183 lines
7.7 KiB
Plaintext
183 lines
7.7 KiB
Plaintext
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<HTML><HEAD>
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<TITLE> Transmission Through a Simulated Channel </TITLE>
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</HEAD><BODY>
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<H1> Transmission Through a Simulated Channel </H1>
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<P>Once a codeword has been found to represent a source message, it
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can be sent through a <I>channel</I>, with the result that certain
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data is received as the output of the channel, which will be related
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to the codeword sent, but with random noise. This software currently
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handles only memoryless binary channels, for which each bit sent
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through the channel results in a separate piece of data being
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received, and the noise affecting one bit is independent of the noise
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affecting other bits.
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<P>For a <I>Binary Symmetric Channel</I> (BSC), each bit sent
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results in a bit being received. The bit received differs from the
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bit sent with some error probability, <I>p</I>, which is the same for
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0 bits and for 1 bits. In other words, the probability distribution
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for the bit received given the bit sent is as follows:
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<BLOCKQUOTE>
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P(receive 1 | send 1) = P(receive 0 | send 0) = 1-<I>p</I><BR>
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P(receive 1 | send 0) = P(receive 0 | send 1) = <I>p</I>
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</BLOCKQUOTE>
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<P>For an <I>Additive White Gaussian Noise</I> (AWGN) channel, the
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data received at each time is equal to the data sent plus Gaussian
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noise with mean zero and some standard deviation, <I>s</I>,
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independently for each bit. For this software, the data sent is -1
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for a 0 bit and +1 for a 1 bit. In other words, the distribution
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of the received data given the bit sent is as follows:
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<BLOCKQUOTE>
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data received | send 1 ~ N(+1,<I>s</I><SUP><SMALL>2</SMALL></SUP>)<BR>
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data received | send 0 ~ N(-1,<I>s</I><SUP><SMALL>2</SMALL></SUP>)
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</BLOCKQUOTE>
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<P>It is typically assumed that the standard deviation of the noise
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varies with the rate at which bits are sent, increasing in proportion
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to the square root of the rate. The error rate obtained from sending
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unencoded bits at rate <I>R</I> will then be the same as is obtained
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using a code that repeats each bit <I>n</I> times, and sends these
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bits at rate <I>nR</I> (assuming optimal decoding of each bit by
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thresholding the sum of the <I>n</I> channel outputs corresponding to
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that bit). Another way of looking at this scaling for <I>s</I> is
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that when bits are send at a lower rate, the receiver will be
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accumulating the channel output for a longer time, with the result
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that the amount of noise will decrease (relative to the signal) as a
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result of averaging.
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<P>To account for this, it is common to compare codes for AWGN
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channels in terms of their bit error rate and the value of
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<BLOCKQUOTE>
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<I>E<SUB><SMALL>b</SMALL></SUB></I> / <I>N<SUB><SMALL>0</SMALL></SUB></I>
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= 1 / 2<I>R</I><I>s</I><SUP><SMALL>2</SMALL></SUP>
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</BLOCKQUOTE>
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at which they operate, where <I>R</I>=<I>K</I>/<I>N</I> is the rate
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of the code, and <I>s</I> is the noise level at which the code
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achieves the quoted bit error rate. Hence, a code operating at a lower
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rate is allowed to assume a lower noise level to make the comparison fair.
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It is common to quote
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<I>E<SUB><SMALL>b</SMALL></SUB></I> /
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<I>N<SUB><SMALL>0</SMALL></SUB></I> in decibels (db), equal to
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10 log<SUB><SMALL>10</SMALL></SUB>(<I>E<SUB><SMALL>b</SMALL></SUB></I>
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/ <I>N<SUB><SMALL>0</SMALL></SUB></I>).
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<P>The <I>Additive White Logistic Noise</I> (AWLN) channel is similar
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to the AWGN channel, except that the noise comes from a logistic rather
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than a Gaussian distribution. The probability density function for the
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noise is
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<BLOCKQUOTE>
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(1/<I>w</I>) exp(-<I>n</I>/<I>w</I>) / [1 + exp(-<I>n</I>/<I>w</I>)]<SUP>2</SUP>
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</BLOCKQUOTE>
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where <I>n</I> is the amount of noise, and <I>w</I> is a width parameter
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for the distribution, analogous to the <I>s</I> parameter for
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Gaussian noise. (However, <I>w</I> is not equal to the standard deviation
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for the logistic distribution, which is
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sqrt(pi<SUP><SMALL>2</SMALL></SUP>/3)<I>w</I>.) <B>Note:</B> Although I've
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named this channel in analogy with the AWGN channel, it does not share
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the properties discussed above regarding how noise levels would be expected
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to change when the data rate changes.
