Large System Analysis of Zero-Forcing Precodingin MISO Broadcast Channels with Limited

FeedbackSebastian Wagner�y, Romain Couillet�z, Dirk T. M. Slocky, and Merouane Debbahz

�ST-ERICSSON, 06904 Sophia-Antipolis, France, Email: fsebastian.wagner, romain.couilletg@stericsson.comyEURECOM, 06904 Sophia-Antipolis, France, Email: fsebastian.wagner, dirk.slockg@eurecom.frzSUPELEC, 91192 Gif sur Yvette, France, Email: fromain.couillet, merouane.debbahg@supelec.fr

Abstract—In this paper we analyze the sum rate of zero-forcing(ZF) precoding in MISO broadcast channels with limited feed-back, transmit correlation and path loss. Our analysis assumesthat the number of transmit antennasM and the number of usersK are large, while their ratio remains bounded. By applying re-cent results from random matrix theory we derive a deterministicequivalent of the signal-to-interference plus noise ratio (SINR)and compute the sum rate maximizing number of users as wellas the limiting sum rate for high signal-to-noise ratios (SNR),as a function of the channel errors and the channel correlationpattern. Simulations show that theoretical and numerical resultsmatch well, even for small system dimensions.

I. INTRODUCTION

The capacity achieving precoding strategy of the GaussianMIMO broadcast channel based on the non-linear dirty-papercoding (DPC) technique [1]. But so far no efficient prac-tical algorithm implementing the optimal DPC scheme hasbeen proposed. Therefore, low complexity linear precodingstrategies have gained a lot of attention since they achievea large portion of the rate region at moderate complexity. Aclassical linear interference mitigating scheme is zero-forcing(ZF) precoding which has first been analyzed in the contextof MIMO broadcast channels in [2].

In this contribution we consider a system where both thenumber of transmit antennas M and the number of users Kare large but their ratio �(M),M=K is bounded. We extendthe models of [3]–[5] by considering imperfect channel stateinformation at the transmitter (CSIT), transmit correlation aswell as different path losses of the users. With the aid ofrecent tools from random matrix theory (RMT), we derivea deterministic equivalent of the signal-to-interference plusnoise ratio (SINR) of ZF precoding which is independentof the individual channel realizations. From the deterministicequivalent of the SINR we determine the sum rate maximizingnumber of users.

Notation: In the following boldface lower-case and upper-case characters denote vectors and matrices, respectively. Theoperators (�)H, tr(�) and, for X of size N�N , Tr(X), 1

N trXdenote conjugate transpose, trace and normalized matrix trace,respectively. The expectation is E[�] and diag(x1; : : : ; xN ) isthe diagonal matrix with elements xi on the main diagonal.

The N�N identity matrix is IN and =[z] is the imaginarypart of z2C.

II. MATHEMATICAL PRELIMINARIES

Definition 1 (Deterministic Equivalent): Let fXN : N =1; 2; : : : g be a set of complex random matricesXN of size N�N . For some functional f we define a deterministic equivalentm�XN

of mXN, f(XN ) as any series m�X1

;m�X2; : : : such

thatmXN

�m�XN

N!1�! 0 (1)

almost surely.In present work we are interested in deterministic equivalentsof expressions of the form

mBN ;QN(z) = TrQN (BN � zIn)

�1 (2)

where QN 2Cn�n is a Hermitian positive definite matrix andBN 2Cn�n is of the type

BN = T1=2N XNRNX

H

NT1=2N (3)

where RN 2 CN�N is nonnegative definite Hermitian,TN 2 Cn�n is diagonal and XN 2 Cn�N is random withindependent and identically distributed (i.i.d.) entries of zeromean and variance 1=n. In the course of the derivations, wewill require the following result,

Theorem 1: [6, Theorem 1] Under the above model forBN where TN and RN , QN have uniformly bounded spectralnorm (w.r.t. N ), as (n;N) grow large with ratio �(N),N=nsuch that 0 < lim infN �(N) � lim supN �(N) <1. DefinemBN ;QN

