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(1)Computation of the Active Distances for Trellis Coded Modulation B. Baumgartner , A. Hof, M. Bossert Dept. of Telecommunications and Applied Information Theory, TAIT University of Ulm, Albert-Einstein-Allee 43, D-89081 Ulm, Germany Email: bernd.baumgartner,axel.hof,martin.bossert @e-technik.uni-ulm.de. Abstract Recently, the concept of active distances, originally introduced for binary convolutional codes, was extended to the case of nonlinear trellis coded modulation. These distance measures allow an improved evaluation of the error correcting capability for trellis codes. Typically, the larger the slope of the active distances the better the performance of the trellis code for low . Moreover, these distance measures are of particular interest for the design of concatenated coding schemes, with respect to the choice of the inner trellis code as well as the interleaver. In this paper we present a matrix based method to compute the active distances for trellis codes in Euclidean space, encoded by an -state finite state machine. The computation is based on a simplified version of the distance generating matrix, which is well known from the calculation of the distance spectrum of trellis codes. In general the distance generating matrix is of size

(2)

(3) . We show that for trellis codes, encoded by an -state FSM, the distance generating matrix can always be reduced to

(4)

(5) !" . Finally, simulation results are presented that confirm the importance of the active distances.. 1. Introduction. Formerly, research for trellis coded modulation (TCM) focused on the optimization of the asymptotic behavior of the code. It is well known that for the additive white Gaussian noise (AWGN) channel the minimum squared Euclidean distance between any two code sequences is decisive for the performance in asymptotic good channels (high ). Since in the 90’s concatenated coding schemes with iterative decoding techniques came up, the requirements for the codes changed. The minimum distance of a code is not longer the only parameter to optimize. For serial code concatenation three main components have to be considered for optimization with respect to the applied decoding scheme: the inner code/encoder, the outer code/encoder and the interleaver, respectively. Especially the interleaver and the inner encoder are crucial for the overall performance. It is clear that for a concatenated coding scheme, the inner code has to work proper in a very bad channel environment, i.e., at low . This means for high noise power, the APP decoder should be able to produce good extrinsic estimates. Therefore, inner codes which are good at low # are preferable for this task. As a result new distance measures have to be defined, that allow a forecast of the performance of codes at low $# . 1 This work was supported by the German research council Deutsche Forschungsgemeinschaft under Grant Bo 867/8. For convolutional codes as well as for coded modulation or continuous phase modulation some work was done on new distance measures, as for instance the minimum Hamming weight/Euclidean distance of nonmerging code sequences of certain lengths. It was H¨ost et al. who introduced the family of active distances for convolutional codes [1] tackling exactly the same problem of finding more powerful distance measures. For example, with the active burst distance an improved description of the error correcting capability is possible. Moreover, for a serially concatenated coding scheme based on binary convolutional codes, the active distances on the one hand allow a proper choice of the inner code and on the other hand they provide a tool for the optimization of the interleaver [2]. Recently, the concept of active distances has been extended to nonlinear trellis coded modulation [3]. It was shown, that the higher the slope of the active burst distance, the better the error correcting capability of the trellis code. Furthermore, the improved error correcting capability typically results in trellis codes with a superior performance at low ## . Thus, trellis codes with large slope of the active burst distance typically provide better extrinsic estimates for low . This fact in turn can be used for a serially concatenated scheme, where the gain of the inner code can be transformed into a gain of the overall concatenated system. Moreover, in [4] the active distances introduced in [3] were computed for the special case of continuous.

(6) phase frequency shift keying (CPFSK), where even a closed form solution for the active burst distance was possible. Using the active burst distance, conditions were derived that determine the minimum number of outer codes for woven coded CPFSK for that product distance can be guaranteed. In this paper, we present a matrix based method to compute the active distances for trellis codes that are encoded via an -state finite state machine (FSM). The method utilizes a simplified version of the so called distance generating matrix. This distance generating matrix is in general of size

(7)

(8) and well known from the matrix based methods used to compute the distance spectrum of a trellis code [5], [6], and [7]. Since the dimensions of the distance generating matrix grow strongly with , we address the state reduction of the distance generating matrix. We show that for trellis codes encoded via an -state FSM, without any code symmetries, this matrix can be reduced to

(9) !"

