new maximum a posteriori joint channel estimation and equal- ization receivers with frequency-selective fast-fading channels. We conclude that pilot-aided PSK ...
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 2, FEBRUARY 2001
DPSK Versus Pilot-Aided PSK MAP Equalization for Fast-Fading Channels Linda M. Davis and Iain B. Collings
Abstract—This letter compares pilot-aided phase-shift keyed (PSK) and differential PSK (DPSK) modulation when using new maximum a posteriori joint channel estimation and equalization receivers with frequency-selective fast-fading channels. We conclude that pilot-aided PSK has superior bit-error rate performance in this case. However, at low signal-to-noise ratio, performance is similar, and DPSK is competitive due to reduced receiver complexity. Index Terms—Communication systems, equalizers, fading channels, MAP estimation.
In contrast to DPSK, our MAP equalizer for absolutely encoded PSK transmission [1] requires additional pilot symbols in order to resolve phase ambiguity, with a consequent penalty in power and bandwidth efficiencies. Our pilot-assisted MAP algorithm is fundamentally different to conventional pilot-assisted demodulation schemes [7], [8]. An important result is that the pilot symbol rate is dominated by the need for resolution of phase ambiguity and not the need to effectively sample the underlying channel response. II. EXPANDED MAP RECEIVER FOR DPSK
I. INTRODUCTION
R
ECENTLY we presented a maximum a posteriori (MAP) equalizer for joint channel estimation and equalization for frequency-selective fast-fading channels [1]. In this letter, we extend the concept to the case of differentially encoded transmission. This is motivated by the development of a MAP demodulator in [2] for a differentially encoded system employing constant amplitude modulation with a flat-fading channel. This in turn owes its roots to [3], and references therein. Here we present a differential MAP equalizer for frequency-selective fading channels. In addition, the transmitted symbols are not required to be constant amplitude for our equalizer. In this letter, we focus on the relative performances of differentially encoded phase-shift keyed (PSK) modulation and comparable pilot-aided absolutely encoded PSK modulation, both employing MAP equalization. To improve the performance of differentially detected (i.e., differential PSK [DPSK]) systems, detectors which extend the observation interval beyond two symbols have been developed (e.g., [4], [5]). The differential MAP equalizer we present in this letter uses this principle. The MAP algorithm is a symbol-by-symbol estimator which uses forward–backward processing taking into account all symbols in a block. As in [1], we expand the state space of the trellis for the purpose of joint channel estimation and equalization. Minimum mean-square-error (MMSE) estimators are used to provide a different channel estimate for each state in the trellis. This is different to per-survivor processing, which uses maximum-likelihood detection [6] and forms channel estimates based on surviving paths.
Consider the transmission of differentially encoded symbols over a frequency-selective (or frequency-flat) fast-fading communication channel. Assume that from the original message sequence , the differential encoder generates the -ary symbols to be transmitted. In this letter, we adopt a discrete equivalent model of the transmission system [9], [1]. The system with notation definitions is shown in Fig. 1. For transmitted symform a set of sufficient statistics bols, the received samples for the receiver, where (1) where The oversampled transmitted sequence is for integer, otherwise. is the is the oversampling rate where is the symbol period and sampling period. The MAP equalizer works at the symbol rate. Assuming for a moment that the channel is perfectly known at the receiver, states where the MAP trellis has . The variance of the additive white Gaussian noise where denotes comis is the two-sided spectral noise density plex conjugate, and represent the set of observations . The [9]. Let observation probability for transition from state at time to state at time is then
(2) Paper approved by W. E. Ryan, the Editor for Modulation, Coding, and Equalization of the IEEE Communications Society. Manuscript received January 25, 1999; revised November 10, 1999. This paper was presented in part at the 1999 IEEE Wireless Communications and Networking Conference (WCNC), New Orleans, LA, 1999. L. M. Davis is with Global Wireless Systems Research, Bell Laboratories, Lucent Technologies, Sydney, Australia. I. B. Collings is with the School of Electrical and Information Engineering, University of Sydney, Sydney, NSW 2006 Australia. Publisher Item Identifier S 0090-6778(01)01301-0.
denotes the -th state at time where channel model, and
represents the
(3) The tilde indicates a hypothesized value. Note that through (3), knowledge of the channel is required in (2).
0090–6778/01$10.00 © 2001 IEEE
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 2, FEBRUARY 2001
Fig. 1.
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Discrete equivalent differentially encoded transmission system with MAP equalizer.
Now, when the channel is not known at the receiver, the state space of the trellis may be expanded to encompass the lowpass nature of the fading to allow channel estimation using MMSE techniques [1]. When additional samples at the sample rate states. Using are used, the expanded trellis has can be estimated from previous linear MMSE estimation, observations and the hypotheses associated with the transition from state to state , using (4) and denotes transpose. where The vector of prediction coefficients is given by (5) where
and
are
hypothesized
versions
of , , are
, and respectively. The channel covariance matrices, and assumed to be known. in (2) is replaced by the prediction The noise variance, error variance
(6) . where Using the MMSE estimates (4) to determine the observation probabilities (2), the forward and backward recursions for the MAP algorithm are performed [10]. Without loss of optimality and with computational advantages, the MAP algorithm is usually implemented in the log-domain. Suboptimal variations may be employed to further reduce complexity [11]. The main modification required in the MAP algorithm for differential detection is in the choice of states for summing probabilities. Of course, for frequency-selective channels and for our will correspond to a particexpanded trellis, more than one . At time , the a posteriori probability for a -ary ular message symbol is given by
(7) are determined by the combination The states for which of the differential encoding rule and the hypotheses and for each state.
