Reduced-State Soft-Input/Soft-Output Algorithms for Complexity Reduction in Iterative and Non-Iterative Data Detection

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1 Reduced-State Soft-Input/Soft-Output Algorithms for Complexity Reduction in Iterative and Non-Iterative Data Detection Xiaopeng Chen and Keith M. Chugg Abstract Soft-input/soft-output (SISO) algorithms have been widely used for iterative detection in various applications since this technique was introduced for decoding Turbo codes. However, the complexity of the SISO algorithms is a major concern in the detector implementation. In this paper, a novel way to simplify the SISO algorithms is proposed based on the concept of state reduction via decision feedback. The resulting complexity reduction is exponential in the number of feedback taps. The proposed low-complexity SISO algorithm can be applied directly in place of the standard SISO (e.g., a full-state forward-backward algorithm). Additionally, thresholding the softoutputs of the reduced-state (RS) SISO can provide a robust and effective alternative to RS sequence detectors. Simulation results are also provided to illustrate the advantages of the RS-SISO. I. INTRODUCTION Currently, there is great interest in soft-input/soft-output (SISO) algorithms [-5] due to the outstanding performance of iterative data detection. Like the Viterbi algorithm (VA) [6], the SISO algorithm is derived for an embedded system that can be modeled as a finite state machine (FSM) [7]. However, instead of making hard decisions as the VA does, the SISO algorithm generates the soft information, which is a reliability measure for the investigated data (random variables). A hard decision can be readily obtained by thresholding the soft information. The key notion carried with the soft information, also its main advantage over a hard decision, is that the soft information can be refined by some ways (e.g., through iteration) to make a later hard decision more reliable [,2]. Because of this advantage, SISOs have been widely used in various application in communications, such as, decoding of (both serial and parallel) Turbo codes [8,9] and other codes [0,], nearcapacity multi-user detection [2], near-optimal two-dimensional data detection [3], the detection of TCM signals in fading channels [4], etc. One of the main concerns of implementing the SISO algorithm, as is the case for the VA, is its complexity (i.e., the number of states). The complexity of a SISO algorithm grows exponentially with the memory length of the FSM, e.g., the number of taps in an inter-symbol interference (ISI) channel or the constraint length of a convolutional code. For a fixed-interval minimum sequence metric (FI-) SISO algorithm [2,3], its computational complexity is approximately twice that of the VA and the number of memory units required is proportional to the product of block length and the number of states of the FSM. For instance, when the FI- algorithm is applied to a 0-tap ISI channel with QPSK signalling, its complexity is dominated by the number of the states of the FSM, i.e., 4 9 =262; 44. This makes the detector based on the FI- algorithm practically infeasible for many applications. By employing the currently-available hard decisions [5,6], Xiaopeng Chen was with CSI, EE Systems Dept., University of Southern California, Los Angeles, CA He is now with Marvell Semiconductor Inc., Sunnyvale, CA Keith Chugg is with CSI, EE Systems Dept., University of Southern California, Los Angeles, CA This work was supported in part by a U.S. Army SBIR contract to ViaSat, Inc. (DAAB07-98-C-K004). the decision feedback technique is widely used for complexity reduction in forward(backward)-only hard-decision algorithms, e.g., RSSE in [7], DDFSE in [8] and RSSE-RTS in [9]. In this paper, the decision feedback technique is applied to SISO algorithms to reduce the complexity. Since the SISO algorithms include a forward and a backward recursion, truncated states are properly defined and survivor paths are generated in both recursions separately. Also, the desired soft output is yielded in a simplified way. The resulting complexity reduction is exponential in the number of feedback taps. We will refer to the proposed algorithms as reduced-state (RS) SISO algorithms. The proposed RS-SISO has the exactly same input/output interface as the standard SISO algorithm [,2]. Therefore, the RS-SISO can be used to replace the standard SISO directly. In addition, the RS-SISO algorithm may feed its own soft output back to its input port (which we refer to as self-iteration) to refine the soft output. Therefore, in the case of sequence detection, the RS-SISO algorithm can also replace a hard-decision algorithm (e.g., VA, DDFSE) and give robust and excellent performance. This paper is organized as follows. The FSM model and notations are given in Section II. Two versions of RS-SISO algorithm are presented in Section III along with the specific concern for iterative applications. Finally, the RS-SISO algorithm is compared numerically to some existing algorithms in two different transmission systems in Section IV. Concluding remarks are contained in Section V. II. AN FSM MODEL AND NOTATIONS We will only consider the time-invariant FSM in this paper. In this case, a FSM can be described by a regular trellis diagram. Such a trellis consists of a set of states S = fs ;S 2 ; ;S N g. The state of the FSM at time k is s k = S i 2 S. The trellis transition t is deterministically driven by a date source that outputs sequence fa ;a 2 ; ;a K g with symbol assumed to be independently drawn from the M-ary alphabet A = fa 0 ;A ; ;A M g. Therefore, in an FSM, each transition t is associated with a starting state s s (t), an ending state s e (t), an input symbol a(t) and a output symbol x(t). In this paper, for the simplicity of presentation, we only consider the FSM whose state is defined as s k = ( L ; L+ ; ; ): 2 Such an FSM is said to have memory length L and the number of state N = M L. Consequently, for a transition t k = ( L ; ; ),we have s s (t k ) = ( L ; ; ), s e (t k ) = ( L+ ; ; ), a(t k )= and the output symbol x k = x(t k ). For a general ISI channel, the function x( ) can be a -to- mapping. For a trellis coded modulation (TCM) scheme, it is usually an n-to- mapping. In this paper, for random variable w, we use W i to denote its possible value and W to denote its sample space. 2 This is a non-recursive FSM without parallel transitions, referred to as a simple FSM in [3]. More discussion about other types of FSM will be given in Section V.

