Multisampling decision-feedback linear prediction

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Multisampling Decision-Feedback Linear Prediction Receivers for Differential Space–Time Modulation Over Rayleigh Fast-Fading Channels Cong Ling, Student Member, IEEE, Kwok Hung Li, Senior Member, IEEE, Alex C. Kot, Senior Member, IEEE, and Q. T. Zhang, Senior Member, IEEE

Abstract—Novel decision-feedback (DF) linear prediction (LP) receivers, which process multiple samples per symbol interval in conjunction with optimal sample combining, are proposed for differential space–time modulation (DSTM) over Rayleigh fast-fading channels. Performance analysis demonstrates that multisampling DF-LP receivers outperform their symbol-rate sampling counterpart in fast fading substantially. In addition, an asymptotically tight upper bound on the pairwise error probability is derived. In view of this bound, the design criterion of DSTM for fast fading is the same as that for block-wise static fading. To avoid the estimation of the second-order statistics of the channel, a polynomial-model-based DF-LP receiver is proposed. It can approach the performance of the optimum DF-LP receiver at high signal-to noise ratios, provided fading is moderate. Index Terms—Differential detection, diversity combining, linear prediction (LP), space–time modulation, time-selective fading.

I. INTRODUCTION

D

IFFERENTIAL space–time modulation (DSTM) [1]–[3] is an extension of the standard single-antenna differential phase-shift keying (DPSK) modulation to multiple-antenna systems. Both of them can work in the presence or in the absence of channel state information. Performance degradation of DSTM due to the unknown channel is 3 dB in signal-to-noise ratio (SNR) over quasi-static fading channels. The theory based on group codes [2], [3] simplifies the design and implementation of DSTM. Specifically, by restricting the group to be Abelian, Hochwald and Sweldens [3] introduced a class of diagonal space–time signals. The early work on signal reception assumed that the fading process keeps constant over two DSTM symbols. However, the land mobile channel is time selective, usually modeled according to Jakes [4]. It has been well known that differential demodulation of DPSK suffers an irreducible floor of bit-error rate (BER) over such channels. A similar error floor was observed for differential detection of DSTM as well [5]–[8]. Specifically, it was found that the effective Doppler shift is

increased times for diagonal DSTM, where stands for the number of transmit antennas. To a large extent, the “faster” fading experienced by DSTM smears its diversity advantage. If fading is adequately fast, increasing the number of transmit antennas actually deteriorates the performance of a differential detector [8]. Decision-feedback (DF) detection based on linear prediction (LP) has been applied in [6]–[8] to reduce the error floor of DSTM over time-selective fading channels. Compared with competing alternatives, such as block detection [9] and sequence detection [10], the DF-LP receiver is characterized by its low complexity. Despite large gains achieved by DF-LP receivers over the differential detector, the performance is still unsatisfactory in fast-fading channels because of the increased effective Doppler shift. Multisampling, usually at the Nyquist rate, is an effective way to improve the performance of LP receivers in fast-fading channels [11]–[14]. When signals are subject to noticeable Doppler spread, one sample per symbol interval is no longer a sufficient statistic. For that reason, the purpose of this paper is to develop new DF-LP receivers for DSTM by sampling the received signal twice or more per symbol interval. The remainder of the paper is divided into five sections. Section II presents the DSTM system model. Multisampling DF-LP receivers are introduced in Section III for diagonal DSTM. Section IV is devoted to the performance analysis of multisampling DF-LP receivers. Numerical results are reported in Section V. Finally, conclusions are drawn in Section VI. Throughout this paper, matrices (vectors) are represented in is the -by- identity (null) bold upper (lower) case. denotes the complex conjugate, transpose, matrix, Hermitian transpose, pseudoinverse, stands for the Krostands for the determinant (trace) necker product, represents the Frobenius of a matrix, and norm. II. SYSTEM MODEL

Paper approved by M.-S. Alouini, the Editor for Modulation and Diversity Systems of the IEEE Communications Society. Manuscript received May 6, 2002; revised December 3, 2002. This work was supported in part by the Singapore Millennium Foundation. C. Ling, K. H. Li, and A. C. Kot are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]; [email protected]; [email protected]). Q. T. Zhang is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2003.814213

