Microsoft Word - Chapter 5-4

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1 22229 现代数字通信与编码理论 December, 2 XDU, Wnter 2 Lecture Notes Capacty-Approachng Coded-Modulaton 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. TCM 码的 (Symbol-Based Nonbnary) MAP 译码算法 Consder a general trells code. Let = ( ) u u, u2,..., u N be an nformaton symbol sequence of length N. Each nformaton symbol s assumed to be a bnary m-tuple,.e., u ( u, u,..., u ) = F, for =,2, N. For convenence, we wll denote u by ts m,,, m 2 decmal representaton;.e., assume u {,,...,2 m U = }. The m-bt symbols are fed nto a TCM encoder, producng a sequence of N modulated symbols, x = ( x x x ),,..., N 2, where x X wth X standng for the sgnal constellaton of sze X = 2 n. We assume that the N symbols x are transmtted over an AWGN channel. Let = = (,,..., ) y y y y2 y N be the receved sequence, where y (=,2, N) s the nosy observaton of x f x s not punctured; otherwse, set y = (mplyng no observaton obtaned). uˆ Fgure 5.9. A trells codng system The receved sequence s fed to the TCM decoder to produce the estmate of transmtted 2 m -ary nformaton symbols. To mnmze the probablty of symbol errors, the symbol-based MAP decoder may be used. Ths decoder computes the APP for every 2 m -ary nformaton symbol;.e., ( N y ) m P u =, for =,,...,2, =,2,..., N. Then t maes decsons accordng to the MAP crteron: N ( ) uˆ = arg max P u = y U

2 where = {,,...,2 m } U. We now descrbe the MAP algorthm n detal by usng the trells dagram. For no loss of generalty of the result, we assume that the trells dagram contans parallel transtons. Refer to Fg Denote by S the set of encoder states and by M = S the number of states. In Fg S =4 s assumed. Let S be the encoder state at tme. A branch at the th secton n the correspondng trells dagram can be specfed by b ( S, u, x, S), where u and x are the nformaton and modulated symbols assocated wth the state transton S S. Denote by B ( s', s ) the set of all the parallel branches connectng S = s' S and S = s S. The non-bnary MAP algorthm can be derved as follows. S S - u /x S S N N y y y + 3 Fgure N N ( = y ) = ( =, = ', = y ) P u P u S s S s ( s', s) B pu ( = S, = s', S = s, y ) N N ( s', s) B p( y ) = (5.9.) where B s the set of transtons S S that are caused by the nput u =. Accordng to N the BCJR algorthm, the probabltes pu ( = S, = s', S = s, y ) can be calculated as where pu ( = S, = s', S = s, y ) = α ( ') s γ ( s', s) β ( s) N 2

3 () s p( S = s, y ) s the forward state metrc; α β () s p( y S = s) s the bacward state metrc; and N + ( ) γ ( s', s) p u =, S = s, y S = s' s the metrc of the trells branch b connectng S - = s to S = s wth u =. N Note that we do not necessarly need the exact values of P( u ) N Snce for a fxed, the vector P( u ) = y, but only ther ratos. = y, beng a probablty vector, must sum to unty, thus by normalzng the sum to unty, we can omt the constant factor use the followng smplfed expresson: N ( ) P u = y α ( s') γ( s', s) β( s) ( s', s) B p( y ) N n (5.9.) and Now we consder the calculaton of the branch metrc γ ( s', s), whch s the ey pont of the symbol-based MAP algorthm, and s also the prmary dfference between the MAP algorthm for bnary turbo codes and the algorthm under consderaton. Usng Bayes rule, the term γ ( s', s) can be wrtten as ( s', s) = p( y S = s, S = s', u = ) P( S = s S = s', u = ) P( u = S = s') γ = p( y x ) P( u = ) P( S = s S = s', u = ) (5.9.2) where p( y x ) s the probablty of recevng y gven the transmtted symbol x, Pu ( ) s the a pror probablty of u =, and PS ( = s S = s', u = ) s or, ndcatng whether or not a state transton S S wth nput u = exsts. Thus, for a trells branch b, we have Defne γ ( s', s) Pu ( ), f y = = py ( x) Pu ( ), otherwse ( y ) γ ( s', s) p S = s, S = s' (5.9.3) 3

4 = b B( s', s) γ ( s', s) whch represent the combned branch metrcs. Then the non-bnary BCJR algorthm can be summarzed as follows. Forward recurson: α() s = α (') s γ(',) s s (5.9.4) s' S Bacward recurson: β ( s') = β( s) γ( s', s) N Output: P( u ) s S (5.9.5) = y α ( s') γ( s', s) β( s) (5.9.6) ( s', s) B The boundary condtons are the same as for the bnary turbo codes. For numercal stablty, t s necessary to control the dynamc range of the lelhood terms computed n (5.9.4) to (5.9.6). Ths can be performed by normalzng the sum of the α ( s) and the β ( s) values to unty at every partcular symbol. Of course, the algorthm can also be mplemented n the log-doman wth reduced computatonal complexty, resultng n the Log-MAP or Max-Log-MAP verson Turbo Trells-Coded Modulaton (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 Ths technque was orgnally proposed by Robertson and Worz n 996 [?] TTCM Encoder A TTCM encoder conssts of the parallel concatenaton of two (or multple) trells-coded modulaton (TCM) schemes n the same fashon as bnary turbo codes. Each component TCM encoder conssts of a convolutonal encoder of rate m/(m+) and a sgnal mapper. The mappng of coded bts to sgnal ponts follows the Ungerboec s rule of mappng by set parttonng. The frst TCM encoder operates on the orgnal nput bt sequence, whle the second TCM encoder manpulates the nterleaved verson of the nput bt sequence. Here, the nterleavng wors on groups of bts nstead of ndvdual bts. See Fg , where a symbol-based odd-even nterleaver s assumed. To avod excessve rate loss, a specal puncturng technque s used: Symbols from a 2 m+ -ary sgnal constellaton are transmtted alternately from the frst and second encoders;.e., the puncturng matrx s gven by P = Thus each nformaton bt par s transmtted n exactly one transmtted symbol wth the party bt alternately chosen from the frst and second encoders. 4

5 TCM encoder Trells encoder Sgnal mapper Symbol nterleaver Puncturng and Multplexng Trells encoder 2 Sgnal mapper Fgure TTCM encoder. The puncturer s used to avod rate loss by enablng the transmsson of sgnals carryng the same nformaton bts only once. Assume that a D-dm constellaton s used. At each step, m nformaton bts are nput to the TTCM encoder and then a modulated symbol selected from a D-dm sgnal constellaton of sze 2 m+ s transmtted. Ths yelds a throughput of 2 m/ D bts per 2-dm symbol. We now consder the above odd-even nterleavng-puncturng rule more 更深入 更一般地. Let L denote the number of component codes used n a parallel concatenated system. Then the above odd-even nterleavng-puncturng rule may be expressed mathematcally va ntroducng the followng defntons. Defnton: A (pseudo-random) nterleaver satsfyng the followng constrant s called a modulo-l nterleaver: π ( ) mod L= mod L, where stands for the symbol poston before nterleavng. Defnton: A modulo-l puncturer s one that punctures all the modulated symbols n the lth component code, except those at poston { mod L= l}. For a TTCM scheme, the modulo-l nterleavng together wth the modulo-l puncturng rule ensure that one and only one modulated symbol carryng the same u s transmtted, and that the punctured symbols are unformly dstrbuted n each component code. It s clear that when L=2, the above modulo-l nterleavng-puncturng rule s equvalent to that used n [3]. As an example, Fg shows an 8-state TTCM encoder for 8-PSK. 5