(10) " . This reduction, to approximately half the size for large , is quite small. Nevertheless, by building up the reduced matrix instead of the non-reduced, the complexity for the following state reduction algorithm can be further decreased. The presented method is a straightforward extension of the calculation of the active distances for convolutional codes which can be found in [8]. Here, the computation is based on a pruned version of the so called transition matrix, whereby this transition matrix is also well known from the computation of the weight enumerator function. The paper is organized as follows. In Section 2, we discuss some fundamentals concerning trellis codes encoded via an -state FSM and the distance generating matrix. We introduce the simplified version of the distance generating matrix which is sufficient for the computation of the active distances. Additionally, due to complexity reasons we address the minimization of the distance generating matrix. In Section 3, we recall the definition of the active distances from [3], restricted to trellis codes that can be encoded with an -state FSM. In Section 4, we derive the expressions for the computation of the active distances for trellis codes based on the simplified distance generating matrix introduced in Section 2. For the reason of comprehension as well as motivation, we have a simple ongoing example, taken from [3], throughout the whole paper. Finally, we draw some conclusions in Section 5.. 2. Trellis Codes and Distance Generating Matrix. With the term trellis code or equivalently trellis coded modulation we refer to a code in Euclidean space that can be encoded with an -state FSM encoder . For example the famous Ungerboeck codes [9] build a subset of all possible trellis codes that can be encoded. . via an -state FSM encoder. Such an encoder can be completely specified by its state diagram or its trellis diagram, where both are timeinvariant and the latter is made up of only one trellis section, that is repeated several times. Another possibility to describe such an encoder is an square matrix, known as transition matrix [10].. . .

(11) "!$# &%(' %. 2.1 Transition Matrix. ) *( . be the state space of Let the FSM encoder with cardinality . Furthermore, let denote the th edge or transition, , that points from state to state . This means that up to parallel transitions can point from one state to another. The entry in matrix is defined by:. - ./& 0 1 2 if / exists *+, 3 (1) if / not exists where 14/& denotes the label of 4/& given by 1/& 567 87 . The 67 are 9 -tuples with components from :/; representing the input, and therefore, 6 !<:/;

(12) . The

(13) 8 /& are =?> -tuples with components from a given signal constellation @BADC representing the output, and therefore, 8 !E@FHG . IJ. Example 1 Consider the trellis coded modulation schemes analyzed in [3]. Given the rate , memory , recursive systematic convolutional encoder with generator matrix. KL. M 1 N O QP S RN the considered signal constellations are the classical WV and an asymmetric version 4ASK constellation @UT V @ T

(14) , both depicted in Fig. 1. By labeling the four A(1). (b) Im 11. (a) 00. − √15. − √35. A(2). (a) (b) 00 11 −1.05 −. Fig. 1.. X. (c) 01. (d) 10. + √15. Im. 2 − 1.052. +. X. + √35 Re (c) (d) 01 10. 2 − 1.052 +1.05. Re. 4ASK constellations and mapping. YZ &9 &[4 \] @UT WV. , as symbols of the 4ASK constellations with depicted in Fig. 1, we get the FSM encoder shown in Fig. 2. The trellis codes encoded with this FSM are denoted by TCM , when constellation is used, and TCM

(15) , when constellation

(16) is used. The encoder state space is

(17) , and therefore, . Since there are no parallel transitions, we have and there is only one input/output pair for each entry of the transition matrix . Both the input. a %b. TV. * . T WV. ^ _` @ T V .

(18) 0/d 0/10. 0/a 0/00. 1/b 1/11. σ2. contain incorrect transitions. This is one of the reasons why the terms good and bad superstates are used in literature rather than the terms correct and incorrect superstates. Applying the afore mentioned classification we can arrange the distance generating matrix based on a state space with the good superstates at the beginning, followed by the bad superstates:. σ1 1/c 1/01. Fig. 2.. FSM encoder. . and the output labels are -tuples. The corresponding transition matrix according to (1) is given by . . 3 P 44Y[ 3 4

(19) \9 S . . After a short discussion about some general things concerning the distance generating matrix [6], [11], we introduce a simplified version of this distance generating matrix, which is sufficient to compute the active distances. To analyze the distance properties of a trellis code, two FSM encoders have to be considered, namely the encoder FSM and the decoder FSM . We consider the two FSMs to be identical and therefore assume perfect maximum likelihood sequence estimation (MLSE). The distance generating FSM can be derived from a combination of the encoder and decoder FSM and is represented by the distance generating matrix . The state space

(20) of the distance generating FSM is given by the Cartesian product of the state space of the encoder and decoder FSMs, i.e.,. TV. . .

