SNR Mismatch and On-Line Estimation in Turbo Decodingy - CiteSeerX

SNR Mismatch and On-Line Estimation in Turbo Decodingy Todd A. Summers [email protected]

Stephen G. Wilson [email protected]

Department of Electrical Engineering University of Virginia Charlottesville, VA 22903 keywords: turbo decoding, SNR estimation, iterative decoding

Abstract Iterative decoding of turbo codes, as well as other concatenated coding schemes of similar nature, requires accurate knowledge of the signal-to-noise ratio of the channel so that proper blending of the a posteriori information of the separate decoders is achieved. In this paper, we study the sensitivity of decoder performance to mis-estimation of the SNR, and propose a simple on-line scheme that estimates the unknown SNR from each code block, prior to decoding. We show this scheme is suciently adequate in accuracy to not appreciably degrade performance.

y

work sponsored by NSF grant NCR-9415996 and NASA/LeRC contract NAG3-1948

1 Introduction Turbo codes and other similar constructions have captured the fancy of the error control coding community in that performance quite near the Shannon capacity limit is attainable with moderate complexity decoding. The technique uses one or more component codes, with separate decoders processing each code to estimate the a posteriori probability of the various message bits. These estimates become prior information to the other decoder(s), which further update the a posteriori estimates. Upon iteration between decoders, it is found that error probability in the range of 10?5 is attainable at SNR within 1 dB of channel capacity, provided the blocklength of the overall code is fairly long, say 20000 bits. The technique is also applicable to shorter blocks; however performance is not so impressive as above. Readers are referred to references [1] through [5] for descriptions of the technique, and for performance analysis. One requirement of the decoding algorithm, which normally uses the MAP algorithm [6] to estimate a posteriori probabilities, is knowledge of the channel signal-to-noise ratio for a Gaussian noise channel. In essence, this is needed to supply the proper combination of prior bit statistics, which are in fact obtained as a posteriori data from previous iterations, and raw channel measurements, which are Gaussian random variables. Though the MAP algorithm requires an SNR parameter to produce the correct MAP estimates, there is little evidence on how sensitive decoder error rate is to mismatch of this parameter. It is also interesting that the SNR is not a requirement in Viterbi decoding without a priori information. Two questions thus come to mind. First, what is the sensitivity of turbo decoder performance to mismatch in the knowledge about SNR? Second, how can SNR be determined eciently on a packet-by-packet basis? This paper addresses both questions, and we show that roughly speaking the required SNR accuracy to prevent large performance degradation is -2 dB and +6 dB. (The iterative decoding algorithm is curiously more tolerant of overestimation of SNR than under-estimation.) Similar results have also been independently 1

reported in [7] and [8]. Regarding the second issue, we show a simple estimator of SNR, based on sums of squared receiver values and sums of their absolute values, can provide accurate estimates for even short blocklengths, and that use of this on-line procedure does not degrade performance appreciably relative to the known-SNR condition. We assume that binary PSK transmission of code bits is performed over an additive white Gaussian noise channel. Coherent demodulation is assumed along with perfect synchronization. Under this assumption, the received data can be represented by

q

rn =  Es + nn =  + nn

(1)

where nn is a Gaussian random variable having zero mean and variance 2 = N0=2, and the two-sided noise spectral density of the channel noise process is N0=2 W/Hz. In this notation, the symbol energy is related to the energy per bit by Es = Eb R, where R is the code rate, typically 1/2, but arbitrary. Hereafter, we use the term SNR to mean Eb =N0. As a numerical comment, it is known that coding schemes with R = 1=2 can achieve reliable communication provided Eb=N0 exceeds 0.18 dB, the channel capacity limit with antipodal modulation. For R = 1=3 codes, the corresponding limit is -0.50 dB. Turbo codes have been demonstrated, at least with blocklength of 65536 bits, to perform within 0.7 dB of these limits at error probability 10?5 . For lower error probability, the performance curves

are somewhat, sometimes called a \ oor".