6 Fgure From [Robertson98] Table 5.9. lsts the generator polynomals of some component TCM codes that can be employed n the TTCM scheme. These polynomals were obtaned by Robertson and Worz usng an exhaustve computer search method. In Table 5.9., m denotes the number of nformaton bts to be encoded from the m nformaton bts contaned n an nput symbol TTCM Decoder The teratve decoder for TTCM s smlar to that used to decode bnary turbo codes, except that there s a dfference n the nature of the nformaton passed from one decoder to the other. Fg llustrates the concept of a pror, a posteror and extrnsc nformaton used n TTCM decoder. 6

7 Fgure Schematc of the component decoders for TTCM codes Note that n a symbol-based nonbnary TTCM scheme, the m systematc nformaton bts and the party bt are transmtted together n the same modulated symbol. Hence, the systematc component (correspondng receved systematc value n bnary turbo decoder) of the non-bnary symbol cannot be separated from the extrnsc component. In ths scenaro the symbol-based nformaton can be splt nto only two components: La and L e&s. Each decoder passes only the latter nformaton to the next component decoder whle the a pror nformaton s removed at each component decoder s output. The complete TTCM decoder operatng n the log doman s depcted n Fg At the begnnng of decodng, the receved symbols are nput to the metrc bloc so as to generate a set of M=2 m+ symbol lelhoods. The selector swtches select the current symbol s relablty metrc produced by the metrc bloc f the current symbol was not punctured by the correspondng encoder. Otherwse puncturng wll be appled where the lelhoods of the varous legtmate symbols at ndex are set to n the log-doman. That s 初始化 L p& s, log py ( x), for un-punctured symbols =, for punctured symbols On ths bass, the TTCM decoder operates accordng to the turbo prncple. Specfcally, for the MAP decoder, f the modulated symbol from the component TCM encoder s punctured at tme, then y = * for decoder, and set L & =. The only nput p s s a pror nformaton L = L from the other component decoder, and ths ncludes the () (2) a e& s systematc nformaton for u. At the output of the MAP decoder, the a pror nformaton s subtracted from the a posteror nformaton, so that the same nformaton s not used more than once n the other component decoder. Thus, the nformaton passed from the decoder to the decoder 2, Le = L La, contans only the extrnsc nformaton on u. If the modulated symbol from the component TCM encoder s not punctured at tme, then L & log py ( x) represents the channel nformaton, and the nformaton passed p s from the decoder to the decoder 2, L = L, contans the nseparable extrnsc as well as e e& s 7

8 systematc nformaton component on u. The decoder 2 operates accordng to the same message-passng prncple as above. () L a Fgure Structure of the TTCM decoder Numercal results Concatenated Two-State TCM (CT-TCM) Schemes Encoder The component encoder of a CT-TCM code conssts of a bnary two-state trells encoder followed by a mult-ary sgnal mapper, see Fg Let a bnary n-tuple d d,, d ) be an nformaton symbol. Let d = {, } be an nput sequence to = (,, n the bnary encoder, producng a coded symbol sequence c={c }. Each c contans a party chec bt q,.e., c = (d, q ), and s mapped to an approprate 2 n+ -ary sgnal constellaton, producng a modulated symbol x. In the followng, c and x={x } are also referred to as unmodulated and modulated codewords, respectvely. d d nformaton symbols = n ( d,,, d, ) bnary encoder coded symbols c =,q ( d ) sgnal mapper modulated symbols x Fg The component encoder structure for the CT-TCM scheme. We assume that the bnary encoder n Fg s characterzed by the two-state trells n Fg (smlar to the tree encoder n [Lpng2]). The party chec bt q s generated by q = q + d g = = d g mod 2, > (5.9.7) 8

9 wth q = d g. Here T g = ( g,, g,,, g, n ) s an ndcaton vector defned by f d, j partcpates n party chec g, j = (5.9.8) otherwse The code n Fg s completely specfed by {g }. = d q g = = q d g d g = = q = q = d g = Fg A two-state trells dagram wth 2 n+ branches n a trells secton. (In ths fgure, n=2.) The Global Encoder of CT-TCM Scheme Fg depcts a global CT-TCM scheme, where M component encoders are concatenated n parallel by M symbol-nterleavers. Modulo-M nterleavers satsfyng the followng constrants are assumed, ( m) π ( ) mod M = mod M, for m =,, M- In order to ncrease spectral effcency, we puncture all the modulated symbols n the mth component code, except those at poston { mod M = m}. Ths, together wth the constrant n (3), ensures that one and only one modulated symbol carryng the same d s transmtted, and that the punctured symbols are unformly dstrbuted n each component code. We wll always assume that a sgnal constellaton of sze 2 n+ s used. Ths yelds a spectral effcency of n bts per symbol. nput () symbol nterleaver π 2-state encoder sgnal mapper () symbol nterleaver π 2-state encoder sgnal mapper output puncturer symbol nterleaver ( M ) π 2-state encoder sgnal mapper Fg A global CT-TCM encoder structure Decoder 9

10 The CT-TCM decoder structure s based on the mult-dmensonal turbo-decoder ncorporatng the BCJR algorthm. The global decoder operatng n the log doman for a CT-TCM code s shown n Fg It conssts of M local APP (a posteror probablty) decoders, each for one component code. The varables nvolved n Fg are the log-lelhood (LL) values, as detaled below. DEC- DEC-m DEC-(M-) Output (m) π - T (m) L a (m) L e - (m) L c APP (m) L [ π ] ( m) Fg The global decoder: T for delay of one teraton and π for nterleavng. (m) L the a posteror LL values for all nformaton symbols after decodng the mth (m) c component code. L the LL values {log p y x ) } based on ndvdual channel observatons of the ( ( m) mth component code. Its elements { L } are calculated as c, L ( m) c, log p( y =, x ), for un - punctured symbols for punctured symbols (m) L a the a pror LL values for all nformaton symbols for the mth component code. It s ntalzed to zeroes, mplyng no a pror nformaton. (m) L e the extrnsc nformaton produced by the mth component code, defned by L = L L. ( m) ( m) ( m) e a The M local decoders operate successvely. nformaton generated by all the local decoders except DEC-m, (m) L a contans the accumulated extrnsc ( m) L a ( m+ ) ( m+ 2) ( M ) () () ( m ) = Le + Le + Le + Le + Le + + Le from the teraton from the prevous teraton current

11 It s used together wth channel observatons L n the next APP decodng of the mth component code. A dscusson on ths global decoder can also be found n [6]. (m) c PC-TCM In ths scheme, two parallel concatenated rate /(+) convolutonal codes use all ther nformaton bts to produce the party bt whch s sent to ther respectve symbol mappers, but the nformaton bts are punctured n such a fashon that one half of them are used n the frst trells code and the other half n the second trells code [Benedetto-Dvsalar96]. Ths s done to lmt the number of ponts n the sgnal constellaton to 2 (/2)+. Also, the nformaton bts transmtted by the two trells encoders are nterleaved by dfferent bt nterleavers (see Fgure 5.9.). In [Fragoul-Wesel], a varaton of ths scheme s presented where a symbol nterleaver s used nstead of bt nterleavers, resultng n mproved BER performance n the waterfall regon. Fgure Multlevel Codes and Multstage Decodng MLC was frst proposed by Ima and Hraawa n 977, wth the dea of protectng each