(21) . T V E T V T V T V ? ! # (' . 1N. . . $ # &. S . (2). 9 . . . 9 . . . . 9 !. . In general the entries in the distance generating matrix are polynomials of certain degree with several dummy bases representing the distance measures under consideration. Moreover, usually the polynomials have coefficients other than taking the probability of the transitions into account [6]. When there are no parallel transitions, the coincidence matrix 6" contains only one zero distance term. In contrast for parallel transitions the coincidence matrix also contains non-zero distance terms, representing . Hence, the coincidence error events of length 7 matrix #" can be decomposed in the part 6"8" that contains correct transitions with zero distance and in the part "89 that contains erroneous transitions with non-zero distances, i.e.,. 3. . Since the first part of the superstate label always corresponds to an encoder state and the last part always to a decoder state, we skip the superscripts and for the state labels in the following. For encoder and decoder FSM with states, the distance generating FSM has

(22) states and hence the distance generating matrix is of size

(23)

(24) . The states of the distance generating FSM are known as superstates or product states. Usually the superstates of the distance generating FSM are classified into good superstates and bad superstates. Good superstates are all superstates with identical label numbers, i.e., , and bad superstates are those with different label numbers, i.e., . Thus, the number of good superstates in the

(25)

(26) distance generating matrix is and the remaining

(27) ! superstates are bad superstates. With identical encoder and decoder FSMs (MLSE), the good superstates are also the correct superstates, when there are no parallel transitions. If there are parallel transitions, good superstates may also. . " # %. Hereby " is the coincidence matrix, $ is the ' ' diverging matrix, % is the ' ' merging matrix and & is the ' ' untouching matrix. Denoting an entry of the distance generating matrix by ( *)+) , we have to compare transitions in terms of distances from the encoder starting in the state and ending in a state *) and a decoder transition starting from the state and ending in the state ) . Therefore we can identify the corresponding submatrix an entry ( ) ) belongs to by ,-- " if 0 1 2 . $ if 0 1 2 -( ) ) -/ % if 32 1 2 & if 34 5 0 . . 2.2 Distance Generating Matrix. TV. P. . . " 4 "8" : "89. For the computation of the active distances it is sufficient to consider a simplified version of the distance generating matrix described above. First of all, the part "8" of the matrix " is skipped. This is because we are interested in error events rather than in correct paths. Second the entries ( *)) of the distance generating matrix are just monomials in ; , representing minimum squared Euclidean distances, instead of polynomials. The reason for this is that the active distances for trellis codes were introduced as code properties and not as encoder properties. Therefore, we just need one dummy base that represents the minimum squared Euclidean distance. And last but not least the coefficients are set because we are not interested in the probabilities to of error events. Thus, it is sufficient to define the entries. 9 . 3.

(28) 9 . ( *)) of the distance generating matrix the following way:. -3. 9 ( ) ) . %a 1 0 1 4 else . ) ) ; . (3). . This ensures that the zero distance terms in " ( 1 2 ) are skipped when there are no parallel transitions ( ). Thus, (

(29) *)+) is the minimum squared Euclidean distance between any pair of output symbols ) from the encoder transition matrix and ) from the decoder transition matrix :. %Q 8 T VV 8 T

(30) ) ) ,

(31) . 8 T() V / &

(32) 8 T *) V 1 . . TV V T. (4) T8 ) V

(33) % 1 0 1 2 8 +T ) V

(34) else 7 The first case in (4) and especially the constraint :. ensures that the zero distance terms are skipped in the case of parallel transitions. So when there are parallel transitions ( ) we have to take the minimum distance of all non-zero distance terms ( 7 ). The active distances can also be defined for some other distance measures than the squared Euclidean distance (see (4)), e.g., the Hamming distance between code symbols [3]. Doing so one simply has to change the underlying distance measure in (4), the rest of the computation remains the same.. %. . Example 2 Now we derive the distance generating matrix for the FSM encoder in Example 1. Its state space is . The first

