TWO EFFICIENT STOPPING CRITERIA FOR ITERATIVE DECODING Wei Jiang, Daoben Li Beijing University of Posts and Telecommunications School of Information Engineering No. 10, Xi Tu Cheng Road, Hai Dian District, Beijing 100876, P. R. China Email: [email protected] ABSTRACT The paper proposes two new stopping criteria for iterative decoding of Turbo codes. The first criterion is based on the mutual information between the logarithm likelihood ratio and the data bits, and the second one is based on the a posteriori error probability of the decoded information bits. Compared to the existing stopping rules, the proposed criteria need not to store any extra data, and can reduce the average number of iterations much closer to the genie stopping rule when the performance degradation is negligible. 1. INTRODUCTION Turbo codes[1] employ iterative decoding to achieve nearShannon-limit error-correction performance. As the number of iterations required for error-free decoding depends on the channel state and the noise, a good stopping criterion can greatly save the iterations on average and improve the throughput of the decoder. In [2], a stopping criterion based on cross entropy (CE) is introduced. This criterion calculates the cross entropy between the a posteriori probability distribution of the output of the two element decoders. It requires the storage of the logarithm likelihood ratio (LLR) output of the first decoder and the calculation is quite complex. Two simplifications to the cross entropy rule, the sign-change-ratio (SCR) criterion and the hard-decision-aided (HDA) criterion are proposed in [3], which still require the storage of the last-iteration data. Another simplification to CE criterion, the sign difference ratio (SDR) criterion, is proposed in [4], which compares the sign change between the LLRs of the output and the input of an element decoder. A comprehensive study of stopping criteria based on the soft decision and the hard decision of the information bits is performed in [5]. Another kind of criteria utilizes the CRC check of the data frame [6]. Most of the existing stopping criteria compare the specific statistics with a threshold. There’s often a trade-off beThis work is supported by National Natural Science Foundation of China (NSFC) under grant no. 90604035.

tween the average number of iterations and the performance degradation. The threshold should be chosen to be able to decrease the average number of iterations as large as possible while keeping the error on a certain low level. In [5], the so-called genie stopping criterion, which stop the iteration when the hard decision is correct, is used for comparison. It’s found that most of the criteria require at least one half iterations more than the genie criterion. In this paper, two new stopping criteria are proposed. They are based on the mutual information between the logarithm likelihood ratio and the data bits, and the a posteriori error probability, respectively. Compared to the conventional stopping rules, the proposed criteria don’t need any storage of the extra data and can reduce the average iteration number much closer to the genie stopping rule while causing negligible performance degradation. Moreover, simulation results show that the proposed criteria can even perform better than the genie stopping criterion on the bit error rate. 2. ITERATIVE DECODING STRUCTURE Denote the frame of data bits of length N as u. Assuming BPSK modulation is used, u is mapped into the symbol sequence x, with its elements xi ∈ {1, −1}, 0 ≤ i < N . The channel is supposed to be AWGN, and the received symbols can be expressed as y = x + n,

(1)

where n is the vector of additive Gaussian noise with mean zero and variance σn2 . Assume channel state information (SCI) 2 Lc = 2 (2) σn is known at the receiver. The typical iterative decoding structure[1] is show in Fig. 1. Two SISO (soft-input soft-output) decoders corresponding to the two element codes are separated by interleavers. The extrinsic information output of the first SISO decoder, Le1 (u), is interleaved and passed to the second

Fig. 1. The structure of iterative decoder SISO decoder as the a priori information. Similarly, the extrinsic information, Le2 (u), is fed back to the first decoder as the a priori information. The iterative decoding goes on until a stopping rule satisfies. 3. TWO NEW STOPPING CRITERIA The stopping criteria test the condition based on the change of the LLRs or the statistics on the LLRs. For example, the HDA criterion[3] stops the iteration if the hard decision based on the LLRs keeps the same between two iterations. The S3 criterion in [5] compares the minimum absolute value of the average LLRs from both SISO decoders with a threshold θ3 . The rule is satisfied if   1 n n |L1 (ui ) + L2 (ui )| ≥ θ3 , (3) min 0≤i
The method of EXIT chart finds that when the iterative decoding converges to the correct decoding, the trajectory of the EXIT chart arrives at the (1, 1) point, i.e., I(E; X) and I(A; X) trend to be one. So the mutual information is a good indicator of the correct decoding, and can be used to decide when to terminate the iterations. It’s more reasonable to use the a posteriori LLRs but not the extrinsic LLRs or the a priori LLRs for the stopping rule. The mutual information between the a posteriori LLRs and the data bits, I(L; X), can be defined the same as (4) except that the distribution of the extrinsic LLRs is replaced with that of the a posteriori LLRs. The stopping criterion based on the mutual information can be stated as: MI (Mutual Information) Criterion: The mutual information between the a posteriori LLRs and the data bits is compared with a threshold. Terminate the iterations if the following condition satisfies: 1 − I(L; X) < θMI ,

