mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2025-10-30 20:40:28 -04:00
343 lines
16 KiB
HTML
343 lines
16 KiB
HTML
|
<HTML><HEAD>
|
||
|
|
|
||
|
|
<TITLE> Decoding Received Blocks </TITLE>
|
||
|
|
|
||
|
|
|
||
|
|
</HEAD><BODY>
|
||
|
|
|
||
|
|
<H1> Decoding Received Blocks </H1>
|
||
|
|
|
||
|
|
Transmitted codewords are decoded from the received data on the basis
|
||
|
|
of the <I>likelihood</I> of the possible codewords, which is the
|
||
|
|
probability of receiving the data that was actually received if the
|
||
|
|
codeword is question were the one that was sent. This software
|
||
|
|
presently deals only with memoryless channels, in which the noise is
|
||
|
|
independent from bit to bit. For such a channel, the likelihood
|
||
|
|
factorizes into a product of likelihoods for each bit.
|
||
|
|
|
||
|
|
For decoding purposes, all that matters is the relative likelihood
|
||
|
|
for a bit to be 1 versus 0. This is captured by the <I>likelihood
|
||
|
|
ratio</I> in favour of a 1, which is P(data | bit is 1) / P(data |
|
||
|
|
bit is 0).
|
||
|
|
|
||
|
|
<P>For a Binary Symmetric Channel with error probability <I>p</I>,
|
||
|
|
the likelihood ratio in favour of a 1 bit is as follows:
|
||
|
|
<BLOCKQUOTE>
|
||
|
|
If the received data was +1: (1-<I>p</I>) / <I>p</I><BR>
|
||
|
|
If the received data was -1: <I>p</I> / (1-<I>p</I>)
|
||
|
|
</BLOCKQUOTE>
|
||
|
|
For an Additive White Gaussian Noise channel, with signals of +1 for a 1 bit
|
||
|
|
and or -1 for a 0 bit, and with noise standard deviation <I>s</I>, the
|
||
|
|
likelihood ratio in favour of a 1 bit when data <I>y</I> was received is
|
||
|
|
<BLOCKQUOTE>
|
||
|
|
exp ( 2y / s<SUP><SMALL>2</SMALL></SUP> )
|
||
|
|
</BLOCKQUOTE>
|
||
|
|
For an Additive White Logistic Noise channel, the corresponding
|
||
|
|
likelihood ratio is
|
||
|
|
<I>d</I><SUB><SMALL>1</SMALL></SUB>/<I>d</I><SUB><SMALL>0</SMALL></SUB>,
|
||
|
|
where
|
||
|
|
<I>d</I><SUB><SMALL>1</SMALL></SUB>=<I>e</I><SUB><SMALL>1</SMALL></SUB>
|
||
|
|
/ (1+<I>e</I><SUB><SMALL>1</SMALL></SUB>)<SUP><SMALL>2</SMALL></SUP> and
|
||
|
|
<I>d</I><SUB><SMALL>0</SMALL></SUB>=<I>e</I><SUB><SMALL>0</SMALL></SUB>
|
||
|
|
/ (1+<I>e</I><SUB><SMALL>0</SMALL></SUB>)<SUP><SMALL>2</SMALL></SUP>,
|
||
|
|
with <I>e</I><SUB><SMALL>1</SMALL></SUB>=exp(-(<I>y</I>-1)/<I>w</I>) and
|
||
|
|
<I>e</I><SUB><SMALL>0</SMALL></SUB>=exp(-(<I>y</I>+1)/<I>w</I>).
|
||
|
|
<BLOCKQUOTE> </BLOCKQUOTE>
|
||
|
|
|
||
|
|
<P>It is usual to consider codewords to be equally likely <I>a
|
||
|
|
priori</I>. This is reasonable if the source messages are all equally
|
||
|
|
likely (any source redundancy being ignored, or remove by a
|
||
|
|
preliminary data compression stage), provided that the mapping from
|
||
|
|
source messages to codewords is onto. Decoding can then be done using
|
||
|
|
only the parity check matrix defining the codewords, without reference
|
||
|
|
to the generator matrix defining the mapping from source messages to
|
||
|
|
codewords. Note that the condition that this mapping be onto isn't
|
||
|
|
true with this software in the atypical case where the code is defined
|
||
|
|
by a parity check matrix with redundant rows; see the discussion of <A
|
||
|
|
HREF="dep-H.html">linear dependence in parity check matrices</A>.
|
||
|
|
This minor complication is mostly ignored here, except by the exhaustive
|
||
|
|
enumeration decoding methods.
