Hop-by-Hop Beamforming for Dual-Hop MIMO AF Relay Networks

Abstract

A comprehensive performance analysis of dual-hop multiple-input multiple-output amplify-forward relay networks with hop-by-hop beamforming is presented. The impact of practical transmission impairments; (i) feedback delays, (ii) channel estimation errors and (iii) spatially-correlated fading on the system performance is studied. Specifically, the amount of performance degradation due to these impairments are quantified analytically and illustrated through numerical results. Numerical results show that these impairments degrade the system performance significantly. The cumulative distribution function of the end-to-end signal-to-noise ratio is derived and used to obtain the moment generating function, the outage probability, and the average symbol error rate (SER) in closed-form. The asymptotic outage probability and average SER are derived to obtain valuable system-design insights such as the diversity order and array gain. Further, our analyses are validated through Monte-Carlo simulations.

Full text

Page 1

Hop-by-hop Beamforming for Dual-hop MIMO AF
Relay Networks
Gayan Amarasuriya, Chintha Tellambura and Masoud Ardakani
Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada T6G 2V4
Email: {amarasur,chintha,ardakani}@ece.ualberta.ca
Abstract—This paper presents a comprehensive performance
analysis of dual-hop multiple-input multiple-output amplify-and-
forward relay networks with hop-by-hop beamforming. The im-
pact of practical transmission impairments; (i) feedback delays,
(ii) channel estimation errors and (iii) spatially-correlated fading
on the system performance is studied. Specifically, the amount of
performance degradation due to these impairments are quantified
analytically. To this end, a general closed-form expression for the
cumulative distribution function of the end-to-end signal-to-noise
ratio is derived and used to obtain the outage probability, and the
average symbol error rate (SER). Further, the asymptotic outage
probability and average SER are derived to obtain valuable
system-design insights such as the diversity order and array
gain. Numerical results are presented to show the detrimental
effect of imperfect channel state information on the system
performance. Further, our analyses are validated through Monte-
Carlo simulations.
I. INTRODUCTION
Cooperative multiple-input multiple-output (MIMO) relay
networks are currently receiving significant research interest
and being investigated for emerging wireless system standards
such as IEEE 802.16m WiMAX and Long Term Evolution
(LTE)-Advanced [1]. Transmit beamforming can be employed
to improve the performance of dual-hop MIMO relay net-
works, particularly because of its robustness against severe
effects of fading [2]. In dual-hop MIMO relay networks, this
robustness is achieved by steering the transmitted signal along
the maximum eigenmodes of the source-to-relay (S → R)
and relay-to-destination (R → D) hops. Nevertheless, a com-
prehensive performance analysis of hop-by-hop beamforming
for dual-hop MIMO amplify-and-forward (AF) relay networks,
which allows the possibility of using all MIMO-enabled ter-
minals and considers practical transmission impairments such
as spatially-correlated fading, channel estimation errors, and
feedback delays, is not available in the literature.
Prior related research: Although the hop-by-hop beam-
forming has already been employed for dual-hop MIMO AF
relay networks [3]–[6], these studies limit the relay (R) to a
single-antenna terminal. In [3], the performance of beamform-
ing for dual-hop (channel-assisted AF) CA-AF relay networks
over independent Rayleigh fading channels is investigated.
Reference [4] extends [3] by considering spatial correlated
fading among antenna elements at the source (S) and R.
Further, in [5], the performance of the system set-up in [3]
is studied over Nakagami-m fading channels. Reference [6]
investigates the performance of beamforming for fixed-gain
AF relay networks over Nakagami-m fading channels.
In addition to the above studies, [7] analyzes the perfor-
mance of dual-hop CA-AF beamforming and its equivalent
systems by using several antenna configurations at S, R and
destination (D). However, in all these system set-ups, at least
one terminal is limited to a single antenna. Moreover, [8]
studies the effect of multiple antennas at S on the outage
