Performance comparison between beamforming and spatial ...

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Performance comparison between beamforming and spatial multiplexing for the downlink in wireless cellular systems Matilde Sánchez-Fernández, Member IEEE, Santiago Zazo, Member IEEE, Reinaldo Valenzuela, Fellow IEEE

Abstract— We compare the performance between beamforming and spatial multiplexing showing in which downlink scenarios the higher performance of spatial multiplexing justify its complexity. We compute performance using readily measurable parameters such as angle spread (AS), antenna separation and signal to noise ratio (SNR). Firstly, a semi-analytical approach relates these measurable parameters with parameters that theoretically characterize beamforming optimality such as the spatial correlation matrix first two eigenvalues and SNR. Secondly, the achieved spectral efficiency is given for beamforming and spatial multiplexing as a function of antenna separation, AS and SNR. Also, a “practical” region is given where beamforming achieves at least 90% of the spectral efficiency of spatial multiplexing. Index Terms— MIMO, angular spread, antenna separation, beamforming, spatial multiplexing.

I. I NTRODUCTION Any MIMO system provides a number of degrees of freedom for transmitting the information that might vary from one to a number dependent on the number of antenna elements. The higher the number of degrees of freedom available, the better from the spectral efficiency point of view. However, how many of them are effectively available is mostly related to spatial correlation with the exception of keyhole channels [1]. Given a number of degrees of freedom available two strategies stand out. Generalized beamforming makes use of only one of the modes available, while Spatial Multiplexing (SM) makes use of all of them. Low complexity is one of the main advantages of the beamforming (BF) approach. However, BF may be suboptimal. Spatial Multiplexing has the ability to approach the maximum channel capacity. However, it is significantly more complex to implement. The first step in comparing these systems is a consistent treatment of correlation models as in [2]. Recent works tackle from different perspectives the suitability of BF in various correlation and SNR scenarios. For example, in [3] it is treated from an optimization of eigenvalues perspective, in [4] and [5] from a reconfigurable antennas point of view and in [6] with an error exponent focus. The aim of this study is to characterize the performance trade-off between BF to SM in terms of readily measurable parameters such as AS, antenna separation and SNR, showing in which scenarios the extra complexity of SM may be justified. Thus, the well-known beamforming optimality claim in low SNR is further quantified here, making concrete these low This work has been supported by projects CYCIT TIC2002-03498 and TEC 2005-07477-c02-02.

SNR values. Our approach is not only based in BF optimality itself, but on a quasi-optimality of BF, where BF achieves at least 90% of the spatial multiplexing approach. Our study assumes perfect channel knowledge at the receiver while the transmitter is only provided with channel distribution information (CDIT), as a more realistic assumption 1 . A low SNR scenario is assumed for a first semi-analytical approach that relates these measurable or “practical” parameters with “ theoretical” parameters that characterize BF optimality such as the spatial correlation matrix first two eigenvalues and SNR [7], [8]. The result in [9] is used to redefine the optimal region for BF in terms of just two parameters: the maximum eigenvalue of the correlation matrix and the SNR. The second approach shows when the extra complexity of SM results in higher performance as compared to BF. The achieved spectral efficiency for BF and SM strategies is obtained as a function of the antenna separation, AS and SNR. Also, the practical region where the best BF strategy equals at least 90% of SM spectral efficiency is calculated. In this region, although BF is not strictly optimal, it provides a reasonable and much simpler alternative. II. S YSTEM M ODEL A. Propagation Model The MIMO channel is described by a channel matrix H, with elements hnm defining the channel fading from the mth transmitting antenna in the Base Station (BS) to the n-th receiving antenna. The number of transmitting and receiving elements is respectively M and N . We assume a stationary, flat fading channel with spatial correlation. From [10], and sampling the wave vector space of the transmitter and receiver into a finite set of planar waves departing from directions {k1 , k2 , .., kL } and {k01 , k02 , .., k0L0 } respectively, H can be written as in equation (1). There, r0n and rm are respectively the position of the n-th receiving element and the m-th transmitting element and S(k0 , k) is a scattering function of the channel, that relates the planar wave emitted from k direction impinging on the receiver on direction k0 . BN and BM are rectangular deterministic matrices dependent on the antenna geometry and 1 It should be noted that conventional BF requires the knowledge of the angle of arrival. We assume this to be comparable to the requirement of knowledge of the channel statistics (covariance). Furthermore, as shown in equation 5 the channel statistics will be dependent on the channel angular spread and the antenna geometry at the transmitter.

