IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 12, DECEMBER 1998 1619
Upper Bounds on the Bit-Error Rate of
Optimum Combining in Wireless Systems
Jack H. Winters, Fellow, IEEE, and Jack Salz, Member, IEEE
Abstract—This paper presents upper bounds on the bit-error
rate (BER) of optimum combining in wireless systems with
multiple cochannel interferers in a Rayleigh fading environment.
We present closed-form expressions for the upper bound on
the bit-error rate with optimum combining, for any number of
antennas and interferers, with coherent detection of BPSK and
QAM signals, and differential detection of DPSK. We also present
bounds on the performance gain of optimum combining over
maximal ratio combining. These bounds are asymptotically tight
with decreasing BER, and results show that the asymptotic gain
is within 2 dB of the gain as determined by computer simulation
for a variety of cases at a
10
0
3
BER. The closed-form expressions
for the bound permit rapid calculation of the improvement with
optimum combining for any number of interferers and antennas,
as compared with the CPU hours previously required by Monte
Carlo simulation. Thus these bounds allow calculation of the
performance of optimum combining under a variety of conditions
where it was not possible previously, including analysis of the
outage probability with shadow fading and the combined effect
of adaptive arrays and dynamic channel assignment in mobile
radio systems.
Index Terms— Bit-error rate, optimum combining, Rayleigh
fading, smart antennas.
I. INTRODUCTION
A
NTENNA arrays with optimum combining combat multi-
path fading of the desired signal and suppress interfering
signals, thereby increasing both the performance and capacity
of wireless systems. With optimum combining, the received
signals are weighted and combined to maximize the signal-to-
interference-plus-noise ratio (SINR) at the receiver. Optimum
combining yields superior performance over maximal ratio
combining, whereby the signals are combined to maximize
signal-to-noise ratio, in interference-limited systems. However,
while with maximal ratio combining the bit-error rate can
be expressed in closed form [1], with optimum combining
a closed-form expression is available only with one interferer
[2], [3]. With multiple interferers, Monte Carlo simulation has
been used [3]–[5], but this requires on the order of CPU hours
even with just a few interferers. Thus the improvement of
optimum combining has only been studied for a few simple
Paper approved by N. C. Beaulieu, the Editor for Wireless Communication
Theory of the IEEE Communications Society. Manuscript received September
21, 1993; revised November 28, 1996. This paper was presented in part at
the 1994 IEEE Vehicular Technology Conference, Stockholm, Sweden, June
8–10, 1994.
J. H. Winters is with AT&T Labs–Research, Red Bank, NJ 07701 USA.
J. Salz, retired, was with AT&T Labs–Research, Crawford Hill Laboratory,
Holmdel, NJ 07733 USA.
Publisher Item Identifier S 0090-6778(98)09388-X.
Fig. 1. Block diagram of an
M
-element adaptive array.
cases, and detailed comparisons (e.g., in terms of outage
probability) have not been done.
In [6], we showed that, with
antenna elements, the
received signals can be combined to eliminate
interferers in the output signal while obtaining an
diversity improvement, i.e., the performance of maximal ratio
combining with
antennas and no interference. However,
this “zero-forcing” solution gives far lower output SINR than
optimum combining in most cases of interest and cannot be
used when
.
In this paper we present a closed-form expression for the up-
per bound on the bit-error rate (BER) with optimum combining
in wireless systems. We assume flat fading across the channel
and independent Rayleigh fading of the desired and interfering
signals at each antenna.
1
Equations are presented for the
upper bound on the BER for coherent detection of quadrature
amplitude modulated (QAM) and binary phase-shift-keyed
(BPSK) signals, and for differential detection of differential
phase-shift-keyed (DPSK) signals. From these equations, a
lower bound on the improvement of optimum combining over
maximal ratio combining is derived.
In Section II we derive the upper bound on the BER. In
Section III we compare the upper bound to Monte Carlo
simulation results. A summary and conclusions are presented
in Section IV.
