Multiuser Transmit Beamforming via Regularized ... - Semantic Scholar

Multiuser Transmit Beamforming via Regularized Channel Inversion: A Large System Analysis Van K. Nguyen

Jamie S. Evans

Defence Science and Technology Organisation Edinburgh, SA 5111, Australia [email protected]

Department of Electrical and Electronic Engineering University of Melbourne Parkville, VIC 3010, Australia [email protected]

Abstract— In this paper, we analyze the performance of multiuser transmit beamforming for the broadcast channel. We focus on transmit beamforming via regularized channel inversion and our analysis is in the large system limit where both the number of users and the number of antennas approach infinity with their ratio held constant. We derive an expression for the signalto-interference-plus-noise ratio in this large system limit. We then use this result to obtain a very simple expression for a locally optimal regularization parameter, that which maximizes the asymptotic signal-to-interference-plus-noise ratio.

I. I NTRODUCTION In this paper we consider the vector broadcast channel where a single transmitter equipped with multiple antennas sends separate and independent messages to multiple users each with a single antenna. Amongst other things, this channel is an appropriate model for the single-cell downlink in a cellular network where inter-cell interference is negligible. There has been a great deal of interest in such channels in recent times with new results on the sum capacity [1], [2], [3] and then the capacity region [4]. In this paper we are concerned with a suboptimal but relatively simple communication method for the vector broadcast channel known as linear precoding or beamforming (see for example [5], [6], [7]). Our primary focus is on transmit beamforming based on regularized channel inversion (RCI) as discussed in [8]. The design of the RCI beamformer involves selecting a regularization parameter. In [8], for the special case where the number of transmit antennas is equal to the number of users, the authors show how to choose the regularization parameter to optimize an approximation to the signal-to-interference-plusnoise ratio (SINR) at the receivers. In this paper we use large system analysis [9], [10], [11] to analyze the SINR of the RCI beamformer and then use the large-system SINR to derive a locally optimal regularizaion parameter. The result is valid for any ratio of transmit antennas to users and reduces to the result in [8] when this ratio is one. Although the optimal regularization parameter is obtained in the large system limit we show that there is very little system throughput loss when the large-system optimal value is employed systems with a small number of users and transmit antennas. The remainder of this paper is organized as follows. In Section II, we introduce the channel model and formulate

an SINR expression for the RCI beamformer. In Section III, we analyze the SINR in the large system limit. Based on the asymptotic SINR, we derive a locally optimal regularization parameter and examine the asymptotic SINR in some limiting cases in Section IV. In Section V, we compare the performance of the RCI beamformer designed based on the asymptotic SINR and one designed based on the actual realization of the SINR in systems with a finite number of users and show that there is minimal throughput loss incurred by employing the asymptotically optimal regularisation parameter. Notation: The following notation is used throughout the paper. Bold upper (lower) letters denote matrices (column vectors); (·)∗ , (·)T and (·)H denote complex conjugate, transpose and Hermitian transpose operations, respectively; || · ||2 stands for the Euclidean norm of a vector; tr(·) denotes the trace operation of a matrix; E[·] denotes the expected value of the expression in the brackets; log and ln denote the logarithm to the base of 2 and the natural logarithm, respectively. II. P RELIMINARIES A. System Model We consider the downlink of a multiuser system, where the base station with N transmit antennas simultaneously sends independent data symbols sk , 1 ≤ k ≤ K, to K users, each with one receive antenna. To reduce the amount of interuser interference at the receiver, the data symbols of all users will be jointly processed by an N ×K transmit beamforming matrix P. The transmitted signals across N transmit antennas are x = Ps, where s = [s1 , s2 , ..., sK ]T , and are constrained to have a fixed transmitted power E[||x||22 ] = Ptr . The receive data at the kth user is given by yk =

N

hk,n xn + wk

(1)

n=1

where xn = [x]n is the signal sent from the nth antenna and wk is a circularly-symmetric complex Gaussian random variable with zero mean and variance σ 2 . The coefficient hk,n is the fading gain for the path between antenna n and user k. It is modeled as a circularly-symmetric complex Gaussian random variable with zero mean and unit-variance. Stacking the received signal from all K users into a vector

978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.

y = [y1 , y2 , ..., yK ]T , we can write y = Hx + w

(2)

where w = [w1 , w2 , ..., wK ] and H is the K × N matrix of downlink channel gains with hk,n as elements. T

B. The Regularized Channel Inversion Beamformer

in (6) is the interference caused by other users to the kth user. With the assumption that the data symbols sent to one user are independent to other users (i.e., E[s∗j sk ] = δj,k ), the energy of the interference term is E[Ik Ik∗ ] = c2

where c is a normalizing constant and α is the regularization parameter. With the total transmit power constraint E[||x||22 ] = Ptr , the normalizing constant is given by trace

