A systematic approach to noncoherent detection for DPSK modulation in single-user correlated diversi

| diversity with discipline CISS 1997, Johns Hopkins University, Baltimore, MD, March 19{21 A Systematic Approach to Noncoherent Detection for DPSK Modulation in Single-User Correlated Diversity Rayleigh Fading Channels with Applications to Post-Combining Decorrelative Multiuser Detection The author is with the Department of Electrical and Computer Engineering at the University of Colorado in Boulder, CO 80309. This work was supported in part by NSF grant NCR9406069. The results of this paper were rst mentioned in a tutorial talk on Noncoherent Multiuser Detection by the author in the IEEE Communication Theory Workshop in Dustin, Florida, April, 1996. A correlated diversity Rayleigh fading channel is de ned as one that admits inter-diversity branch correlation in both the multiplicative fading as well as the additive noise processes and models a variety of diversity communication systems operating over Rayleigh fading channels. In this paper we are interested in applications where receiver simplicity is paramount. The modulation method of choice for such applications is di erential phase shift keying so that noncoherent detection is possible at the receiver without having to implement complex channel estimation algorithms. Previous work in noncoherent detection for DPSK modulation is con ned to ad-hoc strategies of the equal-gain combining type. We introduce a systematic approach for dealing with noncoherent detection for DPSK modulation in correlated diversity Rayleigh fading channels. For slowly fading channels where the fading parameters in the diversity branches can be regarded as being essentially identical over two successive symbol intervals, the generalized likelihood ratio test (GLRT) is shown to yield an excellent solution. It has the advantage of not needing the knowledge of the statistics of the fading processes. A minimum error probability (MEP) detector is also derived that is applicable to slow as well as fast fading channels. This MEP detector requires a limited knowledge of the statistics of the fading processes. Exact bit error rates are obtained for the GLRT and the MEP detectors. A comparative analysis with a recently proposed ad-hoc decorrelating equal gain combiner shows signi cant performance advantages for the GLRT and MEP detectors. Applications to post-combining decorrelative multiuser detection are also considered. In this paper we are concerned with the correlated diversity Rayleigh fading (CDRF) channel. We de ne the CDRF channel as one that admits inter-diversity branch correlation in both the multiplicative fading as well as the additive noise processes. The CDRF channel models a variety of diversity communication systems including multipath diversity, time or frequency diversity, receiver antenna diversity etc. Inter-diversity branch correlation in the fading processes may result in the case of receiver antenna diversity systems for instance, when space limitations (such as on a hand-held telephone) dictate that the antennas be spaced closer to each other than what is required for independent fading. In time or frequency diversity systems, it may arise due to stringent delay or bandwidth constraints, respectively. The inter-diversity branch correlation in the additive noise may result in the case of a multipath diversity system because of bandwidth constraints. When the signalling waveform has a bandwidth that is su ciently large compared to the coherence bandwidth of the channel to result in multiple resolvable paths at the receiver, but however, because the waveform may still be constrained in bandwidth and/or due to the distortion introduced by the channel, the various time translates of the waveform that arrive at the receiver may not be mutually orthogonal. In general, it can be said that the correlated diversity channel results when systems employ . No resource (space, time, bandwidth) can be used as if it were freely available. In very slowly (static) fading channels where the coherence time is on the order of a hundred times the symbol duration, the fading coe cients remain constant over sufciently many symbols to allow for their accurate estimation at the receiver, thereby enabling coherent modulation and detection methods. For such slowly fading, perfectly estimable channels, and under the assumptions that ISI is negligible, and that the time translates of the received signalling waveform are orthogonal, and the fading coe cients in the various paths fade independently, the maximal ratio combiner or the RAKE receiver is optimal in that it minimizes error probability [7] [8]. The extension of the maximal ratio combiner to the correlated diversity channel is simple enough and quite well understood (cf. [5] [8]). The coherent RAKE receiver is therefore stock in the trade of diversity communications engineering. When one relaxes the assumption of the availability of perfect knowledge of the fading coe cients, one of several approaches can be adopted depending on given complexity and performance speci cations. The dynamic evolution of the fading channel states can be modeled by assuming the channel state vector to be a Gauss-Markov process. In the detection of a symbol in the time interval, the conditional mean estimate based on past observations of the matched lter outputs can be obtained by using a decision-directed Kalman lter which delivers not only the channel state estimate but the associated error covariance matrix that contains information regarding the con dence level that one can attribute to the estimates (cf. [12]). The detector then uses the channel state estimate as if it were the correct value but it also optimally incorporates the error covariance information in making the decision on the symbol. This approach has its origins at least as far back as 1969 in Kailath's generalized likelihood formula for the detection of rather general random