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Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981 You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links. 152 9 — Communication over a Noisy Channel The rate of the code is R = K/N bits per channel use. [We will use this definition of the rate for any channel, not only channels with binary inputs; note however that it is sometimes conventional to define the rate of a code for a channel with q input symbols to be K/(N log q).] A decoder for an (N, K) block code is a mapping from the set of length-N strings of channel outputs, A N Y , to a codeword label ˆs ∈ {0, 1, 2, . . . , 2K }. The extra symbol ˆs = 0 can be used to indicate a ‘failure’. The probability of block error of a code and decoder, for a given channel, and for a given probability distribution over the encoded signal P (sin), is: pB = P (sin)P (sout =sin | sin). (9.15) The maximal probability of block error is sin pBM = max P (sout =sin | sin). (9.16) sin The optimal decoder for a channel code is the one that minimizes the probability of block error. It decodes an output y as the input s that has maximum posterior probability P (s | y). P (s | y) = P (y | s)P (s) s ′ P (y | s′ )P (s ′ ) (9.17) ˆsoptimal = argmax P (s | y). (9.18) A uniform prior distribution on s is usually assumed, in which case the optimal decoder is also the maximum likelihood decoder, i.e., the decoder that maps an output y to the input s that has maximum likelihood P (y | s). The probability of bit error pb is defined assuming that the codeword number s is represented by a binary vector s of length K bits; it is the average probability that a bit of sout is not equal to the corresponding bit of sin (averaging over all K bits). Shannon’s noisy-channel coding theorem (part one). Associated with each discrete memoryless channel, there is a non-negative number C (called the channel capacity) with the following property. For any ɛ > 0 and R < C, for large enough N, there exists a block code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is < ɛ. Confirmation of the theorem for the noisy typewriter channel In the case of the noisy typewriter, we can easily confirm the theorem, because we can create a completely error-free communication strategy using a block code of length N = 1: we use only the letters B, E, H, . . . , Z, i.e., every third letter. These letters form a non-confusable subset of the input alphabet (see figure 9.5). Any output can be uniquely decoded. The number of inputs in the non-confusable subset is 9, so the error-free information rate of this system is log 2 9 bits, which is equal to the capacity C, which we evaluated in example 9.10 (p.151). pBM ✻ achievable C ✲ R Figure 9.4. Portion of the R, pBM plane asserted to be achievable by the first part of Shannon’s noisy channel coding theorem.
Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981 You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links. 9.7: Intuitive preview of proof 153 0 1 ✏ ✲ ✏✏✶ B B A ✏ ✲ ✏✏✶ E E DC Y Z✏ ✲Z - ✏✏✶ ✏ ✲ . ✏✏✶ H H I GF 0 1 00 10 01 11 00 10 01 11 0000 1000 0100 1100 0010 1010 0110 1110 0001 1001 0101 1101 0011 1011 0111 1111 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z - A B C D E F G H I J K L M N O P Q R S T U V W X Y Z - 0000 1000 0100 1100 0010 1010 0110 1110 0001 1001 0101 1101 0011 1011 0111 1111 N = 1 N = 2 N = 4 How does this translate into the terms of the theorem? The following table explains. The theorem How it applies to the noisy typewriter Associated with each discrete memoryless channel, there is a non-negative number C. For any ɛ > 0 and R < C, for large enough N, there exists a block code of length N and rate ≥ R The capacity C is log 2 9. Figure 9.5. A non-confusable subset of inputs for the noisy typewriter. Figure 9.6. Extended channels obtained from a binary symmetric channel with transition probability 0.15. No matter what ɛ and R are, we set the blocklength N to 1. The block code is {B, E, . . . , Z}. The value of K is given by 2 K = 9, so K = log 2 9, and this code has rate log 2 9, which is greater than the requested value of R. and a decoding algorithm, The decoding algorithm maps the received letter to the nearest letter in the code; such that the maximal probability of block error is < ɛ. 9.7 Intuitive preview of proof Extended channels the maximal probability of block error is zero, which is less than the given ɛ. To prove the theorem for any given channel, we consider the extended channel corresponding to N uses of the channel. The extended channel has |AX| N possible inputs x and |AY | N possible outputs. Extended channels obtained from a binary symmetric channel and from a Z channel are shown in figures 9.6 and 9.7, with N = 2 and N = 4.
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