned the code into several unique segments based on
the suffix property of unique elements [1 1] and we just decode the
parts. Partitioning the stream is as simple as identifying the [1 1]
pairs that occur, at the terminating ends. This system implements
the standard binary Fibonacci codes, which means that row vectors
can only contain 0 or 1. Ref: http://en.wikipedia.org/wiki/Fibonacci_coding
Ugly O(k.N^2) encoder.Ref: Wikipedia article accessed March, 2006
@url{http://en.wikipedia.org/wiki/Fibonacci_coding},  UCI Data Compression
Book, @url{http://www.ics.uci.edu/~dan/pubs/DC-Sec3.html}, (accessed
October 2006)

@example
@group
fiboenco (10)
    @result{} @{[ 0 1 0 0 1 1]@}
fiboenco (1:4)
    @result{} @{[1 1], [0 1 1], [0 0 1 1], [1 0 1 1]@}
@end group
@end example
See also: fibodeco
@end deftypefn



@node fibosplitstream, filter, fiboenco, Function Reference
@subsection fibosplitstream

@deftypefn {Function File} {} fibosplitstream (@var{code})

Returns the split data stream at the word boundaries
Assuming the stream was originally encoded using @code{fiboenco}
and this routine splits the stream at the points where "11"
occur together & gives us the code-words which
can later be decoded from the @code{fibodeco} This however doesn't
mean that we intend to verify if all the codewords are correct,
and in fact the last symbol in the return list can or can not be
a valid codeword

A example use of @code{fibosplitstream} would be
@example
@group
fibodeco (fibosplitstream ([fiboenco(randint (1, 100, [0, 255]))@{:@}]))
fibodeco (fibosplitstream ([fiboenco(1:10)@{:@}]))
@end group
@end example
See also: fiboenco, fibodeco
@end deftypefn



@node filter, fmdemod, fibosplitstream, Function Reference
@subsection filter

@deftypefn  {Loadable Function} {y =} filter (@var{b}, @var{a}, @var{x})
@deftypefnx {Loadable Function} {[@var{y}, @var{sf}] =} filter (@var{b}, @var{a}, @var{x}, @var{si})

Digital filtering of vectors in a Galois Field. Returns the solution to
the following linear, time-invariant difference equation over a Galois
Field:
@tex
$$
\sum_{k=0}^N a_{k+1} y_{n-k} = \sum_{k=0}^M b_{k+1} x_{n-k}, \qquad
 1 \le n \le P
$$
@end tex
@ifnottex

@smallexample
@group
   N                   M
  SUM a(k+1) y(n-k) = SUM b(k+1) x(n-k)      for 1<=n<=length(x)
  k=0                 k=0
@end group
@end smallexample
@end ifnottex

@noindent
where
@tex
 $a \in \Re^{N-1}$, $b \in \Re^{M-1}$, and $x \in \Re^P$
@end tex
@ifnottex
 N=length(a)-1 and M=length(b)-1
@end ifnottex
An equivalent form of this equation is:
@tex
$$
y_n = -\sum_{k=1}^N c_{k+1} y_{n-k} + \sum_{k=0}^M d_{k+1} x_{n-k}, \qquad
 1 \le n \le P
$$
@end tex
@ifnottex

@smallexample
@group
            N                   M
  y(n) = - SUM c(k+1) y(n-k) + SUM d(k+1) x(n-k)  for 1<=n<=length(x)
           k=1                 k=0
@end group
@end smallexample
@end ifnottex

@noindent
where
@tex
$c = a/a_1$ and $d = b/a_1$
@end tex
@ifnottex
 c = a/a(1) and d = b/a(1)
@end ifnottex

If the fourth argument @var{si} is provided, it is taken as the
initial state of the system and the final state is returned as
@var{sf}.  The state vector is a column vector whose length is
equal to the length of the longest coefficient vector minus one
If @var{si} is not supplied, the initial state vector is set to all
zeros
@end deftypefn



@node fmdemod, fmmod, filter, Function Reference
@subsection fmdemod

@deftypefn {Function File} {} fmdemod (@var{x}, @var{fc}, @var{fs})
Create the FM demodulation of the signal x with carrier frequency fs
Where x is sample at frequency fs
See also: ammod, amdemod, fmmod
@end deftypefn



@node fmmod, gen2par, fmdemod, Function Reference
@subsection fmmod

@deftypefn {Function File} {} fmmod (@var{x}, @var{fc}, @var{fs})
Create the FM modulation of the signal x with carrier frequency fs. Where
x is sample at frequency fs
See also: ammod, fmdemod, amdemod
@end deftypefn



@node gen2par, genqamdemod, fmmod, Function Reference
@subsection gen2par

@deftypefn  {Function File} {@var{par} =} gen2par (@var{gen})
@deftypefnx {Function File} {@var{gen} =} gen2par (@var{par})

Converts binary generator matrix @var{gen} to the parity check matrix
@var{par} and visa-versa. The input matrix must be in standard form
That is a generator matrix must be k-by-n and in the form [eye(k) P]
or [P eye(k)], and the parity matrix must be (n-k)-by-n and of the
form [eye(n-k) P'] or [P' eye(n-k)]

See also: cyclgen, hammgen
@end deftypefn



@node genqamdemod, genqammod, gen2par, Function Reference
@subsection genqamdemod

@deftypefn {Loadable Function} {@var{y} =} genqamdemod (@var{x}, @var{C})
General quadrature amplitude demodulation. The complex envelope
quadrature amplitude modulated signal @var{x} is demodulated using a
constellation mapping specified by the 1D vector @var{C}.
@end deftypefn



@node genqammod, gf, genqamdemod, Function Reference
@subsection genqammod

@deftypefn {Function File} {@var{y} =} genqammod (@var{x}, @var{c})

Modulates an information sequence of integers @var{x} in the range
@code{[0 @dots{} M-1]} onto a quadrature amplitude modulated signal
@var{y}, where @code{M = length (c) - 1} and @var{c} is a 1D vector
specifying the signal constellation mapping to be used. An example of
combined 4PAM-4PSK is

@example
@group
d = randint (1, 1e4, 8);
c = [1+j -1+j -1-j 1-j 1+sqrt(3) j*(1+sqrt(3)) -1-sqrt(3) -j*(1+sqrt(3))];
y = genqammod (d, c);
z = awgn (y, 20);
plot (z, "rx")
@end group
@end example
See also: genqamdemod
@end deftypefn



@node gf, gftable, genqammod, Function Reference
@subsection gf

#endif
  "-*- texinfo -*-
\



@node gftable, gfweight, gf, Function Reference
@subsection gftable

@deftypefn {Function File} {} gftable (@var{m}, @var{primpoly})

This function exists for compatibility with matlab. As the Octave Galois
fields store a copy of the lookup tables for every field in use internally,
there is no need to use this function

See also: gf
@end deftypefn



@node gfweight, golombdeco, gftable, Function Reference
@subsection gfweight

@deftypefn  {Function File} {@var{w} =} gfweight (@var{gen})
@deftypefnx {Function File} {@var{w} =} gfweight (@var{gen}, "gen")
@deftypefnx {Function File} {@var{w} =} gfweight (@var{par}, "par")
@deftypefnx {Function File} {@var{w} =} gfweight (@var{p}, n)

Calculate the minimum weight or distance of a linear block code. The
code can be either defined by its generator or parity check matrix, or
its generator polynomial. By default if the first argument is a matrix,
it is assumed to be the generator matrix of the code. The type of the
matrix can be defined by a flag "gen" for the generator matrix or
"par" for the parity check matrix

If the first argument is a vector, it is assumed that it defines the
generator polynomial of the code. In this case a second argument is
required that defines the codeword length

See also: hammgen, cyclpoly, bchpoly
@end deftypefn



@node golombdeco, golombenco, gfweight, Function Reference
@subsection golombdeco

@deftypefn {Function File} {} golombdeco (@var{code}, @var{m})

Returns the Golomb decoded signal vector using @var{code} and @var{m}
Compulsory m is need to be specified. A restrictions is that a
signal set must strictly be non-negative. The value of code
is a cell array of row-vectors which have the encoded Golomb value
for a single sample. The Golomb algorithm is
used to encode the "code" and only that can be meaningfully
decoded. @var{code} is assumed to have been of format generated
by the function @code{golombenco}. Also the parameter @var{m} need to
be a non-zero number, unless which it makes divide-by-zero errors
This function works backward the Golomb algorithm see
@code{golombenco} for more details on that
Reference: Solomon Golomb, Run length Encodings, 1966 IEEE Trans Info Theory

An example of the use of @code{golombdeco} is
@example
@group
golombdeco (golombenco (1:4, 2), 2)
    @result{} [1 2 3 4]
@end group
@end example
See also: golombenco
@end deftypefn



@node golombenco, hammgen, golombdeco, Function Reference
@subsection golombenco

@deftypefn {Function File} {} golombenco (@var{sig}, @var{m})

