0122229现代数字通信与编码理论 December 10,2011 XDU,Winter 2011 Lecture Notes Capacity-Approaching Coded-Modulation Schemes 5.9级联TCM TCM码可以按Tubo方式并行级联，也可以与其它码构成串行级联系统（例如，在 RS+TCM中做内码，在TCM+STBC中做外码)。 5.9.1TCM码的Symbol-Based Nonbinary)MAP译码算法 Consider a general trellis code.Let u=()be an information symbol sequence of length N.Each information symbol is assumed to be a binary m-tuple,i.e. =)"for =1.2.N.For convenience,we will denote u by its decimal representation,ie.,assume=,."-1.The m-bit symbols are fed into a TCM encoder,producing a sequence of N modulated symbols,x=(x),where xwithstanding for the signal constellation of size2".We assume that the symbols xx are transmitted over an AWGN channel.Let y=y=(y.y.)be the received sequence,where y(k=1,2.N)is the noisy observation of x if x is not punctured;otherwise,sety=*(implying no observation obtained). TCM yk Channel Symbol-based encoder MAP decoder Figure 5.9.1Atrellis coding system The received sequence is fed to the TCM decoder to produce the estimate of transmitted 2-ary information symbols.To minimize the probability of symbol errors,the symbol-based MAP decoder may be used.This decoder computes the APP for every 2"-ary information symbol;i.e.,P(=iy),for i=0,1,."-1,k=1,2.N Then it makes decisions according to the MAP criterion: ig =argmax P(u =ily
1 0122229 现代数字通信与编码理论 December 10, 2011 XDU, Winter 2011 Lecture Notes Capacity-Approaching Coded-Modulation Schemes 5.9 级联 TCM TCM 码可以按 Turbo 方式并行级联，也可以与其它码构成串行级联系统（例如，在 RS+TCM 中做内码，在 TCM+STBC 中做外码）。 5.9.1 TCM 码的(Symbol-Based Nonbinary) MAP 译码算法 Consider a general trellis code. Let u uu u = ( 1 2 , ,., N ) be an information symbol sequence of length N. Each information symbol is assumed to be a binary m-tuple, i.e., ( ) ,0 ,1 , 1 2 , ,., m k k k km uu u u = ∈ − F , for k=1,2,.N. For convenience, we will denote uk by its decimal representation; i.e., assume {0,1,.,2 1} m k u ∈ U = − . The m-bit symbols are fed into a TCM encoder, producing a sequence of N modulated symbols, ( ) 1 2 , ,., N x = x x x , where k x ∈X with X standing for the signal constellation of size | | 2n X = . We assume that the symbols xk are transmitted over an AWGN channel. Let 1 12 ( , ,., ) N N y y = = yy y be the received sequence, where k y (k=1,2,.N) is the noisy observation of k x if k x is not punctured; otherwise, set yk = ∗ (implying no observation obtained). ˆk u Figure 5.9.1 A trellis coding system The received sequence is fed to the TCM decoder to produce the estimate of transmitted 2m-ary information symbols. To minimize the probability of symbol errors, the symbol-based MAP decoder may be used. This decoder computes the APP for every 2m-ary information symbol; i.e., ( ) 1 | , for 0,1,.,2 1, 1,2,., N m Pu i i k N k = = −= y . Then it makes decisions according to the MAP criterion: ˆ arg max | ( 1 ) N k k i u Pu i ∈ = = y U
where u=(01.2m-1 We now describe the MAP algorithm in detail by using the trellis diagram.For no loss of generality of the result we assume that the trellis diagram contains parallel transitions.Refer to Fig.5.9.2.Denote by S the set of encoder states and by M=the number of states.In Fig.5.9.2=4 is assumed.Let S be the encoder state at time k.A branch at the kth section in the corresponding trellis diagram can be specified by b=(SS),where ug and x are the information and modulated symbols associated with the state transition SS Denote by B.(s',s)the set of all the parallel branches connecting S=s'es and S=sES.The non-binary MAP algorithm can be derived as follows. ① ① ② ② ② ② ③ 3) +③ y y Figure5.9.2 Pa=)Pu=S=8=） A,5=功 (5.9.1) p(y) where B is the set of transitions SS that are caused by the input ui.According to the BCJR algorithm,the probabilities p(.=s'S=s.y)can be calculated as p(u=1,S1=S',S=3,y)=4-(s)y(s',s)B(g) where
