It is used together with channel observationsL in the next APP decoding of the mth component code.A discussion on this global decoder can also be found in6] 5.9.4 PC-TCM In this scheme,two parallel concatenated ratek/(+1)convolutional codes use all their information bits to produce the parity bit which is sent to their respective symbol mappers,but the information bits are punctured in such a fashion that one half of them are used in the first trellis code and the other half in the second trellis code [Benede o-Divsalar].This is done transmitted by the two trellis encoders are interleaved by different bit interleavers(see Figure 5.9.11).In [Fragouli-Wesel01].a variation of this scheme is presented where a symbol interleaver is used instead of bit interleavers,resulting in improved BER performance in the waterfall region Figure5.9.11 5.10 Multilevel Codes and Multistage Decoding MLC was first proposed by Imai and Hirakawa in 1977,with the idea of protecting each 11
11 It is used together with channel observations (m) Lc in the next APP decoding of the mth component code. A discussion on this global decoder can also be found in [16]. 5.9.4 PC-TCM In this scheme, two parallel concatenated rate k/(k+1) convolutional codes use all their information bits to produce the parity bit which is sent to their respective symbol mappers, but the information bits are punctured in such a fashion that one half of them are used in the first trellis code and the other half in the second trellis code [Benedetto-Divsalar96]. This is done to limit the number of points in the signal constellation to 2(k/2)+1. Also, the information bits transmitted by the two trellis encoders are interleaved by different bit interleavers (see Figure 5.9.11). In [Fragouli-Wesel01], a variation of this scheme is presented where a symbol interleaver is used instead of bit interleavers, resulting in improved BER performance in the waterfall region. Figure 5.9.11 5.10 Multilevel Codes and Multistage Decoding MLC was first proposed by Imai and Hirakawa in 1977, with the idea of protecting each
address bit of the signal point by an individual binary code C at level i.The individual codes were chosen in such a way that the minimum diatance of the Euclidean space code was maximized.This is now known as balanced distance rule.At the receiver side,each code C is decoded individually starting from the lowest level and taking into account decisions of prior decoding stages.This is known as multistage decoding. mpared ith TCM,the MLC approach provides rates anycode (e.g,block codes,convolutional codes,or concatenated codes)can be used as component code.TCM can be viewed as a special case of MLC using a single convolutional code with a nonbinary output alphabet while higher levels remain uncoded. 5.10.1 Block coded modulation(BCM) BCM is an ex ample of multilevel coded modulation.从TCM的set-partitioning可以看 出，信号label中的不同bit要求不同的保护能力。在BCM中，we provide this by用不 同纠错能力的码编不同bit具体地说，encode the LSB with a powerful code,and encode the MSB with the least powerful code(甚至不编码)。 将label排成一个阵列，forming columns with LSB at the top ccc.c- →x=[xx.x] Vectorsa,b,and e should form三个不同的长为n的码。 As an example,Fig.5.10.1 shows a 3-level BCM code over the 8-PSK signal set. Lo) c0) (8.1,8)repetition code 8-PSK/QPSK (8.7.2)SPC code QPSK/BPSK (8,8,1)trivial code Fig5.10.1 An I-level BCM code over a signal set(M-PSK or M-QAM)with 2 signal points is constructed in the same manner as a 3-level 8-PSK BCM code.For 0si<,let C,be a binary (n,k,d)linear block code of length n,dimension,and minimum Hamming distance di.Let