Consider a multiple-antenna communication system over a flat-fading channel displayed in Fig. 1, where data are sent from transmit antennas to receive antennas. In DSTM, signals matrix whose row indexes repare grouped into an resent different antennas and column indexes represent time in. The matrices are properly normalstants ized such that the average power of each column is one. The total

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Fig. 1. Block diagram of the rate-

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R DSTM.

transmitted power, therefore, does not depend on the number of transmit antennas. transmit antennas and a data rate A DSTM system with b/(channel use) contains different signals. of unitary matrix drawn from a group Each signal is an (under matrix multiplication) [2], [3]. Every bits to be transwhere are mapped to matrix . Before mitted at time instant are differentially encoded in a transmission, the matrices fashion similar to DPSK [2], [3]

that we adopt a pulse such that no intersymbol interference (ISI) is generated at the sampling points otherwise. Such pulses can be designed in a way similar to “partial-response” signals [15] at the expense of some excess bandwidth. An example is the so-called “multi-Nyquist” pulse [16]

(1) In this paper, we only consider diagonal signals of the form [3]

(2) . Optimal values of for 1, where up to 5 are tabulated in [3] through exhaustive search. 2, and For almost all these diagonal DSTM constellations, it is possible to construct Gray mapping to assign information bits to signal [8]. matrix for , The fading processes are assumed to be complex normal and spatially independent. For Jakes’ U-shaped Doppler spectrum, the autocorrelation of the fading process is given by [4] (3) is the zeroth-order Bessel function of the first kind, where and is the maximum Doppler frequency shift. Unlike [5]–[8], we do not restrict the fading process to be constant during a symbol interval. This gives the receiver an opportunity to benefit from multisampling when fading gets faster. Likewise, the are assumed to be independent across both time noises distributed. Because and space, and are identically of the power normalization, the average bit SNR at each receive . antenna is denote the sampling period, where is the Let duration of a phase-shift keying (PSK) symbol, and is the number of samples extracted during each symbol interval. The receiver front end is an ideal low-pass filter with a cutoff fre, which is customary for sampling at faster than quency symbol rate [11], [13], [21]. The noise samples at the output of this filter will be uncorrelated, but the variance is increased to . Pulse shaping is implemented at the transmitter. Suppose

where

is a classical Nyquist pulse. If we select , then the bandwidth of will be , hence, no distortion after passing through limited to , has a half-sine-shape the receiver filter. When , decaying to zero smoothly. spectrum with bandwidth This pulse is akin to the popular raised-cosine pulse with is sufficient with a rolloff factor one. In addition, respect to performance in most situations. Accordingly, we to avoid too much bandwidth expansion in this limit paper, even if the analysis is valid for arbitrary values of . The receiver structure can be extended to the case of no excess bandwidth. In principle, the resulted ISI might be cancelled by maximum-likelihood sequence detection (MLSD) [11], [13], [16]. , the received signal of the Since we consider diagonal th receive antenna at time instant is given by

(4) where the discrete time index is expressed in units of ,1 except , the th diagonal entry of . Define the fading for vector for the th receive antenna during the th symbol interval as . Stacking these vectors for different yields a matrix . All matrices constitute a fading sample matrix of the 1Such usage of the time index should not be confused with that for DSTM supersymbols and subsequent channel and noise matrices, which is always expressed in units of .

MT

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Fig. 2.

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Description of signal sampling with 0

= 2 (a) for DPSK; (b) for diagonal DSTM for the link between a pair of antennas.

that represents the relevant . By defining the received signal fading samples seen by and noise matrix accordingly, we obtain the matrix matrix representation of the signal model

the process fine where

, ,

. De,

(5) When scenario.

