Differential Space–Time Modulation with Eigen-Beamforming for Correlated MIMO Fading Channels Xiaodong Cai, Senior Member, IEEE, and Georgios B

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 4, APRIL 2006 1279 Differential Space–Time Modulation With Eigen-Beamforming for Correlated MIMO Fading Channels Xiaodong Cai, Senior Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE

Abstract—In this paper, joint differential space–time modula- many training symbols, when the channel is rapidly changing tion (DSTM) and eigen-beamforming for correlated multiple-input in a mobile environment, and/or when the number of transmit multiple-output (MIMO) fading channels. While DSTM does not antennas is large. In such cases, unitary space–time modulation require knowledge of each channel realization, the channel’s spatial correlation can be easily estimated without training at the re- [11] and differential space–time modulation (DSTM) [7], [12], ceiver and exploited by the transmitter to enhance the error prob- [14], [16], [31] are well motivated because they bypass CSI ac- ability performance. A transmission scheme is developed here that quisition at the receiver. combines beamforming with differential multiantenna modulation Whenever (even partial) CSI is available at the transmitter, based on orthogonal space–time block coding. Error probability is it should be exploited to further improve the performance of analyzed for both spatially correlated and independent Rayleigh fading channels. Based on the error probability analysis, power multiantenna transmission systems. Since in most cases the loading coefficients are derived to improve performance. The ana- transmitter cannot acquire the CSI perfectly, utilization of par- lytical and simulation results presented here corroborate that the tial CSI at the transmitter has received considerable attention proposed scheme can achieve considerable performance gain in recently. A general statistical model of partial CSI is presented correlated channels relative to DSTM without beamforming. in [18], [33]. Based on this model, a linear transformation was Index Terms—Beamforming, differential optimal transmitter applied to orthogonal STBC to enhance the symbol error rate eigen-beamforming and space modulation, error probability (SER) performance in [18]. Since in frequency-division duplex analysis, optimal transmitter eigen-beamforming and space block codes. (FDD) systems the transmitter obtains channel knowledge from the feedback channel, two cases of partial CSI, termed mean and covariance feedback, were studied to maximize I. INTRODUCTION channel capacity in [33]; capacity maximization based on ECENT advances in wireless communications show that covariance feedback was also investigated for multiple-input R multiantenna systems can support high data rates with low multiple-output (MIMO) systems in [15]. Optimal beamformers error probability [6], [23], [32]. Without any channel knowledge combined with coherent STBC were derived in [34] and [35] to at the transmitter, space–time coding offers an effective fading minimize symbol error probability, based on channel mean and countermeasure, and thereby reduces error probability [1], [29], correlation, respectively. Transmit-beamformer optimization [30]. If the receiver can acquire the channel state information in terms of average signal-to-noise ratio (SNR) and expected (CSI) reliably, coherent detection along with space–time trellis mutual information based on partial CSI was studied in [24]. coding [30], or orthogonal space–time block coding (STBC) [1], Note that all these works assume that perfect or estimated CSI [29], or recent high-rate high-performance space–time coding is available at the receiver. [22] can be employed to enable the available spatial diversity. In this paper, we consider DSTM based on orthogonal STBC, However, channel estimation becomes difficult or requires too which was also investigated in [7], [8], [16], [31] without partial CSI at the transmitter for independent, identically distributed (i.i.d.) fading channels. Since practical multiantenna systems Manuscript received July 29, 2004; revised May 19, 2005. This paper ap- peared in part in the Proceedings of the International Conference on Acoustics, may exhibit strong correlation among fading channels asso- Speech and Signal Processing, 2003, Hong Kong, April 2003. This work ciated with different transmit antennas on the downlink [15], was prepared through collaborative participation in the Communications and [26], we will focus on spatially correlated channels. The Networks Consortium sponsored by the U.S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement channel’s spatial correlations depend on the mobile’s angular DAAD19-01-2-0011. The U. S. Government is authorized to reproduce and position with respect to the transmitter [4], [26]. This implies distribute reprints for Government purposes notwithstanding any copyright that the channel’s spatial correlations will typically change notation thereon. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official slowly, even when the channel coefficients fluctuate relatively policies, either expressed or implied, of the Army Research Laboratory or fast [10]. While knowledge of each channel realization is not the U. S. Government. The associate editor coordinating the review of this available in a differential space–time transmission system, the manuscript and approving it for publication was Dr. Yucel Altunbasak. X. Cai is with the Department of Electrical and Computer Engineering, Uni- channel’s spatial correlations can be estimated at the receiver versity of Miami, Coral Gables, FL 33124 USA (e-mail: [email protected]). without training and be fed back to the transmitter. We will G. B. Giannakis is with the Department of Electrical and Computer Engi- exploit these channel correlations at the transmitter to combine neering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: geor- [email protected]). DSTM with eigen-beamforming. We will also analyze the error Digital Object Identifier 10.1109/TSP.2006.870637 performance of DSTM, which is applicable to both i.i.d. and

