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Optimal Transmitter Eigen-Beamforming and Space-Time Block Coding based on Channel Correlations?

Shengli Zhou and Georgios B. Giannakis

Dept. of ECE, Univ. of Minnesota, 200 Union Street SE, Minneapolis, MN 55455

Abstract—Optimal transmitter designs obeying the water-?lling principle are well-documented, and widely applied when the propagation channel is deterministically known and regularly updated at the transmitter. Because channel state information may be costly or impossible to acquire in rapidly varying wireless environments, we develop in this paper statistical water-?lling approaches for stationary random fading channels. The resulting optimal designs require only knowledge of the channel’s second order statistics that do not require frequent updates, and can be easily acquired. Optimality refers to minimizing a tight bound on the symbol error rate. Applied to a multiple transmit-antenna paradigm, the optimal precoder turns out to be a generalized eigen-beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the channel’s covariance matrix, and with proper power loading across the beams. Coupled with orthogonal space time block codes, two-directional eigen-beamforming emerges as a more attractive choice than conventional one-directional beamforming, with uniformly better performance, and without rate reduction or complexity increase.

I. I NTRODUCTION Multi-antenna diversity is well motivated for wireless communications through fading channels. In certain applications, e.g., cellular downlink, multiple receive antennas may be expensive or impractical to deploy, which endeavors diversity systems relying on multiple transmit antennas. When perfect or partial channel state information (CSI) is made available at the transmitter, multi-antenna systems can further enhance performance and capacity [7]. For slowly time-varying wireless channels, this amounts to feeding back to the transmitter the instantaneous channel estimates [7, 11]. But when the channel varies rapidly it is costly, yet not meaningful, to acquire CSI at the transmitter, because optimal transmissions tuned to previously acquired information become outdated quickly. Designing optimal transmitters based on statistical information about the underlying stationary random channel, is thus well motivated. So long as the channel remains stationary, it has invariant statistics. Through ?eld measurements, or theoretical models, the transmitter can acquire such statistical CSI a priori [8]. Alternatively, the receiver can estimate the channel correlations, and feed them back to the transmitter on line (this is referred to as covariance feedback in [5, 11]). Based on channel covariance information, optimal transmitter design has been pursued in [5, 11] based on a capacity criterion. Focusing on symbol by symbol detection, optimal precoding was designed in [2] to minimize the symbol error rate (SER) for differential BPSK transmissions, and in [4] for PSK based on channel estimation error, and conditional mutual information criteria. In this paper, we design optimal transmit-diversity precoders for widely used constellations, and our performance-oriented

? This work was supported by the NSF Wireless Initiative grant no. 9979443, the NSF grant no. 0105612, and by the ARL/CTA grant no. DAAD1901-2-011.

approach relies on the Chernoff bound on SER. Optimal precoders turn out to be eigen-beamformers with multiple beams pointing to orthogonal directions along the eigenvectors of the channel’s covariance matrix; hence, the name optimal transmitter eigen-beamforming. The optimal eigen-beams are power loaded according to a spatial water-?lling principle. To increase the data rate without compromising the performance, we also propose parallel transmissions equipped with orthogonal space time block coding (STBC) [1, 3, 10] across optimally loaded eigen-beams. Interestingly, coupling optimal precoding with orthogonal STBC leads to a two-directional eigenbeamforming that enjoys uniformly better performance than the conventional one-directional beamforming without rate reduction, and without complexity increase. Notation: Bold upper (lower) letters denote matrices (column vectors); ? ? , ? ? and ? ? denote conjugate, transpose, and Hermitian transpose, respectively; ? stands for the absolute value of a scalar and the determinant of a matrix; E ? stands for expectation, tr ? for the trace of a matrix; Re ? stands for the real part of a complex number; ? denotes the identity matrix of size ? ; ? ?? denotes an all-zero matrix stands for a diagonal matrix with with size ? ? ? ; diag on its diagonal; ? ? denotes the ?th entry of a vector.

