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arXiv:1401.1011v1 [cs.IT] 6 Jan 2014
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 1
Outage Probability of Dual-Hop Multiple Antenna
AF Systems with Linear Processing in the Presence
of Co-Channel Interference
Guangxu Zhu, Caijun Zhong, Member, IEEE, Himal A. Suraweera, Member, IEEE, Zhaoyang
Zhang, Member, IEEE and Chau Yuen Member, IEEE
Abstract—This paper considers a dual-hop amplify-and-
forward (AF) relaying system where the relay is equipped
with multiple antennas, while the source and the destination
are equipped with a single antenna. Assuming that the re-
lay is subjected to co-channel interference (CCI) and addi-
tive white Gaussian noise (AWGN) while the destination is
corrupted by AWGN only, we propose three heuristic relay
precoding schemes to combat the CCI, namely, 1) Maximum ratio
combining/maximal ratio transmission (MRC/MRT), 2) Zero-
forcing/MRT (ZF/MRT), 3) Minimum mean-square error/MRT
(MMSE/MRT). We derive new exact outage expressions as well as
simple high signal-to-noise ratio (SNR) outage approximations for
all three schemes. Our findings suggest that both the MRC/MRT
and the MMSE/MRT schemes achieve a full diversity of N, while
the ZF/MRT scheme achieves a diversity order of N−M, where
Nis the number of relay antennas and Mis the number of
interferers. In addition, we show that the MMSE/MRT scheme
always achieves the best outage performance, and the ZF/MRT
scheme outperforms the MRC/MRT scheme in the low SNR
regime, while becomes inferior to the MRC/MRT scheme in
the high SNR regime. Finally, in the large Nregime, we show
that both the ZF/MRT and MMSE/MRT schemes are capable
of completely eliminating the CCI, while perfect interference
cancelation is not possible with the MRC/MRT scheme.
Index Terms—Dual-hop relaying, amplify-and-forward, co-
channel interference, linear precoding, performance analysis
I. INTRODUCTION
The relay channel was first introduced by Van der Meulen
in 1971 [1]. Later, in the seminal work of [2], Cover and
El Gamal laid foundations to the information-theoretic under-
standing of the relay channel. The attention on relay channels
was recently rekindled as a means to improve the coverage
and link reliability in the context of cooperative wireless
communications systems [3]. Various relaying methods have
been proposed in the literature [3], among which, the amplify-
and-forward (AF) protocol is the most popular one, due to its
simplicity and low-cost implementation. In AF systems, the
relay simply forwards a scaled version of the received noisy
signal from the source to the destination.
Guangxu Zhu, Caijun Zhong and Zhaoyang Zhang are with the Institute
of Information and Communication Engineering, Zhejiang University, China.
(email:caijunzhong@zju.edu.cn).
Himal A. Suraweera is with the Department of Electrical & Electronic
Engineering, University of Peradeniya, Peradeniya 20400, Sri Lanka (email:
himal@ee.pdn.ac.lk)
Chau Yuen is with the Singapore University of Technology and Design, 20
Dover Drive, Singapore 138682 (email: yuenchau@sutd.edu.sg)
To improve the spectrum efficiency, future cellular systems
are likely to adopt a more aggressive frequency reuse strategy,
which will inevitably result in an interference-limited com-
munication environment [4]. When the relay technology is
adopted in cellular systems [5], the interference environment
becomes increasingly complex. Motivated by the need to
understand the performance limitations, a number of works
investigating the impact of co-channel interference (CCI) on
the performance of relay systems have appeared. For example,
[6,7] studied the performance of relay selection for AF
systems with CCI. Assuming Rayleigh fading channels, [8]
examined the outage probability of dual-hop fixed-gain AF
relaying systems with an interference-limited destination, and
[9] studied the outage probability and the average bit error
rate of dual-hop variable-gain AF relaying systems with an
interference-limited relay. Later, a scenario considering the
more general Nakagami-mfading model was investigated in
[10,11]. Moreover, different cases with CCI at both the relay
and the destination nodes have been investigated in [12–14].
More recent works have also investigated the effect of CCI on
single antenna two-way relaying systems for the decode-and-
forward protocol [15] and the AF protocol [16]. However, it
is worth noting that all these prior works deal with the case
where all nodes are equipped with a single antenna.
It is well known that the multiple-input multiple-output
(MIMO) technology provides extra spatial degrees of freedom
which can be efficiently utilized for interference cancellation.
To this end, MIMO has been identified as one of the key
enabling physical layer technologies in wireless standards
such as LTE-Advanced and IMT-Advanced [17]. Despite the
importance, so far only a few papers have investigated the
impact of CCI in MIMO relaying systems [18–20].
In this paper, we consider the scenario with multiple anten-
nas at the relay, and single antenna at the source and desti-
nation. This particular system setup studied in the relay com-
munication literature [18, 21] is applicable in several practical
scenarios where two nodes (e.g., machine-to-machine type
low cost devices) exchange information with the assistance of
an advanced terminal such as a cellular base-station/cluster-
head sensor. At this point, it is important to highlight the
major differences between the current paper and state-of-
the art in the literature. Unlike [18] which only considered
a single interferer, the current paper allows for arbitrary
number of interferers at the relay node. Compared with [19],
which assumed an interference-limited single antenna relay,
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 2
the current work considers a multiple antenna relay. Compared
with [20], which investigated the outage performance of the
scenario with interference-limited multiple antenna relay, the
current work considers the more general setup by taking
into account of the effect of additive white Gaussian noise
(AWGN) at the relay. More importantly, in contrast to [18–
20], where the simple maximum ratio transmission (MRT)
and maximum ratio combining (MRC) schemes were used,
the current paper adopts more sophisticated linear combining
schemes to suppress the CCI.
In the presence of CCI, linear diversity combining schemes
have been widely adopted in the multiple antenna systems
because of the low complexity and good performance [22].
In the same spirit, in this paper we propose a heuristic two-
stage relay processing scheme, i.e., the relay first applies linear
combining methods to suppress the CCI, and then forwards
the transformed signal to the destination by using the MRT
scheme. Three popular linear combining methods, i.e., MRC,
zero-forcing (ZF) and minimum mean square error (MMSE),
are investigated. To the best of the authors’ knowledge, the
analysis of diversity combining schemes for the suppression
of CCI in dual-hop AF relaying systems has not been presented
in the existing literature.
We present a detailed performance analysis of the con-
sidered MRC/MRT, ZF/MRT and MMSE/MRT schemes in
Rayleigh fading channels. Our main contributions are sum-
marized as follows:
•For the MRC/MRT scheme, we derive a new exact
expression involving a single integral for the outage
probability of the system, and present a tight closed-form
outage lower bound. In addition, we obtain a simple high
signal-to-noise ratio (SNR) outage approximation, and
prove that the MRC/MRT scheme achieves a diversity
order of N, where Nis the number of antennas at the
relay.