12 address bt c () of the sgnal pont by an ndvdual bnary code C at level. The ndvdual codes were chosen n such a way that the mnmum datance of the Eucldean space code was maxmzed. Ths s now nown as balanced dstance rule. At the recever sde, each code C j s decoded ndvdually startng from the lowest level and tang nto account decsons of pror decodng stages. Ths s nown as multstage decodng. Compared wth TCM, the MLC approach provdes flexble transmsson rates; any code (e.g., bloc codes, convolutonal codes, or concatenated codes) can be used as component code. TCM can be vewed as a specal case of MLC usng a sngle convolutonal code wth a nonbnary output alphabet whle hgher levels reman uncoded. 5.. Bloc coded modulaton (BCM) BCM s an example of multlevel coded modulaton. 从 TCM 的 set-parttonng 可以看出, 信号 label 中的不同 bt 要求不同的保护能力 在 BCM 中,we provde ths by 用不同纠错能力的码编不同 bt. 具体地说,encode the LSB wth a powerful code, and encode the MSB wth the least powerful code ( 甚至不编码 ) 将 label 排成一个阵列,formng columns wth LSB at the top. a a a an b b b b = n c c c c n [ ] x = x x x n Vectors a, b, and c should form 三个不同的长为 n 的码 As an example, Fg. 5.. shows a 3-level BCM code over the 8-PSK sgnal set. Fg. 5.. An l-level BCM code over a sgnal set (M-PSK or M-QAM) wth 2 l sgnal ponts s constructed n the same manner as a 3-level 8-PSK BCM code. For < l, let C be a bnary (n,, d ) lnear bloc code of length n, dmenson, and mnmum Hammng dstance d. Let 2

13 c c c = ( c, c,..., cn ) ( c, c,..., cn ) () () () () = () () () () = ( c, c,..., cn ) ( l ) ( l ) ( l ) ( l ) be l codewords n C, C,, C l-, respectvely. Formng the followng sequence ( c c c,..., cj cj cj,..., cn cn cn ) c c c = For j < n, we tae () () ( l ) () () ( l ) () () ( l ) () () ( l ) c c c as the label of a sgnal pont a X. Then () () ( l ) j j j ( ) ( () () ( l ) ) ( () () ( l ) ) ( () () ( l ) ) ( () () ( l c c c = ),..., j j j,..., n n n ) f f c c c f c c c f c c c = ( ) x, x,..., xn () () ( ) s a sequence of n sgnals, wth x ( l j f cj cj c j ) = X. Then { c c c c for } ( ) ( ) C C C C C () () ( l ) () () ( l ) ( ) f = f < l s an l-level BCM code over X wth dmenson = l MLC MLC s 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..2 所示 A generc MLC encoder s shown n Fg A bloc of K bnary source data symbols, ( ) u = u, u,..., u, u {,}, 2 K s parttoned nto l blocs: u ( 2 K ) = u, u,..., u, =,... l () () () () of length K wth l = K = K. Each data bloc ENC, producng codewords () = ( c () (),..., c n ) () u s fed nto an ndvdual bnary encoder c of the component code C. For smplcty, we have assumed equal code lengths n at all levels, but n prncple the choce of component codes s arbtrary. The code symbols { () j = cj, < l} c at tme j s mapped to the sgnal () () ( l ) constellaton, producng the transmtted sgnal xj f( j) f ( cj cj c j ) = c = X. 通常, 调 制器使用 Ungerboec s mappng by set-parttonng 完成信号的映射 The code rate of C s gven by 3

14 R K l l = R = = = = n K n (5..) Parttonng of data Sgnal mappng Fgure MLC encoder Informaton-theoretc consderatons: MLC approach (together wth ts MSD procedure) s a straghtforward consequence of the chan rule of mutual nformaton. Consder a MLC scheme wth l levels. Assume that a bnary code s used as a component code at each level, and 调制器使用 Ungerboec s 子集划分方法进行二进制 label 向量 () (, (),, ( l ) c c c ) c = 到信号点 x 的映射 () ( ) At partton level, the subset ( c... c ) () ( ) subsets ( c... c () ( ) X ) and X ( c... c ) at level +: () ( ) ( ) () ( ) ( ) ( l ) ( j) X ( c... c c ) = { x = f( c) c = ( c... c b +... b ), b GF(2), + j< l} X are teratvely dvded nto two further Snce the mappng f(.) s bjectve ndependently of the actual parttonng strategy, the mutual nformaton I(X;Y) between the transmtted sgnal pont x X and the receved sgnal pont () ( l ) y Y s equal to the mutual nformaton I ( c... c ; Y) c {,} l and the receved sgnal pont y Y. From the chan rule for mutual nformaton [Gallager68], we have () ( l ) ( ; Y) = ( c... c ; Y) I X I () ( ; () ) ( ; () ( l ) () ( 2) ) ( ;... l c Y c Y c c Y c c ) between the label vector = I + I + + I (5..2) Ths mples that transmsson of vectors wth bnary dgts () c, < l, over the physcal channel can be separated nto the parallel transmsson of ndvdual dgts c () over l equvalent channels, provded that () ( ) c... c are nown. Ths 基本原理 s llustrated n 4

15 Fg (2) () () c c c () () c c = Fgure 5..3 From the chan rule, the mutual nformaton of the equvalent channel can be calculated as () ( ; ()... ( ) () ( ) () ( ) ( ) ( ) () () ) (... l ;... + ) (... l ;... c Y c c = c c Y c c c c Y c c ) I I I () ( l ) () ( ) where ( c... c ; Y c... c ) I s calculated by averagng over all possble combnatons of () ( ) c... c. ( 在第 层的所有信号子集上求平均 ) 即 I { } ( c ( )... c ( l ) ; Y c ()... c ( ) ) () ( ) ( c ( )... c ( l ) ; Y c ()... c ( = E ) I c... c ) Random varables 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 capacty of the equvalent channel s gven by ( ) ( ) () ( ) () ( ) () ( ) C = ( c ; Y c... c ) = E () ( ) C( X( c... c )) E () ( ) C ( X( c... c c c c c )) ( ) () ( ) () ( ) where C X ( c... c ) denotes the capacty when usng only the subset ( c... c ) { } () ( ) a pror probabltes Pa ( ) / P ( c... c ) X. X wth The capacty C () 将作为第 层分量码的码率设计依据 The total capacty C of a 5

16 multlevel coded modulaton scheme s then l C = C = () Desgn of multlevel codes: The essental pont for the desgn of a MLC scheme s the assgnment of code rates to the ndvdual codng levels. In [Huber99], fves rules are ntroduced for rate desgn. For the capacty desgn rule, the ndvdual rates are chosen to be equal to the capactes of the equvalent channels,.e., () () R = C, =,,..., l c For the choce of FEC codes, powerful probablstc codng methods such as turbo codes and LDPC codes are really needed only at the lower levels. At the hgher levels, the channels become qute clean and the capacty C( Λ / Λ ) approaches log 2( Λ / Λ ), so that the desred redundancy approaches zero. For these levels, algebrac codes and decodng methods may be more approprate. The state of the art n multlevel codng s revewed n [Wachsmann99]. Fgures depct the code rate desgn based on the capacty rule for 8-ASK and 8-PSK MLC schemes. 3 C Capacty (bts/symbol) C C 2 C E s /N (db) Fg 采用 8-ASK 的 MLC 系统各层容量 6