, the foregoing model degenerates to the DPSK

III. DF-LP RECEIVER Using (1), we can rewrite (5) as (6) It is evident that the knowledge of . The optimum LP for detect , , has the form

is required to by past signals

(7) is the -byprewhere is the prediction order and should minimize the mean-square diction coefficient matrix. and . Analogous error (MSE) between to the symbol-rate sampling receiver for DPSK [17], it can be shown that this is equivalent to minimizing

(9) representing are the fading-plus-noise samples, with . This rounding toward minus infinity and particular form of the time index is caused by noncontiguity of the samples in time (see Fig. 2, which shows that the samples are contiguous in time for DPSK, but not for diagonal are to be linearly predicted from the DSTM). samples , each equipped with most recent , namely, a separate linear predictor there are linear predictors in total. The prediction tap mahas an alternative expression trix , where each is a matrix. can be factored as . Furthermore, Then , we define correlation matrices , and that th entries of are independent of the antenna indexes. The these matrices are given by

(8)

and . It where indicates that we need to design the linear predictor for the . fading-plus-noise process Since the fading processes are independent in space, the LP is decoupled between different pairs of antennas. This means that, in essence, only one linear predictor needs to be designed for

(10) is the autocorrelation coefrespectively, where for , ficient of the sampled fading process, and and is zero, otherwise.

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The optimal predictor taps are easily obtained through the [18]. Nevertheless, the Wiener–Hopf equation existence of correlated samples bearing the same information symbol calls for a diversity-combining strategy to collect signal energy effectively. To specify the combiner, we might view this problem as a special case of coherent maximum-ratio combining (MRC), where channel estimation is provided by LP. Since the channel estimate is a vector of complex Gaussian random can be variables, the residual error considered an equivalent colored Gaussian noise, whose corre. As lation matrix equals a result of noise correlation, the conventional MRC is not optimum. A usual approach in such a situation is to incorporate a noise-whitening filter prior to combining [19], [20], which can , where be obtained by the Cholesky factorization is an upper triangular matrix. After noise decorrelation, we . Now have is white within one symbol the equivalent noise interval. The optimal combiner in this situation is MRC, leading to

trices rather than scalar coefficients, when viewed at the PSK symbol level. Another is that a noise-whitening filter is incorporated following the Wiener–Hopf linear predictor. The optimum LP described above requires knowledge of the autocorrelation function of the fading-plus-noise process. A way to estimate the autocorrelation was to make the linear predictor adaptive by means of, e.g., the recursive least-squares (RLS) algorithm [17]. However, this incurs extra computational complexity. An alternative is presented in the remainder of this section that does not rely on the autocorrelation function. There has been “blind” LP for fading channels based on appropriate deterministic models to mimic the temporal fading correlation [14], [21]. It is known that the fading process can be modeled as a polynomial in time due to its bandlimited nature [21]. If we apply a th-order polynomial model, then , where are the coefficients of the polynomial. To derive the predictor taps for prediction of , , it is convenient to view this as an equivalent from . problem of linearly predicting reflecting past samples as We define a matrix

(11)

Then the tap vector for , is given by the , where is the least-squares solution unit vector [21]. Suppose that the modeling error is negligible in comparison with the noise. Note that this is not always true. It is only treated as such for derivation of the noise-whitening filter. Under this condition, the estimation error is mainly due to the background Gaussian noise

where the prediction tap matrix (symmetric) to form a single

has been combined with weighting matrix (12)

This way, the noise-whitening filter does not appear explicitly. For a good linear predictor , the elements of can be small, thereby possibly leading to of large elements. To prevent numerical problems in practical implementation, we suggest the . normalization as Let be decomposed into becomes

. Then (7)

where and denote the Gaussian noise vector asand , respectively. The covariance sociated with matrix may be approximated by

(13) In

DF

receivers,

previously detected matrices replace true signals, yielding the

decision

Because the constant does not affect the decision, it can be dropped to result in a weighting matrix (15)

(14) Compared to symbol-rate sampling DF-LP receivers, there are two distinctions in (14). One distinction is that the multisampling DF-LP receiver uses -by- prediction coefficient ma-

which has the salient feature that it does not depend on channel includes Bin and statistics, either. The predictor , where fading Ho’s predictor [14] as a special case of was modeled as an unknown Doppler frequency and a firstorder Taylor series expansion was applied. However, as will be demonstrated in Section V, the inclusion of the proposed noise-whitening filter translates into improved performance relative to [14]. If the fading rate is moderate, a polynomial model may approximate the fading process accurately. Then the weight matrix (15) has near-optimum performance at high SNR. On the other

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hand, if the channel fades so fast that the polynomial model is no longer accurate, the weight matrix (15) may result in performance loss.