|
||
|
|
|
||
|
|
<P>Assuming equal <I>a priori</I> probabilities for codewords, the
|
||
|
|
probability of correctly decoding an entire codeword is minimized by
|
||
|
|
picking the codeword with the highest likelihood. One might instead
|
||
|
|
wish to decode each bit to the value that is most probable. This
|
||
|
|
minimizes the bit error rate, but is not in general guaranteed to lead
|
||
|
|
a decoding for each block to the most probable complete codeword;
|
||
|
|
indeed, the decoding may not be a codeword at all. Minimizing the bit
|
||
|
|
error rate seems nevertheless to be the most sensible objective,
|
||
|
|
unless block boundaries have some significance in a wider context.
|
||
|
|
|
||
|
|
<P>Optimal decoding by either criterion is infeasible for general
|
||
|
|
linear codes when messages are more than about 20 or 30 bits in
|
||
|
|
length. The fundamental advantage of Low Density Parity Check codes
|
||
|
|
is that good (though not optimal) decodings can be obtained by methods
|
||
|
|
such as probability propagation, described next.
|
||
|
|
|
||
|
|
<A NAME="prprp"><H2>Decoding by probability propagation</H2></A>
|
||
|
|
|
||
|
|
<P>The probability propagation algorithm was originally devised by
|
||
|
|
Robert Gallager in the early 1960's and later reinvented by David
|
||
|
|
MacKay and myself. It can be seen as an instance of the sum-product
|
||
|
|
algorithm for inference on factor graphs, and as an instance of belief
|
||
|
|
propagation in probabilistic networks. See the <A
|
||
|
|
HREF="refs.html">references</A> for details. Below, I give a fairly
|
||
|
|
intuitive description of the algorithm.
|
||
|
|
|
||
|
|
<P>The algorithm uses only the parity check matrix for the code, whose
|
||
|
|
columns correspond to codeword bits, and whose rows correspond to
|
||
|
|
parity checks, and the likelihood ratios for the bits derived from the
|
||
|
|
data. It aims to find the probability of each bit of the transmitted
|
||
|
|
codeword being 1, though the results of the algorithm are in general
|
||
|
|
only approximate.
|
||
|
|
|
||
|
|
<P>The begin, information about each bit of the codeword derived from
|
||
|
|
the received data for that bit alone is expressed as a <I>probability
|
||
|
|
ratio</I>, the probability of the bit being 1 divided by the
|
||
|
|
probability of the bit being 0. This probability ratio is equal to
|
||
|
|
the likelihood ratio (see above) for that bit, since 0 and 1 are
|
||
|
|
assumed to be equally likely <I>a priori</I>. As the algorithm
|
||
|
|
progresses, these probability ratios will be modified to take account
|
||
|
|
of information obtained from other bits, in conjunction with the
|
||
|
|
requirement that the parity checks be satisfied. To avoid double
|
||
|
|
counting of information, for every bit, the algorithm maintains a
|
||
|
|
separate probability ratio for each parity check that that bit
|
||
|
|
participates in, giving the probability for that bit to be 1 versus 0
|
||
|
|
based only on information derived from <I>other</I> parity checks,
|
||
|
|
along with the data received for the bit.
|
||
|
|
|
||
|
|
<P>For each parity check, the algorithm maintains separate
|
||
|
|
<I>likelihood ratios</I> (analogous to, but distinct from, the
|
||
|
|
likelihood ratios based on received data), for every bit that
|
||
|
|
participates in that parity check. These ratios give the probability
|
||
|
|
of that parity check being satisfied if the bit in question is 1
|
||
|
|
divided by the probability of the check being satisfied if the bit is
|
||
|
|
0, taking account of the probabilities of each of the <I>other</I>
|
||
|
|
bits participating in this check being 1, as derived from the
|
||
|
|
probability ratios for these bits with respect to this check.
|
||
|
|
|
||
|
|
<P>The algorithm alternates between recalculating the likelihood
|
||
|
|
ratios for each check, which are stored in the <B>lr</B> fields of the
|
||
|
|
parity check matrix entries, and recalculating the probability ratios
|
||
|
|
for each bit, which are stored in the <B>pr</B> fields of the entries
|
||
|
|
in the sparse matrix representation of the parity check matrix. (See
|
||
|
|
the documentation on <A HREF="mod2sparse.html#rep">representation of
|
||
|
|
sparse matrices</A> for details on these entries.)