2 Besides the knowledge of the FSM structure, the SISO algorithm also requires some soft information as its input. The soft input is the sequence of reliability measures of the input symbol and the output symbol x k. The two main kinds of soft information that widely used are the probability distribution P( ); P(x k ) and its negative-log domain counterpart metric M( ); M(x k ) [2]. Note that the superscript i and o to be used with P( ) and M( ) denote input and output, respectively. We will call the resulting algorithm as the a-posteriori probability (APP) algorithm and minimum sequence metric () algorithm 3, respectively [2,3]. In this paper, we assume that the knowledge of FSM structure and the a-priori soft information are perfectly known. III. LOW COMPLEXITY RS-SISO ALGORITHMS A. A-Posteriori Probability RS-SISO In a RS-SISO, we truncate the memory of the FSM and rebuild the corresponding trellis. The truncated state at time k is defined by v k = f L ; L+; ; g, where L» L and define L 2 = L L. If L = L, then v k = s k, and the following derivation will simply result in the standard SISO algorithm. It will be shown that the standard SISO is M L2 times more complex than the RS-SISO. First, we define a few state sets for future use: F(j) =fi : V i! V j is an allowable forward transitiong () B(i) =fj : V i ψ V j is an allowable backward transitiong(2) C(m) =fj : v k+ = V j is consistent with = A m g (3) As in the standard APP-SISO algorithm, the following key quantities are defined for the APP RS-SISO algorithm recursively as: ff P k (j) = ff i2f(j) k (i)fl f k (i; j) (4) fi P k (i) = fi j2b(i) k+(j)fl b k+ (i; j) (5) with initial values as ff 0 (i) = Pr(v 0 = V i ) and fi K (j) = Pr(v K = V j ). For example, at time k = 0, if the initial state of the FSM is V l, then ff 0 (l) = ;ff 0 (j) = 0;j 6= l. If no knowledge about v 0 is available, then ff 0 (i) = M L. Note that the APP algorithm does not generate survivors naturally. One reasonable definition is: for a state at time k, the forward (or backward) survivor state is the state contributing most in the summation of (4) (or (5)). Other reasonable definitions of the survivor are also applicable. Consequently, associated with each truncated state v k = V i, we can obtain a truncated survivor path in the forward recursion as ~a f k (i) = f~ k L (i); ~ak (i); ; k L+ ~ak k L (i)g, and a backward truncated survivor path as ~a b k (i) =f~ak k (i); ~ak (i); ; k+ ~ak k+l (i)g 2 (Fig. ). Based on the definition of survivor paths, we define the quantities associated with t k (i; j) = (v k = V i ;v k+ = V j ) in both forward and backward recursion as (Fig. ): fl f k (i; j) =Pi [x k = x(~a f k (i);t k(i; j))]p i [ (i; j)] (6) flk b (i; j) =Pi [x k+l2 =x(t k (i; j); ~a b k+ (j))]pi [ (i; j)] (7) where (i; j) is the value of associated with a forward transition V i! V j. When L 2 =0, fl f k (i; j) =flb k (i; j), yielding the standard APP algorithm. 3 The APP algorithm has an equivalent variant operating in the log-domain [,20], called as the log-app algorithm. It may be obtained from the algorithm via a modified min( ) operator [3]. forward survivor path at time k truncated state at time k+ ã k k L ãk L k L L+ truncated state at time k (b) L L+ truncated state at time k+ truncated state at time k completion step at time k x k backward survivor path at time k+ ã k+ k+ ã k+ k+l2 x k+l2 Fig.. The construction of the complete forward and (b) backward transition and completion step at time k in the RS-SISO algorithm. The complexity of the forward and backward recursion in (4) and (5) is primarily determined by the number of states, i.e., M L. Compared to the standard APP-SISO algorithm, the complexity of proposed APP RS-SISO is reduced by M L2 times. Finally, two types of soft output for can be obtained by P o ( = A P m )= ff j2c(m) k(j)fi k (j) (8) P o e ( = A m )=c P o ( = A m )=P i ( = A m ) (9) where c is a normalization constant. Due to the truncation of state, P o (x k ) cannot be yielded directly as in the standard SISO algorithm. However, we can still obtain P o (x k ) approximately. One simple way is: P o e (x k = x(t Q k k )) = l=k L Po e (a l : t k ) (0) which can work fairly well [2]. It is noteable that (0) can be calculated recursively due to the temporal relationship between t k and t k+. P o e ( ) can be viewed as the (approximate) a-posteriori probability normalized to