2 where {0,1,.,2 1} m U = − . We now describe the MAP algorithm in detail by using the trellis diagram. For no loss of generality of the result, we assume that the trellis diagram contains parallel transitions. Refer to Fig. 5.9.2. Denote by S the set of encoder states and by M = |S| the number of states. In Fig.5.9.2 |S|=4 is assumed. Let k S be the encoder state at time k. A branch at the kth section in the corresponding trellis diagram can be specified by 1 ( ,) k kkk b S uxS ≡ − , where uk and k x are the information and modulated symbols associated with the state transition k k 1 S S − → . Denote by ( ', ) Bk s s the set of all the parallel branches connecting 1 ' k S s − = ∈S and k S s = ∈S . The non-binary MAP algorithm can be derived as follows. 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 uk /xk Sk-1 S S k 0 SN yk 1 1 k − y 1 N k+ y Figure 5.9.2 ( 1 11 ) ( ) ( ', ) | , ', | i k N N k kk k ss B Pu i Pu iS s S s − ∀ ∈ = = = == y y ∑ 1 1 ( ', ) 1 ( , ', , ) i ( ) k N kk k N ss B pu iS s S s p − ∀ ∈ = == = ∑ y y (5.9.1) where i Bk is the set of transitions S S k k −1 → that are caused by the input uk=i. According to the BCJR algorithm, the probabilities 1 1 ( , ', , ) N kk k pu iS s S s = == − y can be calculated as 1 1 ( , ', , ) N kk k pu iS s S s = == − y 1( ') ( ', ) ( ) i kk k αγ β s ss s =⋅⋅ − where
a(s)=p(S=s.y)is the forward state metric. B(s)=p(y=s)is the backward state metric;and (s',s)=p(ug=i.S.=s.yS=s')is the metric of the trellis branch b connecting S=s'to S=s with us=i. Note that we do not necessarily need the exact values of P(=ily,but only their ratios. Since for a fixedk,the vector P(y),being a probability vector,must sum to unity, thus by normalizing the sum to unity,we can omit the constant factor p(y in (5.9.1)and use the following simplified expression P(u =ily)>a(s)-n(s'.s)B(s) ( Now we consider the calculation of the branch metric(s's),which is the key point of the symbol-based MAP algorithm,and is also the primary difference between the MAP algorithm for binary turbo codes and the algorithm under consideration. Using Bayes'rule,the term(s'.s)can be written as (s',s)=pyS=3,S-1=S',4=)PS=sS=',4=)P(u=iS-=s) =p(yx)P(u=i)-P(S.=sIS=s'.u=i) (5.9.2) where p()is the probability of receivingy given the transmitted symbol x P() is the a priori probability of ug=i,and P(S=sS=s',=i)is 1 or 0,indicating whether or not a state transition SS with input=i exists.Thus,for a trellis branch b,we have 2s,=P4), ify=* x)P(u). (5.9.3) otherwise Define r(s,s)=p(S=s,yIS-=s) 3
3 α k k k () ( , ) s pS s ≡ = y1 is the forward state metric; βk k N k () ( | ) sp Ss ≡ = y +1 is the backward state metric; and ( ) 1 ( ', ) , , | ' i k k k kk γ s s pu iS s S s ≡ == = − y is the metric of the trellis branch b connecting Sk-1 = s’ to Sk = s with uk = i. Note that we do not necessarily need the exact values of ( | 1 ) N Pu i k = y , but only their ratios. Since for a fixed k, the vector ( ) 1 | N Pu i k = y , being a probability vector, must sum to unity, thus by normalizing the sum to unity, we can omit the constant factor 1 1 ( ) N p y in (5.9.1) and use the following simplified expression: ( ) 1 | N Pu i k = y 1 ( ', ) ( ') ( ', ) ( ) i k i kk k ss B αγ β s ss s − ∀ ∈ ∝ ⋅⋅ ∑ Now we consider the calculation of the branch metric ( ', ) i k γ s s , which is the key point of the symbol-based MAP algorithm, and is also the primary difference between the MAP algorithm for binary turbo codes and the algorithm under consideration. Using Bayes’ rule, the term ( ', ) i k γ s s can be written as ( ', ) i k γ s s 1 11 ( | , ', ) ( | ', ) ( | ') kk k k k k k k k p S sS s u i PS s S s u i Pu i S s = = = =⋅ = = =⋅ = = − −− y 1 ( | ) ( ) ( | ', ) kk k k k k p y x Pu i PS s S s u i = ⋅ =⋅ = = = − (5.9.2) where ( | ) k k p y x is the probability of receiving yk given the transmitted symbol xk, P uk ( ) is the a priori probability of uk = i, and 1 ( | ', ) PS s S s u i kk k = − = = is 1 or 0, indicating whether or not a state transition k k 1 S S − → with input uk = i exists. Thus, for a trellis branch b, we have ( ', ) i k γ s s ( ) , if ( | ) ( ), otherwise k k kk k Pu y p y x Pu ⎧ = ∗ = ⎨ ⋅ ⎩ (5.9.3) Define ( ) 1 ( ', ) , | ' k k kk γ ss pS s S s ≡= = − y
