An Implementation of a Soft-Input Stack Decoder For Tailbiting Convolutional Codes

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1 An Implementation of a Soft-Input Stack Decoder For Tailbiting Convolutional Codes Mukundan Madhavan School Of ECE SRM Institute Of Science And Technology Kancheepuram mukundanm@gmail.com Andrew Thangaraj Department Of Electrical Engineering Indian Institute Of Technology,Madras Chennai andrew@iitm.ac.in Abstract In this paper, we propose a novel method for sequential stack decoding of tailbiting convolutional codes with either a systematic or a non-systematic encoder. The proposed method estimates the tailbiting initial state by running several stack decoders for a limited depth. The estimated initial state is used in a regular stack decoder for completing the decoding. The proposed decoder was simulated in software and implemented on a fixed-point DSP platform for the tailbiting convolutional code (rate-1/2, constraint length 6) in the IEEE e (WiMax) standard. Our simulations show that the proposed method with a finite stack size loses about 1 to 2dB when compared to optimal decoders at bit-error rates of However, the complexity of the proposed method is lesser than that of the optimal decoders at moderate to high signal-to-noise ratios. The DSP implementation can provide a processing rate of more than 120 kbps at a frame-error rate of Introduction Convolutional codes were introduced by Elias in 1955, as an alternative to block codes. In 1967, the Viterbi algorithm [1] was introduced for efficient soft-input decoding of convolutional codes. The Viterbi algorithm works on a trellis representing the codewords of a convolutional code and has now become the de facto standard in the decoding of convolutional codes. However, it has a single disadvantage - its computational complexity increases exponentially with the constraint-length of the convolutional code. Hence, it cannot be practically used with convolutional codes having constraint lengths higher than six. This shortfall of the Viterbi algorithm caused the advent of other sub-optimal algorithms. One such popular algorithm is the ZJ stack algorithm [2]. The stack algorithm is a sequential decoding technique that expands only selected paths that appear most 1 This work was done at IIT Madras during Mukundan Madhavan s summer fellowship. likely in each stage of the trellis. The computational complexity of the stack algorithm is independent of the constraint length, and instead, varies with the signal-to-noise ratio (SNR). So, the stack decoder could be preferable to the Viterbi algorithm for decoding convolutional codes having constraint-lengths higher than six. However, it must be noted that the performance of a stack decoder, in terms of both error and bit rates, deteriorates sharply for lower values of SNR. Tailbiting is an idea that has become popular in recent wireless communication systems. The basic idea of tailbiting is to eliminate the rate loss due to zero termination in a convolutional code. However, tailbiting increases decoder complexity, since the initial state is random and unknown at the decoder. Various algorithms have been proposed for the decoding of a tailbiting convolutional code. Most of these algorithms such as those presented in [3], [4], [5] and [6] are based on the Viterbi algorithm and expand all states in every stage of the trellis. The stack algorithm can also be adapted to estimate the initial state of a tailbiting trellis. Orten [7] has proposed a stack-decoding technique for a systematic tailbiting code. But this method assumes the presence of a sequence of K error-free bits in the received sequence for a convolutional code of constraint length K. This assumption, though reasonable for systematic codes (where the constraint lengths are usually kept low), cannot be made about non-systematic codes, which are primarily used with high constraint length codes necessary for good performance in terms of error-rates [8]. The algorithm we propose for the estimation of the initial state during sequential decoding of a tailbiting convolutional code involves the expansion of the trellis to a limited depth (lesser than the maximum possible depth) 2 K times, once for each of the possible initial states. Then the best paths obtained in each expansion are compared using the value of their path metric, and the initial state giving the highest metric is passed onto a regular ZJ stack decoder as the estimated tailbiting initial state. Simulations with the proposed decoder were carried out using the tailbiting convolutional code (rate-1/2, constraint