12 address bit c (i) of the signal point by an individual binary code Ci at level i. The individual codes were chosen in such a way that the minimum diatance of the Euclidean space code was maximized. This is now known as balanced distance rule. At the receiver side, each code Cj is decoded individually starting from the lowest level and taking into account decisions of prior decoding stages. This is known as multistage decoding. Compared with TCM, the MLC approach provides flexible transmission rates; any code (e.g., block codes, convolutional codes, or concatenated codes) can be used as component code. TCM can be viewed as a special case of MLC using a single convolutional code with a nonbinary output alphabet while higher levels remain uncoded. 5.10.1 Block coded modulation (BCM) BCM is an example of multilevel coded modulation. 从 TCM 的 set-partitioning 可以看 出，信号 label 中的不同 bit 要求不同的保护能力。在 BCM 中，we provide this by 用不 同纠错能力的码编不同 bit.具体地说，encode the LSB with a powerful code, and encode the MSB with the least powerful code (甚至不编码)。 将 label 排成一个阵列，forming columns with LSB at the top. 01 1 01 1 01 1 n n n aa a bb b cc c − − − ⎡⎤ ⎡ ⎤ ⎢⎥ ⎢ ⎥ = ⎣⎦ ⎣ ⎦ a b c " " " ⇒ = x [ x01 1 x x " n− ] Vectors a, b, and c should form 三个不同的长为 n 的码。 As an example, Fig. 5.10.1 shows a 3-level BCM code over the 8-PSK signal set. Fig. 5.10.1 An l-level BCM code over a signal set (M-PSK or M-QAM) with 2l signal points is constructed in the same manner as a 3-level 8-PSK BCM code. For 0 ≤ <i l , let Ci be a binary (n, ki, di) linear block code of length n, dimension ki, and minimum Hamming distance di. Let
co=(c0,c0,c9） c"=(c,c",c） c-=(c-,c-,c） be I codewords in Co.CC,respectively.Forming the following sequence co*c四*.*c-=(C0c.c-,c0c".C-,cc.c） For 0j<n,we takeas the label of a signal point a.Then f(co*c四.*c-）=f(cc".c-）,f(cc".c-）,f(cc.c）》 =(,x-） is a sequence of n signals,with=f(.Then cf(co*Cw**C-）=k*e"*.*e-je0 e for0si< is an /-level BCM code over with dimensionk=k+++ 5.10.2MLG MLC is BCM的推广。分量码不必是等长的二进制线性分组码，也可以是卷积码、 级联码甚至非二进制码.它的编码器的一般结构如图5.l0.2所示。A generic MLC encoder: is shown in Fig.5.10.2.A block of K binary source data symbols,u=()0 is partitioned into /blocks: u0=(49,49,2）,i=01-1 of lnghach daa blcd inniry cncoer ENC producing codewords=)of the component code C For simplicity,we have assumed equal code lengths n at all levels,but in principle the choice of component codes is arbitrary.The code symbols c=at time j is mapped to the signal constellation,.producing the transmitted signal,=fc,)=f(cc"c-）eX.通常，调 制器使用Ungerboeck'smapping by set-partitioning完成信号的映射。 The code rate of C is given by
13 ( ) ( ) ( ) (0) (0) (0) (0) 01 1 (1) (1) (1) (1) 01 1 ( 1) ( 1) ( 1) ( 1) 01 1 , ,., , ,., , ,., n n l ll l n cc c cc c cc c − − − −− − − = = = c c c # be l codewords in C0, C1,., Cl-1, respectively. Forming the following sequence ( ) (0) (1) ( 1) (0) (1) ( 1) (0) (1) ( 1) (0) (1) ( 1) 00 0 11 1 ,., ,., lll l jj j nn n cc c cc c cc c −−− − −− − cc c ∗ ∗∗ = " "" " For 0 ≤ <j n , we take (0) (1) ( 1) l jj j cc c " − as the label of a signal point a ∈X . Then ( )( ) ( ) ( )( ) (0) (1) ( 1) (0) (1) ( 1) (0) (1) ( 1) (0) (1) ( 1) 00 0 11 1 ,., ,., ll l l jj j nn n f fcc c fcc c fc c c −− − − −− − cc c ∗ ∗∗ = " "" " ( ) 01 1 , ,., n x x x = − is a sequence of n signals, with ( ) (0) (1) ( 1) l j jj j x fcc c − = ∈ " X . Then ( ) { ( ) } (0) (1) ( 1) (0) (1) ( 1) ( ) for 0 l li i f f il − − C CC C C " " ∗ ∗ ∗ = ∗ ∗ ∗ ∈ ≤< cc c c is an l-level BCM code over X with dimension 01 1 l kk k k = +++ " − . 5.10.2 MLC MLC is BCM 的推广。分量码不必是等长的二进制线性分组码，也可以是卷积码、 级联码甚至非二进制码。它的编码器的一般结构如图 5.10.2 所示。A generic MLC encoder is shown in Fig.5.10.2. A block of K binary source data symbols, u = ∈ (uu u u 1 2 , ,., , {0,1} K k ) , is partitioned into l blocks: ( ) () () () () 1 2 , ,., , 0,. 1 i i ii i K u = =− uu u i l of length Ki with 1 0 l i i K K − = ∑ = . Each data block ( )i u is fed into an individual binary encoder ENCi, producing codewords ( ) () () () 0 1 ,., iii n c c = − c of the component code Ci. For simplicity, we have assumed equal code lengths n at all levels, but in principle the choice of component codes is arbitrary. The code symbols { } ( ) ,0 i j j c = c il ≤ < at time j is mapped to the signal constellation, producing the transmitted signal ( ) (0) (1) ( 1) ( ) l j j jj j x f fcc c − = c = ∈ " X .通常，调 制器使用 Ungerboeck’s mapping by set-partitioning 完成信号的映射。 The code rate of C is given by