(35) the

(36) good

(37) superstates ' superstates

(38) are followed by

(39) bad superstates. Taking only the minimum distance terms according to (3) and (4) into account, we obtain the following distance generating matrix : + . ^ 5$ & Q . . . . 3. . 3 . . 3. ; ;. %a. " V T " V T . 3 ; ;. W $ V T W $ V T. ;. V T " $ V. ; ; ;. T $ V T W " V. T. ; ;. ;. ;. V T " $ V T W " V T $ V T. . . Since the encoder FSM contains no parallel transitions ( ), the entries in the submatrix " are all zero.. Theorem 1 The distance generating matrix of every trellis code encoded by an -state FSM encoder can be reduced to

(40) !

(41) . *. ^ &

(42) &

(43) & ,+ & ,+ .. Proof Assume the state space of encoder to be ordered as follows,. the. FSM. Furthermore, let the four submatrices " , #$ , % , and & from (2) be subdivided into smaller matrices. For the coincidence matrix 6" we get 3 22201 0 1 0 " ". #". . . ... 10 10. . with the. . & . & .-/-- & . _ .. .. .. . " " _ & -/-- _ _. submatrices. . :9 :9. 9 1 65 (5) The diverging (merging) matrix 6$ ( % ) is subdivided into ( ) submatrices $ 1 % ( %% ), * = : F F ;2<22 10>= 10 10 10>= $ & --/- $ & T ,+ WV

(44) .. .. .. #$5 . . . $ 10 10 $ .

(45) --/- 222 10 T , 10 + WV % % _ /

(46) . ..

(47) .. .. %E 0>= 0 . . % % 10 10>= , T + WV & --/- T , + WV _ 1 ". 87587. 4. .. .. 2.3 State Reduction. The dimensions

(48) of the distance generating matrix grow dramatically with . Already for trellis codes with manageable number of states, e.g., "! , which can be realized by a binary shift register with # memory elements, the distance generating matrix is "% &! '% . Thus, there are of size

(49)

(50) $! " % !

(51) )("('( matrix entries which in general are polynomials. In our case the entries are just monomials, but it is still very important to possibly reduce the. a. 3. state space, and therefore, reduce the dimensions of the distance generating matrix. In [6], [11] a modification of the Huffman-Mealy-Moore reduction [10] is employed for state minimization. The main idea is to find groups of equivalent superstates and combine them to one new equivalent state. An important note about about this reduction scheme is, that you have to apply it first row wise and then column wise to ensure that the result has the minimal number of equivalent states. It may occur, that the state reduction algorithm for columns does not yield any reduction, but applied to rows, it does. By subdividing the distance generating matrix into submatrices one can find conditions in [12] that these submatrices have to fulfill in order to be able to group the corresponding superstates together to equivalent superstates. We use this fact to show that the distance generating matrix for a trellis code encoded via an -state FSM encoder can always be reduced to

(52)

(53) ! even if there are no special code symmetries at all.. . 3. 3. K . . ... with. 1 % F $. 87

(54) 87. 4. 8@ 8A. 8A 8@. 9 1 % F 9 1 F %?5. (6).

(55) and. 87587. 9 % F S (7) 9 F %5 Finally, the untouching matrix #& is subdivided into the matrices &/& % F , : 7 , * = , i.e., + 22 2 10 = 10 10 10 = & &

(56) .

(57) -. --

(58) T , . + WV .. & 10>= 10 .. .. & & 10 10 = ,. T + WV

(59) --- T , +7 V T , + WV %. %. F . . @ A @. A . P. .. .. with. % F &. 875:9 :9(8 7. . @ A. A . @. P 99 /&1 %% FF 99 FF %% S . (8). When the sums of all elements in a row (or a column) of each of the submatrices in (5), (6), (7), and (8) do not depend on the row (column), then the pairs of superstates ( ), 7 2 7 can be grouped together as one equivalent superstate. The new entry is the sum of the rows (columns). This constraint is obviously fulfilled for the submatrices in (5) because they are just scalars. Since the submatrices in (6) are row vectors, it is obvious that the sum of the rows does not depend on the row. The submatrices in (7) are column vectors so the sum of each column does not depend on the column. The missing constraints for columns for the submatrices in (6) and for rows for the submatrices in (7) and for rows as well as columns for the submatrices in (8) are also fulfilled because of identical encoder and decoder FSMs. This can easily be seen from the fact that % % . Since there are