2

  0 0 0 0 0 0 ejk1 r1 ejk2 r1 .. ejkL0 r1 S(k01 , k1 ) S(k01 , k2 ) .. S(k01 , kL ) 0 0 0 0 0 0  ejk1 r2 ejk2 r2 .. ejkL0 r2   S(k02 , k1 ) S(k02 , k2 ) .. S(k02 , kL ) × H =    .. .. .. .. .. .. .. .. jk0L0 r0N jk02 r0N jk01 r0N S(k0L0 , k1 ) S(k0L0 , k2 ) .. S(k0L0 , kL ) e .. e e  −jk r  e 1 1 e−jk1 r2 .. e−jk1 rM  e−jk2 r1 e−jk2 r2 .. e−jk2 rM   = B† SBM × N   .. .. .. .. −jkL r1 −jkL r2 −jkL rM e e .. e 

S is a rectangular matrix whose statistics will be modelled depending on the angular scattering characteristic. Assuming the channel entries are complex Gaussian and that the scatterers are independent, S is completely characterized by the joint power angular spread (PAS). With separable PAS [11], the power distribution within rows is given by the PAS at the receiver while the power distribution among columns matches the PAS at the transmitter. The complete channel can be further simplified as follows, 1

1

H = B†N ΣR2 GΣT2 BM

(2)

where G is a complex Gaussian random matrix with i.i.d components and unit variance. ΣR and ΣT are deterministic diagonal matrices whose main diagonal is shaped with the corresponding PAS. The trace of these matrices is normalized to one. Any implementation of this channel model implies a particular sampling of the wave space defined by the space vector k. Only azimuthal angle will be considered. This is a reasonable starting point given that the elevational angle spread has been measured to be much less than the azimuthal spread for sufficiently large antenna heights [12]. To further define the channel matrix, two uniform linear arrays are assumed with broadside PAS, characterized by M and N and the spacing between antennas in transmission dt and reception dr . 1) Channel correlation: Given the separable PAS assumption from previous section, the spatial correlation between the (i, j)-th and (k, l)-th entries of H leads to a separable spatial correlation, denoted respectively correlation at the transmitter ΘT and at the receiver ΘR . n o RH (i, j; k, l) , E (H)i,j (H)†k,l = (ΘR )i,k (ΘT )j,l (3) where: ΘR =

ª 1 © E HH† = B†N ΣR BN M

(4)

ΘT =

1 © † ª E H H = B†M ΣT BM N

(5)

It should be noted that under the channel model assumed, a closed-form formulation for spatial correlation is possible. The benefit is that this way the correlation matrices are easily built and dependent exclusively on readily measurable and configurable parameters: antenna geometry and angular spread.

    (1)

B. Transmission Strategies Given CDIT, the optimization of the covariance matrix at the transmitter is usually decomposed into separate optimizations for its eigenvectors V and its eigenvalues P [7], [13], [14]. The eigenvectors that maximize spectral efficiency are those of the spatial correlation at the transmitter ΘT . The optimal eigenvalues, in turn, are obtained iteratively [14]. At low SNR [15] and high correlation environments [7] (high disparity between the correlation eigenvalues) the entire power allocation is to the maximal eigenvalue of ΘT , thus a BF solution achieving optimality. III. L OW SNR A NALYSIS : SEMI - ANALYTICAL APPROACH Low SNR regime is the most relevant scenario in mobile communication systems due mainly to mandatory frequency reuse for efficient bandwidth utilization that determines high interference scenarios. A first order approximation is often accepted for capacity expression [15]: µ ¶ µ ¶ S0 Eb Eb Eb = |dB − |dB + ² (6) C No 3dB No No min For CDIT scenario [16]: S0 =

2 Tr {ΘR }

2

2

Tr {ΘR } + Tr {Θ2R }

Eb loge 2 = No min Tr {ΘR } λmax (ΘT )

(7) (8)

Thus, from eqs. 6-8 it can be observed that for capacity analysis at low SNR the key correlation parameters are λmax (ΘT ) varying between 1 (no correlation) © and ª the number of antennas at the transmitter (M ) and Tr Θ2R varying between N (no correlation) and N 2 . From the capacity point of view, the effect of correlation at the transmitter is clearly different to the effect of correlation at the receiver. Higher correlation Eb leading to higher spectral at the transmitter decreases N o min Eb efficiency for a given No operational point. On the other hand, higher correlation at the receiver decreases the slope of the capacity, thus decreasing the spectral efficiency for a given Eb No operational point. In order to quantify the relationship between key correlation parameters at the transmitter for low SNR analysis and the measurable parameters at the transmitter (antenna separation and angular spread), λmax (ΘT ) values are shown in fig. 1. These values are obtained from the computation of the eigenvalues of equation (5) for typical antenna separation and