(z) as in (2). Then, for z2CnR+,

mBN ;QN(z)�m�BN ;QN

(z)N!1�! 0 (4)

almost surely, where m�BN ;QN(z) is defined as

m�BN ;QN(z) = TrQN (c(z)TN � zIn)

�1 (5)

with c(z) = TrRN

�IN +

1

�e(z)RN

��1(6)

and e(z) is the unique solution of

e(z) = TrTN (c(z)TN � zIn)�1 (7)

with positive imaginary part if =[z]>0, of negative imaginarypart if =[z]< 0, and real positive if z < 0. Moreover e(z) isanalytic on CnR+ and of uniformly bounded module on everycompact subset of C nR+. Note that mBN

(z) ,mBN ;IN (z)is the Stieltjes transform [7] of the eigenvalue distribution ofBN .

III. SYSTEM MODEL

Consider the MISO broadcast channel composed of onecentral transmitter equipped with M antennas and of Ksingle-antenna receivers. Assume M > K and narrow-bandcommunication. Denoting yk the signal received by user k, theconcatenated received signal vector y=[y1; : : : ; yK ]T2CK ata given time instant reads

y =pMHx+ n (8)

with transmit vector x 2 CM , channel matrix H 2 CK�Mand noise vector n�CN (0; �2IK). The transmit vector x isobtained by linear precoding x = Gs, where s�CN (0; IK)is the symbol vector and G = [g1; : : : ;gK ] 2 CM�K is theprecoding matrix. The total transmit power is P >0, hence

tr(E[xxH]) = tr(GGH) � P: (9)

In this paper we consider ZF precoding i.e.

G =�pM

�HHH

��1HH (10)

where H is the estimated channel matrix available at thetransmitter and the scaling factor � is set to fulfill the powerconstraint (9). From (9) we then obtain

�2 =P

1M tr

�HHH

��1 =P

mHHH

(0),

P�: (11)

The received symbol yk of user k is given by

yk = �hHk (HHH)�1hksk + �

KXi=1;i 6=k

hHk (HHH)�1hisi + nk

where hHk and hHk denote the kth row ofH and H, respectively.The SINR k;zf of user k can be written in the form

k;zf =jhHk (HHH)�1hkj2

hHk (HHH)�1HH

[k]H[k](HHH)�1hk +��

(12)

where HH

[k]=[h1; : : : ; hk�1; hk+1; : : : ; hK ]2CM�(K�1) and� = P=�2 denotes the signal-to-noise ratio (SNR). The sumrate Rsum is given by

Rsum =

KXk=1

log (1 + k;zf) [nats=s=Hz]: (13)

Under the assumption of a rich scattering environment thecorrelated channel can be modeled as [8]–[10]

H = L1=2X�1=2 (14)

where X 2 CK�M has i.i.d. zero-mean entries of variance1=M , � 2 CM�M is the nonnegative definite correlation

matrix at the transmitter with eigenvalues �1; : : : ; �M , orderedas �1 � : : : � �M , and L = diag(l1; : : : ; lK), with entriesordered as l1 � : : : � lK , contains the user channel gains,i.e. the inverse user path losses. Note that in (14) the entriesof X are not required to be Gaussian. We assume k�k tobe uniformly bounded from above with respect to M , i.e.adding more transmit antennas does not significantly increasethe correlation between them.

Moreover, we suppose that only H, an imperfect estimateof the true channel matrix H, is available at the transmitter.The channel gain matrix L as well as the transmit correlation� are assumed to be slowly varying compared to the channelcoherence time and are assumed to be perfectly known to thetransmitter. We therefore model H as

H = L1=2X�1=2 (15)

with X =p1� �2X+ �Q (16)

where Q2CK�M is the matrix of channel estimation errorscontaining i.i.d. entries of zero mean and variance 1=M , and� 2 [0; 1]. The parameter � reflects the amount of distortionin the channel estimate H. Furthermore, we suppose that Xand Q are mutually independent as well as independent of thesymbol vector s and noise n. A similar model for imperfectCSIT has been used in [11]–[13].