2 SNR Sensitivity and Estimation The ratio = 2=2 = 2Es=N0 is a parameter of the MAP algorithm for each component decoder [9]. Over-stating has the qualitative eect of imbuing the measurements with more value than they deserve, while under-stating uses the prior information about bits in too strong a manner. In order to examine the sensitivity of turbo decoding to SNR mismatch, a series of 2

simulations were performed of log MAP turbo decoders with R=1/3 and several blocklengths at various true SNRs, while varying assessments of SNR from -6 dB to +6 dB relative to the true SNR. A 16-state (37,21) encoder was used. For each condition, Monte Carlo simulations were performed with stopping criterion of 500 bit errors after decoding and at least 10 frames. Due to time constraints, only 100 bit errors were achieved for SNRs of 2 and 2.5 dB. In Figure 1, error probability is plotted versus mismatch for a block size of N =420 bits and a 20x21 interleaver described in [10]. Notice the interesting qualitative eect of mismatch: 1 dB and perhaps 2 dB under-estimation is rather tolerable, but degradation becomes large for greater mismatch. Over-estimation of SNR is less detrimental than under-estimation, tolerating a mismatch of several dB without signi cant degradation. Obviously, when the true SNR is poor, e.g. below the capacity limit, mismatch does not matter|no decoding scheme can hope to be reliable in this region. On the other hand, for SNR in the region where iterative decoding provides impressive performance, the required accuracy is perhaps between -3 dB and +6 dB. Sensitivity has also been studied independently in [7] and [8], whose results seem to parallel those here. Though one could argue that the actual could be learned over a long period of time for a stationary channel condition, we sought an estimator of that could be self-contained on a packet level, anticipating SNR variations from packet-to-packet in a multi-user application. Speci cally, the parameter of interest is the ratio = 2Es =N0 , as opposed to separate estimates of the signal energy and noise variance. Our interest is in a blind algorithm which does not require the transmission of known training symbols. Training symbols would make the estimation problem more straightforward, but would also consume channel eciency. Thus, the problem is as follows: form an estimate of given a sequence of random variables of the form

q

rn =  Es + nn

where the true data polarity is unknown. 3

(2)

It is conceivable that one could formulate a ML estimator for the parameter , based on a block of observations. If tractable, this would lead to a rather cumbersome estimation formula however. Thus, we took a somewhat more heuristic approach to the estimation. Namely, we considered the mean of the square of rn and the mean absolute value of rn. It is not dicult to show that

h i

E rn2

and

= Es +  2

0 0s 11 q E 2 Es AA s E [jrn j] =  e? 22 + Es @erf @  22 : s

(3)

(4)

Then, E [rn2 ] (E [jrnj])2

= q

? 2Es2 e 2

+

 Es  1 + E2s q Es  q Es 2 = f 2 = f ( ) : 2

erf

(5)

22

We de ne the variable z to replace the ratio of expectations, z = E [rn2 ] =(E [jrn j])2. Thus the ratio of two simple statistical computations and the known function f ( ) provide a means to estimate . However, the complexity of the function prevents the determination of a closed-form solution for from the statistics. This diculty may be alleviated by rst evaluating the function on a pointwise basis over the range of 's of interest and then utilizing a simple polynomial function to approximate the relationship between the statistical ratio and . For SNRs of 0-6 dB, z was determined for several points. A simple second-order polynomial was found to approximate the relation with ,  ?34:0516z 2 + 65:9548z ? 23:6184

4

(6)

d ], dB SD[SNR d ], dB -3 dB Mismatches +3 dB Mismatches SNR, dB E[SNR 0.0 0.018 0.76 4 0 0.5 0.40 0.73 3 0 1.0 0.83 0.71 7 0 1.5 1.28 0.66 2 0 2.0 1.74 0.63 3 0 2.5 2.27 0.59 0 0 3.0 2.84 0.55 1 0 Table 1: Performance of the estimator for dierent values of true SNR While the polynomial approximation is suciently accurate, as evidenced by Figure 2, it must be noted that a table look-up is another valid method for determining an estimate of . In fact, Pietrobon [11] uses the same estimators and a table look-up approach. To develop an estimate for z, expectations are replaced with corresponding block averages, yielding z^ = r2=jrj2. Substitution into (6) then gives the desired ^, or, by simple d. conversion, SNR Our polynomial approximation was tested for a block size of 420 (1260 code symbols) d , were and several values of true SNR. The mean and variance of the SNR estimates, SNR determined over 20,000 trials. Particularly poor estimates of greater than 3 dB above or below the true SNR were also tallied. The results are shown in Table 1. Our estimates are biased due to two features of the estimator. First, the estimated is a convex \ transformation of the computed statistic z^ for the region of interest, and Jensen's inequality implies that the mean SNR estimate will be less than the true SNR. Also, some bias is due to the fact that deviations exist between the t of the second-order polynomial estimate and the actual function of the sample averages, as shown in Figure 2. However, despite the bias, very poor estimates are infrequent, as shown by the scarcity of misestimates of at least 3 dB. Note also that the variance of the estimates decreases for increasing SNR 5