Fig. 2. BER versus SNR, binary PSK, p = 3; f rates 1:4, 1:10, and 1:100, and perfect CSI.
T
= 0:05, pilot symbol
Note that as a result of the symmetry of the constellation for DPSK, the state space of the MAP equalizer can be reduced, with the benefit of reduced computations. By combining states which represent the same message sequence, only states are required. III. DPSK VERSUS PILOT-AIDED PSK The insertion of pilot symbols for phase reference in PSK incurs a signal-to-noise ratio (SNR) penalty of dB
(8)
is the ratio of pilot to data symbols transmitted. where Pilot symbols also result in a reduction in the effective bandwidth being used to send the message. An important result we have observed for MAP equalization is that (when the reduction in bandwidth is not an issue), the appropriate rate for inserting pilot symbols is dominated by the need for phase resolution and not the need to effectively sample the fading channel response. Several channel estimates exist (one for each state), and the pilot symbol serves to eliminate some of the possible transitions (and therefore states and paths) in the trellis. This is in contrast to conventional pilot symbol-aided modulation-demodulation (PSAM) schemes [7], [8], where a single channel estimate is formed by interpolating between pilot symbols. For PSAM, the Nyquist sampling criterion must be satisfied; the channel function is effectively being sampled. There is an optimal choice of pilot symbol rate for a given scenario. For the expanded trellis MAP equalizer this is dominated by the SNR. Fig. 2 shows the bit-error rate (BER) versus SNR of
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Fig. 3. BER performance versus E =N , binary DPSK and pilot-aided PSK (1:4), p = 3; f T = 0:05, flat-fading channel.
the information symbols for a flat-fading system with normaland pilot rates of 1:4, 1:10, and ized Doppler , was used in both cases. 1:100. A short predictor length, Note that unlike PSAM (where the pilot insertion rate for this example would need to be higher than 1:9 according to the sampling theorem), there is a graceful degradation in performance at low SNR as the pilot symbol rate is decreased. As the SNR increases, the pilot symbol rate may be reduced, thus the issue of decreased bandwidth is only critical at low SNR. Fig. 3 shows that even with a relatively high pilot symbol rate, the BER performance of the PSK system is better than that of the corresponding DPSK system. At low SNR, the difference in performance is almost negligible, but at high SNR, an advantage of approximately 2 dB is achieved by the PSK system. Taking into account the pilot symbol overhead in SNR given by (8), dB for ), this 2-dB (in this case advantage is comparable to the advantage of PSK over DPSK for (theoretical) systems using perfect channel state information (CSI) as shown in the figure. An important feature of the MAP equalizers presented here and in [1] is their ability to handle frequency-selective fast-fading channels. We demonstrate this point in Fig. 4 for a -spaced equal power two path channel with normalized . The received signal was sampled fading rate . Ordinary DPSK and PSAM are not suitable for at such severe intersymbol interference [8].
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 2, FEBRUARY 2001
Fig. 4. BER performance versus E 0:05, two-path (T spaced) channel.
=N
, binary PSK,
p
= 10; f
T
=
REFERENCES [1] L. M. Davis, I. B. Collings, and P. Hoeher, “Joint MAP equalization and channel estimation for frequency-selective fast-fading channels,” in Proc. IEEE GLOBECOM Communication Theory Mini-Conf., Sydney, Australia, Nov. 1998, pp. 53–58. [2] P. Hoeher and J. Lodge, “Turbo DPSK: Iterative differential PSK demodulation and channel decoding,” IEEE Trans. Commun., vol. 47, pp. 837–843, June 1999. [3] M. J. Gertsman and J. H. Lodge, “Symbol-by-symbol MAP demodulation of CPM and PSK signals on Rayleigh flat-fading channels,” IEEE Trans. Commun., vol. 45, pp. 788–799, July 1997. [4] D. Makrakis and K. Feher, “Optimal noncoherent detection of PSK signals,” Electron. Lett., vol. 26, pp. 398–400, Mar. 1990. [5] D. Divsalar and M. K. Simon, “Multiple symbol differential detection of MPSK,” IEEE Trans. Commun., vol. 38, pp. 300–308, Mar. 1990. [6] R. Raheli, A. Polydoros, and C.-K. Tzou, “Per-survivor processing: A general approach to MLSE in uncertain environments,” IEEE Trans. Commun., vol. 43, pp. 354–364, Feb. 1995. [7] J. K. Cavers, “An analysis of pilot symbol assisted modulation for Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 40, pp. 686–693, Nov. 1991. [8] , “Pilot symbol assisted modulation and differential detection in fading and delay spread,” IEEE Trans. Commun., vol. 63, pp. 561–580, Apr. 1995. [9] J. Proakis, Digital Communications, 3rd ed. New York: McGraw-Hill, 1995. [10] L. R. Bahl, J. Cocke, F. Jelenik, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284–287, Mar. 1974. [11] P. Roberston, P. Hoeher, and E. Villebrun, “Optimal and suboptimal maximum a posteriori algorithms suitable for turbo decoding,” Eur. Trans. Telecommun., pp. 119–125, Mar. 1997.
