226 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 2, FEBRUARY 2001

DPSK Versus Pilot-Aided PSK MAP Equalization for Fast-Fading ChannelsLinda M. Davis and Iain B. Collings

Abstract—This letter compares pilot-aided phase-shift keyed(PSK) and differential PSK (DPSK) modulation when usingnew maximum a posteriori joint channel estimation and equal-ization receivers with frequency-selective fast-fading channels.We conclude that pilot-aided PSK has superior bit-error rateperformance in this case. However, at low signal-to-noise ratio,performance is similar, and DPSK is competitive due to reducedreceiver complexity.

Index Terms—Communication systems, equalizers, fading chan-nels, MAP estimation.

I. INTRODUCTION

RECENTLY we presented a maximuma posteriori(MAP)equalizer forjoint channel estimation and equalization

for frequency-selective fast-fading channels [1]. In this letter,we extend the concept to the case of differentially encodedtransmission. This is motivated by the development of aMAP demodulator in [2] for a differentially encoded systememploying constant amplitude modulation with a flat-fadingchannel. This in turn owes its roots to [3], and referencestherein. Here we present a differential MAP equalizer for fre-quency-selective fading channels. In addition, the transmittedsymbols are not required to be constant amplitude for ourequalizer. In this letter, we focus on the relative performancesof differentially encoded phase-shift keyed (PSK) modulationand comparable pilot-aided absolutely encoded PSK modula-tion, both employing MAP equalization.

To improve the performance of differentially detected (i.e.,differential PSK [DPSK]) systems, detectors which extend theobservation interval beyond two symbols have been developed(e.g., [4], [5]). The differential MAP equalizer we presentin this letter uses this principle. The MAP algorithm is asymbol-by-symbol estimator which uses forward–backwardprocessing taking into account all symbols in a block.

As in [1], we expand the state space of the trellis for the pur-pose of joint channel estimation and equalization. Minimummean-square-error (MMSE) estimators are used to provide adifferent channel estimate for each state in the trellis. This isdifferent to per-survivor processing, which uses maximum-like-lihood detection [6] and forms channel estimates based on sur-viving paths.

Paper approved by W. E. Ryan, the Editor for Modulation, Coding, and Equal-ization of the IEEE Communications Society. Manuscript received January 25,1999; revised November 10, 1999. This paper was presented in part at the 1999IEEE Wireless Communications and Networking Conference (WCNC), NewOrleans, 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.

In contrast to DPSK, our MAP equalizer forabsolutelyen-coded PSK transmission [1] requires additional pilot symbolsin order to resolve phase ambiguity, with a consequent penaltyin power and bandwidth efficiencies. Our pilot-assisted MAPalgorithm is fundamentally different to conventional pilot-as-sisted demodulation schemes [7], [8]. An important result is thatthe pilot symbol rate is dominated by the need for resolution ofphase ambiguity and not the need to effectively sample the un-derlying channel response.

II. EXPANDED MAP RECEIVER FORDPSK

Consider the transmission of differentially encoded symbolsover a frequency-selective (or frequency-flat) fast-fading com-munication channel. Assume that from the original message se-quence , the differential encoder generates the-ary symbols

to be transmitted. In this letter, we adopt a discrete equiva-lent model of the transmission system [9], [1]. The system withnotation definitions is shown in Fig. 1. For transmitted sym-bols, the received samples form a set of sufficient statisticsfor the receiver, where

(1)

The oversampled transmitted sequence is wherefor integer, otherwise. is theoversampling rate where is the symbol period and is thesampling period.

The MAP equalizer works at the symbol rate. Assuming fora moment that the channel is perfectly known at the receiver,the MAP trellis has states where

. The variance of the additive white Gaussian noise

is where denotes com-plex conjugate, and is the two-sided spectral noise density[9]. Let represent the set of observations . Theobservation probability for transition from stateat timeto state at time is then

(2)

where denotes the-th state at time represents thechannel model, and

(3)

The tilde indicates a hypothesized value. Note that through (3),knowledge of the channel is required in (2).

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 2, FEBRUARY 2001 227

Fig. 1. Discrete equivalent differentially encoded transmission system with MAP equalizer.