(60) " such pairs, we get

(61) ' "

(62) !" new superstates.. 1) )] +! # E' . 90 F 9 1 F. . When the underlying code has a special structure, the distance generating matrix can be further reduced. Note, that a trellis code requires comparable few symmetry constraints to result in a reduced distance generating matrix used to compute the active distances (simplified version of the distance generating matrix), compared to the case of the usual distance generating matrix used to compute the distance spectrum. Hence, it is possible that for a given trellis code, the distance generating matrix used to compute the distance spectrum of the code could not be reduced at all. However, the simplified version of the distance generating matrix used to compute the active distances can be reduced. A similar effect was already addressed in [13], where the so called uniform distance property (UDP) and the uniform error property (UEP) were derived. With few symmetry constraints the trellis code fulfills UDP and the distance generating matrix can be reduced to for the computation of the minimum distance of the code. This case is comparable to the one we have. in this paper concerning the simplified version of the distance generating matrix. Only when a trellis code comprises high symmetry constraints, it fulfills UEP and the distance generating matrix can be reduced to for the computation of the distance spectrum. For trellis codes encoded by a convolutional encoder followed by a mapper (as in our example), it is possible to reduce the distance generating matrix to . Hereby, the linearity of the convolutional code together with certain symmetries in the signal constellation and in the way it is partitioned could be used, see [13], [14], and [15]. In [16] the so called geometrically uniform codes were introduced. For this class of codes the distance spectrum from any trellis code sequence to all other code sequences is the same and all the decision regions have the same shape. This property for trellis codes comprises a very high level of symmetry and for this class of codes, the distance generating matrix can also be reduced to . Denoting the result of the state reduction algorithm of the distance generating matrix by , we see that instead of monomials there might be polynomials because during the state reduction process some monomials are added up. Again, for the computation of the active distances we only need the minimum distance term. Therefore, we extract only the terms corresponding to the minimum squared Euclidean distance from the polynomials. Let the entries of be *) ) (9) - ; ) -/-- - ; . 9 Y. RY. .

(63) where are assumed to be ordered like

(64) the distances

(65) . Denoting the distance generating matrix /

(66) consisting of only the minimum distance terms by , the entry ( ) ) of this matrix is extracted by applying the operation

(67) - to *) ) , which extracts the minimum exponent of a polynomial ) ) ). *)) 0 (10) ; 2;. 9 . 9 T. 9 . V. . . Example 3 For our ongoing example, we can reduce the number of states to

(68) !" , which requires no special code symmetry. The resulting distance generating matrix is given by . . . . 3. . . ;. . 3. 3. T " V. ;. 3. W $ V T. ; T V ; T " $ V W " V ;. T. 1 1YZ \ 1YZ &\ 9 [. Grouping the bad superstate pair

(69)

(70) together, we denote the new superstate by . By applying equations (9) and (10) to the distance generating matrix it is to know whether

(71) .

(72) or

(73) necessary the signal

(74) . By checking constellations in Fig. 1 we see that

(75) !

(76) ! holds " for both constellations yielding just the entry ; in the ' lower right corner. Since the distances depend on the signal constellation and in #"%$ & the mapping, we have to split the computation at this. .9 &[ .YZ \ T W V *+Y Z9. .9 &[.

(77) @ T W V. T WV. point into two parts. For the mapping and constellation , the new distance generating matrix yields . . T V . 3. . . . 3. 3. ;.

(78) ;. 3.

(79). . ; . . . ;. ; . @ T

(80) V we get 3 ; 3 ; . (11). For the mapping and constellation . . T

(81) V . . 3. . . ;. 3. . ;. . (12). . ;. The state reduction process in [6] yields a distance generating matrix, having one of the simple symmetry constraints

(82)

(83) or

(84)

(85) on the constellation. As shown by Fig. 1, both of the constraints are fulfilled, and therefore, we can further reduce the distance generating matrices from (11) and (12) to the the following matrices . . ; . (13) ;

(86) ; ; . (14)

(87) ; ; . 1YZ &9 " 1[

(88) \ T V . P. 1YZ &[ U. 3. 9 \. S. 3. S. Hereby the pair of good superstates . is

(89)

(90) grouped together to one equivalent good superstate . TV. 3. P. Active Distances. In this section we recall the definition of the active distances for trellis coded modulation first introduced in [3]. The underlying distance measure is the squared Euclidean distance. Based on the distance generating matrix, the active distances can only be computed for trellis codes encoded with an -state FSM having a timeinvariant trellis. Therefore the state space is fix over time and in the following we will give the definition of the active distances for trellis codes with a timeinvariant trellis representation. Let. ^. `& Z +7 be a path of length 7 that starts at time " in state and ends at time "! 7 in state

(91) , where &

(92) !+^ . The label string .