|
||
|
|
|
||
|
|
<P>Recalculating the likelihood ratio for a check with respect to some
|
||
|
|
bit may appear time consuming, requiring that all possible
|
||
|
|
combinations of values for the other bits participating in the check
|
||
|
|
be considered. Fortunately, there is a short cut. One can calculate
|
||
|
|
<BLOCKQUOTE>
|
||
|
|
<I>t</I>
|
||
|
|
= product of [ 1 / (1+<I>p<SUB><SMALL>i</SMALL></SUB></I>)
|
||
|
|
- <I>p<SUB><SMALL>i</SMALL></SUB></I> /
|
||
|
|
(1+<I>p<SUB><SMALL>i</SMALL></SUB></I>) ]
|
||
|
|
= product of [ 2 / (1+<I>p<SUB><SMALL>i</SMALL></SUB></I>) - 1 ]
|
||
|
|
</BLOCKQUOTE>
|
||
|
|
where the product is over the probability ratios
|
||
|
|
<I>p<SUB><SMALL>i</SMALL></SUB></I> for the other bits participating
|
||
|
|
in this check. Factor <I>i</I> in this product is equal to probability
|
||
|
|
of bit <I>i</I> being 0 minus the probability that it is 1. The terms
|
||
|
|
in the expansion of this product (in the first form above) correspond to
|
||
|
|
possible combinations of values for the other bits, with the result that
|
||
|
|
<I>t</I> will be the probability of the check being satisfied if the bit
|
||
|
|
in question is 0 minus the probability if the bit in question is 1. The
|
||
|
|
likelihood ratio for this check with respect to the bit in question can then
|
||
|
|
be calculated as (1-<I>t</I>)/(1+<I>t</I>).
|
||
|
|
|
||
|
|
<P>For a particular check, the product above differs for different
|
||
|
|
bits, with respect to which we wish to calculate a likelihood ratio,
|
||
|
|
only in that for each bit the factor corresponding to that bit is left
|
||
|
|
out. We can calculate all these products easily by ordering the bits
|
||
|
|
arbitrarily, computing running products of the factor for the first
|
||
|
|
bit, the factors for the first two bits, etc., and also running
|
||
|
|
products of the factor for the last bit, the factors for the last two
|
||
|
|
bits, etc. Multiplying the running product of the factors up to
|
||
|
|
<I>i</I>-1 by the running product of the factors from <I>i</I>+1 on
|
||
|
|
gives the product needed for bit <I>i</I>. The second form of the
|
||
|
|
factors above is used, as it requires less computation, and is still
|
||
|
|
well defined even if some ratios are infinite.
|
||
|
|
|
||
|
|
<P>To recalculate the probability ratio for a bit with respect to a
|
||
|
|
check, all that is need is to multiply together the likelihood ratio
|
||
|
|
for this bit derived from the received data (see above), and the
|
||
|
|
current values of the likelihood ratios for all the <I>other</I>
|
||
|
|
checks that this bit participates in, with respect to this bit. To
|
||
|
|
save time, these products are computed by combining forward and
|
||
|
|
backward products, similarly to the method used for likelihood ratios.
|
||
|
|
|
||
|
|
<P>By including likelihood ratios from all checks, a similar
|
||
|
|
calculation produces the current probability ratio for the bit to be 1
|
||
|
|
versus 0 based on all information that has propagated to the bit so
|
||
|
|
far. This ratio can be thresholded at one to produce the current best
|
||
|
|
guess as to whether this bit is a 1 or a 0.
|
||
|
|
|
||
|
|
<P>The hope is that this algorithm will eventually converge to a state
|
||
|
|
where these bit probabilities give a near-optimal decoding. This is
|
||
|
|
does not always occur, but the algorithm behaves well enough to
|
||
|
|
produce very good results at rates approaching (though not yet
|
||
|
|
reaching) the theoretical Shannon limit.
|
||
|
|
|
||
|
|
|
||
|
|
<P><A NAME="decode"><HR><B>decode</B>: Decode blocks of received data
|
||
|
|
into codewords.
