infobit pairs (d.dd.d.d,d)= 8PSK symbols 00,01,11,10,00,11 0,2,7,5,1,6 0,3,7,4,1,7 .selector output TTOT 0,3,6,4,0,7 00,01,11,10,00,11 11.00100.10 =g44444) equence of infobit pairs 1 6,7,0,3,0,4 8PSK symbols TOTT Figure 5.9.4 From [Robertson98] polynomials of some co ponent TCM codes that can be employed in the TTCM scheme.These polynomials were obtained by Robertson and Worz using an exhaustive computer search method.In Table 5.9.1,denotes the number of information bits to be encoded from the m information bits contained in an input symbol. -PUNCTURED"TCM CODES WTTH BEST MINIMAL DISTANCE FOR S-PSK AND QAM (IN OCTAL NOTATION) Code m H(D)H(D) H2(D) H(D).e/△ 2-dim.8-PSK.8 states 02 04 4-dim.8PSK.states 4dim.8-Psk 16 state 2.dim 72 8 states 11 05 0 2.dim 22 16 states 3 210204 10 2-dim.22 8 states 2 1104 02 2-dim.22.16 states 2 210410 5.9.2.2 TTCM Decoder The iterative dec oder for TTCM is similar to that used to decode binary turbo codes except that i the nature of the informtion passed from one decoder to the other.Fig.5.9.5 illustrates the concept of a priori,a posteriori and extrinsic information used inTTCM decoder. a
6 Figure 5.9.4 From [Robertson98] Table 5.9.1 lists the generator polynomials of some component TCM codes that can be employed in the TTCM scheme. These polynomials were obtained by Robertson and Worz using an exhaustive computer search method. In Table 5.9.1, m denotes the number of information bits to be encoded from the m information bits contained in an input symbol. 5.9.2.2 TTCM Decoder The iterative decoder for TTCM is similar to that used to decode binary turbo codes, except that there is a difference in the nature of the information passed from one decoder to the other. Fig. 5.9.5 illustrates the concept of a priori, a posteriori and extrinsic information used in TTCM decoder
Channel infa [P&S] A posteriori info Non-binary Apriori info [S] MAP extrinsic info [S] Figure 5.9.5.Schematie of the component decoders for TTCM codes Note that in a symbol-based nonbinary TTCM scheme,the msystematic information bits and the parity bit are transmitted together in the same modulated symbol.Hence,the systematic component(corresponding received systematic value in binary turbo decoder)of the non-binary symbol cannot be separated from the extrinsic component.In this scenario the symbol-based information can be split into ony Laand Each deode passes ony the latter information to the ne component decoder while the a priori information is removed at each component decoder's output. The complete TTCM decoder operating in the log domain is depicted in Fig.5.9.6.At the beginning of decoding,the received symbols are input to the"metric"block so as to generate a set of M-2 symbol likelihoods.The selector switches select the current symbol's eliability metric pr duced by the"metric block"if the current syr as not n ured b the corresponding enc oder Othe rwise puncturing will be applied where the pods of the various legitimate symbols at index k are set to 0 in the log-domain.That is flog p(ylx). for un-punctured symbols Lp= 0. for punctured symbols On this basis,the TTCM decoder operates acc ording to the"turbo principle Specifically,for the MAP decoder 1,if the modulated symbol from the component TCM encoder I is punctured at time k,theny+for decoder 1,and set =0.The only input is a priori information=from the other component decoder,and this includes the systematic information for .At the output of the MAP decoder 1,the a priori information is subtracted from the a posteriori information,so that the same information is not used more than once in the other component decoder.Thus,the information passed from the decoder I to the decoder 2,L=LL,contains only the extrinsic information on If the modulated symbol from the component TCM encoder 1 is not punctured at timek, (y)represents the channel information,and the information passed from the decoder I to the decoder 2.L=Lcontains the inseparable extrinsic as well as >