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II. S YSTEM M ODEL Fig. 1 depicts the block diagram of a transmit diversity system with a single receive- and ?? transmit- antennas. In the th is ?rst transmit-antenna, the information-bearing signal × ?? spread (or, precoded) by the code ? of length ? to obtain the chip sequence: ? ? ?? ? ? ? . The transmission channels are ?at ?? × faded (frequency non-selective) with complex fading coef??? . The received samples in the prescients , ence of additive Gaussian noise ? ? are thus given by:

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??

?? ?

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? ??

?

????

??

??

???

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?? ×? ? ?? ? ? ? · ????

(1)

To cast (1) into a convenient matrix-vector form, we de?ne ? ? ? ? ? ? ? (likethe ? ? vectors ? ), the ?? ? channel vector wise for ? ?? , and the ? ? ?? code matrix ? ?? . Eq. (1) can × . Because we then be re-written as: will focus on symbol by symbol detection, we omit the symbol index , and subsequently deal with the input-output model

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???

×·?

(2)

At the receiver, the channel is acquired ?rst to enable maximum ratio combining (MRC) using

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(3)

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transmitter

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Notice that the SNR expression (6) coincides with that of the MRC output for ?? independent channels [9], with ? ? ? × ?? denoting the th subchannel’s SNR. Averaging over the Rayleigh distributed ? , closed form SER expressions are found in [9] for ? -ary phase shift keying (? PSK), and square ? -ary quadrature amplitude modulation (? -QAM) constellations, as:

receiver

Fig. 1. Discrete-time baseband equivalent model

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× ×

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×

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(7)

The MRC receiver maximizes the signal to noise ratio (SNR) ? ? ? . at its output, and yields × ??? For a given precoder , eq. (3) speci?es the optimal receiver in the sense of maximizing the output SNR. The question that arises is how to select an optimal precoder . In the following, we design optimal for random fading channels, based on knowledge of ? channel’s second-order statistics: namely ¨ ? the ¨ ?? E , and ?? E .

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III. O PTIMAL E IGEN -B EAMFORMING Throughout this paper, we adopt the following assumptions: a0) the channel is complex Gaussian distributed, with zero; mean, and covariance matrix is zero-mean, white, complex Gaussian with a1) the noise each entry having variance ?? per real and imaginary dimension, i.e., ?? ?? ? ; a2) the channel and the noise are uncorrelated. ?? ? is available a3) channel correlation information at the transmitter. Our performance metric for optimal precoder design will be symbol error rate (SER). We next derive a closed-form SER expression. The SNR at the MRC output for a ?xed channel re×? E ? ? ? . Dealization is ? E ? ? E × ? as the average energy of the underlying noting × signal constellation, the SNR ? becomes:

? ? ,? ? is the moment where ? ? generating function of the probability density function (p.d.f) ? evaluated at ? ? [9, eq. (24)], and the of ? constellation-speci?c is given respectively, by:

? ??

?? ?

??

?

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??

(8)

? ×?

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? ?

?

× ?? ?

?

and

? ?

? ??? ? ??

(9)

?

? ?

Because

is Rayleigh, ?

??

??

? ??

?? ? has the form: [9]: ? ?? · ? × ?? ???

(10)

A. Chernoff Bound Criterion Our ultimate goal is to minimize the SER in (7), or (8), with respect to . However, direct optimization based on the exact SER turns out to be dif?cult because of the integration involved. Instead, we will design the optimal precoder based on a tight Chernoff bound on SER. Using the de?nite integral form for the Gaussian Q-function, the well-known Chernoff bound can be easily expressed as:

??

?

? ?

To simplify (4), we diagonalize

? ? diag? ? (5) ?? ? denotes the th eigenvalue of where ? is unitary, and ? that is non-negative. Without loss of generality, we assume that ’s are arranged in a non-increasing order: ? ??? ?. Using (5), we can pre-whiten to , so ?? ? , and the entries of are i.i.d with unit varithat ? ??? . Therefore, the SNR of (4) reduces ance: E ? ?? ? ? to ? × ?? Let us now de?ne ? ? ? ? , which is non-negative de?nite, ? ?? , where and thus it can be decomposed as: diag??? ??? ? contains the ?? non-negative eigen?? has values of . Because ? is unitary, the vector ? ? ? ), with covariance matrix i.i.d entries (denoted by ??? . The SNR can then be further simpli?ed to ? × ?? ??? ? ? ? × ?? (6) ? ? ? ?? ?