•For the ZF/MRT scheme, we first obtain the optimal
combining vector maximizing the end-to-end signal-to-
interference-and-noise ratio (SINR) subject to the ZF
constraint, and then derive a new exact closed-form ex-
pression for the outage probability. We also characterize
the high SNR outage behavior and show that it achieves
a diversity order of N−M, where Mis the number of
interferers.
•For the MMSE/MRT scheme, we derive a new exact
expression involving a single integral for the outage
probability, and propose a tight closed-form outage lower
bound. We also characterize the high SNR outage behav-
ior of the MMSE/MRT scheme, and show that it achieves
a diversity order of N.
•Our results suggest that the MMSE/MRT scheme always
attains the best outage performance, and the ZF/MRT
scheme outperforms the MRC/MRT scheme in the low
SNR regime, while the MRC/MRT scheme achieves a
superior outage performance than the ZF/MRC scheme
in the high SNR regime.
•We also look into the large Nregime,1and demonstrate
that in this case, both the ZF/MRT and MMSE/MRT
schemes are capable of completely eliminating the CCI,
while perfect interference cancelation is not possible with
the MRC/MRT scheme.
The remainder of the paper is organized as follows: Section
II introduces the system model. Section III presents a detailed
investigation of the outage probability achieved by the three
different schemes. Numerical results and discussions are pro-
vided in Section IV. Finally, Section V concludes the paper
and summarizes the key findings.
Notation: We use bold upper case letters to denote matrices,
bold lower case letters to denote vectors and lower case letters
to denote scalars. khkFdenotes the Frobenius norm, E{x}
stands for the expectation of the random variable x,∗denotes
the conjugate operator, while Tdenotes the transpose operator
and †denotes the conjugate transpose operator. CN(0,1)
denotes a scalar complex circular Gaussian random variable
with zero mean and unit variance. Ikis the identity matrix
of size k.Γ(x)is the gamma function and Kv(x)is the v-th
order modified Bessel function of the second kind [25, Eq.
(8.407.1)]. Γ (α, x)is the upper incomplete gamma function
[25, Eq. (8.350.2)] and 2F1(a, b;c;z)is the Gauss Hypergeo-
metric Function [25, Eq. (9.100)].
II. SYSTEM MODEL
Let us consider a dual-hop multiple antenna AF relaying
system as illustrated in Fig. 1, where both the source and
destination are equipped with a single antenna, while the
relay is equipped with Nantennas. We consider the scenario
where the relay is subjected to Mindependently but not
necessarily identically distributed co-channel interferers and
AWGN, while the destination is corrupted by AWGN only2.
We also assume that the direct link between the source and
the destination does not exist due to obstacles or path loss
attenuation/severe fading.
h1
S
Ă
R
Ă
hI1
hI2
hIM
h2
D
Fig. 1: System model: S, R and D denote the source, the
relay and the destination, respectively.
In this paper, we consider half-duplex relaying, and hence
a complete transmission occurs in two phases. During the first
1The large Nregime analysis is of great interest due to the advent of
large-MIMO (massive MIMO) technology [23, 24].
2This scenario is also particularly relevant to frequency-division relay sys-
tems [26] where the relay and the destination experience different interference
patterns.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 3
phase, the source transmits the signal to the relay, and the
signal received at the relay is given by
yr=h1x+
M
X
i=1
hIi sIi +n1,(1)
where the N×1vector h1denotes the channel for the
source-relay link. The entries of h1follow identically and
independently distributed (i.i.d.) CN(0,1). The N×1vector
hIi denotes the channel for the i-th interference-relay link, and
its entries follow i.i.d. CN(0,1), and xis the source symbol
satisfying E{xx∗}=Ps. The i-th interference symbol is sI i
with E{sIi s∗
Ii }=PIi ,n1is an N×1vector and denotes the
AWGN at the relay node with E{n1n†
1}=N0I.
In the second phase, the relay node transmits a transformed
version of the received signal to the destination, and the signal
at the destination can be expressed as
yd=h2Wyr+n2,(2)
where h2is a 1×Nvector and denotes the channel for the
relay-destination link, and its entries follow i.i.d. CN(0,1),n2
is the AWGN with E{n∗
2n2}=N0,Wis the transformation
matrix at relay node with E{kWyrk2
F}=Pr.
Combining (1) and (2), the end-to-end SINR of the system
can be computed as
γ=|h2Wh1|2Ps
M
P
i=1 |h2WhIi |2PI i +kh2Wk2
FN0+N0
.(3)
The optimal relay precoder matrix Wmaximizing the end-to-
end SINR γdoes not seem to be analytically tractable, due to
the non-convex nature of the problem. Hence, in this paper,
we propose a heuristic two-stage relay processing strategy,
i.e., the relay first performs some linear combining method to
suppress the CCI, and then forwards the transformed signal
to the destination using the MRT scheme since it maximizes
the SNR of the relay-destination link. Therefore, the heuristic
relay precoder Wis a rank-1 matrix, i.e., W=ωh†
2
kh2kF
w1,
where ωis the power constraint factor, h†
2
kh2kFis used for
matching the second hop channel and w1is a 1×Nlinear
combining vector, which depends on the linear combining
scheme employed by the relay and will be specified in the
following section.
III. OUTAGE PROBABILITY ANALYSIS
In this section, we investigate the outage probability of the
MRC/MRT, ZF/MRT and MMSE/MRT schemes. New exact
analytical expressions are derived for the outage probability of
all three schemes. In addition, simple high SINR approxima-
tions are presented, which provide a concise characterization
of the achievable diversity order of the system, and enable a
performance comparison of the three schemes.
The outage probability is an important performance metric,
which is defined as the instantaneous SINR falls below a pre-
defined threshold γth . Mathematically, it can be expressed as
Pout = Prob (γ < γth).(4)
A. MRC/MRT Scheme
For the MRC/MRT scheme, w1is set to match the first
hop channel, hence, w1=h†
1
kh1kF. To meet the transmit power
constraint at the relay, the constant ω2can be computed as
ω2=Pr
h†
1h1Ps+
M
P
i=1 |h†
1hIi |2PIi
kh1k2
F
+N0
.(5)
Therefore, the end-to-end SINR for the MRC/MRT scheme,
γMRC can be expressed as (6). Now, with the end-to-end SINR
given in (6), we are ready to establish the outage probability of
the MRC/MRT scheme. For notational convenience, we define
ρ1=Ps
N0,ρ2=Pr
N0and ρIi =PI i
N0,i= 1,...,M. We have
the following key result.
Theorem 1: In the presence of interferers at the relay, the
outage probability of the dual-hop AF relaying system with
the MRC/MRT scheme can be expressed as
PMRC
out = 1 −2e−γth
ρ1−γth
ρ2
Γ (N)
N−1
X
m=0 γth
ρ1m1
m!
m
X
j=0 m
j1
ρ2m−jN+j−1
X
k=0 N+j−1
k
×γth
ρ2N+j−1−k(γth + 1) γth
ρ1ρ2k−m+1
2
I(γth),(7)
with I(γth)given in (8), where D=diag(ρI1, ρI2,···, ρI M ),
ρ(D)is the number of distinct diagonal elements of D,
ρIh1i> ρIh2i>··· > ρIhρ(D)iare the distinct diagonal
elements in decreasing order, τi(D)is the multiplicity of ρIhii
and χi,j (D)is the (i, j)−th characteristic coefficient of D.