Let us write the decision statistic into a Hermitian quadratic form , where and

is a

-by-

permutation

matrix

IV. PERFORMANCE ANALYSIS In this section, the performance of multisampling DF-LP receivers for DSTM is analyzed. The analysis is applicable to DF-LP DSTM receivers with arbitrary prediction taps. Let us begin with the derivation for the pairwise error probability for genie-aided DF-LP receivers, where (PEP) the feedback is assumed to be error free. is transmitted, the decision is given by Assuming

(19) The correlation matrix of the complex Gaussian random vector has a partitioned form (20) The characteristic function of

can be expressed as [22]

(21) are the eigenvalues of . where , , and signify their The particular expressions is comprised of dependency on , but not on . Because independent components, the characteristic function of is simply the product of terms (21), raised to the th power. The probability of error is given by [21]

The matrix identity

yields

(22) (16) Noting that the diagonal matrices in the product commute, (16) can be rewritten as

(17) Further, since the DSTM is diagonal, no cross-product term of the receive signals exists in (17). Consequently, the decision statististatistic can be expanded into the summation of cally independent random variables

(18) Equation (18) implies that an error occurs if , where .

and

It is known that the above integral can be expressed as the sum over the right-half comof residues of the function plex plane. However, numerical computation tends to be unstable for high-order poles (that are frequently encountered in multiantenna communications). Hence, the Gauss–Chebyshev quadrature-based integration technique [23], [24] is adopted in this paper to evaluate the integral of (22). Note that once channel statistics and the linear predictor are is governed by . Due to symmetry of fixed, , namely, the PSK constellation, , and thereby, should be invariant when is changed to . Once is known, is either or , where for are the singular values of the matrix [3]. But both values result in the same performance. It implies that the performance of the DF-LP only through the singular value DSTM receiver depends on . The foregoing exact expression (22) for the PEP provides little insight to the dependence on system parameters. In such a situation, a bound on the error probability is more insightful. A Chernoff upper bound was presented in [8] for symbol-rate sampling. However, the Chernoff bound may be loose for many applications.

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For multisampling receivers with optimal LP, we derive an upper bound on the PEP (see the Appendix)

(23) At high SNR, the bound (23) provides tight approximations to the error probability for “good” DF-LP receivers.2 In the ideal , the bound coincides with the true error probacase, as bility. Since typically has small elements, even if nonzero, for a “good” linear predictor at high SNR, the bound is asymptotically close to the true error rate. The smaller the elements of are, the tighter the bound will be. The implication of the bound (23) is twofold. First, the bound depends again on the singular . Larger singular values translate into better perforvalues should be made large for a good linear mance. Second, predictor, as the error probability is inversely proportional to . This new bound is applicable to the symbol-rate sampling receiver as well. It can be checked that, asymptotically, the , new bound differs at least by a factor of 8/3 for and 16/5 for , , respectively, in comparison to the Chernoff bound for the symbol-rate sampling receiver [8]. Owing to the symmetry of the diagonal group constellation, the BER does not depend on which matrix is sent. Hence, the union bound on the BER for genie-aided DF-LP receivers for Gray-mapped DSTM is approximated by [11]

In the ideal case, as , the error event corresponding to the minimum diversity product [3] would be dominant, so that (24) where is the number of error events corresponding to the minimum diversity product. In light of the asymptotic behavior, the issue of DSTM constellation design over fast-fading channels is bypassed. We conclude that if a constellation performs well when coherently detected, hence, having large , it is also capable of performing well in fast-fading channels, provided an effective linear predictor can be constructed such that the elements of are sufficiently small. In other words, the design criterion remains unchanged. If the influence of feedback errors is taken into account, the actual BER of DF-LP receivers is usually, but not always, doubled. By modeling the error propagation as a Markov chain, Schober et al. [25] recently showed that the increment in BER strongly depends on the predictor coefficients (for DPSK and Gaussian channels). Sometimes, there is no increase in BER at all. We prefer to assess the impact of erroneous feedback on the DF-LP receiver of DSTM by computer simulation in Section V. 2The

Chernoff bound has better accuracy at low SNR.

