Joint beamforming and power control in d

Dobre‡ †Department of Electrical and Computer Engineering, Old Dominion University, USA ‡Faculty of Engineering and Applied Science, Memorial University of Newfoundland, Canada Abstract—

Joint Beamforming and Power Control in Downlink Multiuser MIMO Systems

Shiny Abraham†, Dimitrie C Popescu†, and Octavia A Dobre‡

†Department of Electrical and Computer Engineering, Old Dominion University, USA

‡Faculty of Engineering and Applied Science, Memorial University of Newfoundland, Canada

Abstract— In this paper we discuss joint beamforming and

power control for downlink multiuser MIMO systems with

target signal-to-interference+noise ratios (SINR) constraints.

We derive necessary conditions for minimizing total

interfer-ence at downlink receivers and present an algorithm which

adapts the beamforming vectors and powers incrementally

to meet specified SINR targets with minimum powers The

proposed algorithm is illustrated with numerical examples

obtained from simulations.

Index Terms— MIMO systems, downlink, beamforming,

power control, quality of service.

I INTRODUCTION

MIMO wireless systems exploit spatial dimensions in

wireless channels to provide increased capacity and

diver-sity, as well as to mitigate interference [1] This is achieved

through transmit beamforming and receiver combining

techniques [2], [3] which take advantage of the significant

diversity that is available in MIMO systems When users

are expected to meet specified target SINRs at the receiver,

beamforming is complemented with transmitter power

con-trol, and joint beamforming and power control problems

have been discussed in [4], [5] A related problem was

approached in the context of cellular MIMO systems in

[6] where the proposed algorithm exploits network duality

and is implemented using already existing algorithms

In this paper we present a new algorithm for joint

beam-forming and power control in downlink multiuser MIMO

systems with target SINR constraints at the receivers

The proposed algorithm employs incremental updates for

beamforming vectors and transmitted powers that are

de-signed to reduce interference at downlink receivers subject

to specified SINR and beamforming constraints

II SYSTEMMODEL ANDPROBLEMSTATEMENT

We consider the downlink of a multiuser MIMO wireless

system in which the base station is equipped with N

transmit antennas and communicates with K active users

We assume that a given user k has Nk receive antennas

such that Nk≤ N, ∀k The signal transmitted by the base

station is expressed as

x=

K

X

=1

b √p s = SP1/2b (1) where S = [s1 sK] is the N × K matrix of

unit-norm beamforming vectors, b = [b1, , bK]> is the

K-dimensional vector of symbols transmitted to active users,

and P = diag[p1, , pK] is a K × K diagonal matrix containing the transmit powers corresponding to distinct users in the system

The transmitted signal x is received by the K user receivers through distinct MIMO channels characterized by matrices G1, , GK of dimension Nk× N, and is cor-rupted by additive Gaussian noise vectors n1, , nKwith dimension Nk and covariance matrices Wk = E[nkn>

k],

k = 1, , K Thus, the received signal by a given user k

is given by

rk = Gkx+ nk = GkSP1/2b+ nk k = 1, , K (2)

We assume that all channel matrices Gk are known by the base station transmitter and that they are fixed for the entire duration of the transmission

Our goal in this setup is to derive an algorithm by which the base station transmitter performs joint beamforming and power adaptation such that a set of specified target SINRs {γ1, , γK} is achieved by all users

III SINRANDINTERFERENCEEXPRESSIONS

In order to decode the desired signal at a given receiver

k we rewrite the received signal in equation (2) from the perspective of user k

rk= Gkbk√p

ksk

| {z } desired signal

+ Gk

⎛

⎝

K

X

=1, 6=k

b √p s

⎞

⎠ + nk

interference + noise (zk)

(3)

where the interference-plus-noise zk seen by user k has correlation matrix

Zk = E[zkz>k] = Gk

⎛

⎝

K

X

=1, 6=k

sp s>

⎞

⎠ G>

We note that being a correlation matrix Zk is symmetric and positive definite Thus, it can be diagonalized as Zk=

Ek∆kE>

k and we can define the whitening transformation

Tk = ∆−1/2k E>k (5)

In transformed coordinates equation (3) can be written as

˜

rk = Tkrk= TkGkbk√p

ksk+ Tkzk

= G˜kbk√p

where ˜Gk = TkGk is the MIMO channel matrix seen

by user k in new coordinates and wk = Tkzk is

the equivalent white noise term with covariance matrix

E[wkw>

k] = TkZkT>

k = INkequal to the identity matrix

We now apply the SVD to the transformed channel

matrix to obtain

˜

Gk= UkDkV>k (7) Let us denote the rank of user k’s transformed MIMO

channel matrix ρk = rank( ˜Gk) This is equal to the

number of non-zero singular values and satisfies ρk ≤

min(N, Nk) The singular value matrix Dk may be

parti-tioned as

Dk =

∙ ˜

Dk 0ρk×(Nk−ρk)

0N×ρk 0(Nk−ρk)×(Nk−ρk)