Proof: See Appendix I-A.
Theorem 1 presents the exact outage probability of the
MRC/MRT scheme, which is quite general and valid for the
system with arbitrary number of antennas and interferers. For
the special case with a single interferer, Theorem 1 reduces
to the result derived in [18, Theorem 13]. To the best of the
authors’ knowledge, the integral Idoes not admit a closed-
form expression. However, this single integral expression can
be efficiently evaluated numerically using software such as
Matlab or MATHEMATICA, which still provides computa-
tional advantage over a Monte Carlo simulation method.
Alternatively, we can use the following closed-form lower
bound of the outage probability, which is tight across the entire
SNR range, and becomes exact in the high SNR regime.
Corollary 1: In the presence of interferers at the relay, the
outage probability of the dual-hop AF relaying system with
the MRC/MRT scheme is lower bounded by
PlMRC
out = 1 −
ΓN, γth
ρ2
Γ (N)e−γth
ρ1
N−1
X
k=0 γth
ρ1k1
k!
k
X
l=0 k
l
ρ(D)
X
i=1
τi(D)
X
j=1
χi,j (D)Γ (j+l)
Γ (j)ρl
Ihiiρ1
ρ1+ρIhiiγth j+l
.(9)
Proof: See Appendix I-B.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 4
γMRC =|h2|2|h1|2Ps
|h2|2
M
P
i=1 |h†
1hIi|2PI i
|h1|2+h†
2
2
N0+N0
Pr
|h1|2Ps+
M
P
i=1 |h†
1hIi|2PI i
|h1|2+N0
.(6)
I(γth) =
ρ(D)
X
p=1
τp(D)
X
q=1
χp,q(D)ρ−q
Ihpi
(q−1)! Z∞
0
Kk−m+1 2s(γth + 1) γth
ρ1ρ2
(x+ 1)!(x+ 1)k+m+1
2xq−1e−γth
ρ1N0+1
ρIhpixdx. (8)
While Theorem 1 and Corollary 1 provide efficient methods
for evaluating the exact outage probability of the system, these
expressions are quite complicated, and do not allow for easy
extraction of useful insights. Motivated by this, we now look
into the high SNR regime, and derive a simple approximation
for the outage probability, which enables the characterization
of the achievable diversity order of the MRC/MRT scheme.
Theorem 2: In the high SNR regime, i.e., ρ2=µρ1,ρ1→
∞, with µbeing a finite constant, the outage probability of
dual-hop AF relaying system with the MRC/MRT scheme can
be approximated as
PMRC
out ≈
1
µN+
N
X
k=0 N
kρ(D)
X
i=1
τi(D)
X
j=1
χi,j (D)Γ (k+j)
Γ (j)ρk
Ihii
×γth
ρ1N
Γ (N+ 1) +o γth
ρ1N+1!.(10)
Proof: See Appendix I-C.
For the special case where {ρI i}are equal, i.e., ρI i
∆
=ρIfor
any i, (10) reduces to
PMRC
out =1
Γ (N+ 1) "1
µN+
N
X
k=0 N
kΓ (k+M)
Γ (M)ρk
I#
×γth
ρ1N
+o γth
ρ1N+1!.(11)
Theorem 2 indicates that the MRC/MRT scheme achieves
a diversity order of N. Moreover, it implies that the number
of interferers does not affect the achievable diversity order,
it however, causes a detrimental effect on the array gain.
This key observation suggests that, in the presence of strong
CCI, the outage performance of the MRC/MRT scheme will
be significantly affected. Hence, in such a scenario, the
MRC/MRT scheme may not be suitable. Motivated by this,
we now study the performance of more sophisticated linear
combining techniques with superior interference suppression
capability, namely, the ZF/MRT scheme and the MMSE/MRT
scheme.
B. ZF/MRT Scheme
In the ZF/MRT scheme, the relay utilizes the available
multiple antennas to completely eliminate the CCI.3To ensure
this is possible, the number of the antennas equipped at the
relay should be greater than the number of interferers. Hence,
for the ZF/MRT scheme, it is assumed that N > M.
Define an N×Mmatrix HI= [hI1,hI2···hI M ]as the
interference channel matrix, the SINR expression in (3) can
be alternatively expressed as
γZF =ω2kh2k2
F|w1h1|2ρ1
ω2kh2k2
Fw1HIDH†
Iw†
1+ω2kh2k2
Fkw1k2
F+ 1
.
(12)
Hence, the optimal combining vector w1should be the solu-
tion of the following maximization problem
w1= arg max
w1
γZF
s.t. w1HI=0&kw1kF= 1.(13)
The problem in (13) can be solved as follows:
Proposition 1: The optimal combining vector w1is given
by
w1=h†
1P
qh†
1Ph1
,(14)
where P=IN−HIH†
IHI−1
H†
I.
Proof: See Appendix II-A.
Having obtained the optimal combining vector w1, the end-
to-end SINR can be expressed as
γZF =kh2k2
Fh†
1Ph1ρ1ρ2
kh2k2
Fρ2+h†
1Ph1ρ1+ 1.(15)
With the above SINR expression, we now study the outage
probability of the ZF/MRT scheme.
Theorem 3: In the presence of interferers at the relay, the
outage probability of the dual-hop AF relaying system with
3We would like to point out that the performance of ZF scheme in multiple
antenna dual-hop AF systems has been studied in [27], where ZF is applied for
inter-stream interference cancellation. To the best of the authors’ knowledge,
application of ZF for CCI cancellation in dual-hop AF relaying systems has
not been studied.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 5
the ZF/MRT scheme can be expressed as
PZF
out = 1 −2e−γth
ρ1−γth
ρ2
Γ (N)
N−M−1
X
m=0 γth
ρ1m1
m!
m
X
j=0 m
j
×1
ρ2m−jN+j−1
X
k=0 N+j−1
kγth
ρ2N+j−k−1
×(1 + γth)γth
ρ1ρ2k−m+1
2
Kk−m+1 2s(1 + γth)γth
ρ1ρ2!.
(16)
Proof: We start by expressing the end-to-end SINR given
in (15) as
γZF =y2y3ρ1ρ2
y2ρ2+y3ρ1+ 1,(17)
where y2=kh2k2
Fand y3=h†
1Ph1. From [30], the
probability density function (p.d.f.) of y3can be expressed
as
fy3(x) = xN−M−1
(N−M−1)!e−x.(18)
Hence, the outage probability of the ZF/MRT scheme can be
written as
PZF
out = Prob y2y3ρ1ρ2
y2ρ2+y3ρ1+ 1 ≤γth
= Prob y3
y2−γth
ρ2
y2+1
ρ2≤γth
ρ1!.(19)
To this end, invoking the result of [18, Lemma 3] yields the
desired result.