II. U
PPER BOUND DERIVATION
Fig. 1 shows a block diagram of an -element adaptive
array. The complex baseband signal received by the
th
antenna element in the
th symbol interval is multiplied
by a controllable complex weight
and the weighted signals
are summed to form the array output signal
.
1
As shown in [7], the gain of optimum combining is not significantly
degraded with fading correlation up to about 0.5. Thus our bounds, based on
independent fading, are reasonably accurate and useful even in environments
with fading correlation up to this level.
0090–6778/98$10.00 1998 IEEE
1620 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 12, DECEMBER 1998
With optimum combining, the weights are chosen to maxi-
mize the output SINR, which also minimizes the mean-square
error (MSE), which is given by [8]
MSE
(1)
where
is the received interference-plus-noise correlation
matrix given by
(2)
is the noise power, is the identity matrix, and
are the desired and th interfering signal propagation
vectors, respectively, and the superscript
denotes complex
conjugate transpose. Here we have assumed the same average
received power for the desired signal at each antenna (that
is, microdiversity rather than macrodiversity) and that the
noise and interfering signals are uncorrelated, and without
loss of generality, have normalized the received signal power,
averaged over the fading, to
. Note that the MSE varies at
the fading rate.
For coherent detection of BPSK or QAM, the BER is
bounded by [9]
(3)
where now the expected value is taken over the fading
parameters of the desired and interfering signals, and
is the
variance of the BPSK or QAM symbol levels (e.g.,
and for BPSK and quaternary phase-shift keying (QPSK),
respectively). For differential detection of DPSK, assuming
Gaussian noise and interference,
2
the BER is given by [1]
(4)
Thus the BER expression for both cases differs only by a
constant, and we will now consider the term
.
As shown in the Appendix, this term can be upper-bounded by
(5)
where
denotes the determinant of , and is the
th eigenvalue of .
Since (5) is the key inequality in our bound (and is the only
inequality we use in determining the bound for differential
detection of DPSK), let us examine its accuracy. The bound
is tight if
, and since the ’s are proportional
to the interference signal powers, the bound is tight for
large received SINR, i.e., low BER’s. Although for all cases
and thus BER , for
the BER as given by the bound may exceed . Thus with
small received SINR, occasionally BER’s greater than
may
be averaged into the average BER, reducing the tightness of
2
Since the stronger the interference, the more that optimum combining
suppresses it, with the Gaussian assumption we overestimate the probability
of strong interference. Note that this is consistent with the derivation of an
upper bound on the BER.
the bound. Also, note that with only noise at the receiver,
, where is the variance of the noise normalized to
the received desired signal power, and from (4) and (5)
(6)
where
is the received SINR, while the actual BER is
[1]. Thus even without interference, the bound
differs from the actual BER, and this difference increases as
the received SINR decreases.
Let us consider the case of interference only. In this case,
, which is given by (2), may also be expressed as
(7)
where
,
is the th element of , the sum is extended over
all
permutations of the ’s, is the th element of
the permutation of the
’s, the “ ” sign is assigned for even
permutations (i.e., an even number of swapping of
’s in
the permutation), and the “
” sign for odd permutations. Now
(8)
where
is the average power of the th interferer normalized
to the desired signal power, and
(9)
Similarly, from (7), it can be shown that
(10)
where the sum is over all sets of positive integers
and
that exist such that , with .
For example, when
, there are 6 sets of such
that
(see Table I). All sets are of the form
, e.g., for , except for the
set
for .
is an integer coefficient corresponding to the th set
with
antennas. Note that is obtained by summing the
coefficients (
’s) for similar terms in . can
be determined as shown below.
Since
, and when ,
(10) can also be expressed as
(11)
WINTERS AND SALZ: UPPER BOUNDS ON THE BER OF OPTIMUM COMBINING IN WIRELESS SYSTEMS 1621
TABLE I
V
ALUES OF
(
M
)
q
FOR
M
=2
TO
5
where now .