3

1.4

4

analytical simulation

d =λ/2 t

M=4

dt=3λ/4

1.2

dt=λ

3.5

dt=2λ

1

M=3

0.8 λ2(ΘT)

λmax(ΘT)

3

2.5

0.6 M=2 2

0.4

1.5

0.2

0 0

2

4

6

8

10 12 ASD(degrees)

14

16

18

20

Fig. 1. Maximum eigenvalue of the transmittern correlation λo max (ΘT ) for 3λ different ASD values. M = {2, 3, 4} and dt = λ , , λ, 2λ 2 4

angular spread values in the BS (ASD). The angular power spectra at the BS is assumed to have a broadside Laplacian distribution in azimuth [11]. Correlation at the BS decreases either when increasing the angular spread or the distance between radiating elements, or when decreasing the number of radiating elements. Therefore, for a given channel scenario, i.e. fixed ASD and fixed receiver structure, the alternatives to increase capacity are either reducing the separation among antennas [3], [4], [5] or adding more elements. Of course, for practical reasons it is much more convenient to reduce the antenna separation, while possible, than increasing the number of antenna elements. For example, fig. 1 shows that for ASD= 10◦ the capacity achieved for M = 2 and dt = λ outperforms the capacity achieved for M = 3 and M = 4 with dt ≥ 2λ.

1

1.5

2

2.5 λmax(ΘT)

3

3.5

still matches the downlink scenario in terms of AS [11] that we have in mind for this study. In fig. 2 it is shown the analytical relationship and the simulated eigenvalues for ASD≤ 30◦ and dt ≤ 6λ. 20 BO

BO

BO

15

BO BO

10

5 N=1

BO

0 N=2 BO −5

BO

M=4 M=3 M=2

N=3

A. Optimality of Beamforming

N=4

The achieving-capacity region of BF is being so far characterized by means of three parameters, for a fixed receiver structure: the two principal eigenvalues of the transmitter correlation λmax (ΘT ) and λ2 (ΘT ) and SNR. BF is optimal if [7], [8]: 1 ≥ λ2 (ΘT )SNR

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Fig. 2. Maximum eigenvalue of the transmitter correlation λmax (ΘT ) vs second eigenvalue λ2 (ΘT ). M = {2, 3, 4}

SNR (dB)

1

M=4

M=3

M=2

−10

1

1.5

2

λ

2.5 (Θ )

max

3

3.5

4

T

Fig. 3. Beamforming optimality region (BO) for different correlation scenarios defined by SNR and λmax (ΘT ).

Fig. 3 defines, from eq. 9 and the result in [9] a simplified region where BF is optimal (BO) with respect to two parameters instead of the three in [7], [8]. As correlation in the N −1 transmitter increases, i.e λmax (ΘT ) approaches M , the range −N λmax (Θ1 )SNR T 1 − (λmax (ΘT )SNR) e Γ(1 − N, λmax (Θ1T )SNR ) of SNR where BF is optimal increases. Also it is shown here (9) that there is a maximum SNR value, that decreases with N , for which BF is always optimal independently of the correlation where Γ(·) is the Gamma function. However, this formulation does not bring any quantitative scenario [4], [5]. A cross analysis of figs. 1 and 3 allows insight on which antenna separation or angular spread leads to determining practical parameters for the definition of the BS a BF transmission strategy. A joint analysis of λmax (ΘT ) and antenna. For example, given a SNR = 0 dB scenario and λ2 (ΘT ) behavior depending on antenna separation dt and AS M = 4, if N = 1 BF is always optimal no matter which antenna separation or angular spread we have. If N = 2 it is will do so. In [9] it is shown that λ2 (ΘT ) is a function of λmax (ΘT ) for necessary that λmax (ΘT ) ≥ 3 in order for BF to be optimal. channel scenarios with limited angular spread. This limitation Looking at this scenario, and assuming that ASD= 5◦ , any

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IV. B EAMFORMING V S . S PATIAL M ULTIPLEXING : PRACTICAL APPROACH In many downlink outdoor scenarios, the base station height is enough for ASD to be small. The same way, the receiver is most of the times surrounded by many scatterers and high angular spread in the receiver can be considered. The environments for the practical approach in this section assume ASD≤ 20◦ and different values of antenna separation dt . In the receiver no correlation is considered due to high angular spread or antenna separation dr . For spectral efficiency computation, many realizations of channel samples based in the model presented on section II-A are randomly generated. The optimal power allocation (labelled SM) is computed based in the algorithm given in [14]. Also, the BF solution is presented for the transmitter covariance matrix, where all the power available is placed on the maximum eigenvalue of ΘT .