7 Figure 5.9.5. Schematic of the component decoders for TTCM codes Note that in a symbol-based nonbinary TTCM scheme, the m systematic information bits and the parity bit are transmitted together in the same modulated symbol. Hence, the systematic component (corresponding received systematic value in binary turbo decoder) of the non-binary symbol cannot be separated from the extrinsic component. In this scenario the symbol-based information can be split into only two components: La and Le&s. Each decoder passes only the latter information to the next component decoder while the a priori information is removed at each component decoder’s output. The complete TTCM decoder operating in the log domain is depicted in Fig. 5.9.6. At the beginning of decoding, the received symbols are input to the “metric” block so as to generate a set of M=2m+1 symbol likelihoods. The selector switches select the current symbol’s reliability metric produced by the “metric block” if the current symbol was not punctured by the corresponding encoder. Otherwise puncturing will be applied where the likelihoods of the various legitimate symbols at index k are set to 0 in the log-domain. That is 初始化 & , log ( | ), for un-punctured symbols 0, for punctured symbols k k p sk py x L ⎧ = ⎨ ⎩ On this basis, the TTCM decoder operates according to the “turbo principle”. Specifically, for the MAP decoder 1, if the modulated symbol from the component TCM encoder 1 is punctured at time k, then yk = * for decoder 1, and set & 0 L p s = . The only input is a priori information (1) (2) L L a es = & from the other component decoder, and this includes the systematic information for uk. At the output of the MAP decoder 1, the a priori information is subtracted from the a posteriori information, so that the same information is not used more than once in the other component decoder. Thus, the information passed from the decoder 1 to the decoder 2, L LL e a = − , contains only the extrinsic information on uk. If the modulated symbol from the component TCM encoder 1 is not punctured at time k, then & log ( | ) Lps k k py x represents the channel information, and the information passed from the decoder 1 to the decoder 2, L L e es = & , contains the inseparable extrinsic as well as
systematic information component on The decoder2 operates according to the same message-passing principle as above Figure 5.of the TTCM decoder. 5.9.2.3 Numerical results Coucatenated Two-State TCM (CT-TCM)Schemes The component encoder of a CT-TCM code consists of a binary two-state trellis encode followed by a multi-ary signal mapper,see Fig.5.9.7.Let a binary n-tuple d=(dod)be an information symbol.Let d=dk be an input sequence to the binary encoder,producing a coded symbol sequenceEach ccontains a parity check bit(d)and is mapped to a appropriate ary signal constellation. producing a modulated symbol x.In the following,c andx=x are also referred to as unmodulated and modulated codewords,respectively. information coded modulated symbols binary symbols signal symbols d=(d.d)encoder c=(d,94) mapper Fig.5.9.7.The component encoder structure for the CT-TCM scheme. We assume that the binary encoder in Fig.5.9.7 is characterized by the two-state trellis in Fig.5.98(similar to the tree encoder in [Liping20011).The parity check bit is generated by =9+d8-48, mod 2,k>0 (5.9.7)
8 systematic information component on uk. The decoder 2 operates according to the same message-passing principle as above. (1) La Figure 5.9.6 Structure of the TTCM decoder. 5.9.2.3 Numerical results 5.9.3 Concatenated Two-State TCM (CT-TCM) Schemes 5.9.3.1 Encoder The component encoder of a CT-TCM code consists of a binary two-state trellis encoder followed by a multi-ary signal mapper, see Fig. 5.9.7. Let a binary n-tuple ( , , ) dk = dk ,0 " dk ,n−1 be an information symbol. Let } d = {dk , k ≥ 0 be an input sequence to the binary encoder, producing a coded symbol sequence c={ck}. Each ck contains a parity check bit qk, i.e., ck = (dk, qk), and is mapped to an appropriate 2n+1 -ary signal constellation, producing a modulated symbol xk. In the following, c and x={xk} are also referred to as unmodulated and modulated codewords, respectively. signal mapper coded symbols modulated symbols ( ) k k k c = d ,q k x binary encoder information symbols ( , , ) dk = dk ,0 " dk ,n−1 Fig. 5.9.7. The component encoder structure for the CT-TCM scheme. We assume that the binary encoder in Fig. 5.9.7 is characterized by the two-state trellis in Fig. 5.9.8 (similar to the tree encoder in [Liping2001]). The parity check bit qk is generated by qk qk dk gk = + ⋅ −1 ∑= = ⋅ k i i i 0 d g mod 2, k > 0 (5.9.7)