2 length 6, nonsystematic) in the IEEE e (WiMax) standard (this code will be referred to as the WiMax code in the rest of this article). The proposed method with a maximum stack size of 20 performs within 1dB of comparable decoders at a bit-error rate (BER) of The proposed stack decoder has been implemented for the WiMax code on a Blackfin BF-533 processor. The BF-533 is a 32-bit digital signal processor produced by Analog Devices. The processing rate is more than 120 kbps with a stack size of 30 at a frame-error rate (FER) of 10 3, while a processing rate above 300 kbps was observed with a stack size of 10 at a FER of Thus, the stack decoder implementation provides various options for a designer of a system compliant with the WiMax standard. This flexibility in complexity and the trade-off with performance are major advantages of the stack decoder in practice. In Section 2, we discuss the basics of tailbiting convolutional codes. In Section 3, we describe the Stack algorithm and also, elaborate on the proposed algorithm. Section 4 deals with the Blackfin implementation, and we present some simulation results in Section Tailbiting Convolutional Codes Consider a k-bit message m = [m 1, m 2,...m k ] that needs to be transmitted across a channel using a rate-r convolutional code of constraint length K. Assume that the initial state is fixed to be the all-zero state. To ensure that the final state is the all-zero state, a sequence of K zeros is appended to m and encoded to form a code sequence of length k+k R. Now, the rate becomes k n = kr k + K = R 1 + K. k In many applications (such as wireless transmissions), typical values such as k = 50, and K = 6 result in a significant rate-loss due to zero termination. If the same convolutional code is used in a tailbiting configuration, the initial state (before m 1 is encoded) is not constrained to be zero, but is instead, chosen based on the message. However, the final state (after m k has been encoded) is required to be the same as the initial state. Assuming a non-recursive encoder, the final state is easily seen to be [m k K+1,..., m k ]. Therefore, for nonrecursive encoders, in a tailbiting configuration, the initial state is set to be [m k K+1,..., m k ] for a message m = [m 1, m 2,..., m k ]. In an implementation of a tailbiting encoder, to ensure that the initial and final state of the trellis are the same, the message-bits from m k K+1 through m k are initially fed into the encoder. The corresponding outputs obtained are discarded, but the shift-register contents are retained. Therefore, the initial state and the final state will be [m k K+1...m k ], and the coding rate continues to be R. When the initial state is not fixed and known a priori at the decoder, the Viterbi algorithm ceases to produce a Maximum Likelihood (ML) output codeword. The decoders proposed in prior work [5, 6] employ multiple iterations of the basic Viterbi algorithm for decoding a tailbiting code. However, none of the prior methods can be directly extended to stack decoding so as to take advantage of reduced computational complexity offered by a stack decoder at moderately high SNRs. Separate estimation of the initial state appears to be necessary before a stack decoder can be used for tailbiting codes. In this article, we propose a state estimation method especially suited for stack decoding. 3. Stack Decoder for Tailbiting Codes The ZJ stack decoding algorithm uses a storage called the stack to store all expanded paths in a trellis. The word stack simply refers to an array storage that can be accessed at random. Instead of expanding branches from all possible states at a particular depth like the Viterbi decoder, the stack decoder expands only the path that has the highest metric in the stack memory, irrespective of the path s depth [2]. The metric used by the stack decoder is different from that used by the Viterbi algorithm so as to compare paths at different depths. For the sake of convenience, we assume a ratehalf code (R = 1 2 ) with a constraint-length K. Let the code sequence corresponding to the message m = [m 1, m 2,..., m k ] be c = [c 11, c 21,..., c 1k, c 2k ]. Let the sequence received at the decoder, corresponding to this code-sequence, be y = [y 11, y 21,...y 1k, y 2k ], and let the decoded bits be represented by ˆm = [ ˆm 1,..., ˆm k ]. The codewords of the convolutional code can be described as paths in a 2 K -state trellis [8]. Given that the present state is S (0 S 2 K 1) and the input bit is b (b {0, 1}), let N(S, b) denote the next state and E(S, b) denote the two output bits of the rate-1/2 code. The functions N and E completely describe the trellis and the code. The Fano branch metric (FBM) used by the stack Decoder is defined as ( ) p([y1j y 2j ] E(S, b)) FBM(S N(S, b)) = log 1/2 p([y 1j y 2j ]) for a rate-1/2 code at the j-th trellis stage (p( ) denotes a probability density function). For an AWGN channel with BPSK modulation, the metric can be easily computed. The path metric of a path in the trellis is defined to be the sum of the Fano branch metrics of all its branches. The stack used by a ZJ stack decoder contains a set of paths on the trellis arranged in descending order of path metrics. In every iteration, the top path in the stack (with maximum metric) is expanded into two paths by the addition of new branches corresponding to an input of 0 and 1. These two paths are added to the stack, and the stack is sorted again in descending order of path metrics. Algo-