? ? ? ? ? ? ? ? ? ?

??

?

×

??

(4)

as:

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? ?

?

?

?? ? ? ×???

?

? ?????? ?? ? ? ?

(11)

, by observing that the maximum of the integrand for any ? [9]. Likewise, ? ? in (10) peaks at occurs at , and thus the Chernoff bound for the SER in (7) and (8) can be obtained in a unifying form:

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? E× ??

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?? ?

×? ?

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(12)

? ? ?, and takes on constellation-speci?c where ? values as in (9). The upper bound in (12) can also serve (within ?× ??? a scale) as a lower bound of the SER, e.g., ?× ? ?? ?× ??? for QPSK and ?? [12]. The optimal precoder will be chosen to maximize:

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×? ?

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that will turn out to be more convenient. The equivalent constrained optimization problem is simpli?ed to ??? ? ? subject to (14) ? ?

? ? ? ? where diag? ? ?? ? (13) ? ? ? ?? . Since ?? ? ??? is a monotoniwith cally increasing function, we can equivalently optimize the cost ?? ? ? ? ?? ? ??? · ? function ? ? ? × ?? , ?? ? ? ? ?

under the constraint tr ? , i.e., the average transmitted power per symbol is × . is maximized when the matrix The cost function ? ? is diagonal [12]. We subsequently express:

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? ??

? ?

with respect to ? , Differentiating the Lagrangian ? where denotes the Lagrange multiplier, and equating it to zero, we obtain:

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? ?·

?? ? ?? ?? ? ?? ?

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? . Suppose that the with the special notation ? · given power budget × supports ?? non-zero ? ’s. Solving for using the power constraint, we arrive at the optimal loading:

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?? ? ?

× ·

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(15)

?· ??

??

×

??

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(16)

·

eigenvalues implies The non-increasing order of the ? ???? ? , as con?rmed by (16). We ?rst set that: ? ? ?? ? ? . The entry ?? in (16) with ?? ?? , and test if ?? ?? ?? imposes the following lower bound on the required ? ?? ?? ? ?? ? ?? ?? . SNR: × ?? If × is not large enough to afford optimal power allocation ? , eq. (16) suggests that across all ?? beams, causing ?? ? , and set we should turn off the ?? th beam by setting ?? ?? ?? ? ; and so on until we will ?nd the desired ?? . The practical power loading algorithm is summarized as: ?? , calculate ?? ? based only on the ?rst ? S1) For ? channel eigenvalues as ? ?? ? ? ?? ? ? ? (17)

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? ?

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?? ??

? ?

S2) With the given power budget × ensuring that × ?? falls , and obin the interval ?? ? ?? ?·? , set ?·? ?? tain ? ? according to (16) based only on ? ?. Having speci?ed the optimal ? , we have found the optimal ? in (13). The optimal can be factored from (13) as:

?

where the columns of are orthonormal, and the diagonal enare given by (16). We summarize our result as: tries of Theorem 1: Suppose a0)-a3) hold true. The optimum receive?lter ??? is given by (3), and the optimum precoding matrix ? has and formed as in (5), (16) and ??? an arbitrary orthonormal ? ? ?? matrix. Op(13) with timality in ??? refers to maximum-SNR, while optimality in ??? pertains to minimizing the Chernoff bound on the average symbol error rate.

¨

¨ ??

(18)

¨ ? ¨

?