probability by using maximal ratio transmission (MRT) for
the S → R channel, and [9] extends this study to investigate
the effect of feedback delays on MRT beamforming. Although
the system set-ups in [8], [9] employ multiple-antennas at S,
both R and D are single-antenna terminals.
Motivation and our contribution: As mentioned above,
the hop-by-hop beamforming AF system models considered
in [3]–[9] employ at least one single-antenna terminal. Thus,
the achievable diversity and array gains in the relay channel
are not optimal. Further, all the previous analyses except [9]1
make the ideal assumption of the availability of perfect CSI. In
our analysis, this ideal assumption is replaced by more realistic
ones, and the gaps in the performance analysis of beamforming
in MIMO AF relay networks are filled by employing MIMO-
enabled S, R, and D. We provide a comprehensive perfor-
mance analysis, which includes exact and asymptotic end-
to-end (e2e) performance metrics, and the impact of spatial
correlation, outdated channel state information (CSI), and
channel estimation errors on the system performance.
More specifically, in our system model, all three terminals,
S, R and D, are equipped with Ns, Nr and Nd antennas,
respectively. First, a closed-form expression for the cumulative
distribution function (CDF) of the end-to-end SNR (e2e SNR)
is derived and used to derive the outage probability and
average symbol error rate (SER). In order to obtain direct
insights about the valuable system-design parameters such as
the diversity order and array gain, the asymptotic performance
metrics, which are exact in high SNRs, are derived as well. The
impact of spatially-correlated fading among antenna elements
is studied. In particular, an asymptotically exact tight outage
probability lower bound is derived by considering the arbitrar-
ily correlated transmit and receive correlation matrices at each
terminal. In order to quantify the amount of degradation due to
antenna correlation with respect to the uncorrelated antennas,
the asymptotic outage probability and average SER are also
derived for correlated fading case as well. Numerical results
are provided to investigate the degree of degradation of sys-
tem performance due to practical transmission impairments:
(i) spatially-correlated fading, (ii) feedback delays and (iii)
channel estimation errors. Our results shows that the system
1The analysis in [9] is limited and considers the feedback delay effect on
S → R for CA-AF relay networks having multiple-antenna S, and single-
antenna R and D.
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performance degrades significantly due to these impairments.
Notations: Kν(z) is the ν-th order Modified Bessel func-
tion of the second kind [10, Eq. (8.407.1)].
the Gauss Hypergeometric function [10, Eq. (9.14.1)].
II. SYSTEM MODEL
We consider a dual-hop AF relay network with MIMO-
enabled S, R and D having Ns, Nr and Nd antennas,
respectively. The channel matrices from S → R and R → D
are denoted by Hl|2
transmit antenna to the i-th receive antenna is denoted by
hi,j
l
and is assumed to be independent and identical Rayleigh
fading unless otherwise stated; hi,j
the additive noise at the terminals is modeled as complex zero
mean white Gaussian noise. All the terminals operate in the
half-duplex mode, and e2e data transmission takes place in
two time-slots. The direct link from S → D is assumed to be
unavailable due to heavy shadowing and path-loss.
In practical MIMO beamfoming systems, the estimated
channel matrices are generally perturbed by the addition of
Gaussian errors due to channel estimation errors. Further, the
beamforming vectors could be selected by using outdated CSI
matrices due to feedback delays. The channel matrices having
these two practical transmission impairments can be modeled
as follows:
1) Feedback delay: In practice, the feedback channel from
the receiver to transmitter experiences delays. We thus assume
that the first hop beamforming vectors, u1and v1, are selected
by using the τ1-delayed channel matrix H1(t − τ1), and the
second hop beamforming matrices, u2 and v2, by using τ2-
delayed H2(t−τ2). By following a commonly used model, two
channels can be modeled as Hl(t)|2
where ρl is the normalized correlation coefficients between
hi,j
J0(2πflτl), where fl is the Doppler fading bandwidth [11].
Further, Ed,lis the error matrix, incurred by feedback delay,
having mean zero and variance (1 − ρ2