Theorem 3 provides an exact closed-form expression for the
outage probability of the ZF/MRT scheme. This expression
only involves standard mathematical functions and hence, can
be efficiently evaluated. To gain further insights, we now
look into the high SNR regime, and present a simple and
informative approximation for the outage probability.
Theorem 4: In the high SNR regime, i.e., ρ2=µρ1,ρ1→
∞, the outage probability of the dual-hop AF relaying system
with the ZF/MRT scheme can be approximated as
PZF
out =1
Γ (N−M+ 1)γth
ρ1N−M
+o γth
ρ1N−M+1!.
(20)
Proof: See Appendix II-B.
As expected, we see that the interference power does not
affect the outage probability of the ZF/MRT scheme. It is
also interesting to observe that µdoes not affect the outage
probability at high SNR. In addition, Theorem 4 indicates
that the achievable diversity order of the ZF/MRT scheme is
N−M. Compared with the MRC/MRT scheme, which attains
a diversity order of N, the ZF/MRT scheme incurs a diversity
loss of M. This important observation suggests that complete
elimination of CCI may not be the best option in terms of the
outage performance.
C. MMSE/MRT Scheme
The ZF scheme completely eliminates the CCI at the
relay, which however may cause an elevated noise level. In
contrast, the MMSE scheme does not fully eliminate the CCI,
instead, it provides the optimum trade-off between interference
suppression and noise enhancement. In the following, we
study the outage performance of the MMSE/MRT scheme. To
make the analysis tractable, we assume that ρI i ≡ρI,∀i=
1,2,...,M. It is important to note that the equal interference
power assumption adopted to simplify the ensuing analysis
is of practical interest as well. For example, it applies when
the interference sources are clustered together [28] or when
the interference originates from a multiple antenna source
implementing an uniform power allocation policy. In addition,
we will later illustrate numerically that our analytical results
provide very accurate approximations to the outage probability
for scenarios with distinct interference power.
According to the principle of MMSE [29], w1is given by
w1=h†
1h1h†
1+HIH†
I+1
ρI
I−1
.(21)
Also, in order to meet the power constraint at the relay, we
have
ω2=ρ2
|w1h1|2ρ1+
M
P
i=1 |w1hIi |2ρI+kw1k2
F
.(22)
Therefore, the end-to-end SINR can be expressed as
γMMSE =
|w1h1|2ρ1
1+ 1
ρ2kh2k2
FM
P
i=1 |w1hIi |2ρI+kw1k2
F+|w1h1|2ρ1
ρ2kh2k2
F
.
(23)
In order to study the outage probability, the remaining task
is to characterize the distribution of γMMSE . However, the
involved SINR expression given in (23) is difficult to handle.
Hence, we first express the SINR as
γMMSE =ρ2kh2k2
FZ
1 + ρ2kh2k2
F+Z
,(24)
where Zis defined as
Z∆
=|w1h1|2ρ1
M
P
i=1 |w1hIi |2ρI+kw1k2
F.(25)
Now, let us focus on Zfor the moment. The distribution
of the random variable Zhas been studied in [29], where
an exact cumulative distribution function (c.d.f.) expression
was presented. However, the final expression is a piecewise
function, which is not amenable to further processing, and
does not seem to be useful here. To circumvent this difficulty,
we first derive an alternative unified expression for the c.d.f.
of Z.
Proposition 2: The c.d.f. of the random variable Zcan be
expressed as in (26) shown on the top of the next page, where
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 6
FZ(z) = 1 −
ΓN, z
ρ1
Γ (N)+ Γ (M+ 1) e−z
ρ1z
ρ1NN
X
m=m1
ρN−m+1
I2F1M+ 1, N −m+ 1; N−m+ 2; −ρI
ρ1z
Γ (m) Γ (N−m+ 2) Γ (m−N+M),(26)
m1= max(0, N −M) + 1.
Proof: See Appendix III-A.
Having obtained the c.d.f. expression of Z, we are now
ready to study the outage probability of the MMSE/MRT
scheme, and we have the following key result.
Theorem 5: In the presence of interferers at the relay, the
outage probability of the dual-hop AF relaying system with
the MMSE/MRT scheme can be expressed as in (27) shown
on the top of the next page, where I1(γth)can be written as
in (28) also shown on the top of the next page.
Proof: See Appendix III-B.
To the best of the authors’ knowledge, the integral I1does
not admit a closed-form expression. Nevertheless, this single
integral expression can be efficiently evaluated numerically.
Alternatively, we can use the following closed-form lower
bound, which is tight across the entire SNR range, and
becomes exact in the high SNR regime.
Corollary 2: In the presence of interferers at the relay, the
outage probability of the dual-hop AF relaying system with
the MMSE/MRT scheme is lower bounded by
PlMMSE
out = 1 −
ΓN, γth
ρ2
Γ (N)
ΓN, γth
ρ1
Γ (N)
−Γ (M+ 1) e−γth
ρ1γth
ρ1NN
X
m=m1
ρN−m+1
I×
2F1M+ 1, N −m+ 1; N−m+ 2; −ρI
ρ1γth
Γ (m) Γ (N−m+ 2) Γ (m−N+M)
.(29)
Proof: See Appendix III-C.
Having obtained the exact outage probability of the
MMSE/MRC scheme, we now look into the high SNR regime,
and derive simple analytical approximations for the outage
probability of the system.
Theorem 6: In the high SNR regime, i.e., ρ2=µρ1,ρ1→
∞, the outage probability of the dual-hop AF relaying system
with the MMSE/MRT scheme can be approximated as
PMMSE
out ="Γ (M+ 1)
N
X
m=m1A+
1+1
µN!1
Γ (N+ 1)#γth
ρ1N
+o γth
ρ1N+1!,
(30)
where A=ρN−m+1
I
Γ(m)Γ(N−m+2)Γ(m−N+M).
Proof: When ρ2=µρ1,ρ1→ ∞, we have e−γth
ρ1→1,
2F1a, b;c;−ρI
ρ1γth→1. Together with these observations
and with the help of the asymptotic expansion of the incom-
plete gamma function [25, Eq. (8.354.2)], we can easily obtain
the desired result from Corollary 2 after some simple algebraic
manipulations.
Theorem 6 indicates that the MMSE/MRT scheme achieves
a diversity order of N, the same as the MRC/MRT scheme.
Since the MMSE/MRT scheme in general needs more CSI
compared with the MRC/MRT scheme, it is natural to ask
whether the MMSE/MRT scheme always achieves a strictly
better outage performance. This is indeed the case in the high
SNR regime, as shown in the following corollary.
Corollary 3: In the high SNR regime, the outage proba-
bility of dual-hop AF relaying system with the MMSE/MRT
scheme is strictly smaller than that of the MRC/MRT scheme.