17 3 C Capacty(bts/symbol) C C 2 C E s /N (db) (a).9.8 C C C 2 Capacty (bts/symbol) E s /N (db) (b) Fg Capacty of parttoned subsets for 8-PSK modulaton (wth Ungerboec parttonng) 7

18 3 2.5 C C Capacty(bts/symbol) 2.5 C 2 C E s /N (db) Fg Capacty of 8-PSK MLC wth Gray mappng (n the fgure, C =C 2 ) 5..3 多级译码 (MSD) The component codes C are successvely decoded by the correspondng decoder DEC-. At stage, decoder DEC- processes not only the bloc ( y,..., ) N y = yn Y of receved sgnal ponts, but also decson ( j cˆ ), j < of prevous decodng stages j. As long as error-free decsons ( ) ( ) ˆ j j as an mplementaton of the chan rule. c = c are generated by the decoder DEC-j, MSD can be nterpreted y C () ĉ C () ĉ ( ) ĉ 2 C l ( ) ĉ Fgure 5..7 Note that 尽管 capacty 是相同的 for 不同 labelng strateges, 但是对于 MLC/MSD wth fnte code length, 不同 labelng strateges(ungerboec parttonng, mxed parttonng, Gray labelng) 会影响其 BER 性能 In practce, we recommend MLC schemes based on Ungerboec set parttonng, where powerful codes wth soft-decson decodng are employed 8

19 only at lower levels and hard-decson decodng s appled at hgher levels 多层码的迭代译码 () u Encoder for C () c () v u Demultplexer () u ( ) u Encoder for Encoder for C C l () c ( ) c Interleaver Interleaver π π l () v ( ) v Mappng 2 -ary sgnal constellaton x Fgure 5..8 Encoder y () ĉ C W () ĉ C Wl 2 W ( ) ĉ C l Wl (a) 迭代多级译码器 y Demapper for C λ π DEC- L c () ( ) 判决 () cˆ 计算外信息 L c () e( ) W W W+ ~ Wl ~ (b) 第 级分量译码器结构 Fgure 5..9 MLC 的迭代译码 π W 9

20 λ ( ) ĉ Decoder for C W y Demapper λ W λl π π π l Decoder for C Decoder for Cl ( ) ĉ ( ) cˆ l Wl π l (a) () v C () u () v π C () u y ( l ) v π l C l ( l ) u (b) Fgure 5.. 将各级的 demapper 综合在一起的 Decoder (a) 译码器结构框图 (b)normal graph 5. Bt-Interleaved Coded Modulaton (BICM) wth Iteratve Decodng BICM was frst developed by Zehav prmarly for fadng channels wth the am of ncreasng the dversty order of Ungerboec s TCM schemes, but has turned out to be capable of approachng the capacty of hgh-snr AWGN channels as well. Fg. 5.. depcts the coded modulaton schemes for fadng channels. In Fg. 5.. (a), the channel nterleaver s of symbol-based, whch can also be placed between the convolutonal encoder and sgnal mapper. The symbols are nterleaved wth a depth exceedng the coherent tme of the fadng process. Fg. (b) s the orgnal BICM scheme proposed by Zehav. Compared wth the TCM encoder n (a), the BICM n (b) uses ndependent bt nterleavers for all the bts of a symbol rather than a sngle symbol-based nterleaver. The purpose of bt nterleavng s: To dsperse the bursty errors nduced by the correlated fadng and to maxmze the dversty order of the system; To render the bts assocated wth a gven transmtted symbol uncorrelated or ndependent of each other. 2

21 Snce n BICM the bt-based mnmum Hammng dstance (the number of dfferent bts between the erroneous path n the shortest error event and the correct path) s maxmzed, BICM wll gve a lower BER n Raylegh fadng channels than that of TCM that maxmzes the FED. In other words, for the same complexty BICM acheves better code dversty (.e., the mnmum Hammng dstance), whle TCM acheves better mnmum Eucldean dstance. In fact, the three separate bt nterleavers of sze K can be replaced by a sngle bt-nterleaver wth sze of 3K, resultng n a generc BICM scheme as shown n Fg. 5.. (c). In general, a BICM transmtter comprses an encoder for a bnary code C, a bt nterleaver, and a sgnal mapper. The output sequence of the bt nterleaver s segmented nto m-bt blocs, whch are mapped to a 2 m -pont sgnal constellaton usng Gray codng. The BICM decoder mplements the nverse process, as shown n Fg. 5.. (c). Good performance close to capacty can be obtaned f C s a long Turbo code or LDPC code. Conv. encoder Sgnal mapper Channel nterleaver (a) TCM scheme for fadng channels Conv. Encoder rate-2/3 Bt nterleaver Bt nterleaver Bt nterleaver 8-PSK mapper (b) Zehav s BICM for 8-PSK Bnary encoder Bt nterleaver π b Sgnal mapper Equvalent Channel Bnary De-nterleaver decoder π b Demapper (c) A general BICM scheme Fgure BICM-ID BICM can be vewed as a specal type of SCCC, and the nner code s the symbol 2

22 mapper, whch can be consdered as a nonrecursve sngle-state encoder. So the teratve decodng method for SCCC can be appled to BICM. BICM wth teratve decodng (BICM-ID) was proposed by L and Rtcey [L98] for further mprovng the FED of Zehav s BICM scheme. Ths FED mprovement can be acheved wth the ad of combnng SP(set parttonng)-based constellaton labelng, as n TCM, and by nvong soft-decson feedbac from the decoder s output to the demodulator s nput, n order to exchange soft-decson based nformaton between them. The BICM-ID scheme usng soft-decson feedbac s shown n Fg. 5..2, where soft-n soft-out (SISO) detector/decoder s used n the recever modulo. Let C be a bnary code. Assume that t s concatenated wth an N-dm memoryless modulator over a sgnal set N X C of sze X = M = 2 m map, through a bt nterleaver π and a one-to-one bnary labelng f :{,} m X. Denote the code sequence by c. The nterleaved sequence π ( c ) s () () ( m ) segmented nto m-bt blocs denoted by c t = ( ct, ct,..., c t ), whch are mapped nto sgnal ponts x = f( c ) wth c t as the label of x t. The sgnal x t are then transmtted over the t t channel at tme t, resultng n the receved sgnal y t. Let x denote a sgnal pont n X, and b ( x ) the th bt of the label of x. Wth the use of teratve demodulaton/decodng at the recever, the demapper can use the MAP algorthm to compute APPs for each label bt of x t and delver them to the decoder;.e., () P c = b y = P( x y ) p( y x ) P( x ), for =,m where ( ) ( ) ( ) t t t t t t t xt Xb xt Xb () () Xb = { x X b( x) = b} = { xt = f( ct) X ct = b} s the subset of all the sgnals x X whose label has the value b {,} n poston, and P(x t ) s the a pror probablty of transmttng x t. At the frst teraton, P(x t ) s ntalzed to be / M for all values of x t. 对于第二次及以后各次迭代, 考虑到交织 / 解交织, 在图 5..2 中即为 m ( j) ( ) Px ( ) = Pc = b( x) (wth 充分长的交织器 ) t t j t j= m ( j) ( ) Px ( ) = Pv = b( x); I t t j t j= Therefore, m ( () ) ( ) ( ( j ) t = t = t t t = j( t) ) P c b y P y x P c b x ( ) xt Xb j= 22

