1
Bounds on Outage Probabilities for Diversity
Receptions Over Arbitrarily Correlated Rician
Channels
Bingcheng Zhu
∗†
, Julian Cheng
†
, Naofal Al-Dhahir
‡
and Lenan Wu
∗
∗
School of Information Science and Engineering, Southeast University
Nanjing, Jiangsu, China, Emails:
{230109122, wuln}@seu.edu.cn
†
School of Engineering, The University of British Columbia
Kelowna, BC, Canada Email: julian.cheng@ubc.ca
‡
Electrical Engineering Department, The University of Texas at Dallas
Email: aldhahir@utdallas.edu
Abstract—Existing work on exact outage probabilities of di-
versity receptions over arbitrarily correlated Rician channels
involves intractable integrals. Although asymptotic analysis pro-
vides closed-form approximation and can be highly accurate at
large signal-to-noise ratio, it is not known at what signal-to-noise
ratio will the exact outage probability converge to the asymptotic
result. In this paper, we propose a new approach to derive
the asymptotic outage probability of maximal ratio combining,
selection combining and equal gain combining over arbitrarily
correlated Rician channels. The new asymptotic analysis is more
straightforward than the previous approach, and results in
an interesting geometrical explanation for the asymptotic out-
age probabilities. More importantly, asymptotically tight outage
probability bounds of the three most popular linear diversity
receptions are also derived for the arbitrarily correlated Rician
channels. Using these bounds, we can show at what signal-to-
noise ratio the asymptotic results are accurate without resorting
to computer simulations.
I. I NTRODUCTION
Diversity reception is an effective technique to counter fad-
ing in wireless communication systems. Among all the diver-
sity combining techniques, maximal ratio combining (MRC),
selection combining (SC), and equal gain combining (EGC)
are the most widely studied linear combining techniques. Most
existing work have focused on the performance of diversity
reception over independent channels [1]. However, in practical
systems, channels may be correlated due to insufficient spacing
between antennas. Analyzing the performance of diversity
receptions over correlated channels is challenging and usually
leads to multilevel integral or infinite series [2], [3]. For
diversity receptions over arbitrarily correlated channels, to
the best of authors’ knowledge, there exists no closed-form
expressions for the outage probability.
Since deriving the closed-form outage probabilities is dif-
ficult, previous studies primarily focused on the asymptotic
outage probability of diversity over correlated channels [4]–
[6]. Compared to the exact outage probability, asymptotic
analysis provides closed-form results and can be highly ac-
curate at large signal-to-noise ratio (SNR) [4]. However, the
convergence speed of the asymptotic performance varies for
different systems, and it is unknown at what SNR value the
asymptotic results become accurate approximation of the exact
results. To bound the discrepancy of the approximation using
asymptotic expressions, the authors in [7] studied the error
rate bounds for EGC and SC over channels with a specified
correlation structure. However, to the best of the authors’
knowledge, the outage probability bounds for MRC, EGC
and SC over arbitrarily correlated Rician channels are still
unknown.
In this paper, we propose an alternative method to analyze
the asymptotic outage probabilities for MRC, SC and EGC.
In contrast to the existing techniques, such as parameter-
based and series-based asymptotic analysis, the new approach
is more straightforward, resulting in an elegant geometrical
explanation of the asymptotic outage probability expressions.
Based on this technique, we derive both lower and upper
bounds for the outage probabilities of MRC, SC and EGC over
arbitrarily correlated Rician channels, and the derived bounds
are asymptotically tight in large SNR region.
II. S
YSTEM MODEL
We consider N-branch diversity reception over slow
frequency-nonselective Rician fading channels. After coherent
demodulation, matched filtering, and sampling, the complex
output vector of the N links can be expressed as
r = cx + n (1)
where x is the transmitted complex signal; c =[c
1
,c
2
,
...,c
N
]
T
where c
n
is the complex Rician channel coefficient
of the nth branch and where [·]
T
denotes the transpose; n is an
N × 1 vector representing the additive Gaussian white noise.
The probability density function (pdf) of the channel vector c
is
f
c
(c)=
1
π
N
det (R)
exp
−(c − c
L
)
H
A (c − c
L
)
(2)
where c
L
=[c
L,1
,...,c
L,N
]
T
is a complex line-of-sight
(LOS) vector, which is also the expectation of c; R is the
correlation matrix defined as R = E[(c − c
L
)(c − c
L
)
H
]=
E[c
s
c
s
H
] where c
s
= c −c
L
represents the scattering compo-
nents and (·)
H
denotes the Hermitian operator; det (R) is the
2015 International Conference on Computing, Networking and Communications, Wireless Communications Symposium
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