Returns the Golomb coded signal as cell array
Also total length of output code in bits can be obtained
This function uses a @var{m} need to be supplied for encoding signal vector
into a Golomb coded vector. A restrictions is that
a signal set must strictly be non-negative.  Also the parameter @var{m} need to
be a non-zero number, unless which it makes divide-by-zero errors
The Golomb algorithm [1], is used to encode the data into unary coded
quotient part which is represented as a set of 1's separated from
the K-part (binary) using a zero. This scheme doesn't need any
kind of dictionaries, it is a parameterized prefix codes
Implementation is close to O(N^2), but this implementation
*may be* sluggish, though correct.  Details of the scheme are, to
encode the remainder(r of number N) using the floor(log2(m)) bits
when rem is in range 0:(2^ceil(log2(m)) - N), and encode it as
r+(2^ceil(log2(m)) - N), using total of 2^ceil(log2(m)) bits
in other instance it doesn't belong to case 1. Quotient is coded
simply just using the unary code. Also according to [2] Golomb codes
are optimal for sequences using the Bernoulli probability model:
P(n)=p^n-1.q & p+q=1, and when M=[1/log2(p)], or P=2^(1/M)

Reference: 1. Solomon Golomb, Run length Encodings, 1966 IEEE Trans
Info' Theory. 2. Khalid Sayood, Data Compression, 3rd Edition

An example of the use of @code{golombenco} is
@example
@group
golombenco (1:4, 2)
    @result{} @{[0 1], [1 0 0], [1 0 1], [1 1 0 0]@}
golombenco (1:10, 2)
    @result{} @{[0 1], [1 0 0], [1 0 1], [1 1 0 0],
        [1 1 0 1], [1 1 1 0 0], [1 1 1 0 1], [1 1 1 1 0 0],
        [1 1 1 1 0 1], [1 1 1 1 1 0 0]@}
@end group
@end example
See also: golombdeco
@end deftypefn



@node hammgen, helscanintrlv, golombenco, Function Reference
@subsection hammgen

@deftypefn  {Function File} {@var{h} =} hammgen (@var{m})
@deftypefnx {Function File} {@var{h} =} hammgen (@var{m}, @var{p})
@deftypefnx {Function File} {[@var{h}, @var{g}] =} hammgen (@dots{})
@deftypefnx {Function File} {[@var{h}, @var{g}, @var{n}, @var{k}] =} hammgen (@dots{})

Produce the parity check and generator matrices of a Hamming code. The
variable @var{m} defines the [@var{n},@var{k}] Hamming code where
@code{@var{n} = 2 ^ @var{m} - 1} and @code{@var{k} = @var{n} - @var{m}}
@var{m} must be between 3 and 16

The parity check matrix is generated relative to the primitive polynomial
of GF(2^@var{m}). If @var{p} is specified the default primitive polynomial
of GF(2^@var{m}) is overridden. @var{p} must be a valid primitive
polynomial of the correct order for GF(2^@var{m})

The parity check matrix is returned in the @var{m} by @var{n} matrix
@var{h}, and if requested the generator matrix is returned in the @var{k}
by @var{n} matrix @var{g}

See also: gen2par
@end deftypefn



@node helscanintrlv, huffmandeco, hammgen, Function Reference
@subsection helscanintrlv

@deftypefn {Function File} {@var{outdata} =} helscanintrlv (@var{data}, @var{nrows}, @var{ncols}, @var{Nshift})
@var{nrows}-by-@var{ncols}
See also: helscandeintrlv
@end deftypefn



@node huffmandeco, huffmandict, helscanintrlv, Function Reference
@subsection huffmandeco

@deftypefn {Function File} {@var{sig} =} huffmandeco (@var{hcode}, @var{dict})
Decode signal encoded by @code{huffmanenco}

This function uses a dict built from the
@code{huffmandict} and uses it to decode a signal list into a Huffman
list. A restriction is that @var{hcode} is expected to be a binary code

The returned @var{sig} set that strictly belongs in the range @code{[1,N]}
with @code{N = length (@var{dict})}. Also @var{dict} can only be from the
@code{huffmandict} routine. Whenever decoding fails, those signal values a
re indicated by @code{-1}, and we successively try to restart decoding
from the next bit that hasn't failed in decoding, ad-infinitum. An example
of the use of @code{huffmandeco} is:

@example
@group
hd    = huffmandict (1:4, [0.5 0.25 0.15 0.10]);
hcode = huffmanenco (1:4, hd);
back  = huffmandeco (hcode, hd)
    @result{} [1 2 3 4]
@end group
@end example
See also: huffmandict, huffmanenco
@end deftypefn



@node huffmandict, huffmanenco, huffmandeco, Function Reference
@subsection huffmandict

@deftypefn  {Function File} {} huffmandict (@var{symb}, @var{prob})
@deftypefnx {Function File} {} huffmandict (@var{symb}, @var{prob}, @var{toggle})
@deftypefnx {Function File} {} huffmandict (@var{symb}, @var{prob}, @var{toggle}, @var{minvar})

Builds a Huffman code, given a probability list. The Huffman codes
per symbol are output as a list of strings-per-source symbol. A zero
probability symbol is NOT assigned any codeword as this symbol doesn't
occur in practice anyway

@var{toggle} is an optional argument with values 1 or 0, that starts
building a code based on 1s or 0s, defaulting to 0. Also @var{minvar}
is a boolean value that is useful in choosing if you want to optimize
buffer for transmission in the applications of Huffman coding, however
it doesn't affect the type or average codeword length of the generated
code. An example of the use of @code{huffmandict} is

@example
@group
huffmandict (symbols, [0.5 0.25 0.15 0.1], 1)
    @result{} @{[0], [1 0], [1 1 1], [1 1 0]@}
huffmandict (symbols, 0.25 * ones (1,4), 1)
    @result{} @{[1 1], [1 0], [0 1], [0 0]@}

prob = [0.5 0 0.25 0.15 0.1];
dict = huffmandict (1:5, prob, 1);
entropy (prob)
    @result{} 2.3219
laverage (dict, prob)
    @result{} 1.8500

x = [0.2 0.4 0.2 0.1 0.1];
huffmandict (1, x, 0, true)
    @result{} @{[1 0], [0 0], [1 1], [0 1 0], [0 1 1]@}
huffmandict (1, x)
    @result{} @{[0 1], [1], [0 0 1], [0 0 0 0], [0 0 0 1]@}
@end group
@end example

Reference: Dr.Rao's course EE5351 Digital Video Coding, at UT-Arlington
See also: huffmandeco, huffmanenco
@end deftypefn



@node huffmanenco, ifft, huffmandict, Function Reference
@subsection huffmanenco

@deftypefn {Function File} {} huffmanenco (@var{sig}, @var{dict})

Returns the Huffman encoded signal using @var{dict}. This function uses
a @var{dict} built from the @code{huffmandict} and uses it to encode a
signal list into a Huffman list. A restrictions is that a signal set must
strictly belong in the range @code{[1,N]} with @code{N = length (dict)}
Also @var{dict} can only be from the @code{huffmandict} routine
An example of the use of @code{huffmanenco} is

@example
@group
hd = huffmandict (1:4, [0.5 0.25 0.15 0.10]);
huffmanenco (1:4, hd)
    @result{} [1 0 1 0 0 0 0 0 1]
@end group
@end example
See also: huffmandict, huffmandeco
@end deftypefn



@node ifft, intrlv, huffmanenco, Function Reference
@subsection ifft

@deftypefn {Function File} {} ifft (@var{x})

If @var{x} is a column vector, finds the IFFT over the primitive element
of the Galois Field of @var{x}. If @var{x} is in the Galois Field
GF(2^@var{m}), then @var{x} must have @code{2^@var{m} - 1} elements
See also: ifft
@end deftypefn



@node intrlv, inv, ifft, Function Reference
@subsection intrlv

@deftypefn {Function File} {@var{intrlvd} =} intrlv (@var{data}, @var{elements})
Interleaved elements of @var{data} according to @var{elements}
See also: deintrlv
@end deftypefn



@node inv, inverse, intrlv, Function Reference
@subsection inv

@deftypefn {Loadable Function} {[@var{x}, @var{rcond}] =} inv (@var{a})
Compute the inverse of the square matrix @var{a}.  Return an estimate
of the reciprocal condition number if requested, otherwise warn of an
ill-conditioned matrix if the reciprocal condition number is small
@end deftypefn



@node inverse, isequal, inv, Function Reference
@subsection inverse

@deftypefn {Loadable Function} {[@var{x}, @var{rcond}] =} inverse (@var{a})
See inv
@end deftypefn



@node isequal, isgalois, inverse, Function Reference
@subsection isequal

@deftypefn {Function File} {} isequal (@var{x1}, @var{x2}, @dots{})
Return true if all of @var{x1}, @var{x2}, @dots{} are equal
See also: isequalwithequalnans
@end deftypefn



@node isgalois, isprimitive, isequal, Function Reference
@subsection isgalois

@deftypefn {Loadable Function} {} isgalois (@var{expr})
Return 1 if the value of the expression @var{expr} is a Galois Field.
@end deftypefn



@node isprimitive, istrellis, isgalois, Function Reference
@subsection isprimitive

@deftypefn {Loadable Function} {@var{y} =} isprimitive (@var{a})
Returns 1 is the polynomial represented by @var{a} is a primitive
polynomial of GF(2). Otherwise it returns zero.