where θMI is the predetermined threshold, which should be a positive number close to zero. In practice, the evaluation of I(L; X) according to (4) is quite complicated. It’s shown in [8] that the mutual information can also be expressed as  I(L; X) = 1 − E lb(1 + e−L ) Z +∞ (6) =1− p(L|x = +1)lb(1 + e−L )dL. −∞

When the block length N is large enough, the mutual information can be approximated by I(L; X) ≈ 1 −

X Z +∞ 1 I(E; X) = · pE (ε|X = x∗ ) 2 x∗ =−1,1 −∞ 2 · pE (ε|X = x∗ ) × lb dε, pE (ε|X = −1) + pE (ε|X = 1)

N −1 1 X Hb (Pei ) N i=0

N −1 e−|L(ui )|/2 1 X =1− Hb ( |L(u )|/2 ), i N i=0 e + e−|L(ui )|/2

3.1. Mutual Information Criterion The mutual information between the LLRs and the information bits has been employed to design the Extrinsic Information Transfer (EXIT) chart [7]. The mutual information between the extrinsic information and the transmitted data bits can be expressed as

(5)

(7)

where

e−|L(ui )|/2 e|L(ui )|/2 + e−|L(ui )|/2 is the a posteriori error probability of ui , and Pei =

(8)

Hb (p) = −plb(p) − (1 − p)lb(1 − p), (0 ≤ p ≤ 1). (9) (4)

where pE (ε|X = x∗ ) is the probability density of the extrinsic LLRs on the condition that x∗ is transmitted, and x∗ is assumed to be +1 or −1 with equal probability. The mutual information between the a priori LLRs and the data bits, I(A; X), can be expressed similarly.

3.2. Error Probability Criterion Obviously, the iterative decoding can be terminated when the bit error probability is small enough. Based on the LLRs output from the SISO decoder, the average a posteriori error probability of the data bits is N −1 1 X P¯e = Pe . N n=0 i

(10)

Substitute (8) into (10) and let

6.5 Genie HDA S3 MI, Th.=1E−6 MI, Th.=1E−5 MI, Th.=1E−4 PE, Th.=1E−6

∆

(11)

then we get N −1 1 X e−|L(ui )|/2 P¯e = N n=0 e|L(ui )|/2 + e−|L(ui )|/2

= =

1 N

N −1 X

(12)

eai

6 Average Number of Iterations

ai = ln Pei = − |L(ui )| − ln (1 + exp (− |L(ui )|)) ,

5.5

5

4.5

4

3.5

n=0

1 exp [max∗ (ai )] , N

where max∗ (xi ) = ln

X

3 0.4

0.5

0.6

0.7

0.8

0.9

Eb/N0 (dB)

exi .

(13)

Fig. 2. Average number of decoding iterations

i

Note that the calculation of the function ln(1+e−x ), x > 0 in (11) can be implemented by a simple lookup table, and so as the function max∗ in (12) [9]. Furthermore, max∗ can be approximated by the max function. So the complicated calculation of P¯e can be done with a series of simple operations. So the stopping criterion based on the a posteriori error probability can be stated as: PE (Probability of Error) Criterion: The average a posteriori error probability of the decoded bits is compared with a threshold. The iteration is terminated if P¯e < θPe ,

(14)

where θPe is the predetermined threshold, which should be a sufficiently small positive number. Actually, the exponential operation in (12) can be removed by defining ∆

θP′ e = ln(N θPe ).

(15)

The PE criterion is satisfied if max∗ (ai ) < θP′ e .