To determine the
’s, first note that if
then , and (11) becomes
(12)
where the
’s and the ’s can be seen to be closely
related. From [6],
for , and thus the
’s are the coefficients of the th-order polynomial in ,
. This result is not only
useful when all interferers have equal power, but also serves
as a consistency check on our calculated values of
.
The values of
were generated using a computer
program to examine every permutation in (7) for given
. The
number of each type of
term was calculated to
determine
. Tables I and II list these values for – .
Note that only
and terms exist for , and and
terms also exist for . Values for for higher
can also be easily calculated. However, since the amount
of computer time to generate the values of
increases
exponentially with
, our program could only generate these
values in a reasonable amount of computer time for up to
(where a hundred CPU hours on a SPARCstation20
would be required).
From (3), the upper bound on the BER with coherent
detection of BPSK or QAM is now given by
(13)
TABLE II
V
ALUES OF
(
M
)
q
FOR
M
=6
AND
7
and from (4), the upper bound on the BER with differential
detection of DPSK is given by
(14)
For the case of noise with
interferers, consider the noise as
an infinite number of weak interferers with total power equal
to the noise. That is, let
(15)
and let
. Then, , and
(16)
for
. Therefore, with noise, the BER bound is the same
as in (13) and (14), but with
including the noise. In this
case, if we define the received desired signal-to-noise ratio
as
and the th interferer signal-to-noise ratio as
1622 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 12, DECEMBER 1998
, then (14) becomes [similarly for (13)]
(17)
Since
is the bound with maximal ratio combining, the
term in the brackets is the improvement of optimum combining
over maximal ratio combining based on the BER bound.
Defining the gain of optimum combining as the reduction in
the required
for a given BER, from (17), this gain in decibels
is given by
Gain (dB)
(18)
This gain is therefore independent of the desired signal
power (because the bound is asymptotically tight as
).
However, this is the gain of the BER bound with optimum
combining over the BER bound with maximal ratio combining.
Since the required
for a given BER with maximal ratio
combining is less than the bound, the true gain may differ
from (18) and to obtain a bound on the gain, the gain in (18)
must be reduced accordingly. For example, with differential
detection of DPSK, to obtain a bound the gain given in (18)
is reduced by the factor
. Note that as ,
this factor reduces to one and the gain approaches (18). Thus
we will refer to (18) as the asymptotic gain.
III. C
OMPARISON TO EXACT THEORY AND SIMULATION
In this section, we compare the bound to theoretical results
for
and simulation results for .
Fig. 2 compares theoretical results (from [1]–[3]) for the
gain to the asymptotic gain (18) versus BER with coherent
detection of BPSK. Results are generated for
and ,
and
3 and 10 dB. In all cases the gain monotonically de-
creases to the asymptotic gain as the BER decreases. The gain
approaches the asymptotic gain more slowly with decreasing
BER for larger
and also, at low BER’s, the accuracy of the
asymptotic gain decreases with higher
. Thus the accuracy
of the asymptotic gain decreases as the
required for a given
BER with optimum combining decreases, as predicted by the
approximation in Section II.
Fig. 3 compares theoretical and Monte Carlo simulation [5]
results for the gain to the asymptotic gain with
and
, , and 6. Results are plotted versus , where all
interferers have equal power, for coherent detection of BPSK
Fig. 2. Gain versus BER for coherent detection of BPSK—comparison of
analytical results to the asymptotic gain.
Fig. 3. Gain with
M
=2
for 1, 2, and 6 equal-power interferers versus
signal-to-noise ratio of each interferer—comparison of analytical and Monte
Carlo simulation results with coherent detection of BPSK [5] to the asymptotic
gain.
at a BER.
3
In all cases, the asymptotic gain has the
same shape as the gain and is within 1.7 dB for
,
1.0 dB for
, and 0.4 dB for . Since optimum
combining gives the largest gain when the interference power
is concentrated in one interferer and the least gain when the
interference power is equally divided among many interferers,
and represent the best and worst cases for the
gain in an interference-limited cellular system. Thus from the
results in Fig. 3, we would expect the asymptotic gain to be
within 0.4–1.7 dB of the actual gain for all cases in cellular
systems with
.