See also: gf, primpoly
@end deftypefn



@node istrellis, lloyds, isprimitive, Function Reference
@subsection istrellis

@deftypefn  {Function File} {} istrellis (@var{t})
@deftypefnx {Function File} {[@var{status}, @var{text}] =} istrellis (@var{t})

Return true if @var{t} is a valid trellis structure

If called with two output arguments, @var{text} contains a string indicating
a reason if @var{status} is false or an empty string if @var{status} is true

See also: poly2trellis, struct
@end deftypefn



@node lloyds, log, istrellis, Function Reference
@subsection lloyds

@deftypefn  {Function File} {[@var{table}, @var{codes}] =} lloyds (@var{sig}, @var{init_codes})
@deftypefnx {Function File} {[@var{table}, @var{codes}] =} lloyds (@var{sig}, @var{len})
@deftypefnx {Function File} {[@var{table}, @var{codes}] =} lloyds (@var{sig}, @dots{}, @var{tol})
@deftypefnx {Function File} {[@var{table}, @var{codes}] =} lloyds (@var{sig}, @dots{}, @var{tol}, @var{type})
@deftypefnx {Function File} {[@var{table}, @var{codes}, @var{dist}] =} lloyds (@dots{})
@deftypefnx {Function File} {[@var{table}, @var{codes}, @var{dist}, @var{reldist}] =} lloyds (@dots{})

Optimize the quantization table and codes to reduce distortion. This is
based on the article by Lloyd

 S. Lloyd @emph{Least squared quantization in PCM}, IEEE Trans Inform
 Theory, Mar 1982, no 2, p129-137

which describes an iterative technique to reduce the quantization error
by making the intervals of the table such that each interval has the same
area under the PDF of the training signal @var{sig}. The initial codes to
try can either be given in the vector @var{init_codes} or as scalar
@var{len}. In the case of a scalar the initial codes will be an equi-spaced
vector of length @var{len} between the minimum and maximum value of the
training signal

The stopping criteria of the iterative algorithm is given by

@example
abs(@var{dist}(n) - @var{dist}(n-1)) < max(@var{tol}, abs(@var{eps}*max(@var{sig}))
@end example

By default @var{tol} is 1.e-7. The final input argument determines how the
updated table is created. By default the centroid of the values of the
training signal that fall within the interval described by @var{codes}
are used to update @var{table}. If @var{type} is any other string than
"centroid", this behavior is overridden and @var{table} is updated as
follows

@example
@var{table} = (@var{code}(2:length(@var{code})) + @var{code}(1:length(@var{code}-1))) / 2
@end example

The optimized values are returned as @var{table} and @var{code}. In
addition the distortion of the optimized codes representing the training
signal is returned as @var{dist}. The relative distortion in the final
iteration is also returned as @var{reldist}

See also: quantiz
@end deftypefn



@node log, lu, lloyds, Function Reference
@subsection log

@deftypefn {Loadable Function} {} log (@var{x})
Compute the natural logarithm for each element of @var{x} for a Galois
array
@end deftypefn



@node lu, lz77deco, log, Function Reference
@subsection lu

@deftypefn {Loadable Function} {[@var{l}, @var{u}, @var{p}] =} lu (@var{a})
@cindex LU decomposition of Galois matrix
Compute the LU decomposition of @var{a} in a Galois Field. The result is
returned in a permuted form, according to the optional return value
@var{p}.  For example, given the matrix
@code{a = gf ([1, 2; 3, 4], 3)},

@example
[l, u, p] = lu (a)
@end example

@noindent
returns

@example
l =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   1   0
   6   1

u =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   3   4
   0   7

p =

Permutation Matrix

   0   1
   1   0

@end example

Such that @code{@var{p} * @var{a} = @var{l} * @var{u}}. If the argument
@var{p} is not included then the permutations are applied to @var{l}
so that @code{@var{a} = @var{l} * @var{u}}. @var{l} is then a pseudo-
lower triangular matrix. The matrix @var{a} can be rectangular
@end deftypefn



@node lz77deco, lz77enco, lu, Function Reference
@subsection lz77deco

@deftypefn {Function File} {@var{m} =} lz77deco (@var{c}, @var{alph}, @var{la}, @var{n})
Lempel-Ziv 77 source algorithm decoding implementation. Where

@table @asis
@item @var{m}
message decoded (1xN)
@item @var{c}
encoded message (Mx3)
@item @var{alph}
size of alphabet
@item @var{la}
lookahead buffer size
@item @var{n}
sliding window buffer size
@end table
See also: lz77enco
@end deftypefn



@node lz77enco, matdeintrlv, lz77deco, Function Reference
@subsection lz77enco

@deftypefn {Function File} {@var{c} =} lz77enco (@var{m}, @var{alph}, @var{la}, @var{n})
Lempel-Ziv 77 source algorithm implementation. Where

@table @asis
@item @var{c}
encoded message (Mx3)
@item @var{alph}
size of alphabet
@item @var{la}
lookahead buffer size
@item @var{n}
sliding window buffer size
@end table
See also: lz77deco
@end deftypefn



@node matdeintrlv, matintrlv, lz77enco, Function Reference
@subsection matdeintrlv

@deftypefn {Function File} {@var{intrlvd} =} matdeintrlv (@var{data}, @var{nrows}, @var{ncols})
Restore elements of @var{data} with a temporary matrix of size
@var{nrows}-by-@var{ncols}
See also: matintrlv
@end deftypefn



@node matintrlv, minpol, matdeintrlv, Function Reference
@subsection matintrlv

@deftypefn {Function File} {@var{intrlvd} =} matintrlv (@var{data}, @var{nrows}, @var{ncols})
Interleaved elements of @var{data} with a temporary matrix of size
@var{nrows}-by-@var{ncols}
See also: matdeintrlv
@end deftypefn



@node minpol, modmap, matintrlv, Function Reference
@subsection minpol

@deftypefn {Function File} {} minpol (@var{v})

Finds the minimum polynomial for elements of a Galois Field. For a
vector @var{v} with @math{N} components, representing @math{N} values
in a Galois Field GF(2^@var{m}), return the minimum polynomial in GF(2)
representing those values
@end deftypefn



@node modmap, oct2dec, minpol, Function Reference
@subsection modmap

@deftypefn  {Function File} {} modmap (@var{method}, @dots{})
@deftypefnx {Function File} {y =} modmap (@var{x}, @var{fd}, @var{fs}, "ask", @var{m})
@deftypefnx {Function File} {y =} modmap (@var{x}, @var{fd}, @var{fs}, "fsk", @var{m}, @var{tone})
@deftypefnx {Function File} {y =} modmap (@var{x}, @var{fd}, @var{fs}, "msk")
@deftypefnx {Function File} {y =} modmap (@var{x}, @var{fd}, @var{fs}, "psk", @var{m})
@deftypefnx {Function File} {y =} modmap (@var{x}, @var{fd}, @var{fs}, "qask", @var{m})
@deftypefnx {Function File} {y =} modmap (@var{x}, @var{fd}, @var{fs}, "qask/cir", @var{nsig}, @var{amp}, @var{phs})
@deftypefnx {Function File} {y =} modmap (@var{x}, @var{fd}, @var{fs}, "qask/arb", @var{inphase}, @var{quadr})
@deftypefnx {Function File} {y =} modmap (@var{x}, @var{fd}, @var{fs}, "qask/arb", @var{map})

Mapping of a digital signal to an analog signal. With no output arguments
@code{modmap} plots the constellation of the mapping. In this case the
first argument must be the string @var{method} defining one of "ask",
"fsk", "msk", "qask", "qask/cir" or "qask/arb". The arguments following
the string @var{method} are generally the same as those after the
corresponding string in the function call without output arguments
The exception is @code{modmap ("msk", @var{Fd})}

With an output argument, @var{y} is the complex mapped analog signal. In
this case the arguments @var{x}, @var{fd} and @var{fs} are required. The
variable @var{x} is the digital signal to be mapped, @var{fd} is the
sampling rate of the of digital signal and the @var{fs} is the sampling
rate of the analog signal. It is required that @code{@var{fs}/@var{fd}}
is an integer

The available mapping of the digital signal are

@table @asis
@item "ask"
Amplitude shift keying
@item "fsk"
Frequency shift keying
@item "msk"
Minimum shift keying
@item "psk"
Phase shift keying
@item  "qask"
@itemx "qsk"
@itemx "qam"
Quadrature amplitude shift keying
@end table

In addition the "qask", "qsk" and "qam" method can be modified with the
flags "/cir" or "/arb". That is "qask/cir" and "qask/arb", etc are valid
methods and give circular- and arbitrary-qask mappings respectively

The additional argument @var{m} is the order of the modulation to use
@var{m} must be larger than the largest element of @var{x}. The variable
@var{tone} is the FSK tone to use in the modulation

For "qask/cir", the additional arguments are the same as for
@code{apkconst}, and you are referred to @code{apkconst} for the definitions
of the additional variables

