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UPLINK AND DOWNLINK BEAMFORM ING

FOR FADING CHANNELS

Mats Bengtsson and Bj om O ttersten

Signal Processing, S3, Royal Instituteof Technology

100 44 Stockholm, SWEDEN

Tel: 4 6 79084 63, fax: +46 8 790 72 60

Email:

matsb8s3.

kth. e

ABSTRACT

We highlight some issues in the design of beamformers

for transmission and reception in communications systems

with antenna arrays at the base stations. We assume a ra-

dio channel characterized by frequency flat Rayleigh fading

which is correlated (but not coherent) from antenna element

to antenna element.

Different design criteria are considered and we show the

relationship between downlink and uplink formulations and

solutions. The resulting beamformers are typically given

by quadratically constrained optimization problems which

could be expected to give increased robustness to signal can-

cellation. Unfortunately, signal cancellation is still present,

but diagonal loading can be introduced to minimize the prob-

lem.

1.

INTRODUCTION

When antenna arrays are used at the base station of a cellu-

lar system in environments with multipath propagation, the

standard plane wave model used in much of the array pro-

cessing literature is not applicable.

As

an alternative, some

authors have used a fading channel where the fading pro-

cess in independent between the different antenna elements

[15,16] . Since the assumption of independently fading sub-

channels only holds in environments with severe multipath

or

large separation between the antenna elements, we as-

sume here a channel with correlated fading between the an-

tenna elements. Such models can be used to model e.g. the

diffuse scattering caused by reflections close to each mo-

bile, see [ , 8,181. We assume that the channel is constant

within the time frame allocated to one data burst but varies

randomly from burst to burst.

When the antenna array is used as a receiver, i.e. in

uplink mode, the instantaneous channel can be estimated

directly from the received data, whereas in the downlink,

the transmitting beamformer must be based on information

collected in the uplink. Several schemes have been pro-

posed for the transformation from uplink to downlink. In a

Time Division Duplex (TDD) system with sufficiently short

time slots, the downlink channel

is

virtually identical to

the uplink channel, whereas in a Frequency Division Du-

plex (FDD) system, the channel fades independently at the

two duplex frequencies. However, a statistical model of the

downlink channel can be obtained from the collected up-

link data using a physical model [5,18] or model-free tech-

niques [2,9]. In this paper we assume that the statistics of

the downlink channel is given, exactly or in the form of a

noisy estimate.

The maximum Signal to Interference plus Noise Ratio

(SINR) solution has been used for different classes of prob-

lems, see for example

[171.

For the model with local scat-

tering, the uplink scenario is studied in

[3]

and the corre-

sponding downlink formulation is given in [181. Here, we

show that the maximum SINR solution can be interpreted as

quadratically constrained minimum variance beamforming

[7,12,13].

Since the phase of the fading channel fluctuates ran-

domly, a naive treatment of the Minimum Mean Square

Er-

ror (MMSE) problem gives the all-zero solution. Using the

assumption of a coherent receiver, we give two different so-

lutions to the problem. For the downlink, it is not obvious

how to formulate an MMSE problem and interpret the re-

sult. We propose to apply the uplink solution also in the

downlink.

Simulations have been performed to compare the dif-

ferent beamformers and study the sensitivity to modeling

errors. Diagonal loading

is

successfully used to avoid the

problem of signal cancellation.

2. DATAMODEL

2.1.

Uplink

We assume a frequency flat Rayleigh fading channel, where

the baseband data at the antenna array is collected in the

complex valued

x 1

vector xu t )given by

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k = l

where Sk t) is the baseband signal transmitted at the kth

mobile,

n(t) s

spatially and temporally white Gaussian noise

with covariance matrix

and the array response vector

vk

corresponding to mobile C

is complex Gaussian with

v

E N(O,RU,,)

3)

2.2. Downlink

If the baseband signal transmitted at the antenna airay is

xd(t),

hen the signal r k t ) received at the k:th mobile is

given by

where nk ( t ) s temporally white Gaussian noise with vari-

ance affkand the array response vector

vi

corresponding to

mobile is complex Gaussian with

vi E

N(O,Rt,)

.

In general, Rtkand R$k an be full rank matrices, even

though in many applications several of the eigenvalues are

significantly smaller than the noise variance, which means

that numerically, the rank can be considered lower than the

number of antenna elements. Examples of this kind of mod-

els can be found in [

1,8].

3. ALGORITHMS

We consider beamformers of the form 1

( t )

= w*x

t )

or

estimation of the signal from mobile number one. Similarly,

we use beamformers of the form xd(t)= wsl(t) for trans-

mission of the signal

s l ( t )

o the first mobile.

3.1.

Maximal

SINR

In the uplink, the signal to interference plus noise ratio is

given by [3,17]

Define the interference plus noise covariance matrixRIN=

xi=,

tk+ u i 1

Then the maximum

SINR

beamformer

is given by the eigenvector corresponding to the maximum

eigenvalue of the generalized eigenvalue problem

R;,W = XRINW. (7)

It is easy to show that

RIN

in

(7)

can be replaced by

R:

=

Rtl +RINwithout changing the solution. Thus, the max-

imum SINR beamformer can alternatively be characterized

by (up to a scaling)

arg

m a w*RZw (8)

W'R w =l

1

which

is

a Quadratically Constrained Minimum Variance

(QCMV) beamformer, closely related to the linearly con-

strained minimum variance or Capon beamformer [10,141.

One tempting implementation of (8) is to estimate

Rtl

using a training sequence or a blind DOA based method and

use this estimate together with the unstructured estimate

9 )

t=l

in the calculation of w.

