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ENCODING of INFINITE INPUT STREAMS ENCODING The way infinite streams are encoded using convolution codes will be Illustrated on the code CC 1 . An input stream I = (I 0 , I 1 , I 2 , . . .) is mapped into the output stream C = (C 00 , C 10 , C 01 , C 11 . . .) defined by C 0 (x) = C 00 + C 01 x + . . . = (x 2 + 1)I (x) The first shift register input output and C 1 (x) = C 10 + C 11 x + . . . = (x 2 + x + 1)I (x). The first multiplication can be done by the first shift register from the next figure; second multiplication can be performed by the second shift register on the next slide and it holds C 0i = I i + I i+2 , C 1i = I i + I i−1 + I i−2 . That is the output streams C 0 and C 1 are obtained by convolving the input stream with polynomials of G 1 . will multiply the input stream by x 2 + 1 and the second shift register output input will multiply the input stream by x 2 + x + 1. prof. Jozef Gruska IV054 3. Cyclic codes 45/71 ENCODING and DECODING The following shift-register will therefore be an encoder for the code CC 1 prof. Jozef Gruska IV054 3. Cyclic codes 46/71 SHANNON CHANNEL CAPACITY For every combination of bandwidth (W ), channel type , signal power (S) and received noise power (N), there is a theoretical upper bound, called channel capacity or Shannon capacity, on the data transmission rate R for which error-free data transmission is possible. input output streams For so-called Additive White Gaussian Noise (AWGN) channels, that well capture deep space channels, this limit is (so-called Shannon-Hartley theorem): „ R < W log 1 + S « {bits per second} N Shannon capacity sets a limit to the energy efficiency of the code. For decoding of convolution codes so called Viterbi algorithm Is used. prof. Jozef Gruska IV054 3. Cyclic codes 47/71 Till 1993 channel code designers were unable to develop codes with performance close to Shannon capacity limit, that is Shannon capacity approaching codes, and practical codes required about twice as much energy as theoretical minimum predicted. Therefore there was a big need for better codes with performance (arbitrarily) close to Shannon capacity limits. Concatenated codes and Turbo codes have such a Shannon capacity approaching property. prof. Jozef Gruska IV054 3. Cyclic codes 48/71
CONCATENATED CODES - I CONCATENATED CODES - II Let C in : A k → A n be an [n, k, d] code over alphabet A. Let C out : B K → B N be an [N, K, D] code over alphabet B with |B| = |A| k symbols. The basic idea of concatenated codes is extremely simple. Input is first encoded by one code C 1 and the output is then encoded by second code C 2 . To decode, at first C 2 and then C 1 decoding are used. In 1972 Forney showed that concatenated codes could be used to achieve exponentially decreasing error probabilities at all data rates less than channel capacity in such a way that decoding complexity increases only polynomially with the code block length. Concatenation of C out (as outer code) with C in (as inner code), denoted C out ◦ C in is the [nN, kK, dD] code C out ◦ C in : A kK → A nN that maps an input message m = (m 1 , m 2 , . . . , m K ) to a codeword (C in (m ′ 1), C in (m ′ 2), . . . , C in (m ′ N)), where (m ′ 1, m ′ 2, . . . , m ′ N) = C out (m 1 , m 2 , . . . , m K ) In 1965 concatenated codes were considered as infeasible. However, already in 1970s technology has advanced sufficiently and they became standardize by NASA for space applications. super encoder super decoder outer encoder inner encoder noisy channel inner decoder outer decoder super channel prof. Jozef Gruska IV054 3. Cyclic codes 49/71 CONCATENATED CODES - III prof. Jozef Gruska IV054 3. Cyclic codes 50/71 ANOTHER VIEW of CONCATENATED CODES super encoder super decoder super outer encoder encoder inner encoder noisy channel super channel super decoder inner decoder outer decoder Of the key importance is the fact that if C in is decoded using the maximum-likelihood principle (thus showing an exponentially decreasing error probability with increasing length) and C out is a code with length N = 2 n r that can be decoded in polynomial time in N, then the concatenated code can be decoded in polynomial time with respect to n2 nr and has exponentially decreasing error probability even if C in has exponential decoding complexity. outer encoder inner encoder noisy channel super channel Outer code: - (n 2 , k 2 ) code over GF(2 k 1 ); Inner code: - (n 1 , k 1 ) binary code Inner decoder - (n 1 , k 1 ) code Outer decoder - (n 2 , k 2 ) code length of such a concatenated code is n 1 n 2 dimension of such a concatenated code is k 1 k 2 inner decoder outer decoder if minimal distances of both codes are d 1 and d 2 , then resulting concatenated code has minimal distance ≥ d 1 d 2 . prof. Jozef Gruska IV054 3. Cyclic codes 51/71 prof. Jozef Gruska IV054 3. Cyclic codes 52/71
Page 1 and 2: CHAPTER 3: CYCLIC CODES, CHANNEL CO
Page 3 and 4: RINGS of POLYNOMIALS For any polyno
Page 5 and 6: EXAMPLE The task is to determine al
Page 7 and 8: MULTIPLICATION of POLYNOMIALS by SH
Page 9 and 10: POLYNOMIAL CODES BCH CODES and REED
Page 11: CHANNEL (STREAM) CODING II CONVOLUT
Page 15 and 16: DECODING and PERFORMANCE of TURBO C
Page 17 and 18: APPENDIX APPLICATIONS of REED-SOLOM

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