the a-priori probability and is usually called the extrinsic information []. If further iteration is necessary, one should feedback P o e ( ). Otherwise, the hard decision may be made by the rule: ^ = A m if P o ( = A m ) P o ( = A j ) for all 0» j» M. The completion step in (8) is also illustrated in Fig.. Obviously, this RS-SISO algorithm has the exactly same input/output interface as the standard SISO algorithm. It can be observed that when the RS-SISO generates the soft output for, the input information in time interval [k +;k + L 2 ] has not been used (Fig. ). Other completion approaches are feasible for a RS-SISO which use all the observations, but this may result in an algorithm with complexity dominated by the fullstate completion. Regardless of the completion technique used, the RS-SISO is sub-optimal due to the decision feedback used in the forward and backward recursions. The simplified completion scheme defined in (8) (0) results in further performance degradation and complexity reduction. In order to improve the performance of RS-SISO, we introduce the concept of self-iteration for the RS-SISO algorithm. By feeding its output back to its own input port several times, the RS-SISO can fuse the information in the time interval [k +;k + L 2 ] into the final soft output. It will be shown in Section IV that the self-iteration can improve the performance significantly. B. Minimum Sequence Metric RS-SISO In the case of an additive white Gaussian noise (AWGN), we define the metric counterparts to (6) and (7) as f k (i; j)=mi [x k = x(~a f k (i);t k(i; j))] + M i [ (i; j)] ()

3 b k (i; j)=mi [x k+l2 =x(t k (i; j); ~a b k+ (j))] + Mi [ (i; j)] (2) respectively, where the metric M(w) = ln(p(w)). Then, the sequence metric associated with a symbol sequence 2 k L,or the equivalent state sequence vk k2+ can be defined as Λ f (v P k k2+ k )= 2 k=k f k (i; j) (3) Λ b (v k k2+ k )= 2 k=k b k (i; j) (4) and the key quantities for RS-SISO are defined as ffi k (j) = min v k+ :v k+ =V Λ f j (v k+ ) (5) k (i) = min v K k+ :v k+ =V i Λ b (v K k+ ) (6) Consequently, the forward and backward recursion and the completion steps of RS-SISO can be readily written down as: ffi k+ (j) =min i2f(j) [ffi k (i) + f k+ (i; j)] (7) k (i) =min j2b(i) [ k+ (j) + b k+ (i; j)] (8) M o ( = A m )=min j2c(m) [ffi k (j) + k (j)] (9) M o e ( = A m )=M o ( = A m ) M i ( = A m ) (20) M o e (x k = x(t P k k )) = l=k L Mo e (a l : t k ) (2) Since the forward and backward recursion have the same computational structure as the recursion in the VA, the RS-SISO can obtain the survivor paths used in () and (2) just as in the VA. Again, one feeds back M o e ( ) if further iterations are needed, and uses M o ( ) to make the hard decision by the rule: ^ = A m if M o ( = A m )» M o ( = A j ) for all 0» j» M. Compared to the APP version, the version only involves summation and comparison operations. Computationally, it is much simpler than the APP version. By replacing min( ) by min Λ ( ) in the algorithm [,3,20], one directly obtain the corresponding log-app algorithm. However, the way one defines the survivor path in a log-app algorithm must be specified (i.e.,as we have done in Section III-A). The numerical experiments have shown that the SISO algorithm performs almost as well as its APP counterpart [9,3]. Similarly, compared to the standard algorithm, the RS-SISO has the same input/output interface, and a complexity M L2 times smaller. C. Iterative Detection Using RS-SISO An iterative detection network consists of SISO modules and soft information exchange rules and schedules []. The soft information is circulated inside this network several times before the hard decisions are made. Usually, the maximum iteration number can be estimated numerically. For I iterations, the complexity of iterative detector employing the RS-SISO algorithm is proportional to IM L. IV. NUMERICAL SIMULATION Two types of transmission system are used to test the performance of RS-SISO algorithms: an ISI/AWGN channel and a TCM/ISI/AWGN channel. Because of the simplicity and good performance of SISO algorithms, only the results for the RS-SISO will be presented here. It can be expected that the APP RS-SISO should perform no worse than the RS-SISO [3]. A. ISI/AWGN channels Two 2-tap (L =) ISI/AWGN channels are used in this test (Fig. 2). Channel A has equal entries and Channel B is selected