The ? ? ?? optimal precoder in (18) can be interpreted as a generalized beamformer. Different from conventional beamforming that transmits all available power along the channel’s strongest direction (implemented via the ?rst row of ? ), here ?? beams are formed pointing to ?? orthogonal directions ; along the eigenvectors of the channel covariance matrix takes care thus, the name eigen-beamforming. The matrix of power loading across all beams. Notice that more power ? ? ? is distributed to stronger channels since ? ? ??? ?? . ? ?? × is constant ; thus, the Furthermore, power allocation obeys the water-?lling principle. When the system operates at a prescribed power: × ?? ? eigen-beams ?? ? ?? ?·? , it is clear that only ? rank are used, and a diversity order of ? is achieved. Full diversity schemes correspond to ? ?? . Based on (17), one can easily determine what diversity level to be used for the best performance with a given power budget × . We thus have: Corollary 1: The optimal diversity order is ?, when × ?? falls in the interval: ?? ? ?? ?·? , with ?? ? de?ned in (17). Corollary 2: Full diversity schemes are not SER-bound optimal across the entire SNR range; their optimality is ensured only when the SNR is suf?ciently high: × ?? ?? ?? . Notice that apart from requiring it to be orthonormal, so far we left the ? ? ?? matrix unspeci?ed. To fully exploit the ?? is required. On the diversity offered by ?? antennas, ? other hand, the choice ? ?? does not improve performance; it is thus desirable to choose ? as small as possible to increase the transmission rate. When the desired diversity order is ?, as in Corollary 1, we can reduce the ? ? ?? matrix to an ? ? ?? fat matrix ????? ??? , where is any ? ? ? orthonormal matrix, without loss of optimality. This way, we can achieve rate ? for a diversity transmission of order ?. On the other hand, one can a priori force the matrix (and thus ) to be fat with dimensionality ? ?? , which corre, deterministically. Opsponds to setting ·? ?? timal power loading can then be applied to the remaining beams. We will term this scheme (with chosen beforehand to be ? ?? ) an -directional eigen-beamformer. As a consequence of Theorem 1, we then have: ?? , the -directional eigenCorollary 3: With beamformer achieves the same average SER performance as an ?? -directional eigen-beamformer, when × ?? ?? ·? . Two interesting special cases of Corollary 3 arise. The ?rst is conventional one-directional (1D) eigen-beamforming , [5, 7, 11]. As asserted in Corollary 3, the 1D with ?? ? eigen-beamformer will be optimal when × ?? ; i.e., when the ?rst and second eigenvalues ?? ? are disparate enough, or, when × ?? is suf?ciently low. The more interesting case is 2D eigen-beamforming which . The 2D eigen-beamformer is optimal corresponds to . Notice that when × ?? ?? ? ?? ?? ? the optimality condition for 2D beamforming is less restrictive ?? ? , and than that for the 1D beamforming, since ?? ? ?? ? does not depend on ? and ? . As we shall see in Section IV, the 2D eigen-beamforming also achieves the same rate as 1D beamforming, and subsumes the latter as a special case.

B. Optimally Loaded Eigen-beamforming Interpretation

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IV. E IGEN -B EAMFORMING

AND

STBC

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In the system model (2), we transmit only one symbol over ? time slots (chip-periods), which essentially amounts to repetition coding (or a spread-spectrum) scheme. To overcome the associated rate loss, it is possible to send ? symbols ×? ×? simultaneously. Certainly, this would require symbol separation at the receiver. But let us suppose temporarily that the separation is indeed achievable, and each symbol is essentially going through a separate channel identical to the one for × we dealt with in Section III. The optimal precoder will then be

Ant. 1

time

×? , ×?

×?

?

×? ?

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×? ×? ?

?? ? ? ??

??

Ant. Nt

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¨

in (19) is common , designing Because the factor ’s. separable precoders is equivalent to choosing separable Fortunately, this degree of freedom can be afforded by our de’s are only required to sign in Section III because so far the have orthonormal columns. The desired means of data multiplexing that enables symbol separability at the receiver is possible through orthogonal space-time block coding (STBC) [1, 3, 10]. Our combining of optimal eigen-beamforming with STBC is treated next for complex symbols (see [12] for real symbols). Let ×? and ×? denote the real and imaginary part of × , respectively. The following orthogonal STBC designs are available for complex symbols [3, 10]: ¨ ? ×? ×? ? ? , and De?nition: For complex symbols × ? ? ??? matrices ? each having entries drawn from ? , the space time coded matrix

??

??

space?time block coding

power loading

antenna weighting (eigen beamforming)

?? ¨

?

(19)

Fig. 2. The two-directional (2D) eigen-beamformer, ?? ?

?

? ?

