和积译码算法

1、4.3.1 Choice of Encoder ClassThe answer to the first question depends on the application. For example, many applications require that the encoder be systematic so that the data are readily observable in the code stream. As another example, parallel turbo codes (Chapter 7) require that the constituen

2、t convolutional codes be of the recursive type,and usually systematic. For many applications the encoder class matters very little,particularly applications in which the convolutional code is not a constituentcode of a turbo(-like) code. As seen in Example 4.3 and as we shall see in the discussion b

3、elow of the third question, the choice of encoder class can affect the encoder memory .14.3.2 Catastrophic Encoders24.3.3 Minimal Encoders345674.3.4 Design of Convolutional CodesRegarding the fourth question listed in Section 4.3, for reasons affecting decoder complexity, typical values of n and k a

4、re very small: k =1, 2, 3 and n = 2, 3, 4. Typical code rates are 1/2, 1/3, 1/4, 2/3, and 3/4, with 1/2 by far the most common rate.84.4 Alternative Convolutional Code RepresentationsUnder this section, we consider alternative representations of convolutional codes that are useful for many other asp

5、ects of convolutional codes, such asdecoding, code design, and analysis.4.4.1 Convolutional Codes as Semi-Infinite Linear Codes9101112131415161718194.4.2 Graphical Representations for Convolutional Code EncodersThere exist several graphical representations for the encoders of convolutional codes, wi

6、th each of these representations playing different roles.20We start with the finite-state transition-diagram (FSTD), or state-diagram, graphical model for G(D). The encoder realization for G(D) was presented earlier in Figure 4.4(a) and the encoder state is defined to be the contents of the two memo

7、ry elements in the encoder circuit (read from left to right). From that figure and the state definition, we may produce the following state-transition table from which the codes FSTD may be easily drawn, as depicted in Figure 4.8(a).2122We remark that a list of code sequences of any given length may

8、 be obtained from the trellis by tracing through all possible trellis paths corresponding to that length, along the way picking off the code symbols which label the branches (or edges) within each path:234.5 Trellis-Based Decoders4.5.1 MLSD and the Viterbi Algorithm2425262728294.5.2 Differential Vit

9、erbi Decoding30313233344.5.3 Bit-wise MAP Decoding and the BCJR Algorithm353637383940414243444.6 Performance Estimates for Trellis-Based Decoders4.6.1 ML Decoder Performance for Block CodesTo start, we observe that both channels are symmetric in the sense that for the BSC Pr(y = 1|x = 1) = Pr(y = 0|

10、x = 0) and for the BI-AWGN channel p(y|x) = p(y| x), where x and y represent the channel input and output, respectively. Given these conditions together with code linearity, the probabilityof error given that some codeword c was transmitted is identical to the probability of error given that any oth

11、er codeword c was transmitted. Thus, it is convenient to assume that the all-zeros codeword 0 was transmitted in performance analyses. We can therefore bound the probability of codeword error Pcw using the union bound as follows:4546474.6.2 Weight Enumerators for Convolutional CodesBecause digital c

12、ommunication links are generally packet-based, convolutional codes are often used as block codes. That is, the inputs to a rate-k/n convolutional encoder in this case are blocks of K information bits, which produce blocks of N = K(n/k) code bits.Recall that Aw is the number of weight-w codewords. Or, in the context of the codes trellis, it is the number of weight-w paths that diverge from the all-zeros trellis path one or more times in L = K/k trellis stages. Finding Aw is possible