with odgHereg=(g)is an indication vector defined by [1 ifd participates in parity check g0 otherwise (5.9.8) The code in Fig.5.9.8 is completely specified by 91-1 d:8=0 9=1 - 久484=1 0 9=0g=0=0 Fig.5.9.8.Atwo-state trellis diagram with2branches in a trellis section.(In this figure,n.) The Global Encoder of CT-TCM Scheme Fig.5.9.9 depicts a global CT-TCM scheme,where M component encoders are concatenated in parallel by M symbol-interleavers.Modulo-M interleavers satisfying the ts are assumed, π(k)modM=k mod M, for m=0,1.M In order to increase spectral efficiency,we puncture all the modulated symbols in the mth component code,except those at positionk mod M=m).This,together with the constraint in(3),ensures that one and only one modulated symbol carrying the same d is transmitted,and that the punctured symbols are uniformly distributed in each component code We will always assume that a signal constellation of size 2"is used.This yields a spectral efficiency of n bits per symbol. input symbol interleaver2-state encodersignal mapper symbol interleaver 2-state encoder →signal mapper symbol interleaver 2-state encodersignal mapper Fig 5.9.9.A global CT-TCM encoder structure 5.9.3.2 Decoder 9
9 with 0 d0 g0 q = ⋅ . Here T k k k k n (g , g , , g ) = ,0 ,1 " , −1 g is an indication vector defined by ⎩ ⎨ ⎧ = 0 otherwise 1 if , participatesin parity check , k j k j d g (5.9.8) The code in Fig. 5.9.8 is completely specified by {gk}. 1 qk−1 = = 1 k q qk−1 = 0 qk = 0 dk ⋅ gk = 1 dk ⋅ gk = 0 dk ⋅ gk = 0 ⋅ = 1 k k d g Fig. 5.9.8. A two-state trellis diagram with 2n+1 branches in a trellis section. (In this figure, n=2.) The Global Encoder of CT-TCM Scheme Fig. 5.9.9 depicts a global CT-TCM scheme, where M component encoders are concatenated in parallel by M symbol-interleavers. Modulo-M interleavers satisfying the following constraints are assumed, k M k M m ( ) mod mod ( ) π = , for m = 0,1, . M-1 In order to increase spectral efficiency, we puncture all the modulated symbols in the mth component code, except those at position } {k | k mod M = m . This, together with the constraint in (3), ensures that one and only one modulated symbol carrying the same dk is transmitted, and that the punctured symbols are uniformly distributed in each component code. We will always assume that a signal constellation of size 2n+1 is used. This yields a spectral efficiency of n bits per symbol. 2-state encoder (M −1) symbol interleaver π signal mapper puncturer output input 2-state encoder (1) symbol interleaver signal mapper π 2-state encoder (0) symbol interleaver π signal mapper Fig. 5.9.9. A global CT-TCM encoder structure. 5.9.3.2 Decoder
The CT-TCM decoder structure is based on the multi-dimensional turbo-decoder decoders,each for one component code.The variables involved in Fig.5.9.10 are the log-likelihood (LL)values,as detailed below. DEC-0 DEC-mDEC-(M-1)Output L1 L 7(m) APP → Fig.5.9.10.The global decoder"for delay of one iteration and"for interleaving. Lo) the a posteriori LL values for all information symbols after decoding the mth component code. Im) the LL values (log p(y))based on individual channel observations of the mth component code.Its elements are calculated as Lm- flog p(y.x). for un-punctured symbols 10. for punctured symbols Lthe a prioriLL values for all information symbols for the mth component code. It is initialized to zeroes,implying information the extrinsic information produced by the mth component code,defined by L)=-) The Mlocal decoders operate successively.L contains the accumulated extrinsic information generated by all the local decoders except DEC-m L=L+L2.+L-+L0+L四+.+L-
10 The CT-TCM decoder structure is based on the multi-dimensional turbo-decoder incorporating the BCJR algorithm. The global decoder operating in the log domain for a CT-TCM code is shown in Fig. 5.9.10. It consists of M local APP (a posteriori probability) decoders, each for one component code. The variables involved in Fig. 5.9.10 are the log-likelihood (LL) values, as detailed below. Fig. 5.9.10. The global decoder: “T” for delay of one iteration and “π” for interleaving. (m) L the a posteriori LL values for all information symbols after decoding the mth component code. (m) Lc the LL values {log ( | ) k k p y x } based on individual channel observations of the mth component code. Its elements { } ( ) , m Lc k are calculated as ⎩ ⎨ ⎧ = 0, for punctured symbols log ( | ), for un - punctured symbols ( ) , m k k c k p y x L (m) La the a priori LL values for all information symbols for the mth component code. It is initialized to zeroes, implying no a priori information. (m) Le the extrinsic information produced by the mth component code, defined by ( ) ( ) (m) a m m Le = L − L . The M local decoders operate successively. (m) La contains the accumulated extrinsic information generated by all the local decoders except DEC-m, " " iteration from the current iteration from the previous ( ) ( +1) ( +2) ( −1) (0) (1) ( −1) = + + + + + + m e e e M e m e m e m La L L L L L L Output DEC-0 DEC-m DEC-(M-1) (m) La - APP T (m) π ( ) 1 [ ] m − π (m) L (m) Lc - (m) Le