¨

that (23) is nothing but the MRC output for the single symbol transmission studied in Section III; thus, the optimal loading in (16) enables space-time block coded transmissions to minimize the Chernoff bound on SER, but with rate ? ? . The combination of orthogonal STBC with beamforming has also been studied in [6]. However, the focus in [6] is on channel mean feedback [11] for slowly fading channels, while our approach here is tailored for fast fading random channels. For complex symbols, a rate 1 GCOD only exists for ?? . It corresponds to the well-known Alamouti code [1]:

?

?? ?

¨ ?

·

??

?

?×? ×? ? ?

×?

×?

space time

(24)

???

?

??

?

¨ ×? ·

??

?

? ×?

(20)

?? ¨ ? ¨?¨ ??? ??? ??? ¨?¨? ?¨?¨ ???? ???? ? (21) ? ? ? ?? ?? ¨ ¨ ? ? ? For each complex symbol × ×? · ×? , we de?ne two pre¨ ?? , coders corresponding to ¨ ? as: ? ? . The transmitted STBC is now ? ? and ? ?? ? ?? ? ??? ?? (22) ?? ?× · ?× ? ?

At the th detector output, the decision variable is formed by

is termed a generalized complex orthogonal design (GCOD) in variables × ? ? of size ? ? ?? and rate ? ? , if either one of two equivalent conditions holds true: ?? ? i) ?? ?? ?? [10], or, ? ? × ? ii) The matrices ? satisfy the conditions [3]:

? ?

?

where ? equality in (23) can be easily veri?ed by using (21) [3]. Notice

? · Re ? ? ?? ? (23) ? ? ? ?? × · ? ??? has variance ?? ? ? ? ?? ; and the second

Re

?

? ?

, rate orthogonal STBCs exist, while for For ?? ?? , only rate codes have been constructed [10], [3]. Therefore, for complex symbols, the ?? -directional eigenbeamformer of (22) achieves optimal performance with no , and pays a rate penalty when rate loss only when ?? ?? . To tradeoff the optimal performance for a constant rate 1 transmission, it is possible to construct a 2D eigenbeamformer with the Alamouti code applied on the strongest two-directional eigen-beams. Speci?cally, we can construct ? , which a ? ?? matrix 2-d ? ????? ??? achieves the optimal performance as the ?? -directional eigenbeamformer when × ?? ?? ? , as speci?ed in Corollary 3. The implementation of the 2D eigen-beamformer is depicted in Fig. 2. Notice that the optimal scenario for 1D beamforming was speci?ed in [5] from a capacity perspective. The interest in 1D beamforming stems primarily from the fact that it allows scalar coding with linear pre-processing at the transmit-antenna array, and thus relieves the receiver from the complexity required for decoding the capacity-achieving vector coded transmissions [5, 7, 11]. Because each symbol with 2D eigen-beamforming goes through a separate but better conditioned channel, the same capacity-achieving scalar code applied to an 1D beamformer can be applied to a 2D eigen-beamformer as well. Therefore, 2D eigen-beamforming outperforms 1D beamforming even from a capacity perspective, since it can achieve the has only same coded BER with less power. Notice that if one nonzero entry ? , the 2D eigen-beamformer reduces to the 1D beamformer, with ×? and ?×? transmitted during consecu?

? ??

?

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556

30

10

0

25

Distributed power per branch/ noise power (dB)

Optimal loading, QPSK Optimal loading, 16?QAM Equal power loading 1D beamforming

10

?1

20

Symbol Error Rate

10

?2

15

first beam

10

10

?3

5

10

?4

0

second beam

1D Beam forming, Chernoff bound 1D Beam forming, exact SER Equal power loading, Chernoff bound Equal power loading, exact SER Optimal loading, Chernoff bound Optimal loading, exact SER

?5 ?5

0

5

10 15 20 Total Power/ noise power (dB)