to be (c; 2c; ; 2c). Both channels are normalized to have unit power. The transmitter uses the BPSK signaling scheme (i.e., = ± p E b ). The output of the ISI channel is then corrupted by an AWGN n k with Efn 2 k g = N 0=2. For comparison, we also run two hard decision algorithms: the VA [6] and DDFSE [8] on the same channels (Fig. 2(b)). Similar to in the RS-SISO, a truncated state of length L is defined in the DDFSE. (b) (c) 2-tap ISI VA or DDFSE RS-SISO AWGN hard soft hard Thresholding self iteration Fig. 2. The tested ISI/AWGN channel and (b-c) two types of detector RS-SISO with L =2 Channel E b /N 0 =22dB) Channel E b /N 0 =20dB) Number of Iteration Fig. 3. The convergence property of an iterative detector using the RS-SISO algorithm (L =2). We first describe the convergence properties of detectors employing an RS-SISO with self-iteration. As illustrated in Fig. 3, the convergence occurs after 4 5 iterations. In Fig. 4, several algorithms are compared. An index of the complexity is defined as the product of the number of transitions M L+, the self-iteration number I and the recursion number r. For the forward-only VA r =, while r = 2 for the forward/backward SISO. In Fig. 4 and the following figures, this complexity index is shown for each algorithm used. The performance of the VA with 2 = 2; 048 states is presented as a baseline. Note that thresholding the soft-output of the full-state -SISO yields the same data estimates as the VA, namely that of maximum likelihood sequence detection (MLSD), while thresholding the RS -SISO does not yield the same result as the DDFSE using the same L. First, we notice that the DDFSE performs roughly 3 db worse

4 VA RS-SISO (L =2, I = 4) Channel A DDFSE (L =5) RS-SISO (L =2, I = ) Channel B and B' 64 DDFSE (L =5) - B RS-SISO (L =2, I=4) DDFSE (L =5) - B' VA Fig. 4. The performance comparison of various detection algorithms for Channel A. The number attached to each curve is the complexity index of the corresponding algorithm. than VA at a bit error rate (BER) of 0 4 when L =5, i.e., the DDFSE uses 2 5 =32states. Without self-iteration, the RS-SISO with L =2performs only 0.3 db better than the DDFSE with a 4 times smaller complexity. However, if the self-iteration is used, after only 4 iterations, the performance of RS-SISO is improved by.9 db, and is only 0.8 db away from that of the VA. The complexity index indicates that the RS-SISO is 64 times simpler than the VA with a performance degradation of only < db while it outperforms a DDFSE of similar complexity by 2.2 db. Due to the bidirectional recursion of RS-SISO, its robustness to non-minimum phase channels is expected. However, for the DDFSE, the non-symmetric structure (only a forward recursion is used) degrades its performance greatly when the channel is of non-minimum phase. Channel B is such a channel. Channel B 0 is defined as the time-reversed version of Channel B, i.e., f2c; c; ; cg. The simulation results in Fig. 5 clearly show that the DDFSE with L = 5 virtually fails for Channel B but works well for Channel B 0. However, the iterative detector based on RS-SISO with the same complexity (L =2and I =4) performs nearly optimal for both Channel B and B 0. Note that for both Channel B and B 0, either the VA or the RS-SISO performs same. For RS-SISO this is due to its bidirectional architecture; while for VA this is because it is optimal in the sense of MLSD and Channel B and B 0 have the exactly same spectrum. B. An ISI/AWGN channel with TCM signaling An 8-state, rate R = 2=3 Ungerboeck 8-PSK TCM code (Figure 9 in [2]) is used in this test. The 8-PSK signals from the TCM encoder are fed into a block interleaver. The interleaved 8-PSK signals pass through a 5-tap (L = 4) ISI channel with equal entries (normalized to unit power), and the output is corrupted by a white complex circular Gaussian noise n k with Efjn k j 2 g = N 0 =2. This system is illustrated in Fig. 6. It is too complex to implement a Viterbi detector by considering the concatenated TCM encoder and ISI channel as a single FSM. Alternatively, an effective approach is to build a detector Fig. 5. The robustness of the iterative detector using the RS-SISO algorithm. R =2/3 TCM Encoder inner detector Viterbi hard Equalizer SISO Algorithm RS-SISO soft 8-PSK Mapper Interleaver outer detector Viterbi Detector Viterbi Detector soft SISO inner iteration outer iteration 