23 在迭代解调中,demapper 送给译码器的外信息为 () ( t = b yt) () ( t b I) P c m () ( j) P( ct = b; O) = = P( yt xt) P( ct bj( xt) = ) P c ; = ( ) x j= t Xb j 它被译码器看作输入的先验信息 在对数域中, 它可以表示为 () () () ( = ) = log ( = ) log ( = ; ) L c b P c b y P c b I demapper e t t t t The SISO decoder for FEC code s used for generatng the APPs P( u () t ; ) nformaton bts, as well as the extrnsc APPs P( c () t ; ) O of the O ( 应该是纯外信息, 等于 () ( t )/ ( t; ) APP c y P c I, 不包含信道值 )of the coded bts, from the denterleaved probabltes () ( t b; O) P v algorthms. =. It can be mplemented based on the MAP, Max-Log- MAPA and other such u t Bnary c t Bt nterleaver v t Sgnal x t encoder mapper π b Equvalent Channel Pu () ( t ; O) Bnary decoder Pc () ( t ; I) De-nterleaver Pv () ( t ; O) Demapper y t Pc () ( t ; O) Pv () ( t ; I) Interleaver Fgure BICM-ID scheme usng soft-decson feedbac Labelng method It should be ponted that labelng plays a ey role n BICM. The performance of BICM depends on the labelng used. Fg.x shows the process of subset parttonng for each of the three bt postons for both Gray labelng and SP labelng. It s seen that Gray labelng has a lower number of nearest neghbours compared to SP-based labelng. Hence, Gray labelng s a more approprate mappng durng the frst decodng teraton, and so t s adopted by the non-teratve BICM scheme of Fg.x. Wth teratve decodng and SP-based labelng, BICM-ID can acheve a hgh dversty order as well as a hgh FED. 23

24 By usng the densty evoluton approach for BICM, t s demonstrated that there s no one mapper havng the best performance over all SNR regons [Tan5, IEEE Trans. Wreless, vol.4, no.2, 25]. 图 5..4 给出了 6-QAM 星座的 bnary labelng rule. (a) Gray (b) Set parttonng Fgure 5..4 Bnary labelng rules for 6-QAM 24

25 Interleaver desgn: randomly generated More About BICM: Informaton-theoretc aspects [Gallager68, Care98] In Chapter 6 of [Gallager68], Gallager proved a codng theorem for party-chec codes, whch may be addressed as follows. Theorem: Bnary lnear codes can be used to acheve capacty on an arbtrary dscrete memoryless channel. Fgure 5..5 Gallager also demonstrated a smple algorthm to generate codewords wth the probablty dstrbutons requred to acheve the results of the codng theorem. However, the problem s that fndng decodng algorthm s not smple. Fg shows an example, where each channel codeword s a sequence of N ternary ndependent dgts wth the probabltes Q(a)=3/8, Q(b)=3/8 and Q(c)=2/8. We wll call t Gallager mappng later. (a) label x a b c (b) Fgure 5..6 An example for use of bnary codes on a DMC. (a) System model. (b) Mappng rule Now, BICM-ID wth the use of Turbo-le codes provdes an effectve soluton. Jont demappng-decodng greatly reduces the decodng complexty. 25

26 Fgure 5..7 Care et al. provded n [IT998] a comprehensve analyss of BICM n terms of nformaton rates and error probablty, showng that n fact the loss ncurred by the BICM nterface may be very small. Furthermore, ths loss can essentally be recovered by usng teratve decodng. Coded-modulaton capacty for varous sgnal constellaton n the AWGN channel has been dscussed n Chapter 2. Here, we brefly revew the results on the achevable rates for BICM by followng Care. Defne the sets { x b x b b x b } ( j... ) j l b,..., ( ),..., ( ) j b = j = l j j = j j l X X as the subsets of sgnal ponts havng the l bnary labels b,..., j b j n postons l j... j l. Under the assumpton of an nfnte-length nterleaver, capacty and cutoff rate were studed n [Care98]. Ths assumpton yelds a set of m ndependent parallel bnary-nput channels, for whch the correspondng mutual nformaton s the sum of the correspondng rates of each subchannel, and are gven by I P ( Y x ) = ( Y x ) 2 m nd x X YX B ( X; Y) E log = PYX x X ( ) ( ) m PYX Y x x X = m EBY, log = P x YX Y x X B where B s a random varables tang values on {,}. The jont dstrbuton p(b, Y) of the channel- s The BICM capacty s gven by ( ) m p( BY, ) = 2 X pyx y x x BICM C I nd ( X; Y) Snce BICM can be regarded as MLC wth (Gray mappng and) parallel ndependent decodng (.e., decoder at level maes no use of decsons of other levels j B ), t s shown that [Wachsmann-Huber99] ( ( )) C = C C 2 m BICM CM CM X X = b= b where CM C A s the capacty (nformaton rate) for coded modulaton over a general constellaton A. The BICM ( 或 MLC/PDL) approach s smply a suboptmum approxmaton of an 26

27 optmum coded modulaton scheme, ts capacty strongly depends on the partcular labelng of sgnal ponts. Care et al. showed that the gap to an optmum scheme s surprsngly small wth Gray labelng of the sgnal ponts. However, t has been llustrated that the gap ncreases sgnfcantly f the fnte length of the code s taen nto account. 码长越短, 性能差别越大 Fgure 5..8 shows the BICM mutual nformaton for some sgnal constellatons, dfferent bnary labelng rules and unform nputs for the AWGN channel, as a functon of SNR. For llustraton smplcty, we have only plotted the nformaton rate for the Gray and set parttonng bnary labelng rules from Fgure x. Observe that bnary reflected Gray labelng pays a neglgble penalty n nformaton rate, beng close to the coded modulaton capacty. C (bts / channel use) CM BICM(Gray) BICM(SP) 6QAM 8PSK QPSK SNR (db) Fg Coded modulaton and BICM capactes (n bts per channel use) for multple sgnal constellatons wth unform nputs n the AWGN channel. Gray and set parttonng labelng rules correspond to dashed and dash-dotted lnes, respectvely. The thc sold lnes ndcate CM channel capacty. C (bts / channel use) CM BICM (Gray) 32QAM 6QAM 8PSK QPSK SNR (db) Fg

