On Decision-Feedback Detection of Differential Space–Time Modulation in Continuous Fading

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 10, OCTOBER 20041613On Decision-Feedback Detection of Differential Space–Time Modulation in Continuous FadingCong Ling, Student Member, IEEE, Kwok H. Li, Senior Member, IEEE, and Alex C. Kot, Senior Member, IEEEAbstract—We show that linear prediction (LP)-based decisionfeedback detection (DFD) for nondiagonal differential space–time modulation (DSTM) may suffer from a severe performance degradation in continuously fading channels. DSTM constellations that incur no degradation in LP-DFD are identified as those with a diagonal generator. To cater to other constellations, we propose a low-complexity DFD scheme by inserting decision-feedback symbols into the metric of multiple-symbol differential detection. Index Terms—Continuous fading, decision-feedback detection (DFD), differential space–time modulation (DSTM), linear prediction (LP), multiple-symbol differential detection (MSDD).no degradation to diagonal-generator constellations (defined in Section II). To cater to others, we shall propose multiplesymbol differential detection (MSDD)-based DFD (abbreviated as MS-DFD henceforth) as a remedy, by introducing previously detected symbols into the MSDD metric. An extension of this idea to the sequence-detection regime is made in [8]. Nonetheless, it is in the DFD scheme that the improvement is especially pronounced. II. DSTM SYSTEM MODEL transmit antennas and receive antennas, In DSTM with matrices signals are transmitted as whose row indexes represent different antennas, and column indexes represent different time instants. The matrices are properly normalized so that the average power of each column is one. The generator constellation of rate- (bits/channel matrices use) unitary DSTM is comprised of , where , with the -by- identity matrix [2], [4]. Every bits to be trans, mitted at time index are first mapped to matrix and then differentially encoded as (1) where is an initially transmitted unitary matrix. The set of matrices is called the transmission constellation in this letter, as distinct from . DSTM constellations can be classified as diagonal or nondiagonal. The orthogonal design [1] and most group-theoretic constructions [2], [3] yield nondiagonal generator constellations. Diagonal generator constellations are obtained in [3] and [4] by restricting the group to be commutative. We assume a mobile radio channel subject to time-selective fading in accordance with the Jakes model [9]. The , representing the channel between fading process and receive antenna , is assumed to be transmit antenna and spatially independent. The complex normal autocorrelation of the symbol-rate-sampled fading process , where is the zeis given by is the roth-order Bessel function of the first kind, and for normalized Doppler frequency. Likewise, the noises are assumed to be independent across both distributed. The time and space, and are identically average bit signal-to-noise ratio (SNR) at each receive antenna , accordingly. is be the th column of be the -byLet matrix of channel coefficients seen by , and be an -by- noise matrix. The received signals are given by (2)I. INTRODUCTIONSPACE–TIME modulation allows noncoherent detection versus most coherent schemes of space–time coding. In general, it assumes a channel model of block fading; the result for continuous fading is rather limited. This has motivated the development of differential space–time modulation (DSTM) [1]–[4]. Since the standard differential detection suffers from an irreducible error floor as the fading gets faster, linear prediction (LP)-based decision-feedback detection (DFD) has been widely used (see, e.g., [5]–[7]). Up to now, either diagonal constellations or a compound piecewise-constant/changing fading model is assumed in derivation and analysis of LP-DFD (i.e., time variation of the fading process during a DSTM symbol is ignored). Though a diagonal structure certainly facilitates signal design and reception, nondiagonal constellations are generally superior in performance; they are also more desirable in terms of the linear range of transmitting amplifiers [4]. Meanwhile, we know the physical channel in reality fades continuously. Therefore, in this letter, we scrutinize the behavior of LP-DFD for (mainly) nondiagonal DSTM constellations in continuous fading. We shall show that, in contrast to differential phase-shift keying (DPSK), LP-DFD may be mismatched to nondiagonal DSTM. We shall also demonstrate that, interestingly, it incursPaper approved by R. Schober, the Editor for Detection, Equalization, and MIMO of the IEEE Communications Society. Manuscript received September 15, 2003; revised December 12, 2003; April 13, 2004; and April 19, 2004. This work was supported in part by the Singapore Millennium Foundation. This paper was presented in part at the IEEE International Conference on Communications, Anchorage, AK, May 2003, and published in part for the International Conference on Acoustics, Speech, and Signal Processing, Hong Kong, China, April 2003. C. Ling was