Proof: See Appendix III-D.
D. Large NAnalysis
In this section, we look into the large Nregime with fixed
M, and examine the asymptotic behavior of the three proposed
schemes. We have the following key result.
Theorem 7: When N→ ∞, the end-to-end SINR of the
ZF/MRT and the MMSE/MRT schemes converges to
γ∞=ρ2kh2k2
Fρ1kh1k2
F
ρ2kh2k2
F+ρ1kh1k2
F+ 1,(31)
and the corresponding outage probability is given by
P∞
out = 1 −2e−γth
ρ1−γth
ρ2
Γ (N)
N−1
X
m=0 γth
ρ1m1
m!
m
X
j=0 m
j
×1
ρ2m−jN+j−1
X
k=0 N+j−1
kγth
ρ2N+j−k−1
×(1 + γth)γth
ρ1ρ2k−m+1
2
Kk−m+1 2s(1 + γth)γth
ρ1ρ2!.
(32)
Proof: See Appendix IV.
A close observation reveals that the asymptotic SINR γ∞
presented in Theorem 7 is equivalent to the end-to-end SNR
of the same dual-hop AF relaying system without CCI at
the relay, which suggests that, when Nis large, CCI at the
relay has no impact on the ZF/MRT and the MMSE/MRT
schemes. However, this is not the case for the MRC/MRT
scheme. Let us make a careful scrutiny of the interference term
U1=
M
P
i=1 |h†
1hIi |2ρIi
kh1k2
Fin (6). It can be readily observed that U1
is a hyper-exponential random variable which is independent
of N. Hence, for the MRC/MRT scheme, the effect of CCI
persists even in the large Nregime. This implies that in the
presence of CCI, if the relay is equipped with a large number
of antennas, adopting linear diversity combining schemes with
superior interference suppression capability such as ZF/MRT
and MMSE/MRT is preferred over the simple MRC/MRT
scheme.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 7
PMMSE
out = 1 −2e−γth
ρ1−γth
ρ2
Γ (N)
N−1
X
m=0 γth
ρ1m1
m!
m
X
j=0 m
j1
ρ2m−jN+j−1
X
k=0 N+j−1
kγth
ρ2N+j−k−1
×(1 + γth)γth
ρ1ρ2k−m+1
2
Kk−m+1 2s(1 + γth)γth
ρ1ρ2!+e−γth
ρ1−γth
ρ2γth
ρ1NΓ (M+ 1)
Γ (N)
×
N
X
m=m1
ρN−m+1
I
Γ (m) Γ (N−m+ 2) Γ (m−N+M)
N
X
j=0 N
j1
ρ2N−jN+j−1
X
k=0 N+j−1
kγth
ρ2N+j−1−k
I1(γth),(27)
I1(γth) = Z∞
0
e−(1+γth)γth
ρ1ρ2xe−xxk−N2F1M+ 1, N −m+ 1; N−m+ 2; −ρIγth
ρ11 + γth + 1
ρ2xdx. (28)
TABLE I: Comparison of the MRC/MRT, ZF/MRT and MMSE/MRT Schemes
MRC/MRT ZF/MRT MMSE/MRT
CSI requirement h1and h2h1,h2and HIh1,h2,HI, and N0
Antenna number require-
ment None N > M None
Diversity order N N −M N
Impact of interference
power reduce the array gain no impact reduce the array gain
Decay rate of outage proba-
bility vs. N
slow fast fast
E. Comparison of the Proposed Schemes
We now provide a more concrete comparison for the
three different schemes studied. In the preceding analysis,
the channel state information (CSI) requirement to perform
relay precoding was not explicitly revealed. In practice, the
acquisition of CSI involves additional feedback overhead,
which must be considered in the design of wireless systems.
On the other hand, if a large amount of CSI is available at the
transmitting node, more sophisticated transmission schemes
could be designed to improve the transmission efficiency and
to achieve a better performance. Therefore, in order to make
a fair comparison among the three different schemes, the CSI
requirement of each individual scheme must be characterized.
Table I gives a comparison of the MRC/MRT, ZF/MRT and
MMSE/MRT schemes.
IV. NUMERICAL RESULTS AND DISCUSSION
In this section, we present numerical results to validate the
analytical expressions derived in Section III. Note, the integral
expressions presented in Theorem 1 and Theorem 5 are
evaluated numerically with the build-in functions in Matlab,
i.e., the “quad” command, and we choose the default absolute
error tolerance value 1.0×10−6to control the accuracy of the
numerical integration. In all simulations, we set γth = 0 dB,
ρIi = 0 dB,∀i= 1, ..., M ,µ= 1, and all results are obtained
with 108runs.
Fig. 2 shows the outage probability of the dual-hop AF
relaying system with the MRC/MRT scheme for different
Mand N. As illustrated, the analytical results are in exact
0 5 10 15 20 25
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
ρ1 (dB)
Outage Probability
Monte Carlo Simulation
Analytical
low bound
High SNR Approximation
N=3,M=4
N=2,M=3
N=2,M=4
N=3,M=3
Fig. 2: Outage probability of the MRC/MRT relaying system
with different Mand N.
agreement with the Monte Carlo simulations, which demon-
strates the correctness of the analytical expression given in (7).
Also, the proposed lower bound is sufficiently tight across the
entire SNR range of interest, and becomes almost exact in the
high SNR regime (i.e., ρ1≥15 dB), while the high SNR
approximation works quite well even at moderate SNR values
(i.e., ρ1= 15 dB). In addition, we observe that increasing
Nreduces the outage probability by improving the diversity
order of the system, while increasing Mdegrades the outage
performance by reducing the array gain of the system.
Fig. 3 illustrates the outage probability of the dual-hop AF
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 8
0 5 10 15 20 25 30
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
ρ1 (dB)
Outage Probability
Monte Carlo Simulation
Analytical
High SNR Approximation
8 8.5 9 9.5
10−0.9
10−0.8
N=3,M=2
N=4,M=3
N=4,M=2
Fig. 3: Outage probability of the ZF/MRT relaying system
with different Mand N.
relaying system with the ZF/MRT scheme for different Mand
N. It is observed that, for fixed M, increasing the antenna
number Nyields a significant outage improvement, since the
achievable diversity order of the system is N−M. Moreover,
comparing the curves associated with N= 4,M= 3 and
N= 3,M= 2, we observe that, when N−Mis fixed, the
outage probability difference between different M,Npairs is
almost negligible, and disappears in the high SNR regime, as
shown in Theorem 4.
0 5 10 15 20 25
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
ρ1 (dB)
Outage Probability
Monte Carlo Simulation
Analytical
Lower bound
High SNR Approximation
N=3,M=3
N=3,M=4
N=2,M=3
N=2,M=4
Fig. 4: Outage probability of the MMSE/MRT relaying
system with different Mand N.