IV. DETERMINISTIC EQUIVALENT OF THE SINR

In the following we derive a deterministic equivalent �k;zfof the SINR k;zf of user k for ZF precoding. That is, �k;zfis an approximation of k;zf independent of the particularrealizations of X and Q.

In [6] we derived a deterministic equivalent for regularizedZF (RZF) precoding. The same techniques as for RZF, cannotbe applied for ZF, since by removing a row of H the matrixHHH becomes singular. Therefore, we adopt a differentstrategy and derive the SINR k;zf for ZF of user k and� > 1 as k;zf = lim�!0 k;rzf . The result is summarized inthe following theorem.

Theorem 2: Let �>1 and k;zf be the SINR of user k forZF precoding. Then

k;zf � �k;zfM!1�! 0 (17)

almost surely, where �k;zf is a deterministic equivalent of k;zfand given by

�k;zf =1� �2

lk�2 ��� +��

�

(18)

with

�� =1

��cTrL�1 (19)

��� =c2=�c

2

� � c2=�c2TrL�1 (20)

c2 = Tr�2

�IM +

1

�c��

��2(21)

where �c is the unique solution of

�c = Tr�

�IM +

1

�c��

��1(22)

By Jensen’s inequality c2=�c2 � 1 with equality if � = IM .

The computation of (18) involves the evaluation of only onefixed-point equation, given by (22).

Corollary 1: Let �= IM and L= IK then �k;zf takes theexplicit form

�zf , �k;zf =1� �2

�2 + 1�

(� � 1) (23)

Proof of Corollary 1: By substituting �= IM and L=IK into (22), �c is explicitly given by �c = (� � 1)=�. Sincec2=�c

2=1 we have ��= ���=1=(� � 1).Proof of Theorem 2: The SINR k;rzf of user k of RZF

for large (K;M) is given by [6]

k;rzf =l2k(1� �2)m2

A

lk�(1� �2[1� (1 + lkmA)2]) +(�)� (1 + lkmA)2

(24)where A = XHLX+ ���1 and

mA , mA(0) = TrA�1 (25)

� = mA � �Tr��1A�2 (26)

(�) = TrHHH�HHH+ �IM

��2(27)

The underlying strategy is as follows: The terms in the SINRof RZF that depend on �, i.e. mA, � and , are expandedaround � = 0. Subsequently we take the limit k;zf =lim�!0 k;rzf . Finally, we find the deterministic equivalent �k;zf .

We expand mA=TrA�1 around �=0 as follows

mA(a)=

1

�Tr�� 1

�MtrH�HH(HHH + �IK)�1 (28)

(b)� 1

�Tr�� 1

�MtrH�HH

h(HHH)�1 � �(HHH)�2

i(29)

where (a) follows from the matrix inversion lemma (MIL) andin (b) we rewrite the inverse in terms of a Taylor series of order2 around the point �=0. In step (b) it is necessary to assumethat � > 1 to assure that the maximum eigenvalue of matrix(HHH)�1 is bounded for all large M . For �Tr��1A�2 weobtain

�Tr��1A�2 � 1

�Tr�� 1

�MtrH�HH(HHH)�1

+�

MtrH�HH(HHH)�3: (30)

Substituting (29) and (30) into (26) and taking the limit �!0,we obtain

�� = lim�!0

� =1

MtrH�HH

�HHH

��2: (31)

Replacing mA, � and (�) in (24) with (29), (31) and �=(0), respectively, we have

k;zf = lim�!0

k;rzf =1� �2

lk�2 �� +��

: (32)

Now we derive a deterministic equivalent �� and ��� for �and ��, respectively.

