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On Optimal Decoding for Block Source{Channel Coding over Rayleigh Fading Waveform Channels

Mikael Skoglund Internal Report April 1998 Published in Proc. IEEE International Symposium on Information Theory '98 MIT (Cambridge, MA) 1998, p. 11

IR-S3-SB-9831

ROYAL INSTITUTE OF TECHNOLOGY

Department of Signals, Sensors & Systems Signal Processing S-100 44 STOCKHOLM

KUNGL TEKNISKA H?GSKOLAN

Institutionen f?r Signaler, Sensorer & System Signalbehandling 100 44 STOCKHOLM

ISIT 1998, Cambridge, MA, USA, August 16 { August 21

On Optimal Decoding for Block Source{Channel Coding over Rayleigh Fading Waveform Channels

Dept. of Signals, Sensors & Systems Royal Institute of Technology, S-100 44 Stockholm, Sweden Email: skoglund@s3.kth.se

Abstract | We consider optimal soft decoding for block coding over time-continuous channels with fading and AWGN. The decoder maps the received waveform directly into a source vector estimate, without using intermediate detection or channel decoding.

Mikael Skoglund

Concerning the computation of Li , for di erent values of i, it is straightforward to see that l (i) = l and that l (t; i) can be factored as l (t; i) = gi (t) l(t), where l and l do

I. Introduction Consider a communication system conveying a source vector, X 2 Rd , drawn from a stationary source. The system has three parts: the encoder, the channel and the decoder. The encoder, ", maps X into an integer I = "(X) 2 f0; : : : ; N ? 1g, where N = 2L . Let P (i) = Pr(I = i) and c(i) = E XjI = i]. The bits bm (i) 2 f 1g of the index i = "(x) are transmitted over a channel with frequency- at Rayleigh fading and AWGN. When I = i, the received waveform can be modeled as Y (t) = si (t) + W (t), for t 2P , 0; L T ]. Here si (t) = gi (t)RefA(t)ej!c tg and gi (t) = L bm (i)p(t ? mT ), where m=1 A(t) = A1 (t)+jA2 (t) is a zero-mean stationary complex Gaussian process, p(t) is a rectangular pulse of length T , and W (t) is AWGN of spectral density N0 =2. It holds that A1 (t) and A2 (t) are independent, and E A1 (t)A1(u)] = E A2 (t)A2 (u)] = (t ? u), where ( ) = J0 (2 fm ) (Rayleigh fading; fm is the maximum Doppler frequency). Let Y = fY (t) : t 2 g. Then, the decoder maps the observation Y = y into a source vector ^ ^ ^ estimate X = X(y). Note that X(y) is a mapping \directly" ^ ^ from the waveform, y, into an estimate X. Hence, X(y) is soft in the sense that no decisions are taken, contrasting tra^ ditional decision-based decoding, where X is chosen from a nite codebook. Soft decoding for time-discrete channels has been considered before in for example 1{3]. II. Optimal Decoding ^ We adopt the mean-square error criterion E jjX ? Xjj2 . Hence, the optimal decoder, for a given encoder, is the minimum ^ mean-square error (MMSE) estimator X(y) = E XjY = y]. This decoder can be implemented as PN ?1 i=0 ^ (1) Figure 1: SNR dB] vs. CSNR dB]. Solid : MMSE; Dashed : MAP. X(y) = E XjY = y] = PN ?c(i)P (i)Li (y) 1 j=0 P (j )Lj (y ) Upper pair of lines: fm = 50 Hz, lower pair: fm = 100 Hz. ^ where Li (Y ) is the likelihood ratio for optimal testing, based Fig. 1 shows SNR, E jjXjj2 =E jjX ? Xjj2 , versus channel-SNR R E s2 (t)]dt, for fast fading at two di erent on Y , of the hypothesis I = i with respect to signal absence; (CSNR), 2=N0 i Y (t) = W (t) (c.f., 4]). In (1), P (i) and c(i) depend only on fm , and with T = 2 ms. The MMSE decoder performs betthe source and the given encoder. However, the computation ter than decision-based decoding at low CSNRs, and the two of the likelihood ratio functional, Li (y), from the received data schemes perform equal at high CSNRs, since at a high CSNRs Y = y needs some extra elaboration, and is considered next. the MAP-estimate dominates the weighted sum of (1). Let l (i) and l (t; i) be the eigenvalues and eigenfunctions References of the kernel E si (t)si (u)] = gi (t)gi(u) (t ? R ) cos !c (t ? u)] 1] V. Vaishampayan and N. Farvardin, \Joint design of block u 2 (c.f 4] p. 274). Then, letting Yl (i) = Y (t) l (t; i)dt, on source codes and modulation signal sets," IEEE Transactions on Information Theory, vol. 38, no. 4, pp. 1230{1248, July 1992. Li (Y ) can be be computed as (c.f. 4] p. 304) 2] F. H. Liu, P. Ho, and V. Cuperman, \Joint source and channel "Y ! 1 N =2 # 1 coding using a non-linear receiver," in Proc. IEEE Interna1 X l Yl2 (i) : (2) 0 tional Conference on Communications, Geneva, Switzerland, Li (Y ) = exp N 1993, pp. 1502{1507. 0 l=1 l + N0 =2 k=1 k + N0 =2

10 8 6

not depend on i, and are eigenvalues and eigenfunctions to (t ? u) cos !c (t ? u)]. Consequently, the optimal decoder can be implemented as: Initiation : Estimate P (i) and c(i); Solve for l and l (t), and set l (i) = l and l (t; i) = gi (t) l(t); Use : Given Y = y, calculate yl (i) and employ (2) to compute ^ Li (y); Use P (i), c(i) and Li (y) in (1) to nd X(y). III. Simulation Here we investigate the performance of optimal decoding. As reference, we employ decision-based maximum a posteriori ^ ^ ^ (MAP) decoding; XMAP = c(I ), I = arg maxj fP (j )Lj (y)g. Note that because of the uniform phase of the fading, the decoder needs to use the redundancy contained in fP (i)g to distinguish, e.g., between an index, i, and its complement N ? 1 ? i. The optimal approach to introduce such redundancy would be to co-optimize the encoder with the decoder. The encoder would then become a combined source{channel encoder providing redundancy both for error protection and phase recovery, in an optimal fashion. Here, however, we have chosen to investigate a simpler scheme, since the present focus is on the decoding: The encoder has L = 6 and is de ned by the encoder of a 5-bit vector quantizer (trained for, and used on, blocks of d = 6 from a Gauss-Markov source of correlation 0.9) concatenated with a single check-bit channel encoder that puts P (i) = 0 for i 2 fN=2; : : : ; N ? 1g.

SNR [dB]

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45 50 CSNR [dB]

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3] M. Skoglund and P. Hedelin, \Hadamard-based soft decoding for vector quantization over noisy channels," IEEE Transactions on Information Theory, to appear. 4] H. V. Poor, An Introduction to Signal Detection and Estimation, Springer-Verlag, 2nd edition, 1994.

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