For "qask/arb", the additional arguments @var{inphase} and @var{quadr} give
the in-phase and quadrature components of the mapping, in a similar mapping
to the outputs of @code{qaskenco} with one argument. Similar @var{map}
represents the in-phase and quadrature components of the mapping as
the real and imaginary parts of the variable @var{map}
See also: demodmap, dmodce, amodce, apkconst, qaskenco
@end deftypefn



@node oct2dec, pamdemod, modmap, Function Reference
@subsection oct2dec

@deftypefn {Function File} {@var{d} =} oct2dec (@var{c})

Convert octal to decimal values

Each element of the octal matrix @var{c} is converted to a decimal value

See also: base2dec, bin2dec, dec2bin
@end deftypefn



@node pamdemod, pammod, oct2dec, Function Reference
@subsection pamdemod

@deftypefn  {Function File} {@var{y} =} pamdemod (@var{x}, @var{m})
@deftypefnx {Function File} {@var{y} =} pamdemod (@var{x}, @var{m}, @var{phi})
@deftypefnx {Function File} {@var{y} =} pamdemod (@var{x}, @var{m}, @var{phi}, @var{type})

Demodulates a pulse amplitude modulated signal @var{x} into an
information sequence of integers in the range @code{[0 @dots{} M-1]}
@var{phi} controls the initial phase and @var{type} controls the
constellation mapping. If @var{type} is set to "Bin" will result in
binary encoding, in contrast, if set to "Gray" will give Gray encoding
An example of Gray-encoded 8-PAM is

@example
@group
d = randint (1, 1e4, 8);
y = pammod (d, 8, 0, "gray");
z = awgn (y, 20);
d_est = pamdemod (z, 8, 0, "gray");
plot (z, "rx")
biterr (d, d_est)
@end group
@end example
See also: pammod
@end deftypefn



@node pammod, poly2trellis, pamdemod, Function Reference
@subsection pammod

@deftypefn  {Function File} {@var{y} =} pammod (@var{x}, @var{m})
@deftypefnx {Function File} {@var{y} =} pammod (@var{x}, @var{m}, @var{phi})
@deftypefnx {Function File} {@var{y} =} pammod (@var{x}, @var{m}, @var{phi}, @var{type})

Modulates an information sequence of integers @var{x} in the range
@code{[0 @dots{} M-1]} onto a pulse amplitude modulated signal @var{y}
@var{phi} controls the initial phase and @var{type} controls the
constellation mapping. If @var{type} is set to "Bin" will result in
binary encoding, in contrast, if set to "Gray" will give Gray encoding
An example of Gray-encoded 8-PAM is

@example
@group
d = randint (1, 1e4, 8);
y = pammod (d, 8, 0, "gray");
z = awgn (y, 20);
plot (z, "rx")
@end group
@end example
See also: pamdemod
@end deftypefn



@node poly2trellis, primpoly, pammod, Function Reference
@subsection poly2trellis

@deftypefn {Function File} {@var{t} =} poly2trellis (@var{m}, @var{g})

Convert convolutional code generator polynomials into trellis form

The arguments @var{m} and @var{g} together describe a rate k/n feedforward
convolutional encoder.  The output @var{t} is a trellis structure describing
the same encoder with the fields listed below

The vector @var{m} is a k-by-1 array containing the lengths of each of the
shift registers for the k input bits to the encoder

The matrix @var{g} is a k-by-n octal-value matrix describing the generation
of each of the n outputs from each of the k inputs.  For a particular entry
of @var{g}, the least-significant bit corresponds to the most-delayed input
bit in the kth shift-register

The returned trellis structure contains the following fields:

@table @samp
@item numInputSymbols
The number of k-bit input symbols possible, i.e. 2^k

@item numOutputSymbols
The number of n-bit output symbols possible, i.e. 2^n

@item numStates
The number of states in the trellis

@item nextStates
The state transition table for the trellis.  The ith row contains the indices
of the states reachable from the (i-1)th state for each possible input
symbol

@item outputs
A table of octal-encoded output values for the trellis.  The ith row contains
values representing the output symbols produced in the (i-1)th state for each
possible input symbol
@end table

Input symbols, output symbols, and encoder states are all interpreted with
the lowest indices being the most significant bits

References:

    [1] S. Lin and D. J. Costello, "Convolutional codes," in @cite{Error
    Control Coding}, 2nd ed. Upper Saddle River, NJ: Pearson, 2004,
    ch. 11, pp. 453-513

See also: istrellis
@end deftypefn



@node primpoly, prod, poly2trellis, Function Reference
@subsection primpoly

@deftypefn  {Loadable Function} {@var{y} =} primpoly (@var{m})
@deftypefnx {Loadable Function} {@var{y} =} primpoly (@var{m}, @var{opt})
@deftypefnx {Loadable Function} {@var{y} =} primpoly (@dots{}, "nodisplay\")
Finds the primitive polynomials in GF(2^@var{m}).

The first form of this function returns the default primitive polynomial of
GF(2^@var{m}). This is the minimum primitive polynomial of the field. The
polynomial representation is printed and an integer representation of the
polynomial is returned

The call @code{primpoly (@var{m}, @var{opt})} returns one or more primitive
polynomials. The output of the function is dependent of the value of @var{opt}.
Valid values of @var{opt} are:

@table @asis
@item @code{\"all\"}
Returns all of the primitive polynomials of GF(2^@var{m})
@item @code{\"min\"}
Returns the minimum primitive polynomial of GF(2^@var{m})
@item @code{\"max\"}
Returns the maximum primitive polynomial of GF(2^@var{m})
@item @var{k}
Returns the primitive polynomials having exactly @var{k} non-zero terms
@end table

The call @code{primpoly (@dots{}, \"nodisplay\")} disables the output of
the polynomial forms of the primitives. The return value is not affected.

See also: gf, isprimitive
@end deftypefn



@node prod, pskdemod, primpoly, Function Reference
@subsection prod

@deftypefn {Loadable Function} {} prod (@var{x}, @var{dim})
Product of elements along dimension @var{dim} of Galois array.  If
@var{dim} is omitted, it defaults to 1 (column-wise products)
@end deftypefn



@node pskdemod, pskmod, prod, Function Reference
@subsection pskdemod

@deftypefn  {Function File} {@var{y} =} pamdemod (@var{x}, @var{m})
@deftypefnx {Function File} {@var{y} =} pamdemod (@var{x}, @var{m}, @var{phi})
@deftypefnx {Function File} {@var{y} =} pamdemod (@var{x}, @var{m}, @var{phi}, @var{type})

Demodulates a complex-baseband phase shift keying modulated signal
into an information sequence of integers in the range
@code{[0 @dots{} M-1]}. @var{phi} controls the initial phase and
@var{type} controls the constellation mapping. If @var{type} is set
to "Bin" will result in binary encoding, in contrast, if set to
"Gray" will give Gray encoding.  An example of Gray-encoded 8-PSK is

@example
@group
d = randint (1, 1e3, 8);
y = pskmod (d, 8, 0, "gray");
z = awgn (y, 20);
d_est = pskdemod (z, 8, 0, "gray");
plot (z, "rx")
biterr (d, d_est)
@end group
@end example
See also: pskmod
@end deftypefn



@node pskmod, qamdemod, pskdemod, Function Reference
@subsection pskmod

@deftypefn  {Function File} {@var{y} =} pskmod (@var{x}, @var{m})
@deftypefnx {Function File} {@var{y} =} pskmod (@var{x}, @var{m}, @var{phi})
@deftypefnx {Function File} {@var{y} =} pskmod (@var{x}, @var{m}, @var{phi}, @var{type})

Modulates an information sequence of integers @var{x} in the range
@code{[0 @dots{} M-1]} onto a complex baseband phase shift keying
modulated signal @var{y}. @var{phi} controls the initial phase and
@var{type} controls the constellation mapping. If @var{type} is set
to "Bin" will result in binary encoding, in contrast, if set to "Gray"
will give Gray encoding. An example of Gray-encoded QPSK is

@example
@group
d = randint (1, 5e3, 4);
y = pskmod (d, 4, 0, "gray");
z = awgn (y, 30);
plot (z, "rx")
@end group
@end example
See also: pskdemod
@end deftypefn



@node qamdemod, qammod, pskmod, Function Reference
@subsection qamdemod

@deftypefn {Function File} {} qamdemod (@var{x}, @var{m})
Create the QAM demodulation of x with a size of alphabet m
See also: qammod, pskmod, pskdemod
@end deftypefn



@node qammod, qaskdeco, qamdemod, Function Reference
@subsection qammod

@deftypefn {Function File} {} qammod (@var{x}, @var{m})
Create the QAM modulation of x with a size of alphabet m
See also: qamdemod, pskmod, pskdemod
@end deftypefn



@node qaskdeco, qaskenco, qammod, Function Reference
@subsection qaskdeco

@deftypefn  {Function File} {@var{msg} =} qaskdeco (@var{c}, @var{m})
@deftypefnx {Function File} {@var{msg} =} qaskdeco (@var{inphase}, @var{quadr}, @var{m})
@deftypefnx {Function File} {@var{msg} =} qaskdeco (@dots{}, @var{mnmx})