with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798. He is now with the School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, MSW 2308, Australia (e-mail: cling@). K. H. Ling and A. C. Kot are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: ekhli@.sg; eackot@.sg). Digital Object Identifier 10.1109/TCOMM.2004.8364350090-6778/04$20.00 © 2004 IEEE1614IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 10, OCTOBER 2004where the as-by-matrixis obtained by stacking(3) and is a stretched version of -bysquare, but has dimension , which is no longerfor Bearing in mind the differential rule square , we cannot always guarantee (7) because of the . For example, one can easily check the stretching effect of invalidity of the above relation for Alamouti’s code or Hughes’ rate-1 group constellation [2] with (8) It means that we will probably lose the most fundamental property of differential schemes if nondiagonal constellations are transmitted over continuously changing channels! This problem is peculiar to multiantenna differential modulation, in the sense that it never shows up in DPSK, because (7) is always valid for one-dimensional signals (no stretch of a scalar exists). Interestingly, there do exist exceptions in nondiagonal transmission constellations satisfying (7). This is based on the following result, whose proof is given in [10]. , is valid Proposition: The relation if and only if the generator constellation is diagonal. Note that the diagonality of does not necessarily impose the diagonality of . Diagonal can be modified into nondiagonal by choosing a nondiagonal initial matrix . For example, we may choose in (8) for two-antenna DSTM. Generally, to distribute signal powers evenly to transmit antennas, we may use the discrete Fourier transform (DFT) to construct the initial .1 In this way, diagonal generator matrix constellations can be employed for nondiagonal transmission, yet permit the use of LP-DFD in continuous fading. We caution the use of another way to get nondiagonal transmission constellations, which makes a transformation of diag, where is a unitary matrix onal signals as [4]. This transformation has no influence on the performance of LP-DFD in piecewise-constant channels, but it does affect its performance in continuously fading channels. IV. MS-DFD RECEIVER Since the success of LP-DFD is limited to diagonal generator constellations in continuous fading, in this section, we combine DFD and MSDD to obtain a low-complexity MS-DFD receiver. Basically, we feed decisions back into the MSDD metric. Stacking the variables involved in MSDD for a block of symbols yields the notations. . .. . ..... . .(4)III. LIMITATION OF LP-DFD By embedding a linear predictor, LP-DFD makes use of previous observations [5], [6](5) where for denotes the matrix trace, is the prediction order, are predictor taps, andreflects the feedback of previously detected symbols. For diagonal transmission constellations, the strategy (5) is known to work well, provided that the taps are derived from the Wiener–Hopf equations. It is hoped that LP-DFD would not incur much degradation for nondiagonal . In essence, LP-DFD removes past modulation on received signals so that a linear predictor can be applied to the recovered fading-plus-noise samples. More precisely, if decisions are perin (5) can be expressed fectly correct, the argument inside as(9) (6) To suppress modulation prior to time index , we must have the relation With these, we have the block-signal model . MSDD makes a joint decision in favor of that maximizes the conditional probability density function [5]which basically requires that (7)(10)1This will probably increase the scalar constellation size. A normalized Hadamard matrix may be used instead if is a power of 2.MIEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 10, OCTOBER 20041615TABLE I DETECTOR COMPLEXITY COMPARIONalso has the same diag[11, p. 18], we can show that onal/antidiagonal partitioned structure. Therefore, and contains many null entries, and simplification similar to [3, eq. (20)] can be made. V. ERROR PROBABILITY We adopt a genie-aided (i.e., correct feedback) analysis for DFD. Since matrix inversion is involved in MS-DFD, an analytic investigation of error propagation appears rather difficult. Accordingly, its impact will be assessed by simulation in the next section. We start with the evaluation of the pairwise error probability (PEP) in MS-DFD. It turns out that when is nondiagonal, the uniform error-probability criterion is generally not satisfied in continuous fading. Therefore, it is necessary to conwhen dition on the event evaluating the PEP. is the sent matrix at time index . The MS-DFD Suppose if will make a wrong decisionwhere is the autocorrelation matrix for (in the single receive-antenna scenario) when is transmitted. can be expressed as [8] (11) where -bywhose denotes the Hadamard product, is the autocorrelation matrix of a fading process, th entry is given by , and(15) where a wrong decision occurs if . When , (16) Then we introduce detected symbols into , and include the trial symbol to define Defining , we may express in(12)a Hermitian quadratic