1053-587X/$20.00 © 2006 IEEE 1280 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 4, APRIL 2006 spatially correlated channels with or without beamforming. is easily constructed and leads to low-complexity symbol-by- Note that error probability performance of DSTM based on symbol decoding. Specifically, the symbols transmitted in the orthogonal STBC was not analyzed in [7], [8], [16], and [31]. th block are first collected in an orthogonal space–time Our general error probability formula is applicable to DSTM code matrix as follows [7]: with beamforming in correlated channels investigated in this paper and DSTM without beamforming in i.i.d. channels considered in [7], [8], [16], and [31]. Based on our novel error (1) probability analysis, we will determine the optimal power al- location per beam, and thereby enhance the error performance where and are real and imaginary parts of the complex via loading. An approximate expression for the bit error rate symbol , respectively, i.e., , and (BER) was derived in [9] for STBC-based DSTM over i.i.d. matrices and satisfy the following conditions: Rayleigh fading channels. Pairwise error probability and BER were analyzed for the DSTM of [12] and [31], respectively, for i.i.d. Rayleigh fading channels in [5]. Our error performance analysis for STBC-based DSTM is applicable to both i.i.d. and spatially correlated Rayleigh fading channels. (2) The rest of the paper is organized as follows. Section II presents the signal model and differential reception of multi- Drawing from -ary phase-shift keying ( -PSK) constel- antenna systems with DSTM and eigen-beamforming. Error lations, and letting , it follows readily from (1) and (2) probability analysis is carried out and an optimal power loading that the matrix is unitary, i.e., . The code scheme is developed in Section III. Section IV contains analyt- matrix for DSTM can then be written recursively as [7], [12], ical results and simulations to test the merits of the proposed [14] transmission scheme. Finally, conclusions are drawn in Sec- tion V. (3) Notation: Superscripts and stand for transpose, con- jugate, Hermitian transpose, respectively; denotes expec- with . Since is unitary, matrix is unitary too, tation over the random variables within the brackets; by design. When the number of transmit antennas , and represent the real and imaginary parts of , respec- then elements of each row of are transmitted from tively. Column vectors (matrices) are denoted by boldface lower antennas per time slot. For an arbitrary , however, we may (upper) case letters. We will use to denote the iden- not be able to find a square matrix with . In this tity matrix, the trace of , the determinant of , case, we can construct a square code matrix with , and the th entry of . find using (3), and transmit the first columns of from transmit antennas as in [7]. Mathematically, letting , the matrix codeword transmitted over II. DIFFERENTIAL SPACE–TIME MODULATION WITH antennas in the th block is . EIGEN-BEAMFORMING If are i.i.d., it is natural to transmit the codeword Consider a multiantenna transmission system comprising via transmit antennas with equal power. However, when the transmit antennas and receive antennas, signaling over a channel coefficients associated with different transmit antennas Rayleigh flat-fading channel. Suppose that the base station (BS) are correlated, which is of primary interest in this paper, a and the mobile user assume the roles of transmitter and receiver, modulation scheme exploiting this channel correlation at the respectively. Let denote the channel coefficient between transmitter is well motivated. To this end, we will transmit the the th transmit and the th receive antenna, which is mod- codeword along the eigenvectors of the channel correla- eled as a complex Gaussian random variable with zero mean. tion matrix with proper power loaded on each eigenvector. The channel matrix is then represented by . This transmission scheme, termed eigen-beamforming, was In a wireless environment where the BS is elevated and unob- introduced in [35] for coherent STBC over correlated fading structed and the mobile user is surrounded by local scatterers, channels. The eigen-decomposition of can be written as a ray tracing model is often used to describe the channel [4], , where the diagonal matrix contains the [15], [26], [17, p. 60–65]. For this channel model, it has been ordered eigenvalues of and the unitary matrix consists of shown that in practical settings with reasonable antenna spacing, the corresponding eigenvectors. The transmitted signal during channel gains associated with different transmit antennas ex- the th block can then be expressed by the matrix hibit strong correlations, while those associated with different receive antennas are uncorrelated [15]. Let stand for the th (4) column of the channel matrix . Then, and are uncorrelated when , while elements of , , are correlated. We where the diagonal matrix contains power loading coeffi- define the channel correlation matrix as , . cients which will be specified later in Section III-C. In this sec- While DSTM can be implemented using any unitary matrix tion, the only constraint we impose on the power loading coef- constellations [12], [14], we will consider DSTM based on or- ficients is . Using this constraint, we can verify thogonal STBC as in [7], because the orthogonal STBC matrix that , where stands for the energy per CAI AND GIANNAKIS: DSTM WITH EIGEN-BEAMFORMING FOR CORRELATED MIMO FADING CHANNELS 1281 transmitted symbol. We can also write (4) as a recursion initial- for correlated channels. Due to the code structure in (1), the de- ized by [cf. (3)] tector (10) for the codeword reduces to a symbol-by-symbol detector [7] (5)

