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=0 014 ( ', ) ( ', ) 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