Ptr

(HHH

+ αI)

−1

HHH (HHH + αI)

−1

. (4)

To evaluate the SINR of the RCI beamformer we first rewrite the transmit beamforming matrix from (3) as −1 H H P = c HH H + α I Substituting this expression for P into (2), we have −1 H H s+w . y = cH HH H + αI

(5)

Let hk denotes the kth row of H, which is a 1 × N vector containing the fading gains for the paths between N transmit antennas and the receiver of user k. The received signal at the kth user is then given by −1 H H s + wk yk = chk HH H + α I H −1 H = chk H H + α I hk sk +

K

−1 H chk HH H + α I hj sj + wk . (6)

j=k

The first term

−1 H zk := chk HH H + α I hk sk

(7)

in (6) is the desired signal of user k. By observing that H (HH H + νI) = (HH k Hk + νI + hk hk ) and applying the Matrix Inverse Lemma, we have (HH H + νI)−1 hH k =

−1 H (HH hk k Hk + νI) H 1 + hk (Hk Hk + νI)−1 hH k

(8)

where Hk is H with the kth row removed. Using (8), the desired signal energy of the kth user can then be shown to be given by −1 H 2 hk HH hk k Hk + αI ∗ 2 . (9) E[zk zk ] = c H −1 H 1 + hk Hk Hk + αI hk The second term Ik :=

K j=k

−1 H chk HH H + α I hj sj

−1 H hk HH H + α I hi δi,j

i=k j=k

−1 H × hj HH H + α I hk H −1 H 2 = c hk H H + α I Hk Hk H −1 H × H H + αI hk . (11)

In this paper, we are interested in the regularized channel inversion (RCI) transmit beamformer as in [8] −1 (3) P = cHH H HH + α I

c2 =

K K

(10)

Using (8), and defining Ak and Bk as −1 H hk (12) Ak := hk HH k Hk + αI H H −1 H −1 H Bk := hk Hk Hk + αI Hk Hk Hk Hk + αI hk (13) the SINR of the kth user, denoted by SINRk , is then given by SINRk =

c2 Bk

c2 A2k + (1 + Ak )2 σ 2

(14)

where c2 is given in (4). III. L ARGE S YSTEM SINR A NALYSIS We will now analyze the SINR of the RCI beamformer in the large system limit where both the number of users K and the number of transmit antennas N approach infinity with their ratio, β = K/N , being held constant. An approximation of the asymptotic SINR has been derived in [8] for the case K = N . This asymptotic SIR approximation is expressed in terms of the eigenvalues of the random channel matrix HH H. In this paper, we derive an asymptotic SINR expression for the general cases where β can take on any positive value using results from random matrix theory (see [11] for a thorough review). Define ρ = α/N and write ((12)) as −1 1 H 1 Hk Hk + ρI hH (15) Ak = hk k . N N This expression is in standard form for application of largesystem results. Following [12, Theorem 7] for example we see that in the large-system limit where N, K → ∞ with β held constant, Ak converges (almost surely) to

∞ 1 fβ (u)du g(β, ρ) = (16) u+ρ 0 where

(u − a)+ (b − u)+ fβ (u) := (1 − β) δ(u) + (17) 2πu √ √ with a = (1 − β)2 and b = (1 + β)2 . This integral can be evaluated in closed form 1−β 2(1 + β) (1 − β)2 1 +1+ −1 . g(β, ρ) = + 2 ρ2 ρ ρ (18) +


and recognising that d 1 u +ρ = (u + ρ)2 u+ρ dρ

1 u+ρ

β = 1, K = 5

β = 1, K = 15

40

SINR (dB)

40

SINR (dB)

Following a similar approach, one can show that Bk converges to

∞ u fβ (u)du (19) Bk → (u + ρ)2 0

20

0

−20

0

10 20 SNR (dB) β = 1, K = 80

20

0

−20

30

Finally we can rewrite the expression for the normalizing constant (4) as tr

1 N

1

H NH H

H NH H

−2

+ ρI

0

0

10 20 SNR (dB)

−20

30

10 20 SNR (dB)

30

Ptr / σ2 = 5 dB

Ptr / σ2 = 10 dB

4

(22)

Combining the asymptotic expressions for Ak , Bk and c2 with (14), the large system SINR of the kth user is given by

12

3 2 1 0

0

0.5 ρ

SINR(γ, β, ρ)