Applying Theorem 1 we find �� s.t. �� ��M!1�! 0 almost

surely, as�� = m�

HHH(0) =

1

��cTrL�1; (33)

where �c is defined in (22).In order to find ��� notice that, we can diagonalize � in (31)

s.t. �=Udiag(�1; : : : ; �M )UH and still have i.i.d. elementsin the kth column x0k of XU. Denoting C=HHH and C[k]=

H[k]HH

[k] � �kL1=2x0kx

0Hk L

1=2 and applying [14, Lemma 2.2]twice, equation (31) takes the form

�� =1

M

MXk=1

�2kx0Hk L

1=2C�2[k]L1=2x0k

(1 + �kx0Hk L1=2C[k]L

1=2x0k)2: (34)

Applying [14, Lemma 3.1] together with TrC�1[k] �TrC�1

M!1�! 0 [6], we obtain

��� 1

�TrLC�2

1

M

MXk=1

�2k(1 + �k

1�TrLC

�1)2M!1�! 0 (35)

almost surely. To determine a deterministic equivalentmC;L(0)

� for mC;L(0) = TrLC�1, we apply Theorem 1 asfor (33). For TrLC�2 we have

TrLC�2 = mC2;L(z) =@mC2;L(z)

@z z=0= m0C;L(0): (36)

The derivative of mC;L(0)� is a deterministic equivalent of

m0C;L(0), so that applied to (35), we have a deterministic

equivalent ���, s.t. ��� ���M!1�! 0 almost surely, that verifies

��� =c2=�c

2

� � c2=�c2TrL�1 (37)

where �c and c2 are defined in (22) and (21), respectively.Finally, we obtain (18) by substituting � and �� in (32) bytheir respective deterministic equivalents (33) and (37), whichcompletes the proof.

V. ASYMPTOTICALLY OPTIMAL NUMBER OF USERS K

In general, for fixed �, L, � and �2, consider the problemof finding the optimal number of users K?� (or equally�?�=M=K?�), such that the approximated sum rate R�sum,PK

k=1 log(1 + �k;zf) is maximized, i.e.

�?� = argmax�>1

1

�

Zlog

�1 + �k;zf

�dFL(l); (38)

where we suppose that the user channel gains lk are distributedaccording to some probability distribution function FL. By

setting the derivative of (38) w.r.t. � to zero, we obtain theimplicit equation

�

Z @ �k;zf@� dFL(l)

1 + �k;zf=

Zlog

�1 + �k;zf

�dFL(l): (39)

Thus, �?� is the solution to (39).In the special case of �=IM and L=IK , the SINR �k;zf

is given in Corollary 1 and the solution to (39) has an explicitform. For equation (39) we obtain

a�

1 + a(� � 1)= log (1 + a(� � 1)) (40)

where a = 1��2

�2+ 1

�

. Denoting

w(�) =a� 1

a(� � 1) + 1and x =

a� 1

e; (41)

we can rewrite (40) as

w(�)ew(�) = x: (42)

Notice that w(�) =W(x), where W(x) is the Lambert W-function defined as z = W(z)eW(z), z 2 C. Therefore, bysolving w(�)=W(x) we have

�?� =

�1� 1

a

��1 +

1

W(x)

�: (43)

For � 2 [0; 1], � > 1 we have w � �1 and x 2 [�e�1;1).In this case W(x) is a single-valued function. If � = 0, weobtain the results in [5]. Note that only rational values of �are meaningful in practice.

If the transmit antennas are spaced sufficiently apart themajor loss in sum rate is due to path loss, cf. Figure 1.Therefore, it is of interest to characterize the sum rate gapR� between a user distribution FL and equally distant usersL=IK . For a fixed � and with �2=0, we have

R� = K log

�1 + �(� � 1)

1 + ���

�: (44)

Although �� induces a significant loss in sum rate, we stillhave a linear scaling of the sum rate with SNR [dB] because�� is independent of the SNR. Since � has only a minorimpact on R�, for reasonable antenna separations, we obtainfor �=IM and asymptotically high SNR

Rlim� = lim

�!1R� = K log TrL�1: (45)

From (45) we notice that Rlim� is solely depending on the

distribution of the channel gains L. As an example we supposethat the K users are uniformly distributed on a ring of maximaland minimal radius rmax and rmin, respectively. Furthermore,denoting dk the distance from user k to the transmitter,we apply the exponential path loss model (indicated by thenotation L 6=IK) i.e. lk=�d��k where � is chosen s.t. E lk=1for given rmax, rmin and �. This normalization ensures a fair