form . Then the characteristic function of can be determined in terms of the eigenvalues of as [12](13) (17) After taking the logarithm of (10) and discarding an irrelevant term, we finally obtain the symbol-by-symbol MS-DFD rule with Since the fading processes are statistically independent between is simply receive antennas, the characteristic function of given by raised to the th power. Consequently, the conditional PEP in the presence of receive antennas is given by(14) The MS-DFD structure, in general, does not lead to an LP receiver for nondiagonal constellations. It is only equivalent to LP-DFD for diagonal generator constellations. In Table I, we compare the computational complexity per DSTM symbol, evaluated by the number of arithmetic operations required, of various detectors in terms of . Though MS-DFD is computationally simpler than MSDD, it is necessarily more complex than LP-DFD, because it caters for nondiagonal generator constellations. We remark that further simplification of MS-DFD is possible for an important class of nondiagonal constellations, namely, Hughes’ dicyclic codes [3], which are essentially the only nondiagonal full-diversity group codes. Since the dicyclic code matrices are either diagonal or antidiagonal, the correlation matrix has a partition in diagonal/antidiagonal submatrices. By exploiting the property of the inverse of a partitioned matrix(18) which can be calculated efficiently [12]. Then the union bound is invoked to give an upper bound on the error probability. Finally, an average over all the possible codewords is performed, as the uniform error-probability criterion is not fulfilled. This gives the union bound of the bit-error rate (BER) for MS-DFD genie (19) where represents the Hamming distance between two source bit sequences associated with and .1616IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 10, OCTOBER 2004Fig. 1. Performance of DSTM based on the BPSK Alamouti code with LP-DFD and MS-DFD for f T = 0:03; V = 3, and N = 1.Fig. 2. Performance of three-antenna DSTM with LP-DFD and MS-DFD for f T = 0:03; V = 3, and N = 1.The PEP of MSDD can be derived in virtually the same way replaced as the foregoing analysis, with , which is defined as the th element by . Note that this set contains elements. We of the set have the union bound(20) where represents the Hamming distance between two and of length source bit sequences associated with . The range of the inner sum may be narrowed down to major error events, which yields a truncated union bound. It dominant error events for is accepted that there are DPSK signaling in additive white Gaussian noise (AWGN) [13] or fading channels [14]. This approach is adapted to DSTM in this letter by identifying the nearest neighbors to . LP-DFD can be interpreted as mismatched MS-DFD. Thus, its performance can be easily evaluated as MS-DFD [10]. The only difference is that error propagation may be more pronounced, because the error rate of a mismatched detector is expected to be higher. VI. RESULTS AND CONCLUSION Numerical results on the performance of nondiagonal DSTM with MS-DFD and LP-DFD are reported herein. We focus on rate-1 DSTM using one receive antenna. The observation interval of both MSDD and DFD spans four matrix symbols, i.e., . The actual BER of DFD schemes is approximated by doubling that obtained in the genie-aided analysis. Fig. 1 displays the union bounds of DSTM based on the BPSK Alamouti code over a fast-fading channel with , with various detectors. Clearly, LP-DFD suffers from extremely poor performance; its error rate grows at high SNR and eventually flattens at a level even higher than differential detection! This phenomenon, although somewhat counterintuitive, has been validated by computer simulation [10] and is caused by the ignorance of fading variation duringa matrix symbol. On the other hand, MS-DFD maintains good performance consistently. The performance loss is uniformly less than 4 dB, compared with MSDD. The absence of error propagation is widely considered a major advantage of LP-DFD for differential schemes. Lack of a linear-filter structure in MS-DFD may raise doubts if error propagation can be controlled to a similar degree. Fig. 1 indicates that, fortunately, the influence of decision errors is marginal. There is not much difference from LP-DFD in this aspect. The simulated BER agrees with the theoretic curve quite well at high SNR. Totally different behaviors of diagonal/nondiagonal generator constellations with LP-DFD are illustrated in Fig. 2 for three-antenna DSTM. The diagonal generator is [1 1 3] [4], while the nondiagonal generator constellation is obtained as , where is given by a DFT matrix. It is seen that the BER of LP-DFD for the nondiagonal generator constellation again grows at high SNR. In contrast, when MS-DFD is employed, their performance is virtually indistinguishable. It is worth mentioning that even though LP-DFD with a diagonal dB in this example, generator is slightly better at there exist nondiagonal generator constellations with significant performance gain for high rates [4]. 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