this separation to dt = λ2 in order to avoid possible mutual coupling effects [17]. We name this spectral efficiency values BFλ/2 (ASD, SNR) which is obviously dependent on ASD and SNR2 . The “practical 90%” region is defined by the inequality BFλ/2 (ASD, SNR) ≥ 0.9 SM(ASD, SNR, Dt ), where SM(ASD, SNR, Dt ) is the SM spectral efficiency values for a total antenna size of Dt . 20 BFλ/2(ASD,SNR)

18

BFλ/2=0.9 SM(3λ/2) BFλ/2=0.9 SM(9λ/4)

16

BFλ/2=0.9 SM(3λ)

14 ASD (degrees)

antenna separation smaller that λ would achieve the optimality.

BFλ/2=0.9 SM(9λ/2) BF≥ 0.9 SM

12

BFλ/2=0.9 SM(6λ)

10 8 BO

6 4

11

dt=λ

BF SM

10

2

dt=λ/2

0 −10

−5

0

5

10

15

SNR (dB)

Spectral Efficiency (bitps/Hz)

9 8 M=N=4 7

dt=λ

6 5

Fig. 5. Practical region where beamforming spectral efficiency is above 90% of spatial multiplexing capacity. M = N = 4.

dt=λ/2

10 dB

dt=λ/2

M=N=2 10 dB

4

M=N=4

dt=λ/2

20

dt=λ/2

18

BFλ/2(ASD,SNR)

0 dB 3

Fig. 4.

0

2

4

6

8 10 ASD (degrees)

12

14

16

18

16

20

BF and SM spectral efficiency for M = {2, 4} and N = {2, 4}.

Fig. 4 show the spectral efficiency for M = N = 4 and M = N = 2 antennas. The dependence on ASD and dt , the two parameters defining correlation at the transmitter, and SNR is shown. Both transmission strategies show an overlapping region over ASD (BF optimality), that increases as SNR and the antenna separation decreases. Keeping constant the number of transmitting and receiving elements, the behaviour of BF and SM differs. As ASD increases, BF reduces its spectral efficiency and SM increases its spectral efficiency. Regarding dt , higher antenna separations benefit SM and impairs BF. Finally higher SNR benefit both BF and SM. The theoretical optimality region of BF is being described in section III-A in terms of non-directly measurable parameters. Here, we describe the optimality area in terms of ASD and dt , together with a practical region where although BF is not strictly optimal it achieves spectral efficiency values that are not lower than 90% of the optimal SM solution. This area provides a region where BF is still a satisfactory solution given that the spectral efficiency loss is not significant. For BF solutions the best strategy is to place the antennas as close as a practical implementation allows. We have set

14 ASD (degrees)

2

BF

=0.9 SM(λ/2)

λ/2

BFλ/2=0.9 SM(3λ/4)

dt=λ

BF

=0.9 SM(λ)

BF

=0.9 SM(3λ/2)

BF≥ 0.9 SM

λ/2 λ/2

BFλ/2=0.9 SM(2λ)

12 BO

10 8

BF≥ 0.9 SM

6 4 2 −5

0

5

10

15

20

SNR (dB)

Fig. 6. Practical region where beamforming spectral efficiency is above 90% of spatial multiplexing capacity. M = N = 2.

Both regions, the optimal and the “practical 90%” are sensitive to the number of transmitting and receiving elements. From fig. 3 increasing M and N decreases the BF optimality area. The most restricted regions for BF are of particular interest since they are the ones limiting the BF implementation, particularly, regions where M = N . These regions are shown in figs. 5 and 6. Both figures provide the optimal BF region (BO) for dt = λ2 given in terms of ASD and SNR. From 2 For the sake of simplicity, in figs. 6, 5, the dependence with ASD and SNR is dropped.

