 to a regular pattern, a turbo interleaver randomizes the ordering of the data in an irregular manner. The operation of a turbo encoder and decoder can be described as follows. Let the binary input bits {0,1} be represented by bipolar levels {+1,-1} and assigned to the variable d; which may take values d = +1 and For an AWGN channel, the conditional probability density functions f(x and f(x +1) are referred to as likelihood functions. The common hard decision criterion, known as maximum likelihood, selects the symbol dk +1 or dk depending on the intercept point of received signal value Xk and the above conditional probability density functions using a fixed 9.12 Channel coding and modulation Systematic Bits Input Bits Output Bits Parallel to Serial Converter Parity Bits XP1 Constituent Convolutional Encoder 1 Constituent Turbo Interleaver Convolutional Encoder 2 Interleaved Interleaved Bits Parity Bits XP2 Turbo Encoder Structure Output Bits Decision De-interleaver Interleaver Parity Bits XP1 Decoder 1 Decoder 2 Systematic Bits Interleaver Interleaved Parity Bits XP2 Iterative Turbo Decoder Structure FIGURE 9-41 Generic structure of turbo encoder and iterative decoder threshold of A (decision point); thus, dk = +1 if Xk > A; otherwise, dk==1 Another decision rule, known as Maximum a Posteriori (MAP) criterion takes into account the a posteriori probabilities f(d = +1x) and f(d==-1|)) = to construct the hypotheses H and H2 as follows: (9-31) That is, one selects hypothesis H1(d = +1), = if the f(d = = -1|x); otherwise, H2(d = -1) is selected. Using the Bayes' theorem, [66] the above a posteriori probabilities can replaced by their equivalent expressions, yielding; f(xd = +1)p(d = +1) } f(xd = -1)p(d==1) = (9-32) CHAPTER 9 The IEEE 802. 16m Physical Layer (Part I) The likelihood ratio test is constructed as follows: f(xd = +1)p(d==1)", (9-33) f(xd = -1)p(d==1)m If the input bits are independent and identically distributed random variables, the above equation is simplified as follows: f(x|d = +1)") f(x|d = -1) n (9-