Demaps an analog signal using a square QASK constellation. The input signal
maybe either a complex variable @var{c}, or as two real variables
@var{inphase} and @var{quadr} representing the in-phase and quadrature
components of the signal

The argument @var{m} must be a positive integer power of 2. By default the
same constellation as created in @code{qaskenco} is used by @code{qaskdeco}
If is possible to change the values of the minimum and maximum of the
in-phase and quadrature components of the constellation to account for
linear changes in the signal values in the received signal. The variable
@var{mnmx} is a 2-by-2 matrix of the following form

@multitable @columnfractions 0.125 0.05 0.25 0.05 0.25 0.05
@item @tab | @tab min in-phase   @tab , @tab max in-phase   @tab |
@item @tab | @tab min quadrature @tab , @tab max quadrature @tab |
@end multitable

If @code{sqrt (@var{m})} is an integer, then @code{qaskenco} uses a Gray
mapping. Otherwise, an attempt is made to create a nearly square mapping
with a minimum Hamming distance between adjacent constellation points
See also: qaskenco
@end deftypefn



@node qaskenco, qfunc, qaskdeco, Function Reference
@subsection qaskenco

@deftypefn  {Function File} {} qaskenco (@var{m})
@deftypefnx {Function File} {} qaskenco (@var{msg}, @var{m})
@deftypefnx {Function File} {@var{y} =} qaskenco (@dots{})
@deftypefnx {Function File} {[@var{inphase}, @var{quadr}] =} qaskenco (@dots{})

Map a digital signal using a square QASK constellation. The argument
@var{m} must be a positive integer power of 2. With two input arguments
the variable @var{msg} represents the message to be encoded. The values
of @var{msg} must be between 0 and @code{@var{m}-1}. In all cases
@code{qaskenco (@var{M})} is equivalent to @code{qaskenco (1:@var{m}, @var{m})}

Three types of outputs can be created depending on the number of output
arguments. That is

@table @asis
@item No output arguments
In this case @code{qaskenco} plots the constellation. Only the
points in @var{msg} are plotted, which in the case of a single input
argument is all constellation points
@item A single output argument
The returned variable is a complex variable representing the in-phase
and quadrature components of the mapped message @var{msg}. With, a
single input argument this effectively gives the mapping from symbols
to constellation points
@item Two output arguments
This is the same as one output argument, expect that the in-phase
and quadrature components are returned explicitly. That is

@example
c = qaskenco (msg, m);
[a, b] = qaskenco (msg, m);
all (c == a + 1i*b)
    @result{} 1
@end example
@end table

If @code{sqrt (@var{m})} is an integer, then @code{qaskenco} uses a Gray
mapping. Otherwise, an attempt is made to create a nearly square mapping
with a minimum Hamming distance between adjacent constellation points
See also: qaskdeco
@end deftypefn



@node qfunc, qfuncinv, qaskenco, Function Reference
@subsection qfunc

@deftypefn {Function File} {@var{y} =} qfunc (@var{x})
Compute the Q function
See also: erfc, erf
@end deftypefn



@node qfuncinv, quantiz, qfunc, Function Reference
@subsection qfuncinv

@deftypefn {Function File} {@var{y} =} qfuncinv (@var{x})
Compute the inverse Q function
See also: erfc, erf
@end deftypefn



@node quantiz, randdeintrlv, qfuncinv, Function Reference
@subsection quantiz

@deftypefn  {Function File} {@var{qidx} =} quantiz (@var{x}, @var{table})
@deftypefnx {Function File} {[@var{qidx}, @var{q}] =} quantiz (@var{x}, @var{table}, @var{codes})
@deftypefnx {Function File} {[ @var{qidx}, @var{q}, @var{d}] =} quantiz (@dots{})

Quantization of an arbitrary signal relative to a partitioning

@table @code
@item qidx = quantiz (x, table)
  Determine position of x in strictly monotonic table.  The first
  interval, using index 0, corresponds to x <= table(1)
  Subsequent intervals are table(i-1) < x <= table(i)

@item [qidx, q] = quantiz (x, table, codes)
  Associate each interval of the table with a code.  Use codes(1)
  for x <= table(1) and codes(n+1) for table(n) < x <= table(n+1)

@item [qidx, q, d] = quantiz (...)
  Compute distortion as mean squared distance of x from the
  corresponding quantization values
@end table
@end deftypefn



@node randdeintrlv, randerr, quantiz, Function Reference
@subsection randdeintrlv

@deftypefn {Function File} {@var{intrlvd} =} randdeintrlv (@var{data}, @var{state})
Restore elements of @var{data} with a random permutation
See also: randintrlv, intrlv, deintrlv
@end deftypefn



@node randerr, randint, randdeintrlv, Function Reference
@subsection randerr

@deftypefn  {Function File} {@var{b} =} randerr (@var{n})
@deftypefnx {Function File} {@var{b} =} randerr (@var{n}, @var{m})
@deftypefnx {Function File} {@var{b} =} randerr (@var{n}, @var{m}, @var{err})
@deftypefnx {Function File} {@var{b} =} randerr (@var{n}, @var{m}, @var{err}, @var{seed})

Generate a matrix of random bit errors. The size of the matrix is
@var{n} rows by @var{m} columns. By default @var{m} is equal to @var{n}
Bit errors in the matrix are indicated by a 1

The variable @var{err} determines the number of errors per row. By
default the return matrix @var{b} has exactly one bit error per row
If @var{err} is a scalar, there each row of @var{b} has exactly this
number of errors per row. If @var{err} is a vector then each row has
a number of errors that is in this vector. Each number of errors has
an equal probability. If @var{err} is a matrix with two rows, then
the first row determines the number of errors and the second their
probabilities

The variable @var{seed} allows the random number generator to be seeded
with a fixed value. The initial seed will be restored when returning
@end deftypefn



@node randint, randintrlv, randerr, Function Reference
@subsection randint

@deftypefn  {Function File} {@var{b} =} randint (@var{n})
@deftypefnx {Function File} {@var{b} =} randint (@var{n}, @var{m})
@deftypefnx {Function File} {@var{b} =} randint (@var{n}, @var{m}, @var{range})
@deftypefnx {Function File} {@var{b} =} randint (@var{n}, @var{m}, @var{range}, @var{seed})

Generate a matrix of random binary numbers. The size of the matrix is
@var{n} rows by @var{m} columns. By default @var{m} is equal to @var{n}

The range in which the integers are generated will is determined by
the variable @var{range}. If @var{range} is an integer, the value will
lie in the range [0,@var{range}-1], or [@var{range}+1,0] if @var{range}
is negative. If @var{range} contains two elements the integers will lie
within these two elements, inclusive. By default @var{range} is
assumed to be [0:1]

The variable @var{seed} allows the random number generator to be seeded
with a fixed value. The initial seed will be restored when returning
@end deftypefn



@node randintrlv, randsrc, randint, Function Reference
@subsection randintrlv

@deftypefn {Function File} {@var{intrlvd} =} randintrlv (@var{data}, @var{state})
Interleaves elements of @var{data} with a random permutation
See also: intrlv, deintrlv
@end deftypefn



@node randsrc, rank, randintrlv, Function Reference
@subsection randsrc

@deftypefn  {Function File} {@var{b} =} randsrc (@var{n})
@deftypefnx {Function File} {@var{b} =} randsrc (@var{n}, @var{m})
@deftypefnx {Function File} {@var{b} =} randsrc (@var{n}, @var{m}, @var{alphabet})
@deftypefnx {Function File} {@var{b} =} randsrc (@var{n}, @var{m}, @var{alphabet}, @var{seed})

Generate a matrix of random symbols. The size of the matrix is
@var{n} rows by @var{m} columns. By default @var{m} is equal to @var{n}

The variable @var{alphabet} can be either a row vector or a matrix with
two rows. When @var{alphabet} is a row vector the symbols returned in
@var{b} are chosen with equal probability from @var{alphabet}. When
@var{alphabet} has two rows, the second row determines the probability
with which each of the symbols is chosen. The sum of the probabilities
must equal 1. By default @var{alphabet} is [-1 1]

The variable @var{seed} allows the random number generator to be seeded
with a fixed value. The initial seed will be restored when returning
@end deftypefn



@node rank, reedmullerdec, randsrc, Function Reference
@subsection rank

@deftypefn {Loadable Function} {@var{d} =} rank (@var{a})
Compute the rank of the Galois array @var{a} by counting the independent
rows and columns
@end deftypefn



@node reedmullerdec, reedmullerenc, rank, Function Reference
@subsection reedmullerdec

@deftypefn {Function File} {} reedmullerdec (@var{VV}, @var{G}, @var{R}, @var{M})

Decode the received code word @var{VV} using the RM-generator matrix @var{G},
of order @var{R}, @var{M}, returning the code-word C. We use the standard
majority logic vote method due to Irving S. Reed. The received word has to be
a matrix of column size equal to to code-word size (i.e @math{2^m}). Each row
is treated as a separate received word

The second return value is the message @var{M} got from @var{C}

G is obtained from definition type construction of Reed-Muller code,
of order @var{R}, length @math{2^M}. Use the function reedmullergen,
for the generator matrix for the (@var{R},@var{M}) order RM code