28 Comparson wth Multlevel Codng: Whle MLC wth MSD acheves the coded modulaton capacty, t does not acheve the coded modulaton error exponent. Ths s due to the MLC (wth or wthout MSD) error exponent always beng gven by the mnmum of the error exponents of the varous levels, whch results n an error exponent smaller than. Whle BICM suffers from a nonzero, yet small, capacty loss compared to CM and MLC/MSD, BICM attans a larger error exponent, whose loss wth respect to CM s small. Ths loss may be large for an MLC constructon. In general, the decodng complexty of BICM s larger than that of MLC/MSD, snce the codes of MLC are shorter. One such example s a code where only a few bts out of the m are coded whle the rest are left uncoded. In practce, however, f the decodng complexty grows lnearly wth the number of bts n a codeword, e.g., wth LDPC or turbo codes, the overall complexty of BICM becomes comparable to that of MLC/MSD. Summary: Snce ts ntroducton, BICM has been regarded as a pragmatc yet powerful scheme to acheve hgh data rates wth general sgnal constellatons. Nowadays, BICM s employed n a wde range of practcal communcaton systems, such as DVB-S2, Wreless LANs, DSL, WMax, and the 4G cellular systems. 5.2 Q-ary LDPC-coded Modulatons 5.3 Combned Codng and Shapng 从图 5..7 可知, 采用均匀的 2D 信号集, 最好的系统性能最多只能达到 constraned-capacty, 而此极限距离 Shannon 容量限还有 db 左右的差距 To acheve the Shannon-capacty-approachng performance (n the hgh SNR regme) for coded-modulaton systems, t s necessary to use constellaton shapng n conjuncton wth codng. Modern channel codng technques are subjects of the prevous chapters, and the related shapng methods are dscussed below Overvew of Constellaton Shapng Methods 从前面关于成形增益的讨论中, 我们知道信号星座成形基本上有两种技术 : 第一种我们称为几何成形, 使信号星座的几何形状类似于球体 ( 在高维信号空间中 ); 第二种我们称为概率成形, 使发送的信号具有与正态分布相近的分布 在编码调制系统中具体实现时, 这两种成形技术可以有多种实现方法, 但大体上可以分为两大类 : 单层编码与两层编码方法 在两层方案中 [Forney84], 一层为信道编码, 另一层为信源编码 ( 成形编码 ) 图 5.3. 所示是编码与成形结合的两层方案 首先把一个给定的 n- 维信号星座 A 划分成若干个信 号点子集 A, m 二进制数据流 u 按照一定的速率划分为两个子数据流 u (c) 和 u (s) 子数据流 u (c) 进入一个信道编码器, 输出的序列 c=(c,,c j,,c L ) 用来选择信号子集序列 s=(s,,s t,,s N ), 其中 s { A, m} 这一层称为信道编码层 子数据流 u (s) 进入一 t 个成形编码器, 输出的序列用来选择一个特定的信号点序列 x=(x,,x t,,x N ), 其中 x t 28

29 是子集 s t 里的信号点 这一层称为信源编码层 根据不同的信号选择方法, 信号星座中每个点的使用概率或许不一样 信道编码层的作用是在子集序列中引入冗余, 从而使得接收端能够无误地 ( 以尽可能小的错误概率 ) 恢复子集序列 信源编码层的作用是对于给定的速率, 选择信号点使得信号星座的平均能量尽可能小 ; 等价地, 对于给定的平均能量, 选择尽可能多的信号点使得传输速率尽可能高 两层方案的最佳性 ( 从达到信道容量的角度 ) 是基于编码和成形的可分性的 已经证明, 对于高信噪比的系统, 如果信道编码层和信源编码层分开设计并适当地构成一个两层系统, 则信道容量的损失是可以忽略的 [Forney2] 图 所示是编码和成形结合的单层方案 二进制数据序列 u 进入信道编码器, 产生输出码字 c 码字 c 进入成形映射器后输出信号序列 x 在该方案中, 信号映射器具有成形编码功能, 它把均匀等概的二进制序列转化为统计特性与信道特性相匹配的信道符号序列, 使得相应的互信息尽可能地大 在此基础上, 再设计一个 ( 二元或多元 ) 信道码, 以逼近这个互信息, 从而逼近信道容量 单层结构的最佳性是由 Gallager 引理保证的 [Gallager68], 其对所有的信噪比都成立 理论上, 在接收端可以实现最小误比特率 (BER) 译码 / 解映射算法 在实际中, 可以采用迭代的译码 / 解映射算法 这时, 往往要在编码器和映射器之间引入一个随机交织器 u u ( c ) c 划分信道编码选择子集 u ( s ) 信源编码 s 选择信号点 x u c x 信道编码信号映射 图 5.3. 编码与成形结合的两层实现方案 图 编码与成形结合的单层实现方案 如果允许使用变速率的成形编码, 则一个简单的方法是使用前缀码 ( 如 Haffman 编码 ) 这种方法可以认为是概率成形方法, 其出发点是计算最佳的信源分布, 然后利用不等长的编码技术去逼近这个分布 可以证明, 对于给定的平均能量, 使得速率最大的分布是 Maxwell-Boltzmann 分布 [4] 若以此为 信源分布, 则前缀码是最佳的 Lvngston 把低维的信号星座划分成几个大小不等的子星座, 然后利用不等长编码获得成形增益, 可以认为是几何成形方法 变速率编码的优点是其考虑的信号点都来自低维信号星座, 因而映射 / 解映射 ( 无论用算法还是用查表 ) 容易实现 但是, 正如 Forney 指出的那样, 变速率成形的最大问题是数据需要缓存和再同步, 因而限制了其实用范围 固定速率编码的目的是使得信号星座的形状接近于球体 但是, 要逼近信道容量, 信号星座的维数必须足够大 Forney( 针对两层方案 ) 提出了一种基于无限维空间的成形技术 篱笆图成形 (Trells Shapng), 可以很容易地获得 db 左右的成形增益 还有一种实用的技术是 壳映射 (Shell Mappng)[9], 这些可以看作是几何成形方法, 在维数适中的情况下可以获得部分成形增益 在单层方案中, 使用不等星座点间隔的信号星座 ( 如 Sun 和 van Tlborg 提出的不等间隔的 PAM 信号星座, 一个例子见下图 ), 可以看作是几何成形方法 Gallager 提出的多对一映射 [Gallager68] X. Ma 和 L Png 提出的线性叠加映射 [Ma4], 均使得信号的信号点概率分布接近于正态分布, 从而获得成形增益, 它们属于概率成形方法 小结 : 综上所述, 我们可以将信号星座成形方法总结如下 : 29

30 星座成形 几何成形 概率成形 单层实现方案 : 例如使用不等间隔星座, 等选择概率 ; 两层实现方案 : 例如 trells shapng, shell mappng 单层实现方案 :Gallager mappng, 使用 Huffman 编码的 map 两层实现方案 : 暂无具体例子 下面我们着重讨论两层方案中的几何成形与单层方案中的概率成形 二者的联系在于, 高维球体 ( 或类球体 ) 在任一维上的投影近似服从正态分布 二者的区别在于, 前者依靠优化高维信号星座的形状来获得成形增益, 而后者依靠优化低维信号的概率分布来获得成形增益 前者在译码时一般不需要译码器和映射器之间进行迭代, 而后者通常需要实现迭代的译码 - 解映射算法 Gallager Mappng 令 X 表示信道的输入符号,Y 表示信道的输出符号 对于一个给定的信号星座 A, 假定 P * ( x), x A 是信道的最佳输入分布 ( 能使互信息 I( XY ; ) 最大化 ) Gallager 映射器 X 的基本设计方法如下 : x 首先将信道编码器输出的编码比特序列 c 按一定长度进行分组 v, v, v 2,..., 然后采 用多对一的映射方法, 将一个或多个比特序列 v 映射到 A 的每一个星座点上, 使得由 此导致的星座点选择概率 ( 即 X 的实际分布 ) 近可能接近于 P * X ( x ) 图 是一个使用 4-PAM 星座时的 Gallager 映射例子 在接收端, 我们需要采用 BICM-ID 这种软信息迭代联合解调 / 译码方式来进行信号检测 x x 3