3 rithm 1 provides pseudocode for the stack decoder for a tailbiting code assuming that the initial and final state S 0 is known. Though this version of the stack algorithm is well-known, we have included it here for completeness. We have used the following notation. At any iteration, the stack contains M paths P 1, P 2,, P M. The number of branches in P i (or the depth of P i ) is denoted L i, and the final state of P i is denoted F S i for 1 i M. The path metric of P i is denoted P M i. The inputs to the algorithm are the number of trellis stages k, the initial (and final) state S 0, and the trellis description functions N and E. Algorithm 1 Decoder(k,S 0,N,E) 1: {Initialization} 2: M = 1 3: P 1 = S 0 ; P M 1 = 0 4: {Iteration} 5: while L 1 k do 6: P M+1 = [P 1 N(F S 1, 1)] 7: P 1 = [P 1 N(F S 1, 0)] 8: Update path metrics of P 1 and P M+1 9: Sort stack in descending order of path metrics 10: end while 11: {Decision} 12: Output P 1 and P M 1. In short, this algorithm can be seen as expanding the path that appears to be the best, and if that path turns out to be wrong, the decoder backs out, and pursues the second-best path. This process continues till the decoder reaches the full depth k. Some observations about the stack algorithm are given below. 1. The computational complexity of the stack decoder increases with a decrease in SNR because the algorithm chooses a wrong path more often. In contrast, the stack decoder is much more efficient than the Viterbi decoder at higher SNRs. The computational complexity of a stack decoder is about 1/2 K -times that of a Viterbi decoder at high SNRs. 2. At all SNRs, the stack algorithm is ML if the stack size is assumed to be infinite. With a finite stack size, the stack decoder is suboptimal. At sufficiently high SNRs (depending on the code) the performance of a finite-sized stack decoder is very close to that of the optimal decoder. 3. The stack is a very flexible algorithm. The tradeoff between processing rate and error rate can be varied dynamically during decoding by changing the maximum stack size. This provides various options to a communication systems designer Stack Algorithm For Tailbiting Codes To implement a stack decoder for tailbiting codes, we propose to first estimate the initial tailbiting state separately using the stack decoder with lesser decoding depth. Once the initial state is known, the stack decoder is run for the entire length of the trellis. The method proposed here has been implemented on a Analog Devices Blackfin BF-533 processor for the WiMax code of constraint length six. In fact, the implementation of an efficient stack decoder for this code was our primary motivation for devising the proposed algorithm. The pseudo-code in Algorithm 2 describes the first version of the proposed algorithm. The trellis is expanded beginning with each of the 2 K 1 possible states to a depth l, 5K l 6K (K is the constraint length). The initial state that results in the best final metric is used for complete decoding. Since the initial state of the trellis Algorithm 2 Partial Expansion Algorithm 1: {Finding Initial State} 2: bestmetric= 3: beststate= 0 4: for i = 0 to 2 K 1 do 5: Decoder(l,i,N,E) 6: if bestmetric< P M 1 then 7: bestmetric= P M 1 8: beststate= i 9: end if 10: end for 11: {Decoding} 12: Decoder(k,beststate,N,E) affects only the initial few decisions, partial expansion of the trellis is sufficient to decide on the initial state. The two modules, decoder and initial-state estimator could be implemented as separate modules for reasons discussed in Section The Modified Partial-Expansion Algorithm The algorithm in Section 3.1 can be further improved using a modified partial-expansion technique. The modified algorithm for finding the initial state is carried out in two stages. The first stage involves the expansion of all states to a depth d, which is a lot lesser than l, (5K l 6K). In the second stage, only those initial states, which resulted in the top X metrics, will be expanded up to depth l. The value of X and d must be chosen depending upon the processing rate and performance desired. From simulations, we find that typical values for d are chosen to be around 2K and X can be chosen to be around K at moderate to high SNRs (3dB and above for rate-1/2 codes). Algorithm 3 presents pseudocode for the modified partial-expansion stack decoder.