Fig. 4 examines the outage probability of the MMSE/MRT
scheme for different Mand N. It can be readily observed that
the analytical curves are in perfect agreement with the Monte
Carlo simulation results, and the proposed lower bound and
the high SNR approximation are sufficiently tight. In addition,
similar to the MRC/MRT scheme, we see that the MMSE/MRT
scheme achieves a diversity order of N.
Fig. 5 shows the outage probability of the MMSE/MRT
scheme when the equal interference power assumption is no
longer valid. As we can clearly observe, for a given total
interference power, the gap between the analytical results and
0 5 10 15 20
10−4
10−3
10−2
10−1
100
ρ1 (dB)
Outage Probability
ρI=[10 10 10]
ρI=[5 10 15]
ρI=[1 10 19]
ρI=[1 1 1]
ρI=[0.5 1 1.5]
ρI=[0.1 1 1.9]
total power is 30
total power is 3
Fig. 5: Impact of the received interference power distribution
on the outage performance for the MMSE/MRT scheme with
N= 2 and M= 3.
Monte Carlo simulations is sufficiently small, especially for
the low interference power case where the difference is almost
negligible. In addition, we see that the curves associated
with the equal interference power case have the worst outage
performance. Hence, our analytical results could be used to
serve as an tight outage upper bound in case of arbitrary
interference power profiles.
0 5 10 15 20 25
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
ρ1 (dB)
Outage Probability
ρI=10 dB
ρI=0 dB
MMSE/MRT
ZF/MRT
MRC/MRT
Fig. 6: Outage probability of MRC/MRT, ZF/MRT and
MMSE/MRT schemes with N= 3,M= 2 and different ρI.
Fig. 6 compares the outage performance of the three relay
precoding schemes under different cases of interference power,
i.e., weak interference ρI= 0 dB and strong interference
ρI= 30 dB. We observe the intuitive result that, the outage
performance of the MRC/MRT and the MMSE/MRT schemes
degrades when the interference power becomes stronger, while
the outage performance of the ZF/MRT scheme remains the
same regardless of the interference power levels. Comparing
different curves, we see that the MMSE/MRT scheme always
attains the best outage performance, and the ZF/MRT scheme
outperforms the MRC/MRT scheme at the low SNR regime,
especially when the interference power is large.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 9
1 2 3 4 5 6 7 8 9 10
10−18
10−16
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
N
Outage Probability
MRC/MRT
ZF/MRT
MMSE/MRT
Large N approximation
Fig. 7: Outage comparison of the MRC/MRT, ZF/MRT and
MMSE/MRT schemes with ρ1=ρ2= 10 dB, ρI= 0 dB,
M= 2 and different N.
Fig. 7 illustrates the impact of relay antenna number on the
outage performance of the proposed schemes with fixed source
and relay transmit power. We observe that the MMSE/MRT
scheme always has the best outage performance, while the
MRC/MRT scheme outperforms the ZF/MRT scheme when N
is small, and becomes inferior to the ZF/MRT scheme when N
is large. Moreover, the outage decay rate of the ZF/MRT and
MMSE/MRT schemes is almost the same, which is higher than
that of the MRC/MRT scheme. In other words, the minimum
required antenna number Nto achieve a certain outage prob-
ability is smaller for the ZF/MRT and MMSE/MRT schemes
compared with the MRC/MRT scheme.
V. CONCLUSION
We investigated the outage performance of a dual-hop AF
multiple antenna relaying system employing the MRC/MRT,
ZF/MRT and MMSE/MRT schemes in the presence of CCI.
Exact analytical expressions for the outage probability of all
three schemes were derived, which provide a fast and efficient
means of evaluating the performance of the system. Moreover,
simple and informative high SNR outage approximations were
presented, which enable the characterization of the impact
of key parameters, such as relay antenna number N, num-
ber of interferers Mand interference power on the outage
performance of the system. Our findings suggest that both
the MRC/MRT and MMSE/MRT schemes achieve the full
diversity order of N, while the ZF/MRT scheme achieves a
diversity order of N−M. In addition, the MMSE/MRT scheme
always attains the best outage performance, and the ZF/MRT
scheme outperforms the MRC/MRT scheme in the low SNR
regime, while the opposite holds in the high SNR regime.
Finally, we have shown that, in the asymptotically large N
regime, perfect interference cancellation can be achieved by
using the ZF/MRT and the MMSE/MRT schemes.
APPENDIX I
PROOF FOR THE MRC/MRT SCHEME
A. Proof of Theorem 1
We start by expressing the end-to-end SINR given in (6) as
γMRC =y1y2ρ1ρ2
(y2ρ2+ 1) (U1+ 1) + y1ρ1
,(33)
where y1=kh1k2
F,y2=kh2k2
F,U1=
M
P
i=1
yIi ρI i with yIi =
|h†
1hIi |2
kh1k2
F. It is easy to observe that y1and y2are i.i.d. random
variables with the p.d.f.
fyi(x) = xN−1
(N−1)!e−x.(34)
Also, according to [19], yIi ,i= 1,...,M, are i.i.d. expo-
nential random variables with unit variance. Then, random
variable U1follows the hyper-exponential distribution with
p.d.f.
fU1(x) =
ρ(D)
X
i=1
τi(D)
X
j=1
χi,j (D)ρ−j
Ihii
(j−1)!xj−1e−x
ρIhii.(35)
The outage probability of the system can be computed by
PMRC
out = Prob y1y2ρ1ρ2
(y2ρ2+ 1) (U1+ 1) + y1ρ1≤γth
= Prob y1
y2−γth
ρ2
y2+1
ρ2≤γth
ρ1
(U1+ 1)!.(36)
To this end, invoking the result presented in [18, Lemma 3],
we can obtain the following outage probability expression
conditioned on U1,
PMRC
out = 1 −2e−γth
ρ1(U1+1)−γth
ρ2
Γ (N)
N−1
X
m=0 γth
ρ1
(U1+ 1)m1
m!
×
m
X
j=0 m
j1
ρ2m−jN+j−1
X
k=0 N+j−1
k
×γth
ρ2N+j−1−k
Bk−m+1
2Kk−m+1 2√B,(37)
where B=(γth+1)γth
ρ1ρ2(U1+ 1)
The desired result can be obtained by averaging over U1,
along with some simple basic algebraic manipulations.
B. Proof of Corollary 1
The end-to-end SINR can be upper bounded by
γMRC =
y1y2ρ1ρ2
(y2ρ2+ 1) (U1+ 1) + y1ρ1≤min y1ρ1
U1+ 1 , y2ρ2.(38)
Since y1,y2,U1are independent random variables, the outage
probability of the system can be lower bounded by
PlMRC
out = 1 −Prob y1ρ1
U1+ 1 ≥γth Prob (y2ρ2≥γth ).