which represent the combined branch metrics.Then the non-binary BCJR algorithm can be summarized as follows. Forward recursion: a(s)-Ea-(')n.(s.s) (5.94) Backward recursion: B(s)=∑B(s)M(s',s) (5.9.5) Output: (s)ri(s',5)B(5) (5.9.6) The boundary conditionsa are theame s for the binary b coe For merical is necessary to control the dynamic range of the likelihood terms computed in (5.9.4)to (5.9.6).This can be performed by normalizing the sum of the(s)and the B(s)values to unity at every particular k symbol.Of course,the algorithm can also be implemented in the ov-do Log-MAP or Max-Log-MAP version. reduced computational complexity,resulting in the 5.9.2 Turbo Trellis-Coded Modulation (TTCM) TTCM是由两个（或多个）TCM码按照Turbo码的方式级联起来构成的并行级联编 码调制系统，它是标准二元turbo码的直接推广。This technique was originally proposed by Robertson and Worz in 1996[?] 5.9.2.1 TTCM Encoder A TTCM encoder consists of the parallel concatenation of two (or multiple)trellis-coded modulation(TCM)schemes in the same fashion as binary turbo codes.Each component TCM encoder onsists of a。 nvolutional encoder of rate mm)and a signal mapper.The mapping of bitsto signal points follows the Ungerboeck'sr of mapping by set partitioning.The first TCM encoder operates on the original input bit sequence,while the second TCM encoder manipulates the interleaved version of the input bit sequence.Here,the interleaving works on groups of bits instead of individual bits.See Fig.5.9.3,where a symbol-based odd-even interleaver is assumed.To avoid excessive r te loss,a spe puncturing technique is used:Symbols from a ary signal constellation are ansmitted alternately from the first and second encoders;i.e.,the puncturing matrix is given by P=0 01
4 ( ', ) ( ', ) k k i k b B ss γ s s ∀ ∈ = ∑ which represent the combined branch metrics. Then the non-binary BCJR algorithm can be summarized as follows. Forward recursion: 1 ' ( ) ( ') ( ', ) k kk s α αγ s s ss − ∈ = ∑ S (5.9.4) Backward recursion: 1( ') ( ) ( ', ) k kk s β βγ s s ss − ∈ = ∑ S (5.9.5) Output: ( ) 1 | N Pu i k = y 1 ( ', ) ( ') ( ', ) ( ) i k i kk k ss B αγ β s ss s − ∀ ∈ ∝ ⋅⋅ ∑ (5.9.6) The boundary conditions are the same as for the binary turbo codes. For numerical stability, it is necessary to control the dynamic range of the likelihood terms computed in (5.9.4) to (5.9.6). This can be performed by normalizing the sum of the ( ) k α s and the ( ) k β s values to unity at every particular k symbol. Of course, the algorithm can also be implemented in the log-domain with reduced computational complexity, resulting in the Log-MAP or Max-Log-MAP version. 5.9.2 Turbo Trellis-Coded Modulation (TTCM) TTCM 是由两个（或多个）TCM 码按照 Turbo 码的方式级联起来构成的并行级联编 码调制系统，它是标准二元 turbo 码的直接推广。This technique was originally proposed by Robertson and Worz in 1996 [?]. 