-2会- (5.10.1) =0 A)Encoder forC C) u1) -Encoder for Ca c(1) X leu6!S Encoder for C Figure 5.10.2.MLC encoder InformatiohriMLCapproac (ogether ith its MSD Proedure) is a straig a bir nary code is code at each level,and调制器使用Ungerboeck's子集划分方法进行二进制label向量 c=(c,cm,.,c-)到信号点x的映射。 At partition level i,the subset are iteratively divided into two further subsets(co.c-o）andx(coca-1）at leveli1 x(coc-cm）={x=fcc=(coc0b.b-）,b0eGF2),i+1≤j<1} Since the mappingis bijective independently of the actual partitioning strategy,the mutual information()between the transmitted signal pointx and the received signal point y)is equal to the mutual information Y)between the label vector e,l and the received signal point y From the chain rule for mutual information [Gallager68],we have T(X)=T(coc-:y） =T(co:Yy)）+T(cm:yco）++T(c-:ycoc-a） (5.10.2) This implies that transmission of vectors with binary the physical channel can be separated into the parallel transmission of individual digits c overI equivalent channels,provided that-are known.This基本原理is illustrated in
14 1 1 0 0 l l i i i i K K R R n n − − = = = ∑ ∑= = (5.10.1) Partitioning of data Signal mapping Figure 5.10.2. MLC encoder Information-theoretic considerations: MLC approach （together with its MSD procedure） is a straightforward consequence of the chain rule of mutual information. Consider a MLC scheme with l levels. Assume that a binary code is used as a component code at each level, and 调制器使用 Ungerboeck’s 子集划分方法进行二进制 label 向量 ( ) (0) (1) ( 1) , , l cc c − c = " 到信号点 x 的映射。 At partition level i, the subset ( ) (0) ( 1) . i c c − X are iteratively divided into two further subsets ( ) (0) ( 1) . 0 i c c − X and ( ) (0) ( 1) . 1 i c c − X at level i+1: ( ) { ( ) } (0) ( 1) ( ) (0) ( ) ( 1) ( 1) ( ) . ( ) . . , GF(2), 1 i i ii l j c c c x f c cb b b i j l − +− X = = = ∈ +≤ < c c Since the mapping f(.) is bijective independently of the actual partitioning strategy, the mutual information I(X;Y) between the transmitted signal point x∈X and the received signal point y ∈Y is equal to the mutual information ( ) (0) ( 1) . ; l I ccY − between the label vector {0,1}l c∈ and the received signal point y ∈Y . From the chain rule for mutual information [Gallager68], we have ( ) ( ) (0) ( 1) ; . ; l Y ccY − IX I = ( ) ( ) ( ) (0) (1) (0) ( 1) (0) ( 2) ; ; ; . l l c Y c Yc c Yc c − − = + ++ II I " (5.10.2) This implies that transmission of vectors with binary digits ( ) ,0 i c il ≤ < , over the physical channel can be separated into the parallel transmission of individual digits c (i) over l equivalent channels, provided that (0) ( 1) . i c c − are known. This 基本原理 is illustrated in
Fig5.10.3 2)(0) 8-ASK mapper 000001010011100101110111 Equivalent mapper 0 0 0 channel y 0:01010101 Equivalent mapper for=0 0+ 一X十0十一十 channel y :0 1 0 1 Equivalent mapper for=00 channel :0 1 Figure5.10.3 From the chain rule,the mutual information of the equivalent channel ican be calculated as Tcm;yco.c-w）=tc0c-n:yc-）-T(cewc-:yc.cm） where Y is calculated by averaging over all possible combinations ofco.c-).(在第i层的所有信号子集上求平均)。即 io.-y.）=Emz(o,yoe} Random variables Outcome 等效信道的概念是分析与设计编码调制方案的基本工具。 Thus,the capacity of the equivalent channel is given by c=(r)=E[c(([c(x(] where C)denotes the capacity when using only the subset)with a priori probabilities P(a)/P) The capacity C)将作为第i层分量码的码率设计依据。The total capacity C of a 15
15 Fig.5.10.3 . (2) (1) (0) ccc (1) (0) c c = 00 Figure 5.10.3 From the chain rule, the mutual information of the equivalent channel i can be calculated as ( ) ( ) ( ) ( ) (0) ( 1) ( ) ( 1) (0) ( 1) ( 1) ( 1) (0) ( ) ; . . ; . . ; . i i il i i l i c Yc c c c Yc c c c Yc c − − − +− II I = − where ( ) ( ) ( 1) (0) ( 1) . ; . il i c c Yc c − − I is calculated by averaging over all possible combinations of (0) ( 1) . i c c − . (在第 i 层的所有信号子集上求平均)。即 ( ) (0) ( 1) { ( )} ( ) ( 1) (0) ( 1) ( ) ( 1) (0) ( 1) . . ; . . ; . i il i il i c c c c Yc c c c Yc c − −− −− I I = E 等效信道的概念是分析与设计编码调制方案的基本工具。 Thus, the capacity of the equivalent channel is given by ( ) (0) ( 1) (0) ( ) ( ( )) ( ) ( ) ( ) ( ) (0) ( 1) (0) ( 1) (0) ( ) . . ; . . . i i ii i i i cc cc C c Yc c C c c C c c − − − == − ⎡ ⎤⎡ ⎤ ⎣ ⎦⎣ ⎦ XX E E where ( ) ( ) (0) ( ) . i C cc X denotes the capacity when using only the subset ( ) (0) ( ) . i X c c with a priori probabilities { ( )} (0) ( ) ( ) / . i Pa P c c X . The capacity ( )i C 将作为第 i 层分量码的码率设计依据。The total capacity C of a Random variables Outcome