Comparing (3) with (5) reveals that the fundamental differential transmission equation is not changed by the loaded transmit (11) eigen-beamforming matrices . From (5), it is seen that we need to beamform and power load only in the first transmitted Defining , block ; and after the first block, signals will be automatically and , we can also write the decision transmitted along eigen-beams without any beamforming oper- rule in (11) as ation. For clarity, we first consider a single receive antenna (12) . The received samples in the th block can be written in an vector as When receive antennas are employed, the received (6) signal in the th block can be written in an matrix as where contains complex additive white Gaussian noise (AWGN) with mean zero and variance per dimension. (13) We will detect based on and . As it is common to all differential schemes, we assume that the channel variation is Using the fact that the noise is white, we can derive a deci- negligible from one space–time block to the next, even though sion rule similar to (10), which is given by over a large number of blocks the cumulative variation can be indeed fast. For this reason, we omit the time index of in (6) for notational brevity. Defining , we will detect based on . (14) Conditioned on and , the vector is Gaussian with mean zero and covariance matrix where is the th column of in (13) and the symbol-by- symbol detector is easily derived from (14). Since channel gains (7) associated with different receive antennas are uncorrelated, the error probability results for the single receive antenna generalize Hence, the conditional probability density function (pdf) of is easily to multiple receive antennas. For this reason and without given by loss of generality, we will concentrate on the one receive antenna case in the remaining of this paper. (8) III. PERFORMANCE ANALYSIS AND POWER LOADING