5.9.2.1 TTCM Encoder A TTCM encoder consists of the parallel concatenation of two (or multiple) trellis-coded modulation (TCM) schemes in the same fashion as binary turbo codes. Each component TCM encoder consists of a convolutional encoder of rate m/(m+1) and a signal mapper. The mapping of coded bits to signal points follows the Ungerboeck’s rule of mapping by set partitioning. The first TCM encoder operates on the original input bit sequence, while the second TCM encoder manipulates the interleaved version of the input bit sequence. Here, the interleaving works on groups of bits instead of individual bits. See Fig. 5.9.3, where a symbol-based odd-even interleaver is assumed. To avoid excessive rate loss, a special puncturing technique is used: Symbols from a 2m+1-ary signal constellation are transmitted alternately from the first and second encoders; i.e., the puncturing matrix is given by 1 0 0 1 P ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ Thus each information bit pair is transmitted in exactly one transmitted symbol with the parity bit alternately chosen from the first and second encoders
TCM encoder Trellis Signal encoder 1 mappe ymbol Puncturing ave Trellis encoder 2 Signal mappe Figure 5.9.3 TTCM encoder.The puncturer is used to avoid rate loss by enabling the transmission of signals carrying the same information bits only once. Assume that a D-dim constellation is used.At each step,m information bits are input to the TTCM ncode nodulated syn a D-dim s constellation of size2 1 We now consider the above odd-even interleaving-puncturing rule more更深入、更一般 LetL denote the number of component codes used in a parallel concatenated system.Then introducing the following definitions the above odd-even interleav rule may be expressed mathematically via Definition:A (pseudo-random)interleaver satisfying the following constraint is called a modulo-L interleaver: π(k)modL=k mod L, wherek stands for the symbol position before interleaving. Definition:A modulo-L puncturer is one that punctures all the modulated symbols in the /th component code,except those at positionk modL=1. For a TTCM scheme,the modulo-L interleaving together with the modulo-L puncturing rule ensure that one and only one modulated symbol carrying the same uk is transmitted,and that the punctured symbols are uniformly distributed in each component code.It is clear that when L=2,the above modulo-L interleaving-puncturing rule is equivalent to that used in [3]. As an example,Fig5.9.4 shows an 8-state TTCM encoder for 8-PSK
5 Trellis encoder 1 Trellis encoder 2 Signal mapper Signal mapper Symbol interleaver TCM encoder Puncturing and Multiplexing Figure 5.9.3 TTCM encoder. The puncturer is used to avoid rate loss by enabling the transmission of signals carrying the same information bits only once. Assume that a D-dim constellation is used. At each step, m information bits are input to the TTCM encoder and then a modulated symbol selected from a D-dim signal constellation of size 2m+1 is transmitted. This yields a throughput of 2 / m D bits per 2-dim symbol. We now consider the above odd-even interleaving-puncturing rule more 更深入、更一般 地. Let L denote the number of component codes used in a parallel concatenated system. Then the above odd-even interleaving-puncturing rule may be expressed mathematically via introducing the following definitions. Definition: A (pseudo-random) interleaver satisfying the following constraint is called a modulo-L interleaver: π ( ) mod mod k Lk L = , where k stands for the symbol position before interleaving. Definition: A modulo-L puncturer is one that punctures all the modulated symbols in the lth component code, except those at position { | mod } kk L l = . For a TTCM scheme, the modulo-L interleaving together with the modulo-L puncturing rule ensure that one and only one modulated symbol carrying the same uk is transmitted, and that the punctured symbols are uniformly distributed in each component code. It is clear that when L=2, the above modulo-L interleaving-puncturing rule is equivalent to that used in [3]. As an example, Fig.5.9.4 shows an 8-state TTCM encoder for 8-PSK
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