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<HTML><HEAD><TITLE> Transmission Through a Simulated Channel </TITLE></HEAD><BODY><H1> Transmission Through a Simulated Channel </H1><P>Once a codeword has been found to represent a source message, itcan be sent through a <I>channel</I>, with the result that certaindata is received as the output of the channel, which will be relatedto the codeword sent, but with random noise. This software currentlyhandles only memoryless binary channels, for which each bit sentthrough the channel results in a separate piece of data beingreceived, and the noise affecting one bit is independent of the noiseaffecting other bits. <P>For a <I>Binary Symmetric Channel</I> (BSC), each bit sentresults in a bit being received. The bit received differs from thebit sent with some error probability, <I>p</I>, which is the same for0 bits and for 1 bits. In other words, the probability distributionfor the bit received given the bit sent is as follows:<BLOCKQUOTE>P(receive 1 | send 1) = P(receive 0 | send 0) = 1-<I>p</I><BR>P(receive 1 | send 0) = P(receive 0 | send 1) = <I>p</I></BLOCKQUOTE><P>For an <I>Additive White Gaussian Noise</I> (AWGN) channel, thedata received at each time is equal to the data sent plus Gaussiannoise with mean zero and some standard deviation, <I>s</I>,independently for each bit. For this software, the data sent is -1for a 0 bit and +1 for a 1 bit. In other words, the distributionof the received data given the bit sent is as follows:<BLOCKQUOTE>data received | send 1 ~ N(+1,<I>s</I><SUP><SMALL>2</SMALL></SUP>)<BR>data received | send 0 ~ N(-1,<I>s</I><SUP><SMALL>2</SMALL></SUP>)</BLOCKQUOTE><P>It is typically assumed that the standard deviation of the noisevaries with the rate at which bits are sent, increasing in proportionto the square root of the rate. The error rate obtained from sendingunencoded bits at rate <I>R</I> will then be the same as is obtainedusing a code that repeats each bit <I>n</I> times, and sends thesebits at rate <I>nR</I> (assuming optimal decoding of each bit bythresholding the sum of the <I>n</I> channel outputs corresponding tothat bit). Another way of looking at this scaling for <I>s</I> isthat when bits are send at a lower rate, the receiver will beaccumulating the channel output for a longer time, with the resultthat the amount of noise will decrease (relative to the signal) as aresult of averaging.<P>To account for this, it is common to compare codes for AWGNchannels in terms of their bit error rate and the value of<BLOCKQUOTE><I>E<SUB><SMALL>b</SMALL></SUB></I> / <I>N<SUB><SMALL>0</SMALL></SUB></I>= 1 / 2<I>R</I><I>s</I><SUP><SMALL>2</SMALL></SUP></BLOCKQUOTE>at which they operate, where <I>R</I>=<I>K</I>/<I>N</I> is the rateof the code, and <I>s</I> is the noise level at which the codeachieves the quoted bit error rate. Hence, a code operating at a lowerrate is allowed to assume a lower noise level to make the comparison fair.It is common to quote <I>E<SUB><SMALL>b</SMALL></SUB></I> /<I>N<SUB><SMALL>0</SMALL></SUB></I> in decibels (db), equal to10 log<SUB><SMALL>10</SMALL></SUB>(<I>E<SUB><SMALL>b</SMALL></SUB></I>/ <I>N<SUB><SMALL>0</SMALL></SUB></I>).<P>The <I>Additive White Logistic Noise</I> (AWLN) channel is similarto the AWGN channel, except that the noise comes from a logistic ratherthan a Gaussian distribution. The probability density function for thenoise is<BLOCKQUOTE>(1/<I>w</I>) exp(-<I>n</I>/<I>w</I>) / [1 + exp(-<I>n</I>/<I>w</I>)]<SUP>2</SUP></BLOCKQUOTE>where <I>n</I> is the amount of noise, and <I>w</I> is a width parameterfor the distribution, analogous to the <I>s</I> parameter for Gaussian noise. (However, <I>w</I> is not equal to the standard deviationfor the logistic distribution, which is sqrt(pi<SUP><SMALL>2</SMALL></SUP>/3)<I>w</I>.) <B>Note:</B> Although I'venamed this channel in analogy with the AWGN channel, it does not share the properties discussed above regarding how noise levels would be expectedto change when the data rate changes. <P><A NAME="transmit"><HR><B>transmit</B>: Transmit bits through a simulated channel.