because there is less variance in the computed statistic z^ for greater SNR values. For blocklengths of 420 and 4096, simulations were performed using the on-line estimator to provide for the decoding process. In these simulations, all of the received bits were utilized to determine the necessary statistics. Although some latency is added by the use of the entire block, smaller sample sets should provide similar results. The simulations involving on-line estimation were performed on the same sets of data and noise as another pair of simulations which allowed the decoder to use the true value. The performance curves produced by the dierent simulations are shown in Figure 3. Simulation results for the block sizes chosen show no perceptible degradation versus known-SNR decoder performance. This suggests a fraction of a block's data could be used to form estimates, thus lowering computation and latency.

References [1] C. Berrou, A. Glavieux, and P. Thitimajshima. \Near Shannon limit error-correcting coding and decoding: turbo coding". In International Conference on Communications, pages 1064{ 1070, 1993. [2] S. Benedetto and G. Montorsi. \Unveiling turbo codes: some results on parallel concatenated coding schemes". IEEE Transactions on Information Theory, 42(2):409{429, March 1996. [3] P. Robertson. \Illuminating the structure of code and decoder of parallel concatenated recursive systematic (turbo) codes". In Globecom, pages 1298{1303, 1994. [4] D. Divsalar and F. Pollara. \Turbo codes for PCS applications". In International Conference on Communications, pages 54{59, June 1995. [5] W. Ryan. \A turbo code tutorial". preprint: submitted to Globecom 1997. [6] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv. \Optimal decoding of linear codes for minimizing symbol error rate". IEEE Transactions on Information Theory, pages 284{287, March 1974. [7] M. Jordan and R. Michols. \The eects of channel characterization on turbo code performance". Proc. of Milcom 96. [8] M. Reed and J. Asenstorfer. \A novel variance estimator for turbo-code decoding". In International Conference on Telecommunications, pages 173{178, April 1997.

6

[9] J. Hagenauer, E. Oer, and L. Papke. \Iterative decoding of binary block and convolutional codes". IEEE Transactions on Information Theory, IT-42:429{445, March 1996. [10] W. Blackert, E. K. Hall, and S. G. Wilson. \Turbo code termination and interleaver conditions". Electronics Letters, 31(24):2082{2084, November 1995. [11] S. S. Pietrobon. \Implementation and performance of a turbo/map decoder". submitted to Int. J. Satellite Commun., Feb. 1997.

7

0

10

SNR = 0.5dB

−1

10

−2

BER

10

SNR = 1dB

−3

10

SNR = 1.5dB −4

10

SNR = 2dB −5

10

SNR = 2.5dB

−6

10

−6

−4

−2

0 Eb/No Offset

2

4

6

Figure 1: Error probability versus mismatch for several true SNR values.

8

1.55

1.5

Evaluated Ratio

1.45

1.4

1.35

1.3

1.25

1.2

1.15 −3

−2

−1

0

1

2 2Es/No

3

4

5

6

7

Figure 2: True relation between the statistical ratio and SNR (solid) versus polynomial approximation (dashed).

9

−1

10

solid −−−−− Known SNR dashed − − − Estimated SNR

−2

10

−3

N=420, R=1/3, Helical Interleaver

10

−4

BER

10

−5

10

N=4096, R=1/3, Random Block Interleaver

−6

10

−7

10

−8

10

0

0.5

1

1.5 Eb/No, dB

2

2.5

3

Figure 3: Performance curves for the turbo decoder using 1) the SNR determined by on-line estimation and 2) knowledge of the true SNR.

10