If all symbols are equally likely, the optimum receiver is the DSTM based on orthogonal STBC was studied for i.i.d. chan- maximum-likelihood (ML) detector which finds the matrix nels in [7], [8], [16], and [31], but error probability analysis that maximizes the pdf in (8). When the channels associated was not provided. In this section, we will analyze error prob- with different transmit antennas are i.i.d. , the ML ability performance of DSTM, which is applicable to DSTM detector can be derived from (8) as detailed in [14]. If the chan- with beamforming over spatially correlated channels considered nels are correlated and/or , the likelihood function in in this paper and DSTM without beamforming over i.i.d. chan- (8) is dependent on and C ; and for this reason, the ML nels studied in [7], [8], [16], and [31],. For binary phase-shift detector is not practically feasible. keying (BPSK) and quadrature phase-shift keying (QPSK), we Instead of the ML criterion,we will rely on the fundamental will provide formulas to calculate the exact BER. However, it is differential receiver equation difficult to obtain the exact BER or SER in closed form for any -PSK other than BPSK and QPSK. For this reason, we will (9) also derive an approximate SER for any -PSK constellation, which will come handy in deriving the power loading algorithm. to derive our receiver, which leads to the decision rule [7], [12] A. Exact BER for BPSK and QPSK (10) For BPSK constellations, , the decision variable in (12) is simplified as Although (10) coincides with the ML detector in [14] that was derived for i.i.d. channels, it is clearly no longer ML optimum (15) 1282 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 4, APRIL 2006 while for QPSK constellations, , (12) reduces with , and since , to , and are unitary, it is easy to verify that . To proceed, we need the following lemma regarding the eigenvalues of that we prove in Appendix I. (16) Lemma 1: If , then has two distinct eigenvalues given by and If takes 1 values with equal probability, the BER of . is given by Note that we have and . Due to the special structure of the code matrix , theeigenvalues of are (17) determined by , and are not affected by other symbols. The It will be shown later that this error probability is the same for following proposition provides the key result to BER analysis. and , ; thus, it corresponds also to the overall BER. Proposition 1: The matrix is similar to The decision variable can be expressed as (25) (18) Proof: See Appendix II. where Because is a constant matrix, is determined by through and . The key step in proving this proposition (19) is to apply the permutation matrix and obtain (41). It follows from the proof that this proposition is true for any matrix . It is seen from (18) that is a quadratic form of the complex Hence, if beamforming and power loading are not employed, Gaussian random vector . Thus, the Laplace transform of the then , and the proposition still holds. pdf of is given by [25, p. 595] Corollary 1: With , the eigenvalues of are given by

(26) (20) Proof: See Appendix III. where is the th eigenvalue of matrix . Then, the Given these eigenvalues, we are ready to evaluate the BER error probability can be found as [2], [3] using (21). Let us consider the decision variable in (16). Defining (21) (27) where is a pole of , and denotes the residue of at . From (20), we have ,if we can also write in a quadratic form as . . To evaluate the BER in (21), we need to find the eigenvalues Using (1) and (2), we can show that of . (28) Letting , the covariance matrix in (7) can be written as where . If wereplace in and by , replace by , and use (28), then Lemma 1, Proposition 1 and (22) Corollary 1 also hold true for . Therefore, the error probability given by (21) is the overall BER for BPSK or QPSK. From (22), it appears that depends on code matrices If channel gains are i.i.d., then has only two distinct eigen- and , implying that its eigenvalues may depend on all trans- values, each with multiplicity . When is large, it may be mitted symbols. However, it will turn out that the eigenvalues complicated to find the residue of when calculating of are only determined by , which will enable evaluation BER from (21); in this case, a numerical method can be used of the BER in (21). Define the block diagonal matrix to evaluate BER by calculating the inverse Laplace transform , which can be readily verified to be unitary. of [3]. If beamforming and power loading are not used, Then, we have Lemma 1 and Proposition 1 still hold, and we can calculate the eigenvalues of numerically, which in turn enables us to (23) evaluate the BER. If beamforming and equal power loading are employed, then has eigenvalues identical to those without where beamforming and power loading. Therefore, these two cases have identical error probability performance, which implies that (24) beamforming alone does not improve error performance. This CAI AND GIANNAKIS: DSTM WITH EIGEN-BEAMFORMING FOR CORRELATED MIMO FADING CHANNELS 1283 motivates appropriate power loading coefficients to reduce the minimize with respect to . For this reason, we will derive error probability—a subject we will pursue later. power loading coefficients based on the approximate SER of Section III-B. While in (31) offers a good approximation B. Approximate SER for General -PSK of the actual SER at high SNR as shown in Section III-B, we will If and , then and in (12) demonstrate in Section IV that the approximate SER comes also can be expressed as very close to the actual SER at low SNR. Hence, reducing this approximate SER will decrease the actual SER. Similar to [18], [35], we will minimize the Chernoff bound of the approximate SER in (31) which can be found as (29) (32) where , , , , , and where is the eigenvalue of the channel covariance matrix . When the SNR is high, . To select power loading coefficients, we now formulate the and are negligible; thus, can be well following optimization problem: approximated by