2) Channel estimation errors: In general, orthogonal pilot-
assisted channels estimates are perturbed by Gaussian errors
and can be modeled asˆHl
is the error matrix having mean zero and variance σ2
Gaussian entries. Thus, the estimated channel matrixˆHl is
Gaussian distributed with mean zero and variance 1 + σ2
Further,ˆhi,j
l
and hi,j
l
are jointly Gaussian distributed with a
normalized correlation coefficient of 1/
channel matrices can be written in terms of their estimates as
Hl|2
is the Gaussian error matrix having mean zero and variance
σ2
The above two channel models can readily be expressed in
a general form as follows:
Hl|2
where Hl and ˆHl are the actual and estimated channel
matrices, and Elis the error matrix, and all three have zero
mean and unit variance i.i.d. Gaussian entries. The parameters
2F1(α,φ;γ;z) is
l=1. The channel gain from the j-th
l
∼ CN(0,1). Moreover,
l=1= ρlHl(t−τl)+Ed,l,
l(t) and hi,j
l(t−τl), and for Clarke’s fading spectrum, ρl=
l) Gaussian entries.
???
2
l=1= Hl+ ΔHl, where ΔHl
e,l
e,l.
?
1 + σ2
e,l. The actual
l=1= εlˆHl+ Ee,l, where εl = 1/(1 + σ2
e,l), and Ee,l
e,l/(1 + σ2
e,l) [12].
l=1= ϑlˆHl+ ξlEl,
(1)
ϑl and ξl account for the channel estimate quality and can
be explicitly given for the two cases: (i) feedback delays:
ϑl= ρland ξl=?1 − ρ2
In the first time-slot, S transmits the symbol X, having
E??|X|2??=1, and R receives YRas
YR=ˆ uH
1
P1H1ˆ v1X + n1
=ˆ uH
1
P1
where ˆ u1 and ˆ v1 are the transmit precoding and receive
filtering (beamforming) vectors2at S and R, and selected by
using the imperfect channel matrixˆ H1as the first columns of
ˆU1andˆV1, respectively, corresponding to the largest singular
value ofˆ H1. Here P1is the transmit power at S and ˆ nRcan be
considered as the effective Gaussian noise component having
mean zero and variance σ2
is the largest eigenvalue of the Wishart matrixˆHH
second time-slot, R amplifies the received signal YR with a
gain G and forwards to D again by using beamforming. The
signal received at D, YDis thus given by
YD=ˆ uH
where P2is the transmit power at R and ˆ nDis the effective
Gaussian noise component having mean zero and variance
σ2
the beamforming vectors u2and v2are selected by using the
imperfect channel matrixˆ H2as the first columns ofˆU2and
ˆV2. Further,ˆλ2 is the largest eigenvalues of Wishart matrix
ˆH2ˆHH
e2e SNR γeqcan be derived as
ϑ2
P1ˆλ1ϑ2
Now, the relay gain in (4) is set to G1=
and the e2e SNR can be derived as
γeq=
αγ1+ γ2+ β,
where α =
ϑ2
ther, γ1=
P1ξ2
SNRs of S → R and R → D hops having average SNRs
¯ γ1=
P1ξ2
Alternatively, if the relay has the knowledge of the channel
estimation error variance and long-term feedback channel
statistics (i.e., ϑ1 and ξ1), then the relay gain can be set to
G2=
P2/(P1ϑ2
simplified to obtain the e2e SNR as γeq =
γ1 and γ2 are defined as in (5). Further, (5) can handle the
perfect CSI cases when ρl|2
l, and (ii) channel estimation errors:
e,land ξl=
σ2
ϑl= 1/
?
??
1 + σ2
?
?
?
e,l/(1 + σ2
e,l).
???
ϑ1ˆH1+ξ1E1
ˆ v1X +n1
?
=ϑ1
?
P1ˆλ1X +ˆ nR,(2)
ˆ nR= P1ξ2
1+ σ2
1[12]. Further,ˆλ1
1ˆH1. In the
2[GH2ˆ v2YR+n2]=Gϑ1ϑ2
?
P1ˆλ1ˆλ2X +ˆ nD,
(3)
ˆ nD= G2P1ˆλ1ϑ2
1ξ2
2+ G2ˆλ2ϑ2
2σ2
ˆ nR+ G2ξ2
2σ2
ˆ nR+ σ2
2. Again,
2. After some trivial mathematical manipulations, the
γeq=
1P1ˆλ1ϑ2
2σ2
2ˆλ2
1ξ2
2+ˆλ2ϑ2
ˆ nR+ ξ2
2σ2
?
ˆ nR+
P2/(P1ˆλ1+ σ2
σ2
G2
2
.
(4)
1),
γ1γ2
(5)
P2ξ2
1(P2ξ2
P1ϑ2
1+σ2
2ϑ2
1+σ2
2+σ2
1and γ2=
2
2), and β =
P2ξ2
(P1ξ2
2ˆλ2
2+σ2
2(P1ξ2
1+σ2
1+σ2
1)(P2ξ2
1)+σ2
2+σ2
1σ2
2). Fur-
2
1ˆλ1
P2ϑ2
P2ξ2
2are the instantaneous
P1ϑ2
1+σ2
1
1and ¯ γ2=
P2ϑ2
P2ξ2
2
2+σ2
2.
?
1ˆλ1+ P1ξ2
1+ σ2
1). Now, (4) can further be
γ1γ2
γ1+γ2+1, where
???
2. Hereˆ Σ1 andˆ Σ2 are Nr × Ns and
l=1= 1 and σ2
e,l
2
l=1= 0.
2The singular value decompositions ofˆ H1 andˆ H2 are given byˆ H1 =
ˆ U1ˆ Σ1ˆ VH
1
andˆ H2 =ˆ U2ˆ Σ2ˆ VH
Nd×Nrdiagonal matrices having the largest singular values√λ1and√λ2,
as the first elements on the main diagonals, respectively. Further,ˆ U1,ˆ V1,ˆ U2
andˆ V2are unitary square matrices of sizes Nr×Nr, Ns×Ns, Nd×Nd,
and Nr× Nr.
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Page 3

III. PERFORMANCE ANALYSIS
In this section, the performance analysis is presented. First,
the statistics of the e2e SNR are derived and used to derive
the performance metrics.
A. Statistical characterization of the end-to-end SNR
The exact CDF of the e2e SNR (5) is given by (see
Appendix for the proof)
?
×(k)
(¯ γ1)
?
where?
l=|Nr−Nd|
satisfy dl(i,j)=