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<P><A NAME="transmit"><HR><B>transmit</B>: Transmit bits through a
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simulated channel.
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<BLOCKQUOTE><PRE>
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transmit <I>encoded-file</I>|<I>n-zeros received-file seed channel</I>
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</PRE>
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<BLOCKQUOTE>
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where <TT><I>channel</I></TT> is one of the following:
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<BLOCKQUOTE><PRE>
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bsc <I>error-probability</I>
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awgn <I>standard-deviation</I>
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awln <I>width</I>
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</PRE></BLOCKQUOTE>
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</BLOCKQUOTE>
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</BLOCKQUOTE>
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<P>Simulates the transmission of the bits in
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<TT><I>encoded-file</I></TT> through a channel, with the received data
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being stored in <TT><I>received-file</I></TT>. Typically,
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<TT><I>encoded-file</I></TT> will have been produced by the <A
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HREF="encoding.html#encode"><TT>encode</TT></A> program, but it could
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also come from <A HREF="support.html#rand-src"><TT>rand-src</TT></A>
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or another program. If newlines separate blocks in
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<TT><I>encoded-file</I></TT>, these block boundaries will be preserved
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in <TT><I>received-file</I></TT>.
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<P>Alternatively, a count of zeros to transmit can be given, rather
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than a <I>encoded-file</I>. This count can be the product of the
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block size and the number of blocks, written with <TT>x</TT>
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separating these numbers, with no spaces. The
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<TT><I>received-file</I></TT> will mark the block boundaries with
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newlines, assuming a block size of one if a simple bit count is given.
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Note that zero messages are sufficient for assessing the performance
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of a linear code with a symmetrical channel and a symmetrical decoding
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algorithm. <B>Warning:</B> Ties, messages that lead to floating-point
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overflow, and program bugs can easily make a decoding algorithm
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non-symmetrical, so it's best not to test exclusively on zero
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messages. Indeed, it is best not to do this at all unless you
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really need to avoid the time needed to generate and encode random
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messages.
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<P>The transmission will be corrupted by random noise, which will be
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generated pseudo-randomly based on <TT><I>seed</I></TT>. The actual
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random seed used will be <TT><I>seed</I></TT> times 10 plus 3, so that
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the stream of pseudo-random numbers will not be the same as any that
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might have been used by another program.
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<P>The fourth argument specifies the type of channel, currently either
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<TT>bsc</TT> (or <TT>BSC</TT>) for the Binary Symmetric Channel, or
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<TT>awgn</TT> (or <TT>AWGN</TT>) for the Additive White Gaussian
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Noise channel, or <TT>awln</TT> (or <TT>AWLN</TT>) for the Additive White
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Logistic Noise channel. The channel type is followed by an argument
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specifying the characteristics of the channel, as follows:
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<BLOCKQUOTE>
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<P>BSC: The probability that a bit will be flipped by noise - ie, the
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probability that the bit received is an error.
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<P>AWGN: The standard deviation of the Gaussian noise that is added to the
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encodings of the bits.
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<P>AWLN: The width parameter of the logistic distribution for the noise
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that is added to the encodings of the bits.
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</BLOCKQUOTE>
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See the description of <A HREF="channel.html">channel transmission</A>
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for more details.
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<P><B>Examples</B>: The command:
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<UL><PRE>
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<LI>transmit 10x3 rec 1 bsc 0.1
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</PRE></UL>
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will simulate the transmission of 30 zero bits (3 blocks of size 10) through
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a Binary Symmetric Channel with error probability of 0.1. The result will
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be to store something like the following in the file <TT>rec</TT>:
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<BLOCKQUOTE><PRE>
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0000000000
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1000000000
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0100000000
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</PRE></BLOCKQUOTE>
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If an AWGN channel is used instead, as follows:
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<UL><PRE>
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<LI>transmit 10x3 rec 1 awgn 0.5
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</PRE></UL>
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then the file <TT>rec</TT> will contain data such as:
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<BLOCKQUOTE><PRE>
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-1.36 -0.86 -0.80 -1.19 -1.18 -0.64 -0.31 -1.16 -1.56 -0.79
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-2.20 -1.62 -0.53 -1.29 -1.08 -2.05 -0.75 -1.22 -0.81 -0.52
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-0.86 -0.34 -1.10 -1.30 -1.10 -1.20 -0.37 -1.07 -0.22 -1.46
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</PRE></BLOCKQUOTE>
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<HR>
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<A HREF="index.html">Back to index for LDPC software</A>
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</BODY></HTML>
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