Faster code constructions (also easier) exist, but since
finding permutation order of the basis vectors, is important, we
stick with the standard definitions. To use decoder
function reedmullerdec, you need to use this specific
generator function

see: Lin & Costello, Ch.4, "Error Control Coding", 2nd Ed, Pearson

@example
@group
g = reedmullergen (2, 4);
msg = rand (1, 11) > 0.5;
c = mod (msg * g, 2);
[dec_c, dec_m] = reedmullerdec (c, g, 2, 4)
@end group
@end example
See also: reedmullergen, reedmullerenc
@end deftypefn



@node reedmullerenc, reedmullergen, reedmullerdec, Function Reference
@subsection reedmullerenc

@deftypefn {Function File} {} reedmullerenc (@var{MSG}, @var{R}, @var{M})

Definition type construction of Reed-Muller code,
of order @var{R}, length @math{2^M}. This function
returns the generator matrix for the said order RM code

Encodes the given message word/block, of column size k,
corresponding to the RM(@var{R},@var{M}), and outputs a
code matrix @var{C}, on each row with corresponding codeword
The second return value is the @var{G}, which is generator matrix
used for this code

@example
@group
msg = rand (10, 11) > 0.5;
[c, g] = reedmullerenc (msg, 2, 4);
@end group
@end example
See also: reedmullerdec, reedmullergen
@end deftypefn



@node reedmullergen, reshape, reedmullerenc, Function Reference
@subsection reedmullergen

@deftypefn {Function File} {} reedmullergen (@var{R}, @var{M})

Definition type construction of Reed-Muller code,
of order @var{R}, length @math{2^M}. This function
returns the generator matrix for the said order RM code

RM(r,m) codes are characterized by codewords,
@code{sum ( (m,0) + (m,1) + @dots{} + (m,r)}
Each of the codeword is got through spanning the
space, using the finite set of m-basis codewords
Each codeword is @math{2^M} elements long
see: Lin & Costello, "Error Control Coding", 2nd Ed

Faster code constructions (also easier) exist, but since
finding permutation order of the basis vectors, is important, we
stick with the standard definitions. To use decoder
function reedmullerdec, you need to use this specific
generator function

@example
@group
g = reedmullergen (2, 4);
@end group
@end example
See also: reedmullerdec, reedmullerenc
@end deftypefn



@node reshape, ricedeco, reedmullergen, Function Reference
@subsection reshape

@deftypefn {Loadable Function} {} reshape (@var{a}, @var{m}, @var{n})
Return a matrix with @var{m} rows and @var{n} columns whose elements are
taken from the Galois array @var{a}.  To decide how to order the elements,
Octave pretends that the elements of a matrix are stored in column-major
order (like Fortran arrays are stored)

For example,

@example
reshape (gf ([1, 2, 3, 4], 3), 2, 2)
ans =
GF(2^3) array. Primitive Polynomial = D^3+D+1 (decimal 11)

Array elements =

   1   3
   2   4

@end example

The @code{reshape} function is equivalent to

@example
@group
retval = gf (zeros (m, n), a.m, a.prim_poly);
retval(:) = a;
@end group
@end example

@noindent
but it is somewhat less cryptic to use @code{reshape} instead of the
colon operator. Note that the total number of elements in the original
matrix must match the total number of elements in the new matrix
See also: :
@end deftypefn



@node ricedeco, riceenco, reshape, Function Reference
@subsection ricedeco

@deftypefn {Function File} {} ricedeco (@var{code}, @var{K})

Returns the Rice decoded signal vector using @var{code} and @var{K}
Compulsory K is need to be specified
A restrictions is that a signal set must strictly be non-negative
The value of code is a cell array of row-vectors which have the
encoded rice value for a single sample. The Rice algorithm is
 used to encode the "code" and only that can be meaningfully
decoded. @var{code} is assumed to have been of format generated
by the function @code{riceenco}

Reference: Solomon Golomb, Run length Encodings, 1966 IEEE Trans Info Theory

An example of the use of @code{ricedeco} is
@example
@group
ricedeco (riceenco (1:4, 2), 2)
    @result{} [1 2 3 4]
@end group
@end example
See also: riceenco
@end deftypefn



@node riceenco, rledeco, ricedeco, Function Reference
@subsection riceenco

@deftypefn {Function File} {} riceenco (@var{sig}, @var{K})

Returns the Rice encoded signal using @var{K} or optimal K 
Default optimal K is chosen between 0-7. Currently no other way
to increase the range except to specify explicitly. Also returns
@var{K} parameter used (in case it were to be chosen optimally)
and @var{Ltot} the total length of output code in bits
This function uses a @var{K} if supplied or by default chooses
the optimal K for encoding signal vector into a rice coded vector
A restrictions is that a signal set must strictly be non-negative
The Rice algorithm is used to encode the data into unary coded
quotient part which is represented as a set of 1's separated from
the K-part (binary) using a zero. This scheme doesn't need any
kind of dictionaries and its close to O(N), but this implementation
*may be* sluggish, though correct

Reference: Solomon Golomb, Run length Encodings, 1966 IEEE Trans
Info' Theory

An example of the use of @code{riceenco} is
@example
@group
riceenco (1:4)
    @result{} @{[0 1], [1 0 0], [1 0 1], [1 1 0 0]@}
riceenco (1:10, 2)
    @result{} @{[0 0 1], [0 1 0], [0 1 1], [1 0 0 0],
        [1 0 0 1], [1 0 1 0], [1 0 1 1], [1 1 0 0 0],
        [1 1 0 0 1], [1 1 0 1 0]@}
@end group
@end example
See also: ricedeco
@end deftypefn



@node rledeco, rleenco, riceenco, Function Reference
@subsection rledeco

@deftypefn {Function File} {} rledeco (@var{message})

Returns decoded run-length @var{message}. The RLE encoded @var{message}
has to be in the form of a row-vector. The message format (encoded RLE)
is like repetition [factor, value]+

An example use of @code{rledeco} is
@example
@group
message = [1 5 2 4 3 1];
rledeco (message)
    @result{} [5 4 4 1 1 1]
@end group
@end example
See also: rledeco
@end deftypefn



@node rleenco, roots, rledeco, Function Reference
@subsection rleenco

@deftypefn {Function File} {} rleenco (@var{message})

Returns run-length encoded @var{message}. The RLE form is built from
@var{message}. The original @var{message} has to be in the form of a
row-vector. The encoded @var{message} format (encoded RLE) is like
[repetition factor]+, values

An example use of @code{rleenco} is
@example
@group
message = [5 4 4 1 1 1]
rleenco (message)
    @result{} [1 5 2 4 3 1];
@end group
@end example
See also: rleenco
@end deftypefn



@node roots, rsdec, rleenco, Function Reference
@subsection roots

@deftypefn {Function File} {} roots (@var{v})

For a vector @var{v} with @math{N} components, return
the roots of the polynomial over a Galois Field
@tex
$$
v_1 z^{N-1} + \cdots + v_{N-1} z + v_N
$$
@end tex
@ifnottex

@example
v(1) * z^(N-1) + ... + v(N-1) * z + v(N)
@end example
@end ifnottex

The number of roots returned and their value will be determined
by the order and primitive polynomial of the Galois Field
@end deftypefn



@node rsdec, rsdecof, roots, Function Reference
@subsection rsdec

@deftypefn  {Loadable Function} {@var{msg} =} rsdec (@var{code}, @var{n}, @var{k})
@deftypefnx {Loadable Function} {@var{msg} =} rsdec (@var{code}, @var{n}, @var{k}, @var{g})
@deftypefnx {Loadable Function} {@var{msg} =} rsdec (@var{code}, @var{n}, @var{k}, @var{fcr}, @var{prim})
@deftypefnx {Loadable Function} {@var{msg} =} rsdec (@dots{}, @var{parpos})
@deftypefnx {Loadable Function} {[@var{msg}, @var{nerr}] =} rsdec (@dots{})
@deftypefnx {Loadable Function} {[@var{msg}, @var{nerr}, @var{ccode}] =} rsdec (@dots{})
Decodes the message contained in @var{code} using a [@var{n},@var{k}]
Reed-Solomon code. The variable @var{code} must be a Galois array with
@var{n} columns and an arbitrary number of rows. Each row of @var{code}
represents a single block to be decoded by the Reed-Solomon coder. The
decoded message is returned in the variable @var{msg} containing @var{k}
columns and the same number of rows as @var{code}.

If @var{n} does not equal @code{2^@var{m}-1}, where m is an integer, then a
shorten Reed-Solomon decoding is used where zeros are added to the start of
each row to obtain an allowable codeword length. The returned @var{msg}
has these prepending zeros stripped.

By default the generator polynomial used in the Reed-Solomon coding is based
on the properties of the Galois Field in which @var{msg} is given. This
default generator polynomial can be overridden by a polynomial in @var{g}.
Suitable generator polynomials can be constructed with @code{rsgenpoly}.
@var{fcr} is an integer value, and it is taken to be the first consecutive
root of the generator polynomial. The variable @var{prim} is then the
primitive element used to construct the generator polynomial. By default
@var{fcr} and @var{prim} are both 1. It is significantly faster to specify
the generator polynomial in terms of @var{fcr} and @var{prim}, since @var{g}
is converted to this form in any case.