31 图 类似于 BICM, 对于有限码长, 在 Gallager 映射器中采用不同的 labelng 策略, 系统的 BER 性能也会不同 对于 BICM 系统中 Gallager 映射器的优化设计, 我们提出采用下述的 labelng 设计准则 : ) 对于同一组内的不同标号 (label), 设计的时候尽量最小化标号之间的最大汉明距离 ; 2) 对于不同组内的不同标号, 设计的时候同样尽量最小化标号之间的最大汉明距离 Example 5.3. 将 4 比特长符号调制到 8-PAM 星座的 Gallager 映射 图 该映射不但考虑了星座点使用概率近似离散高斯分布, 而且最小化了组内以及组外相邻 标号之间的最大汉明距离 经过这样的优化设计, 不但给系统带来了成形增益, 而且在 解调译码出错的时候, 最大程度地降低了系统的误比特率 3

32 对于 PAM 信号的设计可以直接推广至常用的 QAM 调制 下面给出一个将符号长度 为 5 比特的符号映射到 6-QAM 星座图上的 Gallager 映射器具体图样 Example 将 5 比特长符号调制到 6-QAM 星座的 Gallager 映射 图 在上述 Gallager 映射器所给出的输入分布下, 采用 8-PAM 与 6-QAM 星座的编码调制系统的信道容量分别如图 5.3.7(a) 和 (b) 所示 32

33 (a) 8-PAM (b) 6-QAM 图 不等概输入分布下编码调制系统的容量 下面我们以基于 LDPC 码的 BICM-ID 系统为例, 来讨论有限码长下采用 Gallager 映射器的具体性能 LDPC 码选取了中国移动多媒体广播 (Chna Moble Multmeda Broadcastng, CMMB) 标准中的 /2 码率规则二元 LDPC 码, 该码校验矩阵的行重为 6, 列重为 3, 码长为 926 比特 信号映射器采用图 所示的 6-QAM Gallager 映射器, 假设信道模型为 AWGN LDPC 码译码器采用和积算法 (SPA), 最大译码迭代次数设定为 次 ; 而解映射器与译码器之间的外部迭代次数设定为 5 次 图 显示了各种不同设计 33

34 准则下谱效率为 2.5 bts/s/hz 的 Gallager 映射编码调制系统仿真结果 可以观察到, 当 BER= 5 时, 优化的映射设计方案比未优化的自然映射方案获得的增益超过 db; 与相同谱效率的常规均匀 32-QAM 调制对比, 优化后的 Gallager 映射系统 BER 性能大约要好.25 db BER Conventonal Gallager mappng Conventonal 32-QAM wth teraton Improved Gallager mappng E b /N (db) 图 EXIT Chart 分析 : 下面我们以 EXIT chart 作为分析工具, 从理论的角度来分析前述的 Gallager 映射器设计准则的有效性 EXIT chart 性能分析原理框图如图 所示 I A I B 图 在图 中, 模块 A 包括 DeM 以及 VND, 而模块 B 只包括 CND 一部分 模块 A 输出的互信息记为 IA, 模块 B 输出的互信息记为 IB,DeM 输出的互信息记为 ID 模块 A 的因子图如图 5.3. 所示 34

35 图 5.3. 模块 A 的 Forney 型因子图 应用对 LDPC 码进行 EXIT Chart 分析的类似方法, 我们可以画出满足不同映射准则的 EXIT 曲线, 图 5.3. 给出了 8-PAM Gallager 映射器的分析结果 其中 LDPC 码集合为规 则 LDPC 码集合 C(3, 6) 图 PAM Gallager 映射 EXIT chart Eb=N = 7.9 db 叠加 Mappng 将经单层或多层信道编码后的数据流划分为多层 ( 如 K 层 ) 比特流, 记第 层中的 35

36 二进制比特流为 v = ( v,, v ) 在 t 时刻, 所有 K 层的编码符号 v 组成一个矢量 N K v = ( v, v, v ), 经过信号映射器映射到信号集 A 中的一个信号点 x t A 线形叠加 t t t t 映射器的结构如图 所示 t v α ( 2 v ) u x K v α K v ( 2 K ) 图 叠加映射器结构 在具体映射过程中, 各层编码符号首先被 2-PAM 映射器映射成数据流 α ( 2v ), {,,, K }, 其中 α 是第 层信息所对应的能量 然后, Σ 映射器将所有 K 层信 号进行求和, 形成一种 和 信号 x : 这里,{ α } K 2 K = x = α ( 2v 2 满足 α = Es 如果所有 α 相同, 则映射器称为 type-i sgma-mapper; 否则, = 称为 type-ii sgma-mapper 由 type-i sgma-mapper 产生的信号星座是等间隔 不等概率的, 如图 所示 由 type-ii sgma-mapper 产生的信号星座一般是不等间隔的 ) 图 一个 sgma mappng wth K=3 的例子 In ths case, Px={/8, 3/8, 3/8, /8}. 对应信号集为 A = {-3,-,,3} wth α = α = α2 = 在 type-i sgma-mapper 中, 各层对应的信号能量相同, 采用互信息设计准则, 各层具有不同的码率, 这样就类似于前述的 Multlevel codng 方案 在 type-ii sgma-mapper 中, 所有层可采用码率相同的信道码, 但是每一层对应不同的能量 ; 通过适当调节层间的能量关系, 可达到在接收端各层能正确译码 功率分配方案有多种形式, 可参考文献 [?] 36

37 当各层的信道码相同 ( 包括码率 ) 时, 一种最简单的功率分配方案如下 : 设 t 时刻发送 信号总能量为 E, 则各层信号能量 α 之间的约束关系为 s 2 2 α / + j > K = 2 2 α j = α K α, 2 = E s = K 2, K 3,, () 2 这里我们对信道噪声方差进行了归一化处理, 即 σ = 在传统的多层编码调制系统中, 互信息的链式法则表现为系统中各层之间的速率分配, 而这里表现为各层之间的功率分配 In [Ma-L4], 一种基于叠加映射的多层编码调制方案 s proposed. n () u C () c π () v a ( 2 v () ) Parttonng ( ) u C ( ) c π ( ) v ( 2 ( ) a ) v 图 类似于对多址信道接收信号的处理, 对于接收到的叠加编码信号, 接收端可采用 successve nterference cancellaton (SIC) 或高斯近似方法进行解映射与译码 下面以第 层为例, 说明高斯近似解映射算法 假定各层采用 LDPC 编码, 其译码 结构框图如图 所示 信道接收到的信息 y 和其它层的软输出外信息 Λ 解映射器, 通过高斯近似解映射算法, 得到分量码译码器的软输入信息 L n j\ 一起输入 (a) 37