(93) of such a path is a code sequence 8 V .

(94) . 1 1 Z +7 8 0 W V W 8 + V V T T F G T Z + T ZFHG V +7 with 8 T WV T FHG V ! @F G . The 8 correspond to the 8 from Section 2.1. For simplicity we skip the indices for the starting state and 7 for the ending state as well as the index for parallel transitions. 1

(95) . and just introduce one index " representing the time in stance. Let

(96)

(97) be the squared Euclidean distance between any two sequences of length 7 , + and

(98)

(99)

(100) + ,. 18 W 8 8 8 0 8 Z 7 8 8 W8 Z 7

(101) 18 W8

(102) Z +7 F G T V T V

(103) (15) by a minimal We assume a trellis code represented or equivalently timeinvariant FSM encoder by a minimal timeinvariant trellis . Let !#"$" be the set of all paths .

(104) that start in !$^ and end in '

(105) ! ^ . For each path .

(106) %! !& "$" ' we' denote by !&" ()" * .

(107) the set of all paths +

(108) beginning ' ^ ,/. ^ and ending in state '

(109) ! ^ .R^ , in state ! that differ from the path .

(110) in all edges: ' ! " ( " * 1

(111) 1' 0 ' ' ' $ +

(112) ! ^ ,

(113) 7! ^ 32 ' ) 4 ! # "0% " 7 Definition 1 Let be a trellis code encoded via an state FSM, represented by the minimal timeinvariant trellis . The family of the jth order active Euclidean distances is given by. Y6 5 79

(114) 87 10.

(115). K. :;=<>< ? @BADCEC. :F ; <EG < G H? @ GA C C ;I:;=<>< ?J? ( *.

(116) L + ' '

(117) 0 1

(118) M . , ^ D ^. for 7 7 57N8 . The placeholder O represents four cases with different sets , and ,-- b: 7 , 7 7 %

(119) " . c: " 7 , 7 - O " -/ rc:DP " 7 ,

(120) 7QP , s: 7 7 where 7 denotes the active burst distance, " 7 and " DP 7 denote the active column distance and reverse 7 denotes the column distance, respectively, and active segment distance. *. Y 0 ^ Y 0 _^ Y > 0 ^ Y ^. Z. Y. ^ ^ ` ^ ` ^ ^ b^ ^ b^. Y. Y>. Y. _ F > . First we see that the active distances are independent of the time instance " a possible error event starts due to the timeinvariant trellis. According to equa' '. tion (15),

(121) R L +

(122)

(123) is the squared Eu clidean distance between two code sequences

(124) ' ' and L +

(125) . The inner minimization is over all ' '. possible paths +

(126) that differ from the path

(127) in all edges. The sets , and of states, in which the ' ' paths +

(128) are allowed to start and end, differ with the type of active distance, which can be seen from Fig. ' ' 3. For the active burst distance +

(129) starts and ends in the same state as the considered path

(130) . Hence, , the active burst distance is only defined for 7 7 % which is the length of the shortest detour a path can take in the trellis from a given path. For the active. . . 0 1 . . ^. 1 . . ^. . . . _ F.

(131) active burst SB = σ distance p(σ, σ 0 ). .. .. .. .. 0. p ¯ (¯ σ, σ ¯ ). . S The entries of all the submatrices are again polynomials. By inspecting the submatrix " we see that the. .. .. active column distance. SE = S. active reverse column distance. SB = S. active segment distance. . .. .. . . ^. . 1 !.