10 8 6 4 2 0

1

0

2

where g(β, ρ) is given by (18). From the above SINR expression, it can be seen that in the large system limit the SINR converges to a nonrandom value and is the same for all users. This asymptotic SINR expression depends only on the number of users per transmit antenna β, the (normalized) regularization parameter ρ, and the SNR γ. To illustrate the convergence of the random SINR to the asymptotic limit, we compare the actually realized SINR calculated by (14) to the asymptotic limit (23). For each value of SNR, we obtain one hundred samples of realized SINR for user one by randomly generating the channel matrix gains H. The results are obtained in the case β = 1 and K = 5, 15, 80 and we use the optimal ρ = 1/γ (or equivalently α = K/γ). IV. S ELECTION OF ρ Figure 2 illustrates the asymptotic SINR (23) for different values of ρ at different levels of SNR and β. It is evident from these plots that the value for ρ has a significant impact on the SINR. In this section, we derive a locally optimal ρ that maximizes the asymptotic SINR of the RCI beamformer. To begin, we find a locally optimal ρ that maximizes the asymptotic SINR for the general cases where β can take any real positive value. This process involves taking the derivative of the asymptotic SINR (23) with respect to ρ and equating it

1

Ptr / σ = 20 dB 100

30

Asymptotic SINR

γg (β, ρ) . (23) 2 d γ + (1 + g(β, ρ)) g(β, ρ) + ρ dρ g(β, ρ)

0.5 ρ 2

Ptr / σ = 15 dB

2

Asymptotic SINR

=

0

0

Fig. 1. Comparison of the randomly generated SINR for user 1 with the asymptotic limit (23), for K = 5, 15, 80 and β = 1 with the corresponding ρ = σ 2 /Ptr .

Ptr u (u+ρ)2 fβ (u)du

Ptr . = d g(β, ρ) + ρ dρ g(β, ρ)

0

(21)

and in the large system limit, c2 converges to c2 → ∞

30

20

Asymptotic SINR

c2 =

Ptr 1

20

−20

10 20 SNR (dB) Asymptotic Result

40 SINR (dB)

(20)

Asymptotic SINR

0

d u fβ (u)du = g(β, ρ) + ρ g(β, ρ) . (u + ρ)2 dρ

SINR (dB)

40

we have

∞

0

20 10 0

0

0.1

0.2

ρ

0.3

0.4

50

0

0

0.05

0.1 ρ

0.15

0.2

Fig. 2. Plots of the asymptotic SINR (23) as a function of ρ for different values of β and SNR levels. Solid line: β = 0.5, dash-dot line: β = 1, and dot line: β = 1.5

to zero. Despite a very complicated expression which requires many pages of algebraic manipulations to solve, the optimal ρ turns out to be remarkably simple. It is given by ρopt = β/γ .

(24)

This optimal value reduces to the same value derived in [8] for the specific case β = 1. V. P ERFORMANCE C OMPARISON Since ρopt is derived based on the analysis in the large system limit, it is of interest to study the performance of the RCI beamformer in a finite-size system. To begin, we will examine how close ρopt is to the optimal ρ of a finitesize system, denoted by ρF S , which is determined based on the realization of H. When K is small, the SINR varies


K = 4, β = 0.8

20 0 0.06 0.08

100 50 0 0.06 0.08

0.1 0.12 0.14 ρ FS

100 0 0.06 0.08

800

Mean = 0.084 −6 Variance = 1. 3 x 10 Frequency

200

K = 32, β = 0.8 1000 Frequency

Frequency

300

Mean = 0.088 −6 Variance = 7 x 10

600 400

100

0

0.1 0.12 0.14 ρ FS

K = 16, β = 0.8 400

Mean = 0.0056 Variance = 7.24 x 10−5 200

0 0.005 0.01 Normalized Throughput Difference K = 16, β = 0.8 600 Mean = 5.4 x 10−4 −7 Variance = 1.14 x 10 400

200

200 0.1 0.12 0.14 ρFS

Frequency

40

150

K = 8, β = 0.8 300

Mean = 0.0019 −6 Variance = 3.2 x 10

200

100

0

Frequency

60

K = 4, β = 0.8 300

Mean = 0.0954 −5 Variance = 6.13 x 10 Frequency

Frequency

80

K = 8, β = 0.8 200

Mean = 0.1088 Variance = 7.08 x 10−4 Frequency

100

0 0.005 0.01 Normalized Throughput Difference K = 32, β = 0.8 1000 Mean = 1.4 x 10−4 800 Variance = 5 x 10−9 600 400 200

0 0.06 0.08

0.1 0.12 0.14 ρFS

Fig. 3. Histograms of the optimal ρ in a finite-size system based on the criterion of maximizing the system throughput at the SNR γ = 10dB and β = 0.8.