5-tap ISI Hamming Euclidean Euclidean AWGN Fig. 6. The tested TCM/ISI/AWGN channel and (b-d) three types of detector. (hard or soft) for each subsystem separately [,4]. As shown in Fig. 6(b), one can use the VA as both the inner and outer processor, but a poor performance is expected because the TCM code is decoded with hard decisions. Replacing the inner VA by a SISO (Fig. 6(c)) can improve the overall performance since more reliable Euclidean is used at the outer Viterbi detector. Moreover, replacing the outer VA by a SISO and using iterative detection, the overall performance can approach that of the optimal performance [3,4]. However, even by treating the subsystems separately, the complexity of the inner detector is prohibitive. For example, in the above system, the inner FSM has 8 4 = 4; 096 states and 8 5 = 32; 768 transitions. Therefore, we consider the RS-SISO for the inner SISO. Due to the sub-optimality of the RS-SISO and the concatenated detection structure [3,4], both the self-iteration of the inner RS- SISO (so called inner iteration) and the outer iteration (see Fig. 6(d)) becomes necessary for an effective detector. Therefore, besides the number of outer iterations I o as for general iterative schemes, there is another design parameter for this specific detector: the number of the inner iterations I i. Before feeding the soft information on the 8-PSK symbols to the outer SISO, the inner RS-SISO may conduct several inner iterations to improve the (b) (c) (d)

5 quality of the soft output. In this case, the complexity index is redefined as M L+ I o I i r. Consequently, a tradeoff between the complexity and the performance exists. Fig. 7 shows the simulation results of various detection schemes. There is a 5 db gain in E b =N 0 at a BER of 0 4 by replacing the VA with a -SISO at the inner stage. When replacing the inner VA by a RS-SISO with L =2(I i =), only 0.3 db gain is obtained. This means that without any performance degradation, the detector complexity is reduced by 32 times. Two types of iterative detector using the RS-SISO algorithm are tested. Both detector converges in only 4-5 iterations (results are not shown here). One detector uses an RS-SISO with L =(8 states), and 3 inner iterations (I i =3). The hard decisions are made after five outer iterations (I o =5). Compared to the SISO VA scheme, the performance degradation is only. db while the complexity saving is 34 times. In order to obtain better performance, the other iterative detector uses L =2 and I i =3. After four outer iterations (I o =4), a 0.3 db gain is obtained over the SISO-VA approach while the complexity saving is roughly 5 times. For comparison, note that in this application any reduced complexity hard-decision processor (e.g., RSSE [7], DDFSE [8]) will perform worse than the VA VA scheme in Fig RS-SISO - SISO (L =, I i =3, I o =5) RS-SISO - VA (L =2, I i =) VA - VA RS-SISO - SISO SISO - VA (L =2, I i =3, I o =4) Fig. 7. The performance comparison of various detection algorithms for the TCM/ISI channel. V. CONCLUDING REMARKS In this paper, the RS-SISO algorithm for the simple FSM was presented and shown to provide both significant complexity reduction and excellent performance. It is conceptually straightforward to extend this algorithm to any FSM to which the concept of state reduction is applicable. It is notable that there exist FSMs with very special structure (i.e., many transitions, but only few outputs), e.g., a TCM encoder, such that it may be able to calculate P o (x k ) or M o (x k ) very efficiently if such soft output is required. Also, besides the feedback technique used in this paper, other state reduction (e.g., Ungerboeck-like set partitioning principle [7,2]) techniques may be employed to derive other forms of the RS-SISO. Due to the sub-optimality of the RS-SISO algorithm, the self-iteration is necessary in many cases. The flexibility of RS-SISO provides many options to trade the complexity/performance. Since the proposed RS-SISO has the same input/output interface as the standard SISO, it can be widely applied to many applications other than those presented in this paper. REFERENCES [] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, Soft-input softoutput modules for the construction and distributed iterative decoding of code networks, European Trans. 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