38 y Λ j\ 解映射器 L n LDPC 码译码器 - L out Λ (b) 图 第 层译码器框图 高斯近似解映射算法如下 : 设信道噪声 接收端的接收信号以及第 层的编码输出信号分别表示为随机变量 W Y 和 V, 则有 : Y K = = α ( 2V ) W (2) + 现在考虑对信号 V 进行解映射, 则 (2) 式可以改写为 ~ Y = α ( 2V ) W (3) + ~ 其中 W = W + α ( 2V ) 不失一般性, 我们总假设信道噪声的方差 σ 2 = ; 而且将 其它各层信息以及信道噪声一起近似看成是一种高斯噪声, 即 W ~ 近似看作满足高斯分布 ~ 2 ~ ~ W W W ~ N( μ, σ ), 其中 则有 μ = α( 2μ ), σ ~ = + 4 V α W σ V ~ W P (a) 2 ( a) ( a) μ = (), σ = P () P () V V V V V ( e) ( y α ( 2m) μ ~ ) ( ) exp W P V m 2 2 σ ~ W 2 (4) 其中 m 取值为 或 根据 (4) 式可知, 在对数域中, 第 层译码的软输入信息可表示为 : L n 2α ( y μ = (5) σ ~ ) W 2 ~ W Trells Shapng 5.4 Implementaton Consderatons and Applcaton Examples 38

39 Coded-modulatons have been wdely used n varous modern communcaton systems, ncludng voce modems as typcal examples for TCM applcatons. Other applcatons nclude moble communcatons, DVB, dgtal TV, and etc Multdmensonal TCM Example: The V.34 Modem Standard (V.34, V.92, Pragmatc TCM) In ths subsecton, we dscuss the TCM scheme used for the V.34 voce-band modem standard. Ths modem s capable of transmttng up to 33.6 bps over the standard telephone system. Some of the aspects related to the error correcton codng are summarzed as follows [Forney96]. The modem s adaptve n the symbol rate t employs and the sze of the constellaton. It s capable of sendng at symbol rates of 24, 2743, 28, 3, 32, or 3429 symbols/second. (These pecular-loong choces are ratonal multples of the basc rate of 24 symbols/second.) The symbol rate s selected by a lne probng sequence employed durng ntalzaton whch determnes the avalable bandwdth. The modem s capable of transmttng varable numbers of bts per symbol. At the hghest rate, 8.4 bts/symbols are carred. Rate s establshed at ln ntalzaton, and rate adjustments can occur durng data transmsson. Adaptve precodng and decson feedbac equalzaton s also employed to compensate for channel dsperson. Shapng va shell mappng s employed, whch provdes modest gans n addton to the codng gans. Adaptve trells codng s employed. The man code s a 6-state four-dmensonal trells code. Ths code provdes 4.66 db of gan and s rotatonally nvarant. However, two other trells codes are also ncluded: a 32-state four-dmensonal code wth 4.5 db of gan and a 64-state four dmensonal code provdng 4.7 db of gan. V.34 modem 的编码调制系统框图如图 5.4. 所示. The trells encoder for the modem s shown n Fgure 5.4.2, wth the correspondng convolutonal encoder n Fgure

40 Fgure 5.4. V.34 Encoder bloc dagram Fgure V.34 Bloc dagram of trells encoder Fgure V.34 6-state convolutonal encoder TCM Schemes n Deep Space Applcatons 4

41 CCSDS 建议了 a 4D 8-PSK trells-coded modulaton for use n bandwdth-constraned communcatons between remote satelltes and Earth statons. 它可以实现分数谱效率 Example: Assumng the bt rate R b of nput data equal to Mbps, the 4D-8PSK-TCM channel symbol rate s 5 Msps for 2 bts/channel-symbol or 4 Msps for 2.5 bts/channel-symbol. The 4D-8PSK trells-coded modulator conssts of a seral to parallel converter, a dfferental coder, a 64 状态 3/4 码率的 convolutonal coder, a constellaton mapper and an 8PSK modulator. Spectral effcences of 2, 2.25, 2.5, and 2.75 bts/channel-symbol are acheved wth four possble archtectures of the constellaton mapper. 谱效率为 2.5b/s/Hz 时的编码器结构如图 所示 Carrer phase ambguty s resolved by the use of a dfferental coder located pror to the trells encoder. The output swtch addresses successvely one of the four symbols ( Z () - Z (3) ) from the constellaton mapper to the 8PSK modulator. Fgure Coder and Mapper Implementaton at 2.5 Bts/Channel-Symbol The dfferental coder depcted n fgure s used to elmnate phase ambguty on carrer synchronzaton. 在 2.5 bts/channel-symbol 时,the nput bts to dfferental coder are w3, w7 and w, and the output bts are x3, x7, and x. The structure of the modulo-8 adder s shown n fgure 5.4.5; t s applcable to both the coder mapper and dfferental coder. 4

42 图 modulo-8 adder 卷积编码器的结构如图 所示, 其连接多项式为 (8 进制 ):h3=5, h2=24, h=6, h=3. 图 状态 convolutonal encoder The constellaton mapper prncple s gven n fgures for the effcency of 2.5 bts/channel-symbol. The correspondence between the sgnals Z () at the nput of the modular and the 8-PSK phase states of the constellatons follows a natural mappng (.e.,,, 2,, 7). 42

43 Fgure 5.4.7: Constellaton Mapper for 2.5 Bts/Channel-Symbol References [] E. Zehav, 8-PSK trells codes for a Raylegh channel, IEEE Transactons on Communcatons, vol. 4, no. 5, pp , 992. [2] G. Care, G. Tarcco, and E. Bgler, Bt-nterleaved coded modulaton, IEEE Transactons on Informaton Theory, vol. 44, no. 3, pp , 998. [3] U. Wachsmann, R. F. H. Fscher, and J. B. Huber, Multlevel codes: Theoretcal concepts and practcal desgn rules, IEEE Transactons on Informaton Theory, vol. 45, no. 5, pp , July 999. [4] J. Tan and G. Stuber, Analyss and desgn of symbol mappers for teratvely decoded BICM, IEEE Transactons on Wreless Communcatons, vol. 4, no. 2, pp , 25. [5] J. L. Massey, Codng and modulaton n dgtal communcatons, n Internatonal Zurch Semnar Dgtal Communcatons, Zurch, Swtzerland, 974. [6] A. Martnez, A. Gull en F`abregas, G. Care, and F. Wllems, Bt-nterleaved coded modulaton n the wdeband regme, IEEE Transactons on Informaton Theory, vol. 54, no. 2, pp , December 28. [7] A. Martnez, A. Gull en F`abregas, G. Care, and F. Wllems, Btnterleaved coded modulaton revsted: A msmatched decodng perspectve, submtted to IEEE Transactons on Informaton Theory, [8] X. L, A. Chndapol, and J. A. Rtcey, Bt-nterleaved coded modulaton wth teratve decodng and 8 PSK sgnalng, IEEE Transactons on Communcatons, vol. 5, no. 8, pp , 22. [9] X. L and J. A. Rtcey, Trells-coded modulaton wth bt nterleavng and teratve decodng, IEEE Journal on Selected Areas n Communcatons, vol. 7, no. 4, pp. 43

44 75 724, 999. [] X. L and J. A. Rtcey, Bt-nterleaved coded modulaton wth teratve decodng, IEEE Communcaton Letters, vol., no. 6, pp. 69 7, November 997. [] R. F. Fscher, Precodng and Sgnal Shapng for Dgtal Transmsson. USA, New Yor, NY: John Wley & Sons, Inc.,