Fig. 3. Union bounds on the BER of genie-aided multisampling DF-LP 0:03. receivers for DSTM over a fading channel with f T

=

V. NUMERICAL RESULTS AND DISCUSSION In this section, numerical results for the performance of is used to represent the BER unless DSTM are reported. otherwise stated. We concentrate on rate-1 DSTM [3, Table I] using one reception antenna. The third-order LP is used . throughout, i.e., The performance of the coherent multisampling receiver is included as a benchmark, since it might be unfair to compare with a symbol-rate sampling coherent receiver. We might view a coherent receiver as the extreme case of the LP receiver, where the fading gains are somehow predicted perfectly. Hence, the PEP of the coherent receiver can be evaluated in the same way as described in (19)–(22), except that the correlation matrix is changed to

When , it can be checked that multisampling results in no improvement for the coherent or noncoherent receiver. However, multisampling and symbol-rate sampling reincreases. The performance ceivers behave differently as is the of a coherent symbol-rate sampling receiver for , since the fading gains are mutually same as that for independent for diagonal DSTM. In contrast, the performance of a coherent multisampling receiver improves with increasing . That is, it can benefit from the implicit time diversity of fast fading. Likewise, the multisampling DF-LP receiver can benefit from the implicit diversity, as well. Figs. 3 and 4 depict the union bounds on the BER of twosample DF-LP receivers for DSTM over fading channels with and , respectively. The two figures show that, though the performance of symbol-rate sampling DF-LP , irreducible error floor receivers is acceptable for . In the latter case, the performance appears when of symbol-rate sampling receivers actually deteriorates with an increasing number of transmit antennas. The advantage of multisampling is obvious, as significant improvement is achieved

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Fig. 4. Union bounds on the BER of genie-aided multisampling DF-LP 0:1. receivers for DSTM over a fading channel with f T

=

over symbol-rate sampling. No error floor is observed for the range of SNR considered. The performance of the two-sample DF-LP receiver can improve with increasing number of transmit antennas if the fading is not too fast. However, it is worth noting that increasing from two to three only worsens the performance of the two, unless the SNR is very sample DF-LP receiver for high. This may be explained by referring to Fig. 2. Since the samples for DSTM are noncontiguous (with the spacing proportional to ), the correlation between the samples is weaker is increased to three. Multisampling will not be as efwhen . Likewise, we expect that multisampling fective as for is less effective for larger . On the other hand, it will be most effective for DPSK. For instance, the performance difference between the coherent multisampling receiver and the DF-LP re(DBPSK) in Fig. 3. However, ceiver is less than 5 dB for and 10 dB for , respectively, the gap is 7.5 dB for at high SNR. The tightness of the upper bound (23) is demonstrated in Fig. 5 and various values of over a fading channel with for . It is seen that the bound gets tighter with increasing and , the bound converges to the SNR. When . exact PEP at high SNR. The bound is a bit loose when This is because the effective Doppler shift is 0.18 in this case, and the linear predictor is no longer “good” for so fast fading. Fig. 6 demonstrates the performance of the polynomial for model-based multisampling DF-LP receiver with and . The order of the polynomial is . We compare the performance of the optimum DF-LP receiver, the polynomial model-based receiver with and without the noise decorrelator, and with the modification suggested in [14] (25)

This modification only balances the SNR for the predicted samples, so the performance is not as good as our proposed noisewhitening filter. As clearly shown in Fig. 6, the polynomial

Fig. 5. Tightness of the upper bound relative to the exact PEP for genie-aided two-sample DF-LP receiver of DSTM over a fading channel with f T = 0:06.

Fig. 6. Performance of the polynomial model-based two-sample DF-LP receiver for two-antenna DSTM over a fading channel with f T = 0:03.

model-based DF-LP receiver with noise decorrelation performs close to the optimum one at high SNR. Without the noise decorrelator, the performance loss is more than 5 dB asymptotically. Little improvement is observed if the modification in (25) is applied. Simulation results are also included in Fig. 6 to assess the impact of erroneous feedback. To check the accuracy of the analytic results assuming correct feedback, the exact theoretic BER for this two-antenna constellation is shown in Fig. 6, rather than the union bound. Since this constellation comprises four ma, 0, 1, 2, 3, with the generator trices [3], it equates to a DQPSK scheme with two-fold diversity combining. Analogous to the analysis of DQPSK [17], its performance, in case of correct feedback, has an exact form

where

,

.