(30) where . Based on (30), it is ar- gued in [7] and [31] that differential modulation incurs 3-dB subject to (33) penalty in SNR relative to orthogonal STBC with coherent detection, since the noise power is doubled. However, SER for- Using the Lagrange multiplier method, we can find power mulas for coherent modulation cannot be readily applied to cal- loading coefficients as follows: culate an approximate SER for differential modulation based on (30), since the phase of is disturbed by the noise and we do not know whether the real and the imaginary parts of (34) are correlated or not. The following fact which we prove in Ap- pendix IV leads to a useful approximate SER expression. Fact 1: and are uncorrelated and have iden- where is the number of beamformers that tical variance . transmit signals, given the transmitted power budget . Note Based on this fact, we see that the signal model (30) is the that the solution in (34) is similar to the optimal power loading same as that of -PSK modulation with maximum ratio com- for coherent STBC of [35] except for a factor 2 in the second bining (MRC) [27, p. 266]. This implies that we can approxi- term, because the in (32) is 3 dB larger than its mately compute the SER using [27, p. 271] counterpart in the coherent STBC of [35]. For the selection of and detailed description of the power-loading algorithm, we refer the reader to [35]. (31) Orthogonal STBC is well appreciated for its low decoding complexity, coding, and diversity gains [29]. While the works in [7], [8], [16], and [31] show that orthogonal STBC can be where , and is the moment-gener- modified to facilitate differential modulation and detection, our ating function (MGF) of the random variable .For work here reveals that the loaded eigen-beamforming derived in Rayleigh fading, a closed SER formcan be obtained from (31) [35] for coherent STBC, can also be used in a differential STBC by using the partial fraction expansion of the MGF. Since the in- setup. stantaneous SNR of is , the approximate SER derived from (30) is 3 dB worse than that of coherent modula- IV. SIMULATIONS AND NUMERICAL RESULTS tion with the same transmitted power. For BSPK and QPSK, we We consider a linear array of equispaced (by ) can also consider only, and derive an approximate BER. antennas at the transmitter, and antenna at the receiver. This approach will be used in Section IV to calculate the ap- We assume that the direction of departure is perpendicular to proximate BER for QPSK. the transmitter antenna array. Let be the wavelength of the transmitted signal, and denote the angle spread. When is C. Power Loading small, channel’s spatial correlation can be calculated from the Here, we will derive power loading coefficients contained in “one-ring” channel model as [26] the diagonal matrix in (4). Our goal is to reduce error probability by exploiting the knowledge of . Although exact BER expressions for BPSK and QPSK constellations of Section III-A (35) are useful for benchmarking performance, they are difficult to 1284 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 4, APRIL 2006

Fig. 2. Simulated and exact BERs of QPSK with known . Fig. 1. Exact and approximate BERs of QPSK with known .