<BLOCKQUOTE><PRE>transmit <I>encoded-file</I>|<I>n-zeros received-file seed channel</I></PRE><BLOCKQUOTE>where <TT><I>channel</I></TT> is one of the following:<BLOCKQUOTE><PRE>bsc <I>error-probability</I>awgn <I>standard-deviation</I>awln <I>width</I></PRE></BLOCKQUOTE></BLOCKQUOTE></BLOCKQUOTE><P>Simulates the transmission of the bits in<TT><I>encoded-file</I></TT> through a channel, with the received databeing stored in <TT><I>received-file</I></TT>. Typically,<TT><I>encoded-file</I></TT> will have been produced by the <AHREF="encoding.html#encode"><TT>encode</TT></A> program, but it couldalso come from <A HREF="support.html#rand-src"><TT>rand-src</TT></A>or another program. If newlines separate blocks in<TT><I>encoded-file</I></TT>, these block boundaries will be preservedin <TT><I>received-file</I></TT>.<P>Alternatively, a count of zeros to transmit can be given, ratherthan a <I>encoded-file</I>. This count can be the product of theblock size and the number of blocks, written with <TT>x</TT>separating these numbers, with no spaces. The<TT><I>received-file</I></TT> will mark the block boundaries withnewlines, assuming a block size of one if a simple bit count is given.Note that zero messages are sufficient for assessing the performanceof a linear code with a symmetrical channel and a symmetrical decodingalgorithm. <B>Warning:</B> Ties, messages that lead to floating-pointoverflow, and program bugs can easily make a decoding algorithmnon-symmetrical, so it's best not to test exclusively on zeromessages. Indeed, it is best not to do this at all unless you really need to avoid the time needed to generate and encode randommessages.<P>The transmission will be corrupted by random noise, which will begenerated pseudo-randomly based on <TT><I>seed</I></TT>. The actualrandom seed used will be <TT><I>seed</I></TT> times 10 plus 3, so thatthe stream of pseudo-random numbers will not be the same as any thatmight have been used by another program.<P>The fourth argument specifies the type of channel, currently 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 GaussianNoise channel, or <TT>awln</TT> (or <TT>AWLN</TT>) for the Additive WhiteLogistic Noise channel. The channel type is followed by an argumentspecifying the 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 that is 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 details.<P><B>Examples</B>: The command:<UL><PRE><LI>transmit 10x3 rec 1 bsc 0.1</PRE></UL>will simulate the transmission of 30 zero bits (3 blocks of size 10) througha Binary Symmetric Channel with error probability of 0.1. The result willbe to store something like the following in the file <TT>rec</TT>:<BLOCKQUOTE><PRE>000000000010000000000100000000</PRE></BLOCKQUOTE>If an AWGN channel is used instead, as follows:<UL><PRE><LI>transmit 10x3 rec 1 awgn 0.5</PRE></UL>then the file <TT>rec</TT> will contain data such as:<BLOCKQUOTE><PRE> -1.36 -0.86 -0.80 -1.19 -1.18 -0.64 -0.31 -1.16 -1.56 -0.79 -2.20 -1.62 -0.53 -1.29 -1.08 -2.05 -0.75 -1.22 -0.81 -0.52 -0.86 -0.34 -1.10 -1.30 -1.10 -1.20 -0.37 -1.07 -0.22 -1.46</PRE></BLOCKQUOTE><HR><A HREF="index.html">Back to index for LDPC software</A></BODY></HTML>?? 快捷鍵說明
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