(132)

(133) min . 1 . ^. 1 . K. Y 7 M . Computing the Active Distances. The family of active distances defined in the previous section can be computed via a matrix. based method using the simplified version of reduced. the state distance generating matrix ,. , with see Section 2. Here ' ( ' ) is the number of good (bad) superstates after the state reduction algorithm. A similar concept on the computation of active distances has been used for convolutional codes [8].. . F . Y 7 Y " 7 Y P " 7 Y > 7 . Active distances. . . . 5 . . = 3 -/-- --- 3 0 F , + F . Hence, corresponds. to the all-ones row vector of . With the operation

(134) - length from (10) the family of active distances for trellis codes is computed by. column (reverse column) distance, the starting (ending) ' '. state of path +

(135) is the same as for path

(136) and the ending (starting) state is from the set . Concerning ' ' the active segment distance, the path +

(137) is allowed to start and end in any state from the set . In the case of nonlinear trellis codes not every code sequence

(138) yields the same result for the inner minimization. Hence, we have to take the minimum over all paths

(139) 3!&"$" . The outer minimization can be skipped, if the result of the inner minimization is invariant with the chosen path

(140) . This holds for example for the class of geometrically uniform codes [16]. The minimum squared Euclidean distance m

(141) in of a trellis code is given by the minimum of the active burst distance. 4. F. =. SE = S. .. .. $ # &. . .. .. SB = S. Fig. 3.. . SE = σ 0. .. .. P " %5 . distances of the polynomial entries correspond exactly to the distances that should be minimized to compute the active burst distance. Furthermore, the matrices " together with $ ( " together with % ) contain all the distances that should be minimized to calculate the active column (reverse column) distance. Finally, contains all the distances that the whole matrix should be minimized to compute the active segment. distance. Let denote a row vector of length with. successive ones followed by successive zeros:. .. .. .. .. . ¯ SS SE (p(σ, σ 0 )) P SB = σ. . and For error events of length 7 we compute subdivide the resulting matrix again according to (2) into four submatrices:. SE = σ 0.

(142) . .

(143) .

(144) . .

(145) . - -

(146) - -

(147) - -

(148) - -

(149) . . . Y. Y. (16) (17) (18) (19). Y>. From the definition as well as the computation of the " 7 ! active distances we see that 7 ! 7 and 7 )P " 7 ! 7 holds.. Y. Y. Y>. Example 4 Computing the active distances based on the distance generating matrices in (13) and (14), we first have a look at error events of length 7 . We obtain: . . . ; ;

(150). ; ; ;. . . ; ;

(151)

(152) ; ; :;. . T WV TV . P P. S. S. T V > Y 3 Y < V Y < T Y ] Y D ] 3 Y > ] T WV T V V 3 W. V T T TV. ] 3 ] 3. and by applying (16)-(19) we compute the active distances for trellis code TCM to ! , " " , DP ! and and ! , for the second trellis code TCM

(153) to " , DP " ! and . In Fig. 4 the result for the active burst distance for larger error event lengths 7 is depicted. The minimum

(154) and

(155)

(156) of both codes are ! . On distances min min of code TCM is much the other hand, the slope smaller than that of code TCM

(157) . The simulation. Y. Y.

(158) active Euclidean burst distances. 35. TCM(1) (2) TCM. 30 25 20. (2). slope λ =2.0. 15 10. (1). slope λ =0.8. 5. 2(1) 2(2) =δ =4.0 min min. δ. 0 0. Fig. 4.. 5 10 error event length j. 15. Active burst distances. results in Fig. 5 show that both codes have the same asymptotic (in terms of # ) behavior, which is due to the equivalent minimum squared Euclidean distances. In contrast to this, the trellis code TCM

(159) with the higher slope has the better error correcting capability and outperforms code TCM for low $ . This gain for trellis code TCM

(160) can be. TV. T WV V T. 0. 10. TCM(1)/SCTCM(1) TCM(2)/SCTCM(2). −1. ≈ 0.5 dB. SER/BER. 10. −2. 10. ≈ 0.4 dB. TCM, SER. −3. 10. SCTCM, BER. −4. 10. Fig. 5.. −4. −2. 0. 2 4 Eb/N0 [dB]. 6. 8. SER/BER of TCM/SCTCM. transformed into a gain of a serially concatenated coding scheme. Therefore, we use the trellis codes and TCM

(161) as inner codes for serially TCM concatenated TCM (SCTCM and SCTCM

(162) ) with random interleaving. The outer code is the rate " , memory binary convolutional code