¸ (8)

where ˜Dkis a ρk×ρkdiagonal matrix containing the

non-zero singular values and the non-zero matrices have appropriate

dimensions

We premultiply by U>

k in equation (6) to obtain

¯rk= DkVk>skbk√p

and we define ¯sk = V>kskand ¯wk = U>kwk We can then

rewrite equation (9) as

¯

rk= Dk¯skbk√p

The partition (8) on the singular value matrix Dk induces

the following partition on the transformed beamforming

vector ¯sk corresponding to user k

¯

sk =

∙

¯

sk1

¯

sk2

¸

(11) where ¯sk1 has dimension ρk× 1 and ¯sk2 has dimension

(Nk− ρk) × 1 Taking into account the partitions in (8)

and (11) we note that the last (N −ρk) components in the

¯

rk vector can be safely ignored as they will be equal to

zero Furthermore, the last (N − ρk) components ¯sk2 of

the transformed beamforming vector should be set to zero

( ¯sk2= 0(Nk−ρk)×1) in order to ensure that no transmitted

signal energy will be wasted on those dimensions of the

transformed channel matrix with zero singular values

Thus, we focus only on those dimensions corresponding

to strictly positive singular values and reduce

dimensional-ity to the rank of the transformed channel matrix by taking

the first ρk elements in ¯rk to obtain

ˆrk= [Iρk0]¯rk= ˜Dk˜skbk√p

where ˜sk= ¯sk1and ˜wk = [Iρ k0] ¯wk We invert the channel

in equation (12) to obtain the equivalent expression

ˆ

rk,inv= ˜D−1k ˆrk = ˜skbk√p

k

| {z } desired signal

+ D˜−1k w˜k

| {z } interf.+noise

(13)

The decision variable for user k obtained by matched filtering is ˆbk= ˜s>

kˆrk,inv and implies that the interference experienced by user k is

ik= ˜s>kD˜−2k ˜sk (14) and its corresponding SINR is γk= pk/ik

IV JOINTBEAMFORMING ANDPOWERCONTROLFOR

DOWNLINKMIMO SYSTEMS

Since the base station transmitter has knowledge of all the user beamforming vectors, transmit powers, and MIMO channel matrices, we will obtain the beamforming and power update equations by solving the constrained minimization of the sum of interference functions, that is

min

⎧

⎨

⎩

˜s1, , ˜sK

p1, , pK

⎫

⎬

⎭

I =

K

X

k=1

ik subject to

γk = γk∗ and ˜s>

k˜sk= 1, k = 1, , K (15)

In order to solve this constrained optimization problem

we define the corresponding Lagrangian function L(˜s1, , ˜sK, p1, , pK, λ1, , λK, ξ1, , ξK) =

= I +

K

X

k=1

λk

µ

pk

ik − γk∗

¶ +

K

X

k=1

ξk(˜s>k˜sk− 1)

(16) where λk and ξk are Lagrange multipliers associated the constraints in equation (15) The necessary conditions for minimizing the Lagrangian (16) are obtained by differen-tiating with respect to the corresponding variables, ˜sk, pk, and multipliers λk, ξk, k = 1, , K and by equating the corresponding partial derivatives to zero Differentiating with respect to the beamforming ˜sk leads to the eigen-value/eigenvector equation

∂Lk

∂˜sk

= 0 =⇒ D˜−2k ˜sk= νk˜sk (17) where νkis expressed in terms of the Lagrange multipliers

as well as user power pk and beamforming vector ˜sk The exact expression of νk is not relevant and any eigenvector

of ˜D−2k will satisfy the necessary condition (17) However,

a meaningful choice for updating user k’s beamforming vector is the eigenvector xkcorresponding to the minimum eigenvalue of ˜D−2k since for a given power value pk this minimizes the term ik corresponding to user k in the sum interference function I In order to avoid potential steep changes in the beamforming vector that may not be tracked

by the receiver due to the minimum eigenvector being far away in signal space from the current beamforming vector, we will use an incremental update that adapts the user beamforming vector in the direction of the minimum eigenvector xk defined by:

˜

sk(n + 1) = ˜sk(n) + mβxk(n)

k˜sk(n) + mβxk(n)k (18)

where β is a parameter that limits how far in terms

of Euclidian distance the updated beamforming vector

can be from the old one and m = sgn[˜s>

k(n)xk(n)]

This update corresponds to an incremental interference

avoidance update [7] that implies a decrease in the user

interference function ik and an increase in the user SINR

Upon adaptation of user k beamforming vector, the

value of its corresponding effective interference function

is given by the expression:

i0k(n) = ˜sk(n + 1)>D˜−2k (n)˜sk(n + 1)

≤ ˜sk(n)>D˜−2

Differentiating now the Lagrangian (16) with respect to

the multiplier λk we obtain another necessary condition

for the constrained optimization problem (15)

∂Lk

∂λk

= pk

ik − γk∗= 0 (20) which indicates that, given the interference function ik the

transmitted power corresponding to user k should match

its target SINR, that is

p∗k = γk∗ik k = 1, , K (21)

Thus, given the value of the effective interference function

i0k(n) after beamforming update in equation (19) the power

value matching the desired target SINR is

p0k(n) = γk∗˜sk(n + 1)>D˜−2k (n)˜sk(n + 1) (22)