4 Algorithm 3 Modified Algorithm 1: {Stage I: Expansion of all states up to depth d} 2: for i = 0 to 2 K 1 do 3: Decoder(d,i,N,E) 4: finalmetric(i) = P M 1 5: end for 6: Sort finalmetric in descending order and store top X initial states in beststate(i), 1 i X 7: {Stage II: Expansion of top X states up to depth h} 8: bestmetric= 9: for all S 0 in beststate do 10: Continue Decoder(h,S 0,N,E) 11: if bestmetric< P M 1 then 12: bestmetric= P M 1 13: initstate= S 0 14: end if 15: end for 16: Decoder(k,initstate,N,E) FER dB,30 5dB,30 6dB,30 FER Vs Processing Rate;Points marked with SNR(dB),Stack Size 3dB,20 5dB,20 6dB,20 5dB,10 6dB, Processing Rate kbps 5dB,4 6dB,4 Figure 2: Operating points for the DSP implementation. 4. Implementation The proposed decoder is to be implemented as two modules - (1) state-estimation module and (2) the main decoder module. In this way, once the tailbiting initial state estimation of the packet is complete, the initial state can be passed on to the decoder module along with the input bits as shown in Fig. 1. Input Buffer Initial State Estimator Stack Decoder Output Figure 1: Two-stage implementation of the proposed decoder. The two modules could be implemented in different devices to facilitate pipe-lined operation. The first module could even be implemented in a non-dsp platform (such as FPGA) for reducing delay by exploiting parallelism inherent in the initial state estimation method. Simulations in C were used to determine the optimal values of d and X in the initial state estimation process. Note that the values will change with SNR. For the 64-state WiMax code, the correct initial state was among the entries in a list with X = 4 for a depth d = 10 for SNRs above 3dB. The second module was completely implemented in BF-533. The various operating points of the implementation are shown in Fig. 2. We see that a FER of 10 3 is possible with a stack size of 20 and a processing rate of (more than) 120 kbps. 5. Simulation Results In this section, we present simulation results to compare the performance and computational complexity of the proposed stack decoder with other decoders for tailbiting convolutional codes proposed in the literature. In Fig. 3, we have provided several plots of BER versus SNR for the rate-1/2 WiMax code. The BER plot for Algorithm 2 has been shown for two depths - l = 40 and l = 50 - for initial state estimation. Also shown is the BER plot for Algorithm 3 (with values of d and X optimised for each SNR). For comparison, we have plotted the performance of a Viterbi decoder aided by a (hypothetical) genie that provides the tailbiting initial state. The BER plot for the decoder proposed in [5] is also shown. We notice that the loss in performance because of our initial state estimation method and finite-sized stack decoding is roughly 1 db at a BER of 10 4 for Algorithm 2 and 2 db for Algorithm 3. The stack size was chosen to be 150 for the initial state estimation and 300 for the stack decoding at SNRs greater than 4dB. In Fig. 4, we have presented plots to compare computational complexity of the proposed decoder. The chosen measure of complexity is the number of expanded states of the trellis for decoding one information bit. For instance, in the genie-aided Viterbi algorithm, this number is 2 K. We have assumed that the computations required per state expansion is a constant across the various decoders under consideration. For stack decoders, the number of expanded states per information bit is a decreasing function of the SNR. From Fig. 4, we observe that the complexity of Algorithm 2 with depth 40 is about half of that of the decoder proposed in [5] at sufficiently high SNRs. The modified partial expansion is one-fourth as

5 Bit Error Rates For Different Decoders Computational Complexity 10 1 Partial Expansion Depth 40 Partial Expansion Depth 50 Wang And Bhargava Genie Viterbi Modified Partial Expansion Wang And Bhargava Modified Partial Expansion Partial Expansion Depth 50 Partial Expansion Depth 40 Bit Error Rate 10 3 States Extended Per Bit Signal To Noise Ratio(dB) Signal To Noise Ratio (db) Figure 3: Simulation results and comparison of BER for the WiMax code. Figure 4: Simulation results and comparison of computational complexity for the WiMax code. complex as the decoder of [5]. 6. Conclusion In conclusion, we see that the proposed stack decoding algorithm for tailbiting convolutional codes provides various useful trade-offs between BER performance and implementation complexity that are not available in previously proposed decoders. The loss in performance because of finite stack size is seen to be within reasonable limits given the gain in computational complexity. These decoders are good candidates for implementation in cellular base stations that require a flexible architecture. 7. References [1] G. D. Forney Jr., The Viterbi Algorithm, Proc. IEEE, Vol. 61, No. 3, pp , Mar [2] F.Jelinek, A Fast Sequential Decoding Algorithm Using A Stack, IBM J. Res. Dev, vol. 13, pp , Nov [3] Richard V. Cox, Carl-Erik W. Sundberg, An Efficient Adaptive Circular Viterbi Algorithm for Decoding Generalized Tailbiting Convolutional Codes, IEEE Trans. Veh. Tech., Vol. 43, No. 1, pp , Feb [4] H. H. Ma and J. K. Wolf, On tailbiting convolutional codes, IEEE Trans. Comm., vol. COM-34, pp , Feb [5] Qiang Wang and Vijay.K.Bhargava, An Efficient Maximum Likelihood Decoding Algorithm for Generalised Tailbiting Convolutional Codes Including Quasicyclic codes, IEEE Trans. Comm., Vol.37, No.8, pp , Aug [6] R. Y. Shao, Shu Lin and M. P. C. Fossorier, Two decoding algorithms for tailbiting codes, [7] Pal Orten, Sequential Decoding of Tailbiting Convolutional Codes for Hybrid ARQ on Wireless Channels, Proc. 49th IEEE Veh. Tech. Conf., Vol. 1, pp , May [8] Shu Lin and Daniel Costello, Error Control Coding, Second Edition, Prentice Hall, NJ, USA, 2004.