(39)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 10
Conditioned on U1,Prob y1ρ1
U1+1 ≥γthcan be computed as
Prob y1ρ1
U1+ 1 ≥γth = 1 −Prob y1<γth
ρ1
(U1+ 1)
=
ΓN, U1+1
ρ1γth
Γ (N).(40)
The next step is to average over the distribution of U1. After
some algebraic manipulations, we arrive at
Prob y1ρ1
U1+ 1 ≥γth =e−γth
ρ1
N−1
X
k=0 γth
ρ1k1
k!
k
X
l=0 k
l
ρ(D)
X
i=1
τi(D)
X
j=1
χi,j (D)Γ (j+l)
Γ (j)ρl
Ihiiρ1
ρ1+ρIhiiγth j+l
.(41)
Now, we look at the second part, Prob (y2ρ2≥γth), which
can be computed as
Prob (y2ρ2≥γth ) = Prob y2≥γth
ρ2
=e−γth
ρ2
N−1
X
m=0 γth
ρ2m
m!=
ΓN, γth
ρ2
Γ (N).(42)
To this end, substituting (41) and (42) into (39) yields the
desired result.
C. Proof of Theorem 2
Starting from (39), conditioned on U1, the outage probabil-
ity of the MRC/MRT scheme can be lower bounded by
PlMRC
out = 1 −
ΓN, γth
ρ2
Γ (N)
ΓN, U1+1
ρ1γth
Γ (N).(43)
Then, invoking the asymptotic expansion of incomplete
gamma function [25, Eq. (8.354.2)], we have
ΓN, γth
ρ2
Γ (N)= 1 −γth
ρ1N
Γ (N+ 1)µN+o γth
ρ1N+1!.(44)
Similarly, we get
ΓN, U1+1
ρ1γth
Γ (N)=
1−1
Γ (N+ 1)(U1+ 1)Nγth
ρ1N
+o γth
ρ1N+1!.
(45)
Hence, the outage lower bound conditioned on U1can be
approximated as
PlMRC
out =
γth
ρ1N
Γ (N+ 1) 1
µN+ (U1+ 1)N+o γth
ρ1N+1!.(46)
To this end, the remaining task is to compute the expectation
of (1 + U1)N. Applying the binomial expansion,
(1 + U1)N=
N
X
k=0 N
kU1k,(47)
and averaging over U1, we get
EU1n(1 + U1)No=
N
X
k=0 N
kρ(D)
X
i=1
τi(D)
X
j=1
χi,j (D)ρIhii−j
(j−1)!
×Z∞
0
xk+j−1e−x
ρIhiidx
=
N
X
k=0 N
kρ(D)
X
i=1
τi(D)
X
j=1
χi,j (D)Γ (k+j)
Γ (j)ρk
Ihii.(48)
Substituting (48) into (46) yields the desired result.
APPENDIX II
PROOF FOR THE ZF/MRT SCHEME
A. Proof of Proposition 1
Substituting w1HI=0and |w1|= 1 into (12), we have
γZF =ω2kh2k2
F|w1h1|2ρ1
ω2kh2k2
F+ 1 .(49)
It should be noted that the power constraint constant ωis
also dependent on the combining vector w1via the following
relationship
ω2=ρ2
|w1h1|2ρ1+ 1.(50)
Hence, we have
γZF =kh2k2
Fρ2|w1h1|2ρ1
kh2k2
Fρ2+ (|w1h1|2ρ1+ 1).(51)
Now, it is easy to show that γZF is an increasing function with
respect to |w1h1|2. Therefore, the original optimization can be
alternatively formulated as
w1= arg max
w1|w1h1|2
s.t. w1HI=0&kw1kF= 1.(52)
To this end, the desired result can be obtained by following
the similar lines as in the proof of [30, Proposition 1].
B. Proof of Theorem 4
In the high SNR regime, the end-to-end SINR γZF can be
tightly bounded by
γZF =y2y3ρ1ρ2
y2ρ2+y3ρ1+ 1 ≤min (y3ρ1, y2ρ2).(53)
Hence, the outage probability of the system can be tightly
lower bounded by
PZF
out ≥Prob (min (y3ρ1, y2ρ2)≤γth ).(54)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 11
Due to the independence of random variables y2and y3, the
outage lower bound can be written as
PZF
out = 1 −Pr ob y3≥γth
ρ1Prob y2≥γth
ρ2
= 1 −
ΓN−M, γth
ρ1
Γ (N−M)
ΓN, γth
ρ2
Γ (N).(55)
Invoking the asymptotic expansion of incomplete gamma
function [25, Eq. (8.354.2)], we have
PZF
out =γth
ρ1N
Γ (N+ 1)1
µN
+γth
ρ1N−M
Γ (N−M+ 1)
+o γth
ρ1N−M+1!.(56)
It is obvious that the first item in (56) is negligible when
compared with the second item in (56). Therefore, the desired
result can be obtained after some simple algebraic manipula-
tions.
APPENDIX III
PROOF FOR THE MMSE/MRT SCHEME
A. Proof of Proposition 2
The random variable Zcan be alternatively expressed as
Z=ρ1
ρI
w1h1h†
1w†
1
w1M
P
i=1
hIi h†
Ii w†
1+w11
ρIIw†
1
,(57)
where Idenotes the unit matrix.
Now define R=HIH†
I+1
ρII, then the MMSE combining
vector w1can be written by
w1=h†
1h1h†
1+R−1
.(58)
Hence, we have
Z=ρ1
ρI
h†
1h1h†
1+R−1
h1h†
1h1h†
1+R−1
h1
h†
1h1h†
1+R−1
Rh1h†
1+R−1
h1
.(59)
Applying the well-known matrix inversion lemma, (59) can be
simplified as
Z=ρ1
ρI
h†
1R−1h1.(60)
To this end, invoking the results presented in [29, Eq.(11)],
the c.d.f. of Zcan be expressed as
FZ(z) = 1 −e−z
ρ1
N
X
m=1
Am(z)
(m−1)!z
ρ1m−1
,(61)
where
Am(z) =
1N≥M+m,
1+
N−m
P
i=1 (M
i)ρI
ρ1zi
1+ ρI
ρ1zMN < M +m. (62)
To obtain a unified expression for the c.d.f. of Z, we find it
convenient to give a separate treatment for the following two
cases: 1) N≥M, and 2) N≤M.
1) N≥M:Noticing that
1 +
N−m
P
i=1 M
iρI
ρ1zi
1 + ρI
ρ1zM= 1 −
M
P
i=N−m+1 M
iρI
ρ1zi
1 + ρI
ρ1zM,(63)
the c.d.f. of Zcan be written as (64) shown on the top of the
next page, which can be further simplified as
FZ(z) = 1 −e−z
ρ1
N
X
m=1
1
(m−1)!z
ρ1m−1
−
N
X
m=N−M+1
e−z
ρ1
(m−1)!z
ρ1m−1
M
P
i=N−m+1 M
iρI
ρ1zi
1 + ρI
ρ1zM
|{z }
S1
.