By default the parity symbols are placed at the end of the coded message.
The variable @var{parpos} controls this positioning and can take the values
@code{"beginning\"} or @code{\"end\"}. If the parity symbols are at the end, the message is
treated with the most-significant symbol first, otherwise the message is
treated with the least-significant symbol first.
See also: gf, rsenc, rsgenpoly
@end deftypefn



@node rsdecof, rsenc, rsdec, Function Reference
@subsection rsdecof

@deftypefn  {Function File} {} rsdecof (@var{in}, @var{out})
@deftypefnx {Function File} {} rsdecof (@var{in}, @var{out}, @var{t})

Decodes an ASCII file using a Reed-Solomon coder. The input file is
defined by @var{in} and the result is written to the output file @var{out}
The type of coding to use is determined by whether the input file is 7-
or 8-bit. If the input file is 7-bit, the default coding is [127,117]
while the default coding for an 8-bit file is a [255, 235]. This allows
for 5 or 10 error characters in 127 or 255 symbols to be corrected
respectively. The number of errors that can be corrected can be overridden
by the variable @var{t}

If the file is not an integer multiple of the message size (127 or 255)
in length, then the file is padded with the EOT (ASCII character 4)
character before decoding

See also: rsencof
@end deftypefn



@node rsenc, rsencof, rsdecof, Function Reference
@subsection rsenc

@deftypefn  {Loadable Function} {@var{code} =} rsenc (@var{msg}, @var{n}, @var{k})
@deftypefnx {Loadable Function} {@var{code} =} rsenc (@var{msg}, @var{n}, @var{k}, @var{g})
@deftypefnx {Loadable Function} {@var{code} =} rsenc (@var{msg}, @var{n}, @var{k}, @var{fcr}, @var{prim})
@deftypefnx {Loadable Function} {@var{code} =} rsenc (@dots{}, @var{parpos})
Encodes the message @var{msg} using a [@var{n},@var{k}] Reed-Solomon coding.
The variable @var{msg} is a Galois array with @var{k} columns and an arbitrary
number of rows. Each row of @var{msg} represents a single block to be coded
by the Reed-Solomon coder. The coded message is returned in the Galois
array @var{code} containing @var{n} columns and the same number of rows as
@var{msg}.

The use of @code{rsenc} can be seen in the following short example.

@example
m = 3; n = 2^m -1; k = 3;
msg = gf ([1 2 3; 4 5 6], m);
code = rsenc (msg, n, k);
@end example

If @var{n} does not equal @code{2^@var{m}-1}, where m is an integer, then a
shorten Reed-Solomon coding is used where zeros are added to the start of
each row to obtain an allowable codeword length. The returned @var{code}
has these prepending zeros stripped.

By default the generator polynomial used in the Reed-Solomon coding is based
on the properties of the Galois Field in which @var{msg} is given. This
default generator polynomial can be overridden by a polynomial in @var{g}.
Suitable generator polynomials can be constructed with @code{rsgenpoly}.
@var{fcr} is an integer value, and it is taken to be the first consecutive
root of the generator polynomial. The variable @var{prim} is then the
primitive element used to construct the generator polynomial, such that
@tex
$g = (x - A^b) (x - A^{b+p})  \cdots (x - A ^{b+2tp-1})$.
@end tex
@ifnottex

@var{g} = (@var{x} - A^@var{b}) * (@var{x} - A^(@var{b}+@var{prim})) * ... * (@var{x} - A^(@var{b}+2*@var{t}*@var{prim}-1)).
@end ifnottex

where @var{b} is equal to @code{@var{fcr} * @var{prim}}. By default @var{fcr}
and @var{prim} are both 1.

By default the parity symbols are placed at the end of the coded message.
The variable @var{parpos} controls this positioning and can take the values
@code{"beginning\"} or @code{\"end\"}.
See also: gf, rsdec, rsgenpoly
@end deftypefn



@node rsencof, rsgenpoly, rsenc, Function Reference
@subsection rsencof

@deftypefn  {Function File} {} rsencof (@var{in}, @var{out})
@deftypefnx {Function File} {} rsencof (@var{in}, @var{out}, @var{t})
@deftypefnx {Function File} {} rsencof (@dots{}, @var{pad})

Encodes an ASCII file using a Reed-Solomon coder. The input file is
defined by @var{in} and the result is written to the output file @var{out}
The type of coding to use is determined by whether the input file is 7-
or 8-bit. If the input file is 7-bit, the default coding is [127,117]
while the default coding for an 8-bit file is a [255, 235]. This allows
for 5 or 10 error characters in 127 or 255 symbols to be corrected
respectively. The number of errors that can be corrected can be overridden
by the variable @var{t}

If the file is not an integer multiple of the message size (127 or 255)
in length, then the file is padded with the EOT (ASCII character 4)
characters before coding. Whether these characters are written to the
output is defined by the @var{pad} variable. Valid values for @var{pad}
are "pad" (the default) and "nopad", which write or not the padding
respectively

See also: rsdecof
@end deftypefn



@node rsgenpoly, scatterplot, rsencof, Function Reference
@subsection rsgenpoly

@deftypefn  {Function File} {@var{g} =} rsgenpoly (@var{n}, @var{k})
@deftypefnx {Function File} {@var{g} =} rsgenpoly (@var{n}, @var{k}, @var{p})
@deftypefnx {Function File} {@var{g} =} rsgenpoly (@var{n}, @var{k}, @var{p}, @var{b}, @var{s})
@deftypefnx {Function File} {@var{g} =} rsgenpoly (@var{n}, @var{k}, @var{p}, @var{b})
@deftypefnx {Function File} {[@var{g}, @var{t}] =} rsgenpoly (@dots{})

Creates a generator polynomial for a Reed-Solomon coding with message
length of @var{k} and codelength of @var{n}. @var{n} must be greater
than @var{k} and their difference must be even. The generator polynomial
is returned on @var{g} as a polynomial over the Galois Field GF(2^@var{m})
where @var{n} is equal to @code{2^@var{m}-1}. If @var{m} is not integer
the next highest integer value is used and a generator for a shorten
Reed-Solomon code is returned

The elements of @var{g} represent the coefficients of the polynomial in
descending order. If the length of @var{g} is lg, then the generator
polynomial is given by
@tex
$$
g_0 x^{lg-1} + g_1 x^{lg-2} + \cdots + g_{lg-1} x + g_lg
$$
@end tex
@ifnottex

@example
@var{g}(0) * x^(lg-1) + @var{g}(1) * x^(lg-2) + ... + @var{g}(lg-1) * x + @var{g}(lg)
@end example
@end ifnottex

If @var{p} is defined then it is used as the primitive polynomial of the
Galois Field GF(2^@var{m}). The default primitive polynomial will be used
if @var{p} is equal to []

The variables @var{b} and @var{s} determine the form of the generator
polynomial in the following manner
@tex
$$
g = (x - A^{bs}) (x - A^{(b+1)s})  \cdots (x - A ^{(b+2t-1)s})
$$
@end tex
@ifnottex

@example
@var{g} = (@var{x} - A^(@var{b}*@var{s})) * (@var{x} - A^((@var{b}+1)*@var{s})) * ... * (@var{x} - A^((@var{b}+2*@var{t}-1)*@var{s}))
@end example
@end ifnottex

where @var{t} is @code{(@var{n}-@var{k})/2}, and A is the primitive element
of the Galois Field. Therefore @var{b} is the first consecutive root of the
generator polynomial and @var{s} is the primitive element to generate the
polynomial roots

If requested the variable @var{t}, which gives the error correction
capability of the Reed-Solomon code
See also: gf, rsenc, rsdec
@end deftypefn



@node scatterplot, shannonfanodeco, rsgenpoly, Function Reference
@subsection scatterplot

@deftypefn  {Function File} {} scatterplot (@var{x})
@deftypefnx {Function File} {} scatterplot (@var{x}, @var{n})
@deftypefnx {Function File} {} scatterplot (@var{x}, @var{n}, @var{off})
@deftypefnx {Function File} {} scatterplot (@var{x}, @var{n}, @var{off}, @var{str})
@deftypefnx {Function File} {} scatterplot (@var{x}, @var{n}, @var{off}, @var{str}, @var{h})
@deftypefnx {Function File} {@var{h} =} scatterplot (@dots{})

Display the scatter plot of a signal. The signal @var{x} can be either in
one of three forms

@table @asis
@item A real vector
In this case the signal is assumed to be real and represented by the vector
@var{x}. The scatterplot is plotted along the x axis only
@item A complex vector
In this case the in-phase and quadrature components of the signal are
plotted separately on the x and y axes respectively
@item A matrix with two columns
In this case the first column represents the in-phase and the second the
quadrature components of a complex signal and are plotted on the x and
y axes respectively
@end table

Each point of the scatter plot is assumed to be separated by @var{n}
elements in the signal. The first element of the signal to plot is
determined by @var{off}. By default @var{n} is 1 and @var{off} is 0

