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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. 578 48 — Convolutional Codes and Turbo Codes 1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 received z4 〈d z3 〈d z2 〈d z1 〈d ✲ ⊕ ✻ z0 ✲ t (b) ✻ ✲ t (a) ⊕ ⊕ ✛ ❵ s ❄ ⊕ ✲ ❄ ⊕ ✲ ❄ ✲ ✲ 21 1, 37 8 0 1 1 0 1 0 1 0 0 0 0 1 1 1 1 0 In general a linear-feedback shift-register with k bits of memory has an impulse response that is periodic with a period that is at most 2 k − 1, corresponding to the filter visiting every non-zero state in its state space. Incidentally, cheap pseudorandom number generators and cheap cryptographic products make use of exactly these periodic sequences, though with larger values of k than 7; the random number seed or cryptographic key selects the initial state of the memory. There is thus a close connection between certain cryptanalysis problems and the decoding of convolutional codes. 48.3 Decoding convolutional codes The receiver receives a bit stream, and wishes to infer the state sequence and thence the source stream. The posterior probability of each bit can be found by the sum–product algorithm (also known as the forward–backward or BCJR algorithm), which was introduced in section 25.3. The most probable state sequence can be found using the min–sum algorithm of section 25.3 (also known as the Viterbi algorithm). The nature of this task is illustrated in figure 48.6, which shows the cost associated with each edge in the trellis for the case of a sixteen-state code; the channel is assumed to be a binary symmetric channel and the received vector is equal to a codeword except that one bit has been flipped. There are three line styles, depending on the value of the likelihood: thick solid lines show the edges in the trellis that match the corresponding two bits of the received string exactly; thick dotted lines show edges that match one bit but mismatch the other; and thin dotted lines show the edges that mismatch both bits. The min–sum algorithm seeks the path through the trellis that uses as many solid lines as possible; more precisely, it minimizes the cost of the path, where the cost is zero for a solid line, one for a thick dotted line, and two for a thin dotted line. ⊲ Exercise 48.2. [1, p.581] Can you spot the most probable path and the flipped bit? Figure 48.6. The trellis for a k = 4 code painted with the likelihood function when the received vector is equal to a codeword with just one bit flipped. There are three line styles, depending on the value of the likelihood: thick solid lines show the edges in the trellis that match the corresponding two bits of the received string exactly; thick dotted lines show edges that match one bit but mismatch the other; and thin dotted lines show the edges that mismatch both bits.
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. 48.4: Turbo codes 579 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 transmit 1 1 1 0 1 0 1 0 0 0 0 1 1 1 1 0 source 1 1 1 1 0 0 1 1 1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 transmit 1 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 source 1 1 1 1 0 0 1 0 Figure 48.7. Two paths that differ in two transmitted bits only. 1111 Figure 48.8. A terminated trellis. Unequal protection A defect of the convolutional codes presented thus far is that they offer unequal protection to the source bits. Figure 48.7 shows two paths through the trellis that differ in only two transmitted bits. The last source bit is less well protected than the other source bits. This unequal protection of bits motivates the termination of the trellis. A terminated trellis is shown in figure 48.8. Termination slightly reduces the number of source bits used per codeword. Here, four source bits are turned into parity bits because the k = 4 memory bits must be returned to zero. 48.4 Turbo codes An (N, K) turbo code is defined by a number of constituent convolutional encoders (often, two) and an equal number of interleavers which are K × K permutation matrices. Without loss of generality, we take the first interleaver to be the identity matrix. A string of K source bits is encoded by feeding them into each constituent encoder in the order defined by the associated interleaver, and transmitting the bits that come out of each constituent encoder. Often the first constituent encoder is chosen to be a systematic encoder, just like the recursive filter shown in figure 48.6, and the second is a non-systematic one of rate 1 that emits parity bits only. The transmitted codeword then consists of When any codeword is completed, the filter state is 0000. ✲ C1 ✲ ✎☞ ✲ ✲ π C2 ✲ ✍✌ Figure 48.10. The encoder of a turbo code. Each box C1, C2, contains a convolutional code. The source bits are reordered using a permutation π before they are fed to C2. The transmitted codeword is obtained by concatenating or interleaving the outputs of the two convolutional codes.
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