(163)

(164) . with generator matrix Decoding is organized iteratively with iterations using symbol-by-symbol APP algorithms for the inner and outer code. From the results for the serially concatenated schemes SCTCM and SCTCM

(165) one can see that there is a significant gain of ! dB in the water fall region. This emphasizes the importance of the active burst distance for TCM as well as SCTCM.. T V. I . TV. K . T WV. TV. .N O. N. T V. 5. 3 N N. 3 TV. Conclusion. After motivating the active distances, we presented a method to compute the active distances for trellis codes encoded by an -state FSM. The method is based. on a simplified version of the well known distance generating matrix used to compute the distance spectrum of trellis codes. We showed that without any symmetry constraints on the trellis code, even the nonsimplified distance generating matrix can be reduced to

(166) "

(167) " . The simplified distance generating matrix used to compute the active distances can be further reduced with quite few symmetry constraints on the trellis code compared to the usually known distance generating matrix used to compute the distance spectrum. The presented method to compute the active distances is easy to implement. Therefore, good trellis codes in terms of the active distances can be searched in order to improve the performance of trellis codes for low # . Such codes typically give better extrinsic estimates for low # which lead to an improved performance of a concatenated coding scheme concerning the waterfall region.. References [1] S. H¨ost, R. Johannesson, K. Sh. Zigangirov, and V. V. Zyablov. Active Distances for Convolutional Codes. IEEE Transactions on Information Theory, vol. 45, pp. 658–669, March 1999. [2] A. Huebner, R. Jordan, and J. Grill. On Permutations with Second Order Separations between Cascaded Convolutional Encoders. In Proc. IEEE Int. Symposium on Information Theory, p. 484, Lausanne, Switzerland, June 2002. [3] B. Baumgartner, M. Bossert, and V. V. Zyablov. On Active Distances for Nonlinear Trellis Coded Modulation. In Proc. 3rd Int. Symposium on Turbo Codes and Related Topics, pp. 103– 106, Brest, France, September 2003. [4] S. Kempf, S. Shavgulidze, and M. Bossert. Woven Coded Continuous Phase Frequency Shift Keying. In Proc. 5th Int. ITG Conference on Source and Channel Coding, Erlangen, Germany, January 2004. [5] E. Biglieri. High-Level Modulation and Coding for Nonlinear Satellite Channels. IEEE Transactions on Communications, vol. COM-32, pp. 616–626, May 1984. [6] W. Zhang, C. Schlegel, and M. J. Miller. Reduced State Distance Spectrum Computation for General Trellis Codes. In Proc. Int. Symposium on Information Theory and its Applications, pp. 431–435, Sydney, Australia, November 1994. [7] C. Schlegel. Trellis Coding. IEEE Press, 1995. [8] R. Jordan. Design Aspects of Woven Convolutional Coding. PhD thesis, University of Ulm, April 2002. [9] G. Ungerboeck. Channel Coding with Multilevel/Phase Signals. IEEE Transactions on Information Theory, vol. IT-28, pp. 55– 67, January 1982. [10] A. Gill. Introduction to the Theory of Finite-State Machines. New York: McGraw-Hill Book Company, 1962. [11] W. Zhang. Finite State Systems in Mobile Communications. PhD thesis, University of South Australia, February 1996. [12] Y.-J. Liu, I. Oka, and E. Biglieri. Error Probability for Digital Transmission Over Nonlinear Channels with Application to TCM. IEEE Transactions on Information Theory, vol. 36, pp. 1101–1110, September 1990. [13] E. Biglieri and P. J. McLane. Uniform Distance and Error Probability Properties of TCM Schemes. IEEE Transactions on Communications, vol. 39, pp. 41–53, January 1991. [14] E. Zehavi and J. K. Wolf. On the Performance Evaluation of Trellis Codes. IEEE Transactions on Information Theory, vol. IT-33, pp. 196–202, March 1987. [15] M. Rouanne and D. J. Costello, Jr.. An Algorithm for Computing the Distance Spectrum of Trellis Codes. IEEE Journal on Selected Areas in Communications, vol. 7, pp. 929–940, August 1989. [16] G. D. Forney, Jr.. Geometrically Uniform Codes. IEEE Transactions on Information Theory, vol. 37, pp. 1241–1260, September 1991..

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