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

Rlimsum

�2=0

�2=0:1

� [dB]

sum

rate

[bits

/s/H

z]

�=IM , L=IK�=IM , L 6=IKUCA (d = �=2), L 6=IK

Fig. 1. ZF, sum rate vs. SNR with M = 32, � = 2 simulation results areindicated by circle marks with error bars indicating one standard deviation ineach direction.

comparison to the scenario L= IK . Assuming rmax� rmin,we obtain

Rlim� � K log

�4

�2 � 4

�+ (�� 2)K log

�rmax

rmin

�: (46)

Therefore, the sum rate gap Rlim� increases with � and

log rmax for a fixed rmin.

VI. NUMERICAL RESULTS

In our simulations all results are averaged over 10,000independent Rayleigh block-fading channel realizations.

The transmit correlation is assumed to depend only on thedistance dij , i; j = 1; 2; : : : ;M between antennas i and jplaced on a uniform circular array (UCA). Thus, (�)ij =J0(2�dij=�) [15], where J0 is the zero-order Bessel functionof the first kind and � is the signal wavelength. To ensure that�M grows slower than O(M), we suppose that the distancebetween adjacent antennas d = di;i+1 is independent of M ,i.e. as M grows the radius of the UCA increases.

Furthermore, we consider that the users are distributeduniformly on a ring with rmax=500m and rmax=35m with�=3:5, [16] (“Suburban Macro”) and � s.t. TrL=1.

Figure 1 compares the sum rate performance of the approx-imated sum rate to Monte-Carlo simulations. We observe, thatthe expressions derived for large (K;M) lie approximatelywithin the band of two standard deviations of the simulationresults even for finite (K;M). Therefore, the approximationderived in Theorem 2 are accurate and can be applied toconcrete optimization problems for the multi-user downlinkchannel. For high SNR, the sum rate loss due to path loss isgiven by (46), Rlim

� �75 [bits/s/Hz], corresponding well to thesimulation results.

Figure 2 compares the optimal number of users K?� in(43) to the optimal number of users K? obtained from Monte-Carlo simulations. More precisely K? is the number of users

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

18

20

22

24

M=16

M=32

UCA (d= �=2), L 6=IK

�=IM , L=IK

� [dB]

optim

alnu

mbe

rof

activ

eus

ers

K?�=M=�?�, (43)K? from exhaustive search

Fig. 2. ZF, sum rate maximizing number of active users vs. SNR with�2=0:1.

K<M that maximizes the ergodic sum rate, when distributeduniformly over the ring defined above. It can be observed thatK?� predicted by the asymptotic results does fit well evenfor finite dimensions. Moreover, introducing correlation andpath loss leads to larger dispersion of the optimal K over theselected SNR range.

Figure 3 depicts the impact of the number of served userson the ergodic sum rate. It can be observed that K?� achievesmost of the sum rate even for finite (K;M) and thus, is a goodchoice for the user allocation at the transmitter. Moreover,we observe that adapting the number of users is beneficialcompared to a fixed K. From Figure 2 we identify K=8 as agood choice (for �2=0:1) and, as expected, the performanceis optimal in the medium SNR regime and suboptimal at lowand high SNR. The situation changes by adding correlationand path loss. Since K=8 is highly suboptimal for low andmedium SNR (cf. Figure 2) we observe a significant loss insum rate in this regime.

VII. CONCLUSION

In this paper we derived deterministic equivalent of theSINR of ZF precoding by applying recent results from randommatrix theory. These approximations are shown to be veryaccurate even for finite dimensions and thus provide usefultools for many engineering applications. In particular theapproximated sum rate enabled us to derive expressions forthe optimal number of users in the cell and to characterize theimpact of a spatial user density on the achievable sum rate.

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5

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�=IM , L=IK

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� [dB]

sum

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