(65)
Now, utilizing the following key observation,
M
P
i=NM
iρI
ρ1zi
1 + ρI
ρ1zM=Γ (M+ 1)
Γ (N+ 1) Γ (M+ 1 −N)ρI
ρ1
zN
×2F1M+ 1, N ;N+ 1; −ρI
ρ1
z.(66)
S1can be alternatively expressed as
S1=
Γ (M+ 1) 2F1M+ 1, N −m+ 1; N−m+ 2; −ρI
ρ1z
Γ (N−m+ 2) Γ (m−N+M)ρI
ρ1z−N+m−1.
(67)
Finally, noticing that
e−z
ρ1
N
X
m=1
1
(m−1)!z
ρ1m−1
=
ΓN, z
ρ1
Γ (N),(68)
and after some algebraic manipulations, we obtain (69) shown
on the top of the next page.
2) N≤M:Similarly, the c.d.f. of Zcan be alternatively
expressed as
FZ(z) = 1 −e−z
ρ1
N
X
m=1
1
(m−1)!z
ρ1m−1
−
N
X
m=1
e−z
ρ1
(m−1)!z
ρ1m−1
M
P
i=N−m+1 M
iρI
ρ1zi
1 + ρI
ρ1zM.(70)
Then, following the same lines as in the derivation of the
N≥Mcase, we get (71) shown on the top of the next page.
To this end, the desired result can be obtained by appropriately
choosing certain parameters.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 12
FZ(z) = 1 −e−z
ρ1
N−M
X
m=1
1
(m−1)!z
ρ1m−1N
X
m=N−M+1
e−z
ρ1
(m−1)!z
ρ1m−1
1−
M
P
i=N−m+1 M
iρI
ρ1zi
1 + ρI
ρ1zM
,(64)
FZ(z) = 1 −
ΓN, z
ρ1
Γ (N)+ Γ (M+ 1) e−z
ρ1z
ρ1NN
X
m=N−M+1
ρN−m+1
I
2F1M+ 1, N −m+ 1; N−m+ 2; −ρI
ρ1z
Γ (m) Γ (N−m+ 2) Γ (m−N+M).
(69)
FZ(z) = 1 −
ΓN, z
ρ1
Γ (N)+ Γ (M+ 1) e−z
ρ1z
ρ1NN
X
m=1
ρN−m+1
I
2F1M+ 1, N −m+ 1; N−m+ 2; −ρI
ρ1z
Γ (m) Γ (N−m+ 2) Γ (m−N+M).(71)
B. Proof of Theorem 5
Starting from the end-to-end SINR presented in (24), the
outage probability of the system can be expressed as
PMMSE
out = Prob ρ2y2Z
(ρ2y2+ 1) + Z≤γth
= Prob Zy2−γth
ρ2
y2+1
ρ2≤γth!.(72)
Then, applying the method used in [18, Lemma 3], and
utilizing the c.d.f. expression of Zgiven in Proposition 2 and
the p.d.f of y2, the outage probability of the system is given
by
PMMSE
out = 1 − I2+γth
ρ1NΓ (M+ 1)
Γ (N)
N
X
m=m1
ρN−m+1
II3
Γ (m) Γ (N−m+ 2) Γ (m−N+M),(73)
where
I2=Z∞
γth
ρ2
ΓN, Cγth
ρ2
Γ (N)
tN−1
Γ (N)e−tdt, (74)
and
I3=Z∞
γth
ρ2
e−C γth
ρ2e−ttN−1CN
2F1M+ 1, N −m+ 1; N−m+ 2; −ρIγth
ρ1Cdt, (75)
with C=t+/ρ2
t−γth/ρ2.
To this end, after some tedious algebraic manipulations,
solving the integrals I2and I3yields the the desired result.
C. Proof of Corollary 2
The end-to-end SINR can be upper bounded by
γMMSE =ρ2y2Z
(ρ2y2+ 1) + Z≤min (Z, ρ2y2).(76)
Hence, the outage probability can be lower bounded by
PMMSE
out ≥Prob (min (Z, y2ρ2)≤γth ).(77)
Due to the independence of y2and Z, the outage probability
lower bound can be computed as
PlMMSE
out (γth) = 1 −Prob (Z≥γth ) Pro b y2≥γth
ρ2.
(78)
To this end, invoking the c.d.f. of Zgiven in Proposition 2
and the p.d.f of y2, the desired result can be obtained after
some simple algebraic manipulations.
D. Proof of Corollary 3
To prove the statement, we only need to show that AMRC >
AMMSE, where
AMRC =1
Γ (N+ 1) "1
µN+
N
X
k=0 N
kΓ (k+M)
Γ (M)ρk
I#,(79)
and
AMMSE =
N
X
m=m1
Γ (M+ 1) ρN−m+1
I
Γ (m) Γ (N−m+ 2) Γ (m−N+M)+
1+1
µN!1
Γ (N+ 1) (80)
A close observation shows that both AMRC and AMMSE have
the a common item 1+1
µN1
Γ(N+1) . Hence, to proof
AMRC > AMMSE, we only need to show A1> A2, where
A1=
N
X
k=1
Γ(M+k)ρk
I
Γ(N−k+ 1)Γ(k+ 1)Γ(M),(81)
and
A2=
N
X
m=m1
Γ (M+ 1) ρN−m+1
I
Γ (m) Γ (N−m+ 2) Γ (m−N+M).(82)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. XX, XX 201X 13
Define t= min(N, M ), after some tedious algebraic manipu-
lations, A2can be alternatively expressed as
A2=
t
X
k=1
Γ (M+ 1) ρk
I
Γ(N−k+ 1)Γ (k+ 1) Γ (M−k+ 1) .(83)
Comparing (83) and (81), it is easy to show that A1> A2,
which completes the proof.
APPENDIX IV
PROOF OF THEOREM 7
In the asymptotic large Nregime, the law of large number
holds, and we have
1
Nh†
1hIi = 0,and 1
NH†
IHI=IM.(84)
For the ZF/MRT scheme, starting from (15), and with the
help of (84), we have
h†
1Ph1=
h†
1h1−1
Nh†
1HI1
NH†
IHI−1
H†
Ih1(85)
=h†
1h1−1
Nh†
1HIH†
Ih1(86)
=
h†
1h1−1
N
M
X
i=1
h†
1hIi h†
Ii h1
=h†
1h1(87)
=kh1k2
F.(88)
Substituting (88) into (15), we have
γZF =kh2k2
Fkh1k2
FPsPr
Prkh2k2
FN0+N0kh1k2
FPs+N0=γ∞.(89)
For the MMSE/MRT scheme, with the help of (84), and
applying the well-known Woodbury matrix identity, we have
R−1=ρII−ρI
NHIH†
I.Hence, Zcan be expressed as
Z=ρ1
ρIρIh†
1h1−ρI
Nh†
1HIH†
Ih1=ρ1kh1k2
F.(90)
Substituting (90) into (24), we have
γMMSE =ρ2kh2k2
Fρ1kh1k2
F
1 + ρ2kh2k2
F+ρ1kh1k2
F
=γ∞.(91)
Given the asymptotic SINR, the desired outage probability
can be computed by following the similar lines as in the proof
of Theorem 3.
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