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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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