Since the value p0

k(n) may not be close to the current power value pk(n) of user k and in order to avoid abrupt

variations we will use a “lagged” power update given by

pk(n + 1) = (1 − μ)pk(n) + μp0k(n) (23)

with 0 < μ < 1 a suitably chosen constant We note that,

the smaller the μ constant is, the more pronounced the

lag in the power update is and the smaller the incremental

power change will be

The proposed algorithm for joint beamforming and

power control in downlink multiuser MIMO systems uses

the beamforming and power update equations (18) and (23)

and is formally stated here:

1) Input Data:

• Beamforming and power matrices S, P,

down-link channel matrices Gk, target SINR values

γ∗

k, and noise covariance matrices Wk, k =

1, , K

• Constants β, μ, and tolerance

2) FOR each user k = 1, , K DO

a) Apply the whitening transformation (5)

fol-lowed by the SVD (7), compute corresponding

˜

D−2k (n), and determine its minimum

eigenvec-tor xk(n)

b) Update user k’s transformed beamforming

vec-tor using equation (18)

c) IF ρk < Nkappend a zero vector of appropriate dimension and obtain the actual beamforming vector sk= Vk˜sk

d) Update user k’s power using equation (23) 3) IF change in sum interference function I is larger than specified tolerance then GO TO Step 2 ELSE STOP: a fixed point has been reached

Numerically, a fixed point of the algorithm is reached when the beamforming and power updates result in changes of the sum interference functions I that are smaller than the specified tolerance We note that the algorithm is guaran-teed to converge to a fixed point since at each update the corresponding iterations go in the direction of a stationary point of the constrained optimization problem (15) with convex cost function and convex variable sets [8] We also note that, as it is the case with incremental algorithms in general, the convergence speed of the algorithm depends

on the values of the beamforming and power increments specified by the algorithm constants β and μ

V SIMULATIONSRESULTS

We illustrate the proposed algorithm for a system with

N = 10 transmit antennas at the base station and K = 5 active users with Nk = 4 antennas each and white noise with covariance matrix Wk = 0.1I4 at all receivers The power matrix is initialized to P = 0.1I5 while the beamforming matrix S and the user channel matrices are initialized randomly The algorithm parameters are set to

β = 0.02, μ = 0.01, = 0.02, and the target SINRs are initialized to γ∗= {5, 4, 3, 2, 1}

In the first experiment we simulate the algorithm for fixed number of active users with variable target SINRs Once the beamforming vectors and powers that meet the specified targets are obtained using the proposed algorithm, user 5 increases its target SINR from 1 to 2.5 As a result of this change the algorithm starts updating user beamforming vectors and powers until a new fixed point that meets the new specified set of target SINRs is reached, when user 5 decreases target SINR to 1.75 and initiates new updates for beamforming vectors and power The algorithm adjusts their values until a new fixed point is reached where the target SINRs are once again met for all users The variation

of user SINRs and powers for this experiment are plotted

in Figure 1 from which we note that each time a change in the target SINR of user 5 occurs there is a sharp change

in all the users’ SINR and power values which is then compensated by the algorithm

In the second experiment we start from the same ini-tializations as before (including the same initial target SINRs) but after convergence to the fixed point where specified target SINRs are met, user 5 becomes inactive Its corresponding beamforming vector and power are dropped from S and P matrices which determines the algorithm to update the beamforming vectors and powers for remaining users until a new fixed point is reached where the target

0 0.5 1 1.5 2

x 104 0

1

2

3

4

5

6

7

Updates

user1 user2 user3

user5

(a) SINR variation

x 104 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Updates

user1 user2 user3 user4 user5

(b) Power variation

Fig 1 Tracking variable SINRs experiment.

SINRs of active users γ∗= {5, 4, 3, 2} are satisfied Then,

a new user becomes active in the system and its (randomly

initialized) beamforming vector and power are added to

the S and P matrices under new user 5 with new target

SINR equal to 0.5 This determines the algorithm to update

again all beamforming vectors and powers until a new

fixed point is reached where the target SINRs for all users

are satisfied The variation of user SINRs and powers for

this experiment are plotted in Figure 2 from where we

note that, similar to the previous experiment, each time a

change in the number of active users in the system occurs

there is a sharp change in all active users’ SINR and power

values which is compensated by the proposed algorithm

VI CONCLUSIONS

In this paper we presented a new algorithm for joint

beamforming and power control in downlink MIMO

sys-tems with target SINRs at the receivers The proposed

al-gorithm uses incremental updates for beamforming vectors

and powers that reduce the total interference in the system,

and can be used for tracking changing target SINRs and/or

variable number of active users

0 2000 4000 6000 8000 10000 12000 14000 16000 0

1 2 3 4 5 6 7 8 9

Updates

user1 user2 user3

user5

(a) SINR variation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Updates

user1 user3 user5

(b) Power variation

Fig 2 Tracking variable number of active users experiment.

ACKNOWLEDGMENT

This work was supported in part by the National Sci-ences and Engineering Research Council of Canada

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