Now, when the channel is not known at the receiver, the statespace of the trellis may beexpandedto encompass the lowpassnature of the fading to allow channel estimation using MMSEtechniques [1]. When additional samples at the sample rateare used, the expanded trellis has states. Usinglinear MMSE estimation, can be estimated from previousobservations and the hypotheses associated with the transitionfrom state to state , using

(4)

where and denotes transpose.The vector of prediction coefficients is given by

(5)

where and are hypothesized versions of, and ,

respectively. The channel covariance matrices,and , areassumed to be known.

The noise variance, in (2) is replaced by the predictionerror variance

(6)

where .Using the MMSE estimates (4) to determine the observation

probabilities (2), the forward and backward recursions for theMAP algorithm are performed [10]. Without loss of optimalityand with computational advantages, the MAP algorithm is usu-ally implemented in the log-domain. Suboptimal variations maybe employed to further reduce complexity [11].

The main modification required in the MAP algorithm for dif-ferential detection is in the choice of states for summing proba-bilities. Of course, for frequency-selective channels and for ourexpanded trellis, more than one will correspond to a partic-ular . At time , thea posterioriprobability for a -arymessage symbol is given by

(7)

The states for which are determined by the combinationof the differential encoding rule and the hypothesesandfor each state.

Fig. 2. BER versus SNR, binary PSK,p = 3; f T = 0:05, pilot symbolrates 1:4, 1:10, and 1:100, and perfect CSI.

Note that as a result of the symmetry of the constellation forDPSK, the state space of the MAP equalizer can be reduced,with the benefit of reduced computations. By combiningstates which represent the same message sequence, only

states are required.

III. DPSK VERSUSPILOT-AIDED PSK

The insertion of pilot symbols for phase reference in PSKincurs a signal-to-noise ratio (SNR) penalty of

dB (8)

where is the ratio of pilot to data symbols transmitted.Pilot symbols also result in a reduction in the effective band-width being used to send the message.

An important result we have observed for MAP equaliza-tion is that (when the reduction in bandwidth is not an issue),the appropriate rate for inserting pilot symbols is dominated bythe need for phase resolution and not the need to effectivelysample the fading channel response. Several channel estimatesexist (one for each state), and the pilot symbol serves to elimi-nate some of the possible transitions (and therefore states andpaths) in the trellis. This is in contrast to conventional pilotsymbol-aided modulation-demodulation (PSAM) schemes [7],[8], where a single channel estimate is formed byinterpolatingbetween pilot symbols. For PSAM, the Nyquist sampling crite-rion must be satisfied; the channel function is effectively beingsampled.

There is an optimal choice of pilot symbol rate for a given sce-nario. For the expanded trellis MAP equalizer this is dominatedby the SNR. Fig. 2 shows the bit-error rate (BER) versus SNR of

228 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 2, FEBRUARY 2001

Fig. 3. BER performance versusE =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 normal-ized Doppler and pilot rates of 1:4, 1:10, and1:100. A short predictor length, , was used in both cases.Note that unlike PSAM (where the pilot insertion rate for thisexample would need to be higher than 1:9 according to the sam-pling theorem), there is a graceful degradation in performanceat low SNR as the pilot symbol rate is decreased. As the SNRincreases, the pilot symbol rate may be reduced, thus the issueof 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 ofthe corresponding DPSK system. At low SNR, the difference inperformance is almost negligible, but at high SNR, an advantageof approximately 2 dB is achieved by the PSK system. Takinginto account the pilot symbol overhead in SNR given by (8),(in this case dB for ), this 2-dBadvantage is comparable to the advantage of PSK over DPSKfor (theoretical) systems using perfect channel state information(CSI) as shown in the figure.

An important feature of the MAP equalizers presentedhere and in [1] is their ability to handle frequency-selectivefast-fading channels. We demonstrate this point in Fig. 4 fora -spaced equal power two path channel with normalizedfading rate . The received signal was sampledat . Ordinary DPSK and PSAM are not suitable forsuch severe intersymbol interference [8].

Fig. 4. BER performance versusE =N , binary PSK,p = 10; f T =

0:05, two-path (T spaced) channel.

REFERENCES

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[10] L. R. Bahl, J. Cocke, F. Jelenik, and J. Raviv, “Optimal decoding of linearcodes for minimizing symbol error rate,”IEEE Trans. Inform. Theory,vol. IT-20, pp. 284–287, Mar. 1974.

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