However, as will be illustrated in

Section 4, this can easily lead to problems with signal can-

cellation. Just as for the Capon beamformer, several meth-

ods

can be used to avoid this problem. One solution is to

use a structured estimate also for RU,,as suggested i? [113,

another is to use diagonal loading, i.e., to replace Ri by

RU,

PI,

see e.g.

[6]

For the downlink, a design criterion similar to (7) can

be derived as the beamformer that gives maximum signal

power at the desired mobile while keeping the total power

transmitted to all other users below a certain threshold. The

details can be found

in

[18]. Similarly, the criterion (8) can

motivated as the beamformer that transmits the minimum

total power to all users while keeping the power transmitted

to the desired user at a fixed level.

3.2. MMSE

A direct application of the data model (1)-(3) on the Mini-

mum Mean Square Error (MMSE) criterion would give the

all-zero solution, since

E[v ]

= 0 which results in

E[s;(t)x (t)] = 0. However, this is only a problem with

the mathematical treatment, since a coherent detector can

track the phase of the signal. In the traditional plane wave

models, this problem is handled mathematically fixing one

element of the array response vector to 1. This procedure

cannot be directly used on the fading channel, so we give

two alternative solutions to the problem.

Perform an eigenvalue decomposition of

Rtl ,

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Then, each realization of vy can be written as

m

; + +

0.9

+

+

+

k = l

where

Pk

are independent Rayleigh distributed random vari-

ables with E[lpk12]

=

x k and

k

are independent uniformly

distributed over [0 ,27~] .

Now, let

(t)=

e-j41xu(t), then the MMSE beam-

former for s I C ) given ( t )s

w = E[jis;(t)] c (R ,)-'el (13)

The same

w

can be applied to

xu

t ) f combined with a co-

herent detector. Note that normalizing ( t ) lso by a factor

l / p l would give a data vector with infinite variance.

An even better solution is to first consider the condi-

tional MMSE solution given a specific realization of vy,

Weond. It is easy to show, using the matrix inversion lemma,

that

QCMV true

x

MMSEP true

and since the phase and the amplitude of the beamformer is

irrelevant, we select the beamformer as

w = arg maxE,; [ ~ w * w ~ ~ n d ~ ~ ]

w w * =l

0.8-

0.7

0.6

w

B 0.5

80.4-

z

0.3

argmaxw*RFkR:,RFhw.

(15)

w w * = l

+ + + + +

p-

O 6

* o

6

-

U

-

B O

o

8

-

o

-

o

Q

- B o O o O ~ ~

With this criterion, the optimal beamformer is thus given by

the principal eigenvector of

R;hR:,RF;,

4. NUMERICAL ILLUSTRATIONS

We have studied an interference limited scenario with two

mobiles subject to local scattering with rectangular angular

distribution and a spread angle of 00 =

3 .

We assume a

uniform linear array with A

=

1 / 2 wavelengths element

separation, thus Rvk s given by [4]

The signal of interest is kept at DOA

61 = 10'

while the

10

dB stronger interferer is moved between -20 and

8 .

The

SNR is 10 dB compared to the signal of interest.

Data is processed in bursts of N

=

100 symbols. The

channel is constant within each burst but fades indepen-

dently from burst to burst. The covariance matrices

R,,

and R, are estimated from the last

10

bursts in order to av-

erage over the fading.

We define a normalized MSE estimate as

The following algorithms were evaluated:

QCMV. estim.

MMSEl estim.

O o

0.121

5

10

15 20 25

30

Source separation degrees

Figure 1: Estimated MSE and SINR, using the algorithms

without diagonal loading.

QCMV

The QCMV formulation

(8)

of the maximum SINR

beamformer.

MMSEl The eigen-decomposition based solution

(13)

to

the MMSE problem.

MMSEZ

The best matched beamformer solution (15) to

the MMSE problem.

BF The traditional beamformer without any interference

suppression, given by the principal eigenvector of R,, .

Both the true and estimated covariance matrices were used.

Figure 1 shows the pFrformance of the originalalgorithms,

whereas in Figure2,

R,, (R,)

was replaced by

R,+100u:I,

(R,

+ 1OOu:I) in the examples using the estimated covari-

ance, in order to decrease the signal cancellation problems.

All plotted results are averaged from 500 data bursts.

5. CONCLUSIONS AND SUMMARY

We have illustrated the use of the maximum SINR and MMSE

criteria in the design of beamformers for fading channels.

The maximum SINR beamformer can be given a mini-

mum variance formulation which is easier to calculate from

measurement data. The MMSE formulation causes some

problems for the particular data model and we have sug-

gested two different solutions.

For

a point source scenario,

both solutions reduce to the traditional MMSE beamformer.

As is shown in Figure 1, the algorithms suffer from sig-

nal cancellation problems. However, when the data covari-

ance matrix is regularized, the suggested algorithms im-

prove significantly, as illustrated in Figure 2, although the

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09}

0.4 -

0.3

0.2

0.1

2

-

QCMV true

+ QCMV estim.

5 10 15 20 25 30

Source

separation,

degrees

Ob

Figure 2: Estimated MSE and SINR, using the algorithms

with diagonal loading.

theoretical performance drops. The only algorithm that does

not react positively to regularization is the second MMSE

solution (15) which performs similarly to the traditional beam-

former when used with estimated channels (not included in

the graphs for clarity). With known channels, MMSE2 per-

forms slightly better than MMSE1.

The design of downlink beamformers is really a multi-

objective optimization problem, but the uplink solutions can

be given reasonable motivations also for the downlink prob-

lem.

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