(16)

4. SIMULATION RESULTS The performance of the two stopping criteria proposed in this paper is evaluated by numerical simulations. The genie criterion, the HDA criterion and the S3 criterion are also simulated for comparison. The code used is a rate 1/3, (23,33,1784) Turbo code, which is proposed for the Consultative Committee for Space Data Systems (CCSDS) standard[10], and is also the code used in [5]. The threshold of S3 criterion θ3 = 7.25, as is suggested by [5]. The threshold of the MI criterion θMI ∈ {1E − 4, 1E − 5, 1E − 6},

and the threshold of the PE criterion θPe = 1E − 6, which are chosen in a trial-and-error manner. And the maximum number of iterations is set to be 20. Fig. 2 shows the average number of decoding iteration with different stopping criteria. It can be seen that the genie criterion performs the best and the HDA criterion consumes the most iterations. The proposed MI criterion and the PE criterion reduce more iterations than HDA and S3 criteria. The MI criterion with threshold 1E − 4 only require about 0.3 iterations more than the genie criterion on average. The frame error rate (FER) and the bit error rate (BER) performance of the Turbo code applying different stopping criteria are illustrated in Fig. 3 and Fig. 4 respectively. It can be seen from Fig. 3 that there is no obvious difference on FER for criteria S3, HDA, and MI with threshold 1E − 6, which are all close to the genie rule. This means that these criteria cause negligible performance degradation while improving the throughput of the iterative decoder greatly. On the other hand, the MI criterion with threshold 1E − 5 and 1E − 4, and the PE criterion with threshold 1E − 6, increase the FER by a little. An interesting phenomenon observed from Fig. 4 is that all the tested criteria perform closely on BER, which means that little bit error is caused by the stopping rules on average. Even though the MI criterion with threshold 1E−5 and 1E − 4 have relatively poor performance on FER, they can perform a little better on BER than the genie criterion. This can be explained by the fact that the Turbo decoding cannot always reduce bit errors in a frame by iterations due to the limitation of the block length and the interleaver. There are cases that the iterative decoding increases the number of error bits when the bit error number has already been very small. As the proposed criteria are based on the statistics about the bits, they can keep the BER on a low level.

−1

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10 −2

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Bit Error Rate

Frame Error Rate

−4

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Genie HDA S3 MI, Th.=1E−6 MI, Th.=1E−5 MI, Th.=1E−4 PE, Th.=1E−6

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Genie HDA S3 MI, Th.=1E−6 MI, Th.=1E−5 MI, Th.=1E−4 PE, Th.=1E−6

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Fig. 3. Frame error rate performance 5. CONCLUSION Two new efficient stopping criteria for iterative decoding, the MI criterion and the PE criterion, are proposed in this paper. The criteria are based on the mutual information between the LLRs and the transmitted data bits, and the a posteriori error probability of the bits, respectively. The proposed criteria don’t need any extra storage, and terminate the iteration as long as the mutual information, or the a posteriori error probability, is below a predetermined threshold. Compared to the existing stopping criteria, such as the HDA criterion and the S3 criterion, the method introduced in this paper saves more iterations. Furthermore, these criteria increase little frame error probability compared with the genie stopping rule, and the MI criterion can perform even better on BER. A drawback of them, especially for the MI criterion, is that they are relatively complicated than some existing criteria. Simplification methods are to be found in the future work. 6. REFERENCES

10

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Fig. 4. Bit error rate performance tions on Communications, vol. 47, no. 8, pp. 1117– 1120, 1999. [4] Y. Wu, B. D. Woerner, and W. J. Ebel, “A simple stopping criterion for Turbo decoding,” IEEE Communications Letters, vol. 4, no. 8, pp. 258–260, 2000. [5] A. Matache, S. Dolinar, and F. Pollara, “Stopping rules for Turbo decoders,” in TMO Progress Report 42-142, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, 2000. [6] A. Shibutani, H. Suda, and F. Adachi, “Reducing average number of Turbo decoding iterations,” Electronics Letters, vol. 35, no. 9, pp. 701–702, 1999. [7] S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Transactions on Communications, vol. 49, no. 10, pp. 1727– 1737, 2001. [8] J. Hagenauer, “The EXIT chart- Introduction to extrinsic information transfer in iterative processing,” in EUSIPCO, 2004.

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