The string @var{str} is a plot style string (example "r+"),
and by default is the default gnuplot point style

The figure handle to use can be defined by @var{h}. If @var{h} is not
given, then the next available figure handle is used. The figure handle
used in returned on @var{hout}
See also: eyediagram
@end deftypefn



@node shannonfanodeco, shannonfanodict, scatterplot, Function Reference
@subsection shannonfanodeco

@deftypefn {Function File} {} shannonfanodeco (@var{hcode}, @var{dict})

Returns the original signal that was Shannon-Fano encoded. The signal
was encoded using @code{shannonfanoenco}. This function uses
a dict built from the @code{shannonfanodict} and uses it to decode a signal
list into a Shannon-Fano list. Restrictions include hcode is expected to be a binary code;
returned signal set that strictly belongs in the @code{range [1,N]},
with @code{N = length (dict)}. Also dict can only be from the
@code{shannonfanodict (...)} routine. Whenever decoding fails,
those signal values are indicated by -1, and we successively
try to restart decoding from the next bit that hasn't failed in
decoding, ad-infinitum

An example use of @code{shannonfanodeco} is
@example
@group
hd = shannonfanodict (1:4, [0.5 0.25 0.15 0.10]);
hcode = shannonfanoenco (1:4, hd)
    @result{} hcode = [0 1 0 1 1 0 1 1 1 0]
shannonfanodeco (hcode, hd)
    @result{} [1 2 3 4]
@end group
@end example
See also: shannonfanoenco, shannonfanodict
@end deftypefn



@node shannonfanodict, shannonfanoenco, shannonfanodeco, Function Reference
@subsection shannonfanodict

@deftypefn {Function File} {} shannonfanodict (@var{symbols}, @var{symbol_probabilites})

Returns the code dictionary for source using Shannon-Fano algorithm
Dictionary is built from @var{symbol_probabilities} using the
Shannon-Fano scheme.  Output is a dictionary cell-array, which
are codewords, and correspond to the order of input probability

@example
@group
cw = shannonfanodict (1:4, [0.5 0.25 0.15 0.1]);
assert (redundancy (cw, [0.5 0.25 0.15 0.1]), 0.25841, 0.001)
shannonfanodict (1:5, [0.35 0.17 0.17 0.16 0.15])
shannonfanodict (1:8, [8 7 6 5 5 4 3 2] / 40)
@end group
@end example
See also: shannonfanoenc, shannonfanodec
@end deftypefn



@node shannonfanoenco, sqrt, shannonfanodict, Function Reference
@subsection shannonfanoenco

@deftypefn {Function File} {} shannonfanoenco (@var{hcode}, @var{dict})

Returns the Shannon-Fano encoded signal using @var{dict}
This function uses a @var{dict} built from the @code{shannonfanodict}
and uses it to encode a signal list into a Shannon-Fano code
Restrictions include a signal set that strictly belongs in the
@code{range [1,N]} with @code{N = length (dict)}. Also dict can only be
from the @code{shannonfanodict} routine
An example use of @code{shannonfanoenco} is

@example
@group
hd = shannonfanodict (1:4, [0.5 0.25 0.15 0.10]);
shannonfanoenco (1:4, hd)
    @result{} [0 1 0 1 1 0 1 1 1 0]
@end group
@end example
See also: shannonfanodeco, shannonfanodict
@end deftypefn



@node sqrt, sum, shannonfanoenco, Function Reference
@subsection sqrt

@deftypefn {Loadable Function} {} sqrt (@var{x})
Compute the square root of @var{x}, element by element, in a Galois Field
See also: exp
@end deftypefn



@node sum, sumsq, sqrt, Function Reference
@subsection sum

@deftypefn {Loadable Function} {} sum (@var{x}, @var{dim})
Sum of elements along dimension @var{dim} of Galois array.  If @var{dim}
is omitted, it defaults to 1 (column-wise sum)
@end deftypefn



@node sumsq, symerr, sum, Function Reference
@subsection sumsq

@deftypefn {Loadable Function} {} sumsq (@var{x}, @var{dim})
Sum of squares of elements along dimension @var{dim} of Galois array
If @var{dim} is omitted, it defaults to 1 (column-wise sum of squares)

This function is equivalent to computing
@example
gsum (x .* conj (x), dim)
@end example
but it uses less memory
@end deftypefn



@node symerr, syndtable, sumsq, Function Reference
@subsection symerr

@deftypefn  {Function File} {[@var{num}, @var{rate}] =} symerr (@var{a}, @var{b})
@deftypefnx {Function File} {[@var{num}, @var{rate}] =} symerr (@dots{}, @var{flag})
@deftypefnx {Function File} {[@var{num}, @var{rate} @var{ind}] =} symerr (@dots{})

Compares two matrices and returns the number of symbol errors and the
symbol error rate. The variables @var{a} and @var{b} can be either:

@table @asis
@item Both matrices
In this case both matrices must be the same size and then by default the
return values @var{num} and @var{rate} are the overall number of symbol
errors and the overall symbol error rate
@item One column vector
In this case the column vector is used for symbol error comparison
column-wise with the matrix. The returned values @var{num} and @var{rate}
are then row vectors containing the number of symbol errors and the symbol
error rate for each of the column-wise comparisons. The number of rows in
the matrix must be the same as the length of the column vector
@item One row vector
In this case the row vector is used for symbol error comparison row-wise
with the matrix. The returned values @var{num} and @var{rate} are then
column vectors containing the number of symbol errors and the symbol error
rate for each of the row-wise comparisons. The number of columns in the
matrix must be the same as the length of the row vector
@end table

This behavior can be overridden with the variable @var{flag}. @var{flag}
can take the value "column-wise", "row-wise" or "overall". A column-wise
comparison is not possible with a row vector and visa-versa
@end deftypefn



@node syndtable, systematize, symerr, Function Reference
@subsection syndtable

@deftypefn {Loadable Function} {@var{t} =} syndtable (@var{h})
Create the syndrome decoding table from the parity check matrix @var{h}.
Each row of the returned matrix @var{t} represents the error vector in
a received symbol for a certain syndrome. The row selected is determined
by a conversion of the syndrome to an integer representation, and using
this to reference each row of @var{t}.
See also: hammgen, cyclgen
@end deftypefn



@node systematize, vec2mat, syndtable, Function Reference
@subsection systematize

@deftypefn {Function File} {} systematize (@var{G})

Given @var{G}, extract P parity check matrix. Assume row-operations in GF(2)
@var{G} is of size KxN, when decomposed through row-operations into a @var{I} of size KxK
identity matrix, and a parity check matrix @var{P} of size Kx(N-K)

Most arbitrary code with a given generator matrix @var{G}, can be converted into its
systematic form using this function

This function returns 2 values, first is default being @var{Gx} the systematic version of
the @var{G} matrix, and then the parity check matrix @var{P}

@example
@group
g = [1 1 1 1; 1 1 0 1; 1 0 0 1];
[gx, p] = systematize (g);
    @result{} gx = [1 0 0 1; 0 1 0 0; 0 0 1 0];
    @result{} p = [1 0 0];
@end group
@end example
See also: bchpoly, biterr
@end deftypefn



@node vec2mat, wgn, systematize, Function Reference
@subsection vec2mat

@deftypefn  {Function File} {@var{m} =} vec2mat (@var{v}, @var{c})
@deftypefnx {Function File} {@var{m} =} vec2mat (@var{v}, @var{c}, @var{d})
@deftypefnx {Function File} {[@var{m}, @var{add}] =} vec2mat (@dots{})

Converts the vector @var{v} into a @var{c} column matrix with row priority
arrangement and with the final column padded with the value @var{d} to the
correct length. By default @var{d} is 0. The amount of padding added to
the matrix is returned in @var{add}
@end deftypefn



@node wgn, , vec2mat, Function Reference
@subsection wgn

@deftypefn  {Function File} {@var{y} =} wgn (@var{m}, @var{n}, @var{p})
@deftypefnx {Function File} {@var{y} =} wgn (@var{m}, @var{n}, @var{p}, @var{imp})
@deftypefnx {Function File} {@var{y} =} wgn (@var{m}, @var{n}, @var{p}, @var{imp}, @var{seed})
@deftypefnx {Function File} {@var{y} =} wgn (@dots{}, @var{type})
@deftypefnx {Function File} {@var{y} =} wgn (@dots{}, @var{output})

Returns a M-by-N matrix @var{y} of white Gaussian noise. @var{p} specifies
the power of the output noise, which is assumed to be referenced to an
impedance of 1 Ohm, unless @var{imp} explicitly defines the impedance

If @var{seed} is defined then the randn function is seeded with this
value

The arguments @var{type} and @var{output} must follow the above numerical
arguments, but can be specified in any order. @var{type} specifies the
units of @var{p}, and can be "dB", "dBW", "dBm" or "linear". "dB" is
in fact the same as "dBW" and is keep as a misnomer of Matlab. The
units of "linear" are in Watts

The @var{output} variable should be either "real" or "complex". If the
output is complex then the power @var{p} is divided equally between the
real and imaginary parts

See also: randn, awgn
@end deftypefn



@bye