25

30

35

0

5

10

15 SNR

20

25

30

Fig. 3. Optimal vs. equal power loading

Fig. 4. SER vs

10

0

×

?? : QPSK

V. N UMERICAL R ESULTS We consider a uniform linear array with ?? antennas at the transmitter, and a single antenna at the receiver. We assume that the side information including the distance between the transmitter and the receiver, the angle of arrival, and the angle spread are all available at the transmitter. Let be the wavelength of a narrowband signal, ? the antenna spacing, and the angle spread. We assume that the angle of arrival is perpendicular to the transmitter antenna array (“broadside” as ? (see [12] for additional , and in [8]), and ? setups). The correlation coef?cients among the antennas are then calculated by [8, eq. (6)]. Fig. 3 shows the optimal power allocation among different beams, for both QPSK and QAM constellations. Notice that the choice of how many beams are retained depends on the constellation-speci?c SNR thresholds. For QPSK, we dB, and ?? ? dB. Since can verify that ?? ? , the threshold ?? ? for 16-QAM is ?? ?? ? ? ? dB higher for QPSK; we observe that dB ?? higher power is required for 16-QAM before switching to the same number of beams as for QPSK. Figs. 4 and 5 depict the exact SER, and the Chernoff bound for: optimal loading, equal power loading, and 1D beamforming. Since the considered channel is highly correlated, only ? beams are used in the considered SNR range for optimal loading. Therefore, the 2D eigen-beamformer is overall optimal for this channel in the considered SNR range, and its performance curves coincide with those of the optimal loading. Figs. 4 and 5 con?rm that the optimal allocation outperforms both the equal power allocation, and the 1D beamforming. The small gap between the Chernoff bound and the exact SER in Figs. 4 and 5 justi?es our approach that pushes down the Chernoff bound to minimize the exact SER.

Symbol Error Rate

tive time-slots, as con?rmed by (24). This leads to the following conclusion: Corollary 4: The 2D eigen-beamformer includes 1D beamformer as a special case and outperforms it uniformly, without rate reduction and without complexity increase. Corollary 4 suggests that 2D eigen-beamformer is more attractive than 1D beamformer, and deserves more attention.

10

?1

10

?2

1D Beam forming, Chernoff bound 1D Beam forming, exact SER Equal power loading, Chernoff bound Equal power loading, exact SER Optimal loading, Chernoff bound Optimal loading, exact SER

10

?3

0

5

10

15 SNR

20

25

30

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Fig. 5. SER vs

×

?? : 16-QAM

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R EFERENCES

[1] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE JSAC, vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [2] J. K. Cavers, “Optimized use of diversity modes in transmitter diversity systems,” in Proc. of VTC, 1999, vol. 3, pp. 1768–1773. [3] G. Ganesan and P. Stoica, “Space-time block codes: a maximum SNR approach,” IEEE Trans. on Info. Theory, pp. 1650–1656, May 2001. [4] G. B. Giannakis and S. Zhou, “Optimal transmit-diversity precoders for random fading channels,” in Proc. of Globecom, 2000, pp. 1839–1843. [5] S. A. Jafar and A. Goldsmith, “On optimality of beamforming for multiple antenna systems with imperfect feedback,” in Proc. of IEEE ISIT, June 2001, pp. 321–321. [6] G. J¨ ngren, M. Skoglund, and B. Ottersten, “Combining transmit beamo forming and orthogonal space-time block codes by utilizing side information,” Proc. of 1st IEEE Sensor Array and Multichannel. Signal Proc. Workshop, Boston, MA, Mar. 2000, pp. 153–157. [7] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Ef?cient use of side information in multiple-antenna data transmission over fading channels,” IEEE JSAC, vol. 16, pp. 1423–1436, Oct. 1998. [8] D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correlation and its effect on the capacity of multi-element antenna systems,” IEEE Trans. on Communications, vol. 48, no. 3, pp. 502–513, Mar. 2000. [9] M. K. Simon and M.-S. Alouini, “A uni?ed approach to the performance analysis of digital communication over generalized fading channels,” Proc. of the IEEE, vol. 86, no. 9, pp. 1860–1877, Sept. 1998. [10] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. on Information Theory, vol. 45, no. 5, pp. 1456–1467, July 1999. [11] E. Visotsky and U. Madhow, “Space-time transmit precoding with imperfect feedback,” IEEE Trans. on Info. Theory, Sept. 2001. [12] S. Zhou, and G. B. Giannakis, “Optimal transmitter eigen-Beamforming and space-time block coding based on channel correlations,” IEEE Trans. on Info. Theory, submitted September 2001.

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