3
This BER was used because the results in [5] were obtained for this BER.
As shown in [5], the gain does not change significantly for BER’s between
10
0
2
and
10
0
3
, the range of interest in most mobile radio systems.
WINTERS AND SALZ: UPPER BOUNDS ON THE BER OF OPTIMUM COMBINING IN WIRELESS SYSTEMS 1623
Fig. 4. Gain versus
M
with two and six equal power interfer-
ers—comparison of Monte Carlo simulation results with coherent detection
of BPSK [3] to the asymptotic gain.
Now, consider the lower bound on the gain obtained from
the BER bound (17), as compared to the asymptotic gain.
Without interference, differential detection of DPSK with
maximal ratio combining and
requires 13.3 dB
(theoretically [10]) for a
BER, while the BER bound
(17) gives
13.5 dB. Thus the lower bound on the gain
(from (17)) at a 10
BER is 0.2 dB less than the asymptotic
gain for any interference scenario—in particular, the lower
bound on the gain is 0.2 dB less than the results shown in
Fig. 3. Similarly, coherent detection of BPSK with maximal
ratio combining and
requires 11.1 dB for a 10
BER, while the BER bound (13) gives 15.0 dB. Thus the
bound is most accurate with differential detection of DPSK
and low BER’s.
Fig. 4 compares Monte Carlo simulation results [3] for the
gain to the asymptotic gain for
and . Results are plotted
versus
with 3 dB for all interferers and coherent
detection of BPSK at a 10
BER. Again the asymptotic
gain has the same shape as the simulation results. The cases
include both many more interferers than antennas and many
more antennas than interferers, but in all cases the asymptotic
gain is within 1.8 dB of simulation results.
IV. C
ONCLUSIONS
In this paper we have presented upper bounds on the bit-
error rate (BER) of optimum combining in wireless systems
with multiple cochannel interferers in a Rayleigh fading envi-
ronment. We presented closed-form expressions for the upper
bound on the bit-error rate with optimum combining, for any
number of antennas and interferers, with coherent detection of
BPSK and QAM signals, and differential detection of DPSK.
We also presented bounds on the performance gain of optimum
combining over maximal ratio combining and showed that
these bounds are asymptotically tight with decreasing BER.
Results showed that the asymptotic gain is within 2 dB of
the gain as determined by computer simulation for a variety
of cases at a 10
BER. These cases include interference
scenarios that cover the range of worst to best cases for the
gain of optimum combining in cellular systems with
.
The bound is most accurate with differential detection of
DPSK and high SINR, corresponding to low BER and a
few antennas. Because of the 2-dB accuracy, the bound is
most useful where the optimum combining improvement is
the largest, which is the case of most interest. The closed-
form expression for the bound permits rapid calculation of
the improvement with optimum combining for any number
of interferers and antennas, as compared with the CPU hours
previously required by Monte Carlo simulation. These bounds
allow calculation of the performance of optimum combining
under a variety of conditions where it was not possible
previously, including analysis of the outage probability with
shadow fading and the combined effect of adaptive arrays and
dynamic channel assignment in mobile radio systems.
A
PPENDIX
Diagonalizing by a unitary transformation , we obtain
(19)
where
denotes an matrix with nonzero
elements only on the diagonal, or
(20)
and
(21)
Let
(22)
Then
(23)
and
(24)
Since with independent, Rayleigh fading at each antenna,
the elements of
are independent and identically distributed
(i.i.d.) complex Gaussian random variables, the elements of
are also i.i.d. complex Gaussian random variables with the
same mean and variance. Furthermore, the
’s are indepen-
dent of the
’s. Thus we can average over the desired and
interfering signal vectors separately, i.e.,
(25)
![Fig. 4. Gain versusM with two and six equal power interferers—comparison of Monte Carlo simulation results with coherent detection of BPSK [3] to the asymptotic gain.](/figures/fig-4-gain-versusm-with-two-and-six-equal-power-interferers-1p0mouyp.webp)