LING et al.: MULTISAMPLING DECISION-FEEDBACK LINEAR PREDICTION RECEIVERS

The influence of using detected symbols as feedback on the BER of multisampling receivers is small. As evidenced in . Similar Fig. 6, the resultant BER is uniformly less than and . For phenomena were observed for other values of example, there is nearly no increase in BER at high SNR for . It is seen that, however, the polynomial model-based DF-LP receiver exhibits performance loss at low SNR. This is caused by noise enhancement associated with the polynomial-based linear predictor. Although a noise decorrelator is inserted after the linear predictor, the design of the linear predictor itself does not consider the existence of noise. It is sufficient to consider the fading process alone to reduce the error floor, which appears at high SNR, but a performance penalty is inevitable at low SNR where noise predominates the performance. This is analogous to the decorrelation detector in multiuser detection, which also suffers from noise enhancement at low SNR. To avoid excess noise enhancement, the linear predictor should be redesigned to take noise into account. But this requires the knowledge of the terms in the polynomial do not play SNR. Moreover, the equally important roles. This, in turn, requires knowledge of the powers of . Avoidance of noise enhancement at low SNR is an open subject of future research.

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After this decomposition, can be written in another form, , where . Bearing in mind the principle of [18], the orthogonality for optimal LP has a simple partitioned form correlation matrix of

where trix of

is the autocorrelation ma. The eigenvalues of satisfy

(27) , hence, the In this form, all partitions are functions of will be related to those of . Note that eigenvalues of the partitions are commutative, and are invertible unless is an or , too. It can be shown that is eigenvalue of or . Hence, the identity never an eigenvalue of for the determinant of a partitioned matrix

under such circumstances implies that

VI. CONCLUSIONS Multisampling DF-LP receivers have been proposed for diagonal DSTM as well as DPSK over fast Rayleigh fading channels. The novelty was manifested in the properties that the receiver uses prediction matrices rather than scalars, and a noise decorrelator is incorporated. The proposed multisampling receiver is new, even for DPSK. In addition, a DF-LP receiver based on the polynomial model of the fading process was proposed to bypass estimation of the second-order statistics of the channel. Performance analysis applicable to arbitrary prediction taps was carried out. Moreover, an upper bound on the PEP was derived for optimum multisampling DF-LP receivers, which is increasingly tight for smaller prediction error. A useful conclusion drawn from the bound is that the design criterion of DSTM constellation does not need any change in fast-fading channels. Numerical results showed that significant improvement was achieved by the multisampling receiver. Computer simulation results are in good agreement with theoretic analysis.

After canceling common terms, we arrive at

(28) Recalling the Hermitian nature of an autocorrelation matrix, can be diagonalized as , where is real and diag, onal, and is unitary. Then, since we have

Let the

APPENDIX

, then the eigenvalues of solutions to the equations

are

(29)

We derive the upper bound for multisampling DF-LP receivers with optimal LP in this Appendix. The derivation relies on the asymptotic eigenanlysis in [26] and [27]. An as defined in (19) and (20) seems eigenanalysis for appearing in the difficult. Instead, we split using the Cholesky decomposition decision statistic

(26)

Define

. The eigenvalues are given by

which apparently occur in positive–negative pairs. Because the or from characteristic function deduced either from is the characteristic function of , they must have the same expression. Then the two sets of eigenvalues must be equal. Such a structure of eigenvalues has important consequences. It is easy to check that the poles of the characteristic function

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conform to the structure of [27, eq. (25)]. Accordingly, an upper bound is given by [27, eq. (33)]

(30) Unlike noncoherent schemes in block-wise static fading or coherent schemes [26], asymptotic tightness is not always guaranteed here. It depends on the linear predictor. Note that this upper ’s increase. If the linear predictor is bound gets tighter as good, i.e., has small elements, the eigenvalues ’s will be large. Then the bound will be tight. Otherwise, it will not be tight, even at high SNR. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their critical comments, which greatly improved the quality of the paper. C. Ling also wishes to thank X. Wu, Peking University, Beijing, China, for helpful discussions. REFERENCES [1] V. Tarokh and H. Jafarkhani, “A differential detection scheme for transmit diversity,” IEEE J. Select. Areas Commun., vol. 18, pp. 1169–1174, July 2000. [2] B. L. Hughes, “Differential space–time modulation,” IEEE Trans. Inform. Theory, vol. 46, pp. 2567–2578, Nov. 2000. [3] B. M. Hochwald and W. Sweldens, “Differential unitary space–time modulation,” IEEE Trans. Commun., vol. 48, pp. 2041–2052, Dec. 2000. [4] W. C. Jakes, Microwave Mobile Communications. Piscataway, NJ: IEEE Press, 1993. [5] C. B. Peel and A. L. Swindlehurst, “Performance of unitary space–time modulation in a continuously changing channel,” in Proc. ICC’01, Helsinki, Finland, June 2001, pp. 2085–2088. [6] R. Schober and L. H.-J. Lampe, “Noncoherent receivers for differential space–time modulation,” in Proc. Globecom’01, San Antonio, TX, Nov. 2001, pp. 1127–1131. [7] C. Ling and X. Wu, “Linear prediction receiver for differential space–time modulation over time-correlated Rayleigh fading channels,” in Proc. ICC’02, New York, NY, Apr.–May 2002, pp. 788–791. [8] R. Schober and L. H.-J. Lampe, “Noncoherent receivers for differential space–time modulation,” IEEE Trans. Commun., vol. 50, pp. 768–777, May 2002.

[9] D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun., vol. 38, pp. 300–308, Mar. 1990. [10] J. H. Lodge and M. L. Moher, “Maximum likelihood sequence estimation of CPM signals transmitted over Rayleigh flat-fading channels,” IEEE Trans. Commun., vol. 38, pp. 787–794, June 1990. [11] X. Yu and S. Pasupathy, “Innovations-based MLSE for Rayleigh fading channels,” IEEE Trans. Commun., vol. 43, pp. 1534–1544, Feb.-Apr. 1995. [12] G. M. Vitetta and D. P. Taylor, “Maximum-likelihood decoding of uncoded and coded PSK signal sequences transmitted over Rayleigh flatfading channels,” IEEE Trans. Commun., vol. 43, pp. 2750–2758, 1995. , “Multisampling receiver for uncoded and coded PSK signal se[13] quences transmitted over Rayleigh flat-fading channels,” IEEE Trans. Commun., vol. 44, pp. 130–133, Jan. 1996. [14] L. Bin and P. Ho, “Data-aided linear prediction receiver for coherent DPSK and CPM transmitted over Rayleigh flat-fading channels,” IEEE Trans. Veh. Technol., vol. 48, pp. 1229–1236, July 1999. [15] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995, pp. 548–550. [16] B. D. Hart and D. P. Taylor, “Extended MLSE receiver for the frequency-flat, fast-fading channel,” IEEE Trans. Veh. Technol., vol. 46, pp. 381–389, May 1997. [17] R. Schober and W. H. Gerstacker, “Decision-feedback differential detection based on linear prediction for MDPSK signals transmitted over Ricean fading channels,” IEEE J. Select. Areas Commun., vol. 18, pp. 391–402, Mar. 2000. [18] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [19] M. Stojanovic and Z. Zvonar, “Differentially coherent diversity combining techniques for DPSK over fast Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 49, pp. 1928–1933, Sept. 2000. [20] M. W. Mydlow, S. Basavaraju, and A. Duel-Hallen, “Decorrelating detector with diversity combining for single user frequency-selective Rayleigh fading multipath channels,” Wireless Pers. Commun., vol. 3, pp. 175–193, 1996. [21] D. K. Borah and B. D. Hart, “A robust receiver structure for time-varying, frequency-flat Rayleigh fading channels,” IEEE Trans. Commun., vol. 47, pp. 360–364, Mar. 1999. [22] M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques. New York: McGraw-Hill, 1966. [23] M. K. Simon and M.-S. Alouini, Digital Communication Over Fading Channels. New York: Wiley, 2000. [24] E. Biglieri, G. Caire, G. Taricco, and J. Ventura-Traveset, “Simple method for evaluating error probabilities,” Electron. Lett., vol. 32, pp. 191–192, Feb. 1996. [25] R. Schober, Y. Ma, and S. Pasupathy, “On the error probability of decision-feedback differential detection,” IEEE Trans. Commun., vol. 51, pp. 535–538, Apr. 2003. [26] M. Brehler and M. K. Varanasi, “Asymptotic error probability analysis of quadratic receivers in Rayleigh-fading channels with applications to a unified analysis of coherent and noncoherent space–time receivers,” IEEE Trans. Inform. Theory, vol. 47, pp. 2383–2399, Sept. 2001. [27] S. Siwamogsatham, M. P. Fitz, and J. Grimm, “A new view of performance analysis of transmit diversity schemes in correlated Rayleigh fading,” IEEE Trans. Inform. Theory, vol. 48, pp. 950–956, Apr. 2002.

Cong Ling (A’01–S’02) was born in Anhui Province, China, in 1974. He received the B.S. and M.S. degrees in electrical engineering from the Nanjing Institute of Communications Engineering, Nanjing, China, in 1995 and 1997, respectively. He is currently working toward the Ph.D. degree at the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. From 1998 to 2001, he was a Lecturer at the Nanjing Institute of Communications Engineering. His research interests are in the general area of wireless communications, with emphasis on spread spectrum, coding and iterative processing, and space–time communication. Mr. Ling was awarded the Singapore Millennium Scholarship in 2002 by the Singapore Millennium Foundation, Singapore. He is a Student Member of the IEEE Communications Society and IEEE Information Theory Society.

LING et al.: MULTISAMPLING DECISION-FEEDBACK LINEAR PREDICTION RECEIVERS

Kwok Hung Li (S’87–M’89–SM’99) received the B.Sc. degree in electronics from the Chinese University of Hong Kong in 1980, and the M.Sc. and Ph.D. degrees in electrical engineering from the University of California, San Diego, in 1983 and 1989, respectively. Since December 1989, he has been with the Nanyang Technological University, Singapore. He is currently an Associate Professor in the Division of Communication Engineering. He has served as Program Director of the M.Sc. (Communications Engineering) program since 1998. His research interest has centered on the area of digital communication theory with emphasis on spread-spectrum communications, mobile communications, coding, and signal processing. He has published more than 80 research papers in journals and conference proceedings. Dr. Li served as the Chairman of IEEE Singapore Communication Chapter from 1999 to 2001 and is still an active member within the Chapter. He was also the General Co-Chair of the Third International Conference on Information, Communications, and Signal Processing (ICICS’01) in Singapore.

Alex C. Kot (S’85–M’89–SM’98) was educated at the University of Rochester, Rochester, NY, and at the University of Rhode Island, Kingston, where he received the Ph.D. degree in electrical engineering in 1989. He was with the AT&T Bell Company in New York. Since 1991, he has been with the Nanyang Technological University (NTU), Singapore, where he is Head of the Information Engineering Division. His research and teaching interests are in the areas of signal processing for communications, signal processing, watermarking, and information security. Dr. Kot served as the General Co-Chair for the Second International Conference on Information, Communications, and Signal Processing (ICICS) in December 1999, the Advisor for ICICS’01 and ICONIP’02. He received the NTU Best Teacher of the Year Award in 1996 and has served as the Chairman of the IEEE Signal Processing Chapter in Singapore. He is the General Co-Chair for the IEEE ICIP 2004 and is currently an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY.

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Q. T. Zhang (S’84–M’85–SM’95) received the B.Eng. degree from Tsinghua University, Beijing, China, the M.Eng. degree from South China University of Technology, Guangzhou, China, both in wireless communications, and the Ph.D. degree in electrical engineering from McMaster University, Hamilton, ON, Canada in 1986. He held a research position and Adjunct Assistant Professorship at McMaster University. In January 1992, he joined the Spar Aerospace Ltd., Satellite and Communication Systems Division, Montreal, QC, Canada, as a Senior Member of Technical Staff. At Spar Aerospace, he participated in the development and manufacturing of the Radar Satellite (Radarsat). He was subsequently engaged in the development of the advanced satellite communication systems for the next generation. He joined Ryerson Polytechnic University, Toronto, ON, Canada, in 1993 and became a Full Professor in 1999. In 1999, he took a one-year sabbatical leave at the National University of Singapore, and is now with the City University of Hong Kong. His research interest includes modulation and transmission technologies on various fading channels with current focus on OFDM/CDMA, space–time coding/modulation, and wireless UWB communications. Dr. Zhang is presently an Associate Editor for the IEEE COMMUNICATIONS LETTERS.