|
||
|
|
|
||
|
|
<BLOCKQUOTE><PRE>
|
||
|
|
decode [ -f ] [ -t | -T ] <I>pchk-file received-file decoded-file</I> [ <I>bp-file</I> ] <I>channel method</I>
|
||
|
|
</PRE>
|
||
|
|
<BLOCKQUOTE>
|
||
|
|
where <TT><I>channel</I></TT> is one of:
|
||
|
|
<BLOCKQUOTE><PRE>
|
||
|
|
bsc <I>error-probability</I>
|
||
|
|
|
||
|
|
awgn <I>standard-deviation</I>
|
||
|
|
|
||
|
|
awln <I>width</I>
|
||
|
|
</PRE></BLOCKQUOTE>
|
||
|
|
and <TT><I>method</I></TT> is one of:
|
||
|
|
<BLOCKQUOTE><PRE>
|
||
|
|
enum-block <TT><I>gen-file</I></TT>
|
||
|
|
|
||
|
|
enum-bit <TT><I>gen-file</I></TT>
|
||
|
|
|
||
|
|
prprp <TT>[-]<I>max-iterations</I></TT>
|
||
|
|
</PRE></BLOCKQUOTE>
|
||
|
|
</BLOCKQUOTE>
|
||
|
|
</BLOCKQUOTE>
|
||
|
|
|
||
|
|
<P>Decodes the blocks in <TT><I>received-file</I></TT>, which are
|
||
|
|
assumed to be have been received through the specified channel. The
|
||
|
|
results written to <TT><I>decoded-file</I></TT> are the specified
|
||
|
|
decoding method's guesses as to what bits were sent through the
|
||
|
|
channel, given what was received. The probability of each bit being a
|
||
|
|
1, as judged by the decoding method being used, is written to
|
||
|
|
<TT><I>bp-file</I></TT>, if given.
|
||
|
|
|
||
|
|
<P>A newline is output at the end of each block written to
|
||
|
|
<TT><I>decoded-file</I></TT> and <TT><I>bp-file</I></TT>. Newlines in
|
||
|
|
<TT><I>received-file</I></TT> are ignored. A warning is displayed on
|
||
|
|
standard error if the number of bits in <TT><I>received-file</I></TT>
|
||
|
|
is not a multiple of the block length.
|
||
|
|
|
||
|
|
<P>A summary is displayed on standard error, giving the total number
|
||
|
|
of blocks decoded, the number of blocks that decoded to valid
|
||
|
|
codewords, the average number of iterations of the decoding algorithm
|
||
|
|
used, and the percent of bits that were changed from the values one
|
||
|
|
would guess for them based just on their individual likelihood ratios.
|
||
|
|
|
||
|
|
<P>If the <B>-t</B> option is given, a line of information regarding each block
|
||
|
|
decoded is written to standard output, preceded by a line of headers.
|
||
|
|
The information for each block is as follows:
|
||
|
|
<BLOCKQUOTE>
|
||
|
|
<TABLE>
|
||
|
|
<tr align="left" valign="top">
|
||
|
|
<td> <B>block</B> </td>
|
||
|
|
<td>The number of the block, from zero</td></tr>
|
||
|
|
<tr align="left" valign="top">
|
||
|
|
<td> <B>iterations</B> </td>
|
||
|
|
<td>The number of "iterations" used in decoding. What exactly an iteration
|
||
|
|
is depends on the decoding method used (see
|
||
|
|
<A HREF="decode-detail.html">here</A>).</td></tr>
|
||
|
|
<tr align="left" valign="top">
|
||
|
|
<td> <B>valid</B> </td>
|
||
|
|
<td>Has the value 1 if the decoding is a valid codeword, 0 if not.</td></tr>
|
||
|
|
<tr align="left" valign="top">
|
||
|
|
<td> <B>changed</B> </td>
|
||
|
|
<td>The number of bits in the decoding that differ from the bit that would
|
||
|
|
be chosen based just on the likelihood ratio for that bit. Bits whose
|
||
|
|
likelihood ratios are exactly one contribute 0.5 to this count.</td></tr>
|
||
|
|
</TABLE>
|
||
|
|
</BLOCKQUOTE>
|
||
|
|
The file produced is is suitable for
|
||
|
|
reading into the S-Plus or R statistics packages, with a command such as
|
||
|
|
<BLOCKQUOTE><PRE>
|
||
|
|
data <- read.table(<I>file</I>,header=T)
|
||
|
|
</PRE></BLOCKQUOTE>
|
||
|
|
|
||
|
|
<P>If instead the <B>-T</B> option is given, detailed information on
|
||
|
|
the process of decoding each block will be written to standard output.
|
||
|
|
For a description, see the <A HREF="decode-detail.html">documentation
|
||
|
|
on detailed decoding trace information</A>.
|
||
|
|
|
||
|
|
<P>The type of channel that is assumed is specified after the file
|
||
|
|
name arguments. This may currently be either <TT>bsc</TT> (or
|
||
|
|
<TT>BSC</TT>) for the Binary Symmetric Channel, or <TT>awgn</TT> (or
|
||
|
|
<TT>AWGN</TT>) for the Additive White Gaussian Noise channel, or
|
||
|
|
<TT>awln</TT> (or <TT>AWLN</TT>) for the Additive White Logistic Noise
|
||
|
|
channel. The channel type is followed by an argument specifying the
|
||
|
|
assumed characteristics of the channel, as follows:
|
||
|
|
<BLOCKQUOTE>
|
||
|
|
<P>BSC: The probability that a bit will be flipped by noise - ie, the
|
||
|
|
probability that the bit received is an error.