significantly from one user to another. The maximal SINR of each user also occurs at different value of ρ. Hence, there is no single value ρ that can maximize the SINRs of all users. In this case, two natural criteria that can be used to determine ρF S are: i) to maximize the system throughput or ii) to maximize the worst SINR. In this paper, we choose the first option. the value which maximizes the system Thus, ρF S refers to K throughput Csum := k=1 log(1 + SINRk ). For finite K, the SINRk of individual user in the system can be found by using (14). Fig. 3 shows histograms of 1000 realizations of ρF S as the number of users K is increased. All the plots are obtained at the SNR γ = 10dB and β is fixed at 0.8. All the fading gains are randomly generated according to the independent and identically distributed Rayleigh fading channels. At the SNR of 10dB and β = 0.8, the optimal ρ based on the asymptotic analysis is ρopt = 0.08. From Fig. 3, one can see that many realizations of ρF S are very close to the asymptotic value ρopt . Fig. 4 plots the histograms of the system throughput difference between using N ρopt and N ρF S as the regularizing constant in the RCI beamformer (3). The difference is normalized by dividing by the system throughput of the RCI beamformer that uses N ρF S . These simulation results demonstrate that there is very little system throughput loss when ρopt is used instead of ρF S . Even when the number of users is as small as K = 4, the mean system throughput difference is still less than 0.6%. VI. C ONCLUSIONS In this paper, we derived an asymptotic SINR expression of the RCI beamformer in the large system limit. The asymptotic SINR only depends on the user-to-antennas ratio, the regularization parameter and the SNR. Based on the asymptotic SINR of the RCI beamformer, we derived a closed-form expression for a locally optimal regularization parameter. Simulation results showed that there is little performance degradation in terms of system throughput when it is used in systems

0

0 0.005 0.01 Normalized Throughput Difference

0

0 0.005 0.01 Normalized Throughput Difference

Fig. 4. Histograms of the normalized system throughput difference at the SNR γ = 10dB and β = 0.8.

with a finite number of users. Finally, we note although we assumed all users to suffer equal large-scale path-loss, it is straightforward to extend the large-system SINR analysis to the case where users have unequal path-loss and receiver noise power. R EFERENCES [1] W. Yu and J. Cioffi, “Sum capacity of a Gaussian vector broadcast channel,” Proc. IEEE Int. Symp. Inf. Theory, Jul. 2002, pp. 498. [2] P. Viswanath and D. Tse, “Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality,” IEEE Trans. Inf. Theory, vol. 49, pp. 1912-1921, Aug. 2003. [3] S. Vishwanath, N. Jindal, and A. Goldsmith, “Duality, achievable rates, and sum capacity of Gaussian MIMO broadcast channels,” IEEE Trans. Inf. Theory, vol. 49, pp. 2658-2668, Aug. 2003. [4] H. Weingarten, Y. Steinberg, and S. Shamai (Shitz), “The capacity region of the Gaussian multiple-input multiple-output broadcast channel,” IEEE Trans. Inf. Theory, vol. 52, pp. 3936-3964, Sep. 2006. [5] H. Boche, M. Schubert, and E. A. Jorswieck, “Throughput maximization for the multiuser MIMO broadcast channel,” in Proc. International Conference of Acoustics, Speech and Signal Processing, vol. 4, pp. 808811, Apr. 2003. [6] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Signal Process., vol. 52, pp. 461-471, Feb. 2004. [7] A. Wiesel, Y. C. Eldar, and S. Shamai (Shitz), “Linear precoding via conic optimization for fixed MIMO receivers,” IEEE Trans. Signal Process., vol. 54, pp. 161-176, Jan. 2006. [8] C. B. Peel, B. M. Hochwald and A. L. Swindlehurst “A vectorperturbation technique for near-capacity multianntenna multiuser communication - Part I: Channel inversion and regularization,” IEEE Trans. Commun., vol. 53, pp. 195-202, Jan. 2005. [9] S. Verdu and S. Shamai (Shitz), “Spectral efficiency of CDMA with random spreading,” IEEE Trans. Inform. Theory, vol. 45, pp. 622-640, Mar. 1999. [10] D. N. C. Tse and S. V. Hanly, “Linear multiuser receivers: Effective interference, effective bandwidth and user capacity,” IEEE Trans. Inform. Theory, vol. 45, pp. 641-657, Mar. 1999. [11] A. M. Tulino and S. Verdu, “Random matrix theory and wireless communications,” Foundations and Trends in Commun. and Inform. Theory, vol. 1, no. 1, pp. 1-182, 2004. [12] J. S. Evans and D. N. C. Tse, “Large system performance of linear multiuser receivers in multipath fading channels,” IEEE Trans. Inf. Theory, vol. 46, pp. 2059-2078, Sep. 2000.