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fig. 3 there is a SNR threshold where BF is always optimal independently of the correlation scenario, i.e., independently of the ASD and dt values, this is represented by the vertical dash-dot line. In both cases it can be observed that decreasing Dt increases the area where BF is within 90% of capacity. For high Dt the region in some cases overlap with the BF optimal region, meaning in this case that the solution is achieving optimality. For M = 4 most of the region for which BF is within 90% of SM performance, is below 0 dB of SNR. For M = 2 this value is increased to 7 dB of SNR. Thus, in systems where capacity is limited by out of cell interference, BF may be a very attractive alternative for most locations, providing almost all the spectral efficiency for a large fraction of locations. V. C ONCLUSION A semi-analytical approach has been undertaken to characterize the behavior of correlation in terms of measurable parameters. It has been shown the dependence on the main eigenvalue of the spatial correlation at the transmitter with AS, dt and M . Also, a much simpler characterization of the BF optimality region is being provided for realistic outdoor scenarios. From a more practical point of view the dependence of spectral efficiency in terms of the measurable parameters is being presented, identifying the “low SNR” regions where BF leads to an optimal or quasi-optimal solution. For most locations in systems where capacity is limited by out of cell interference, BF may be a very attractive alternative, since it performs very close to optimality. ACKNOWLEDGMENT The authors would like to thank G. Foschini, D. Chizhik, A. Lozano and for their helpful discussions. We are also grateful to the reviewers for their useful suggestions. R EFERENCES [1] D. Chizhik, G. Foschini, M. Gans, and R. Valenzuela, “Keyholes, correlations, and capacities of multielement transmit and receive antennas,” IEEE Transactions on Wireless Communications, vol. 1, pp. 361 – 368, April 2002. [2] A. Sayeed, “Deconstructing multiantenna fading channels,” IEEE Transactions on Signal Processing, vol. 50, pp. 2563 – 2579, October 2002. [3] G. Barriac and U. Madhow, “Space-time communication for OFDM with implicit channel feedback,” IEEE Transactions on Information Theory, vol. 50, pp. 3111 – 3129, December 2004. [4] A. Sayeed, V. Raghavan, and J. Kotecha, “Capacity of space-time wireless channels: a physical perspective,” in IEEE Information Theory Workshop, pp. 434 – 439, 2004. [5] A. Sayeed and V. Raghavan, “The Ideal MIMO Channel: Maximizing Capacity in Sparse Multipath with Reconfigurable Arrays,” in IEEE International Symposium on Information Theory, 2006. [6] X. Wu and R. Srikant, “MIMO Channels in the Low-SNR Regime: Communication Rate, Error Exponent and Signal Peakiness,” in IEEE Information Theory Workshop, pp. 428–433, October 2004. [7] S. Jafar and A. Goldsmith, “Transmitter optimization and optimality of beamforming for multiple antenna systems,” IEEE Transactions on Wireless Communications, vol. 3, pp. 1165 – 1175, July 2004. [8] E. Jorswieck and H. Boche, “Channel capacity and capacity-range of beamforming in MIMO wireless systems under correlated fading with covariance feedback,” IEEE Transactions on Wireless Communications, vol. 3, pp. 1543 – 1553, September 2004.

[9] M. Sánchez-Fernández, S. Zazo, and R. Valenzuela, “Simplifying the beamforming optimality region for practical MIMO scenarios,” IEEE Communication Letters, vol. 10, November 2006. [10] D. Chizhik, “Slowing the time-fluctuating MIMO channel by beam forming,” IEEE Transactions on Wireless Communications, vol. 3, pp. 1554 – 1565, September 2004. [11] K. Pedersen, P. Mogensen, and B. Fleury, “A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments,” IEEE Transactions on Vehicular Technology, pp. 437 – 447, March 2000. [12] H. Xu, M. Gans, N. Amitay, R. Valenzuela, T. Sizer, R. Storz, D. T. M. McDonald, and C. Tran, “MIMO channel capacity for fixed wireless: measurements and models,” in IEEE Vehicular Technology Conference (Fall VTC’01), vol. 2, pp. 1068 – 1072, October 2001. [13] E. Visotsky and U. Madhow, “Space-Time Transmit Precoding With Imperfect Feedback,” IEEE Transactions on Information Theory, vol. 47, pp. 2632–2639, September 2001. [14] A. Tulino, A. Lozano, and S. Verdú, “Capacity-Achieving Input Covariance for Single-user Multi-Antenna Channels,” IEEE Transactions on Wireless Communications, vol. 5, pp. 662–671, March 2006. [15] S. Verdú, “Spectral efficiency in the wideband regime,” IEEE Transactions on Information Theory, vol. 48, pp. 1319 – 1343, June 2002. [16] A. Tulino, A. Lozano, and S. Verdú, “Impact of antenna correlation on the capacity of mutiantenna channels,” IEEE Transactions on Information Theory, vol. 51, pp. 2491 – 2509, July 2005. [17] C. Waldschmidt, S. Schulteis, and W. Wiesbeck, “Complete RF system model for analysis of compact MIMO arrays,” IEEE Transactions on Vehicular Technology, vol. 53, pp. 579 – 586, May 2004.