Since the channel’s spatial correlation is considered as a slowly varying effect similar to shadowing [21], it re- mains invariant in our simulations. We will consider two channels with different correlations: channel 1 has and , while channel 2 has and . Channels are normalized so that . For channel 1, the eigenvalues of are contained in ; and for channel 2, we have . While channel 1 is highly correlated, channel 2 is less correlated and provides more diversity. QPSK and 8-PSK constellations will be adopted. Since we use a 4 4 orthogonal STBC, the spectral efficiency is 1.5 b/s/Hz for QPSK and 2.25 b/s/Hz for 8-PSK. In all plots, the SNR is defined as SNR . Fig. 1 displays analytical results of BER performance for QPSK. The correlation matrix of channel 1 has two very small eigenvalues: if we use equal power loading, the power transmitted along two eigenvectors corresponding to these two Fig. 3. Simulated and approximate SERs of 8-PSK with known . small eigenvalues is wasted in the SNR region of practical interest. Hence, the optimal power loading outperforms equal of successive blocks in each run but is independently gener- power loading by more than 3 dB in the SNR region of interest. ated from run to run. Hence, our simulations are valid for fast Channel 2 is less correlated; thus, the performance gap between fading channels as long as the channel variation is negligible optimal and equal power loading is smaller, but still noticeable. in two consecutive blocks, even though over a large number of Recall that the performance of DSTM with beamforming and blocks the cumulative variation can be indeed fast, which is a re- equal power loading is the same as that of DSTM without quirement common to all differential schemes. The error prob- beamforming. Hence, these BER curves demonstrate clearly ability is obtained by averaging the total number of errors over the advantage of combining DSTM with optimally loaded the number of runs. It is seen that the simulation results match beamforming. Both exact and approximate BERs are shown very well with the analytical BER. Fig. 3 depicts SER results in the figure, where it is observed that the approximate BER obtained from simulations and the approximate SER calculated curve is very close to the exact BER curve even in the low from the analytical formula in (31) for 8-PSK. Similar to the SNR region, which justifies the optimal power loading scheme error probability results in Figs. 1 and 2, we confirm again that derived from the approximate BER. optimal power loading offers considerable performance gains Fig. 2 compares simulated against exact BER for QPSK. In for both channels. The approximate SER curve almost coincides simulations, is assumed perfectly known at the transmitter. with the simulated SER curve, which again corroborates that it We ran simulations times. In each run, we generate two con- is reasonable to derive the power loading scheme from the ap- secutive space–time coding blocks: in the first block , proximate SER. and in the second block , where is a randomly gener- While the downlink channel’s spatial correlations can be ated orthogonal STBC matrix. The channel is fixed over a pair estimated from the uplink channel under certain conditions CAI AND GIANNAKIS: DSTM WITH EIGEN-BEAMFORMING FOR CORRELATED MIMO FADING CHANNELS 1285

Fig. 4. BER performance of QPSK with estimated . Fig. 5. SER performance of 8-PSK with estimated .

[20], we can also estimate channel correlations at the receiver and feed them back to the transmitter. Since spatial correlations change with time very slowly, they only require a low-rate feedback channel, which is often feasible in practice. When channel correlations are not available at the transmitter, no beamforming is employed and the transmitted block is , with . At the receiver end, from the detected codeword , we can calculate , and a channel estimate can be obtained as . Note that this channel estimate is not normalized by , but this will not affect power loading scheme since the transmitter can normalize eigenvalues of the estimated correlation matrix before calculating power loading coefficients. We form every blocks, where is chosen sufficiently large so that two consecutive samples are uncorrelated. After collecting estimates , we average them to estimate the channel correlation as: . Once the transmitter Fig. 6. BER comparison between coherent STBC and DSTM with optimally acquires this channel correlation, signals are transmitted with loaded eigen-beamforming. loaded eigen-beamforming according to (4). This channel correlation estimate is biased by the noise variance. Since the is worth to emphasize here that coherent STBC requires CSI at noise variance is easy to measure at the receiver when no data the receiver, and thus, it is not applicable to fast fading chan- are transmitted, an unbiased channel correlation estimate can be nels; whereas DSTM does not require CSI at the receiver, and written as . Note that thus, it performs well in fast fading channels. Both coherent this channel estimate is only used for estimating the channel STBC and DSTM use QPSK. When we calculate the BER of correlation, but not for detecting transmitted symbols. While coherent STBC, we assume that the receiver knows the channel the error in may degrade the channel estimates, it will not perfectly. If we take into account the channel estimation error cause significant error in because of the averaging involved and the transmitted power consumed by training, the BER of in estimating . This is confirmed by the simulation results in coherent STBC will be higher. On the other hand, since DSTM Figs. 4 and 5, which depict the error probability of QPSK and does not require CSI at the receiver, the BER curve in Fig. 6 is its 8-PSK, when estimates are used. In estimating , we use actual BER. We see that for channel 1, the proposed differential , and the noise variance is not subtracted. Comparing modulation scheme actually outperforms the coherent STBC. Figs. 4 and 5 with Figs. 2 and 3, we see that when the SNR is For channel 2, the error performance gap between these two moderate-high, using instead of does not degrade error schemes will be very small after taking into account the channel performance. estimation in coherent STBC. Training in coherent STBC also Since certain practical systems, e.g., 3GPP WCDMA, employ incurs a data rate loss; hence, DSTM offers higher data rates. coherent STBC without beamforming [13, p. 97], we compare Of course, beamforming and optimal power loading can also be the BERof DSTM with optimally loaded eigen-beamforming employed with coherent STBC to improve error probability per- with that of coherent STBC without beamforming in Fig. 6. It formance as in [35]. However, our results here demonstrate that 1286 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 4, APRIL 2006

