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The Art of Error Correcting Coding phần 7

Tái chế có thể được thực hiện trong một cách có hệ thống để giảm thiểu số lượng tính toán. Đặc biệt, như đã đề cập tại mục 7.1, với truyền nhị phân trên một kênh AWGN, có không cần phải tính toán khoảng cách Euclide mà là mối tương quan giữa các từ mã được tạo ra và trình tự nhận được sắp xếp lại. | SOFT-DECISION DECODING 155 Reprocessing can be done in a systematic manner to minimize the number of computations. In particular as mentioned in Section 7.1 with binary transmission over an AWGN channel there is no need to compute the Euclidean distance but rather the correlation between the generated code words and the reordered received sequence. Thus the computation of the binary real sequence x is not needed. Only additions of the permuted received values zi with sign changes given by the generated Ũ are required. Example 7.5.1 Consider the binary Hamming 7 4 3 code with generator matrix 1 0 0 0 1 0 1 0 10 0 111 G 00101 10 0 0 0 1 0 1 1 Suppose that the received vector after binary transmission over an AWGN channel the outputs of a matched filter is given by r 0 5 0 3 1 3 -0 1 0 7 0 6 1 5 OSD with order-1 reprocessing is considered next. The permuted received vector based on reliability values is y n1 r 1 5 1 3 0 7 0 6 0 5 0 3 -0 1 with III 5 6 2 7 3 4 1 . The permuted generator matrix based on reliability values is 10101 0 0 1 0 0 0 1 1 1 10 110 1 0 0 10 0 10 1 G H1 G 0 1110 0 0 G 0 0 10 0 11 V 0 0 1 0 0 1 0 0 0 1110 Therefore G1 nfiG G with n2 I 1 2 3 4 5 6 7 the identity permutation . As a result n2 y 1 5 1 3 0 7 0 6 0 5 0 3 -0 1 The corresponding hard-decision vector is z 0 0 0 0 0 0 1 and the k 4 most reliable values are M0 0 0 0 0. The initial code word is as follows Ũ0 Ũ0G1 0 0 0 0 0 0 0 The decoding algorithm for order-1 reprocessing is summarized in the table below. The metric used is the correlation discrepancy wt ĩ f y lyiI 156 SOFT-DECISION DECODING I u t VC Ằ ze z 0 0000 0000000 0.1 1 1000 1000111 2.3 2 0100 0100101 1.8 3 0010 0010011 1.0 4 0001 0001110 1.4 The smallest metric corresponds to Í 0 and it follows that the decoded code word is vHD n 1 n 1 v0 0 0 0 0 0 0 0 . 7.6 Generalized minimum distance decoding In 1966 Forney 1966b introduced GMD decoding. The basic idea was to extend the notion of an erasure by dividing the received values into .