|
||
|
|
|
||
|
|
<P>AWGN: The standard deviation of the Gaussian noise added to the
|
||
|
|
encodings of the bits.
|
||
|
|
|
||
|
|
<P>AWLN: The width parameter of the logistic distribution for the noise
|
||
|
|
that is added to the encodings of the bits.
|
||
|
|
</BLOCKQUOTE>
|
||
|
|
See the description of <A HREF="channel.html">channel transmission</A>
|
||
|
|
for more about these channels.
|
||
|
|
|
||
|
|
<P>Following the channel specification is a specification of the
|
||
|
|
decoding method to use. The <TT>enum-block</TT> and <TT>enum-bit</TT>
|
||
|
|
methods find the optimal decoding by exhaustive enumeration of
|
||
|
|
codewords derived from all possible source messages. They differ in
|
||
|
|
that <TT>enum-block</TT> decodes to the most likely codeword, whereas
|
||
|
|
<TT>enum-bit</TT> decodes to the bits that are individually most
|
||
|
|
probable. These methods require that a file containing a
|
||
|
|
representation of a generator matrix be given, to allow enumeration of
|
||
|
|
codewords. If the parity check matrix has no redundant rows, any
|
||
|
|
valid generator matrix will give the same decoding (except perhaps if
|
||
|
|
there is a tie). If redundant rows exist, the generator matrix should
|
||
|
|
specify the same set of message bits as the generator matrix that was
|
||
|
|
used for the actual encoding, since the redundancy will lead to some
|
||
|
|
codeword bits being fixed at zero (see <A HREF="dep-H.html">linear
|
||
|
|
dependence in parity check matrices</A>).
|
||
|
|
|
||
|
|
<P>The <TT>prprp</TT> decoding method decodes using <A
|
||
|
|
HREF="#prprp">probability propagation</A>. The maximum number of
|
||
|
|
iterations of probability propagation to do is given following
|
||
|
|
<TT>prprp</TT>. If a minus sign precedes this number, the maximum
|
||
|
|
number of iterations is always done. If no minus sign is present, the
|
||
|
|
algorithm stops once the tentative decoding, based on bit-by-bit
|
||
|
|
probabilities, is a valid codeword. Note that continuing to the
|
||
|
|
maximum number of iterations will usually result in
|
||
|
|
at least slightly different bit probabilities (written to
|
||
|
|
<TT><I>bp-file</I></TT> if specified), and could conceivably change
|
||
|
|
the decoding compared to stopping at the first valid codeword, or
|
||
|
|
result in a failure to decode to a valid codeword even though one was
|
||
|
|
found earlier.
|
||
|
|
|
||
|
|
<P>If the <B>-f</B> option is given, output to <TT><I>decoded-file</I></TT>
|
||
|
|
is flushed after each block. This allows one to use decode as a server,
|
||
|
|
reading blocks to decode from a named pipe, and writing the decoded block
|
||
|
|
to another named pipe.
|
||
|
|
|
||
|
|
|
||
|
|
<P><A NAME="extract"><HR><B>extract</B>: Extract the message bits from a block.
|
||
|
|
|
||
|
|
<BLOCKQUOTE><PRE>
|
||
|
|
extract <I>gen-file decoded-file extracted-file</I>
|
||
|
|
</PRE></BLOCKQUOTE>
|
||
|
|
|
||
|
|
<P>Given a file of codewords in <TT><I>decoded-file</I></TT> (usually,
|
||
|
|
decoded blocks output by <A HREF="#decode"><TT>decode</TT></A>), and a
|
||
|
|
generator matrix from <TT><I>gen-file</I></TT> (needed only to
|
||
|
|
determine where the message bits are located in a codeword), this
|
||
|
|
program writes the message bits extracted from these codewords to the
|
||
|
|
file <TT><I>extracted-file</I></TT>.
|
||
|
|
|
||
|
|
<P>A newline is output at the end of each block written to
|
||
|
|
<TT><I>extracted-file</I></TT>. Newlines in
|
||
|
|
<TT><I>decoded-file</I></TT> are ignored. A warning is displayed on
|
||
|
|
standard error if the number of bits in <TT><I>decoded-file</I></TT>
|
||
|
|
is not a multiple of the block length.
|
||
|
|
|
||
|
|
<HR>
|
||
|
|
|
||
|
|
<A HREF="index.html">Back to index for LDPC software</A>
|
||
|
|
|
||
|
|
</BODY></HTML>
|