E > 0, there exists a channel code C = {c1, CM} of length n over Ax and a channel decoding scheme g such CEC yeA":g(y)#0 1) M > 2[Rn] where Y = (Y1, Yn) and X= (X1, Xn). This is realized with maximum a posteriori decoding, where the 2) If codeword Ci is sent over the channel and word y is received word y=(y1,- yn) is mapped to a channel received, then Pr{g(y)#ci} < E for all i = 1, .,M. codeword C = (c1, cn) that maximizes the probability Pr{X = c|Y = y}. In practice, however, one uses maximum-likelihood decoding, where y is mapped to 3.4.2.3 Source-channel coding a channel codeword C that maximizes the probability Pr{Y = y|X = c}. It is easy to see that maximum- Suppose now that the output of the channel decoding likelihood decoding is equivalent to maximum a poste- scheme is mapped to a word of length k over a riori decoding when all channel codewords are generated reproduction alphabet A. The average distortion of the with the same probability. Another important decoding resulting transmission system is EEd(U, V)], where the random vector U describes a word of k successive Channel protection fundamentals CHAPTER 3.4 source symbols, the random vector V describes the 2. One can show [15] that for this channel reliable corresponding word of k reconstructed symbols, and E transmission is possible as long as the rate of transmission denotes the expectation operator. is smaller than the capacity The source-channel coding theorem [46] says what a system can achieve in terms of average distortion and C bits per transmission, rate of transmission. Theorem 3.4.4 (Shannon's source-channel coding theorem). Given a discrete memoryless source charac- where P is a constraint on the expected value of the terized by rate-distortion function R(D), a discrete random variable X2. If we denote by R the rate of memoryless channel characterized by capacity C > 0, any transmission, by Es = P the symbol energy, and by Eb = D > Dmin, and any r < C/R(D), there exist for sufficiently E/R the energy per bit, then the condition R < C gives