and by the deﬁnition of ,wehave (37) Combining (36) and (37), we obtain (38) Since ,if is an eigenvalue of , then must be an eigenvalue of ; also, and have the same corresponding eigenvector . Multiplying (38) by from the right gives , which implies

(39) Solving this equation, we obtain the two distinct eigenvalues given in Lemma 1.

APPENDIX II Fig. 7. BER of QPSK versus antenna spacing. PROOF OF PROPOSITION 1 Let columns of matrices and consist of eigen- in highly correlated channels, the proposed differential modu- vectors of corresponding to eigenvalues and lation scheme has better or comparable error probability per- , respectively; and define , and formance, relative to the conventional coherent STBC without . Then, we have . beamforming. Recall that and , we infer that if Fig. 7 depicts the BER of QPSK versus antenna spacing for is an eigenvector of corresponding to eigenvalue , typical rural angle spread and urban angle spread it is also an eigenvector of corresponding to eigenvalue environments [28], where we use the channel correlation ; thus, we have . Since , computed from (35) and SNR 18 dB. When the angle spread we can also write this equation as .Defining is small, the channels are highly correlated for a wide range and using (23), we obtain of antenna spacings; and thus, the eigen-beamforming and the optimal loading achieve considerable performance gain. On the other hand, when the angle spread is relatively large, the channel (40) correlation is small after the antenna spacing increases beyond . In this case, the channel correlation matrix is approximately Define a permutation matrix , proportional to the identity matrix and beamforming does not where is a 3 3 matrix whose antidiagonal elements are all improve performance. ones while all other elements are zeros, and stands for Kro- necker product. Note that since is a permutation matrix, we have . It can be shown that V. C ONCLUSION We have analyzed the error probability performance of differential space–time modulation based on orthogonal space–time block coding. Our performance analysis is useful for both spa- (41) tially independent and correlated channels. While an exact BER and thus analysis was carried out for BPSK and QPSK, an approximate SER was derived for all -PSK constellations. Our performance analysis results guided our novel concatenation of differential space–time modulation with eigen-beamforming and derivation of optimal power loading to enhance the error prob- (42) ability performance in correlated channels. Both analysis and Combining (23) and (42), we obtain simulations illustrated that our scheme can achieve considerable performance gain in correlated channels relative to differential (43) space–time modulation without beamforming. Because and are unitary, and , the matrix is similar to the matrix . APPENDIX I PROOF OF LEMMA 1 APPENDIX III PROOF OF COROLLARY 1 From (1) and (2), it can be shown that Since is similar to , matrices and have the same (36) eigenvalues, and it suffices to find the eigenvalues of . We will CAI AND GIANNAKIS: DSTM WITH EIGEN-BEAMFORMING FOR CORRELATED MIMO FADING CHANNELS 1287 use the following identity regarding the determinant of a block ; and similarly, the variance of , , partitioned matrix [19, p. 46] and equals .

if (44) REFERENCES

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[24] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Efficient use Georgios B. Giannakis (F’97) received the Diploma of side information in multiple-antenna data transmission over fading degree in electrical engineering from the National channels,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1423–1436, Technical University of Athens, Greece, in 1981 Oct. 1998. and the M.Sc. degree in electrical engineering, the [25] M. Schwartz, W. R. Benett, and S. Stein, Communication Systems and M.Sc. degree in mathematics, and the Ph.D. degree Techniques. Piscataway, NJ: IEEE Press, 1996. in electrical engineering from the University of [26] D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading corre- Southern California (USC), Los Angeles, in 1983, lation and its effect on the capacity of multi-element antenna systems,” 1986, and 1986, respectively. IEEE Trans. Commun., vol. 48, no. 3, pp. 502–513, Mar. 2000. After lecturing for one year at USC, he joined [27] M. K. Simon and M.-S. Alouini, Digital Communication over Fading the University of Virginia, Charlottesville, in 1987, Channels: A Unified Approach to Performance Analysis. New York: where he became a Professor of electrical engi- Wiley, 2000. neering in 1997. Since 1999, he has been a Professor with the Department of [28] R. A. Soni, R. M. Buehrer, and R. D. Benning, “Intelligent antenna Electrical and Computer Engineering at the University of Minnesota, Min- system for CDMA2000,” IEEE Signal Process. Mag., vol. 19, no. 4, pp. neapolis, where he now holds an ADC Chair in Wireless Telecommunications. 54–67, Jul. 2002. His general interests span the areas of communications and signal processing, [29] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–time block estimation and detection theory, time-series analysis, and system identifica- codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, pp. tion—subjects on which he has published more than 220 journal papers, 380 1456–1467, Jul. 1999. conference papers, and two edited books. His current research focuses on [30] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for transmitter and receiver diversity techniques for single- and multiuser fading high data rate wireless communication: Performance criterion and code communication channels, complex-field and space–time coding, multicarrier, construction,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp. 744–765, Mar. ultra-wide-band wireless communication systems, cross-layer designs, and 1998. sensor networks. [31] V. Tarokh and H. Jafarkhani, “A differential detection scheme for Dr. Giannakis is the (co-)recipient of six paper awards from the IEEE Signal transmit diversity,” IEEE J. Sel. Areas Commun., vol. 18, no. 7, pp. Processing (SP) Society and the IEEE Communications Society (1992, 1998, 1169–1174, Jul. 2000. 2000, 2001, 2003, and 2004). He also received the SP Society’s Technical [32] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Achievement Award in 2000. He served as Editor-in-Chief for the IEEE SIGNAL Telecommun., vol. 10, pp. 585–595, Nov. 1999. PROCESSING LETTERS, as Associate Editor for the IEEE TRANSACTIONS ON [33] E. Visotsky and U. Madhow, “Space–time transmit precoding with SIGNAL PROCESSING and the IEEE SIGNAL PROCESSING LETTERS, as Secretary imperfect feedback,” IEEE Trans. Inf. Theory, vol. 47, no. 6, pp. of the SP Conference Board, as Member of the SP Publications Board, as 2632–2639, Sep. 2001. Member and Vice-Chair of the Statistical Signal and Array Processing Tech- [34] S. Zhou and G. B. Giannakis, “Optimal transmitter eigen-beamforming nical Committee, as Chair of SP for the Communications Technical Committee, and space–time block coding based on channel mean feedback,” IEEE and as a Member of the IEEE Fellows Election Committee. He has also served Trans. Signal Proc., vol. 50, no. 10, pp. 2599–2613, Oct. 2002. as a Member of the IEEE SP Society’s Board of Governors, the Editorial Board [35] , “Optimal transmitter eigen-beamforming and space-time block for the PROCEEDINGS OF THE IEEE, and the steering committee of the IEEE coding based on channel correlations,” IEEE Trans. Inf. Theory, vol. 49, TRANSACTIONS ON WIRELESS COMMUNICATIONS. no. 7, pp. 1673–1690, Jul. 2003.

Xiaodong Cai (S’00–M’01–SM’05) received the B.S. degree from Zhejiang University, China, the M.Eng. degree from the National University of Singapore, Singapore, and the Ph.D. degree from the New Jersey Institute of Technology, Newark, all in electrical engineering. From February 2001 to June 2001, he was a Member of Technical Staff at Lucent Technologies, Whippany, NJ. From July 2001 to October 2001, he was a Senior System Engineer at Sony technology center, San Diego, CA. From November 2001 to July 2004, he was a Postdoctoral Research Associate in the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis. Since August 2004, he has been an Assistant Professor in the Department of Electrical and Computer Engineering, University of Miami, Coral Gables, FL. His research interests lie in the areas of communication theory, signal processing, and networking. His current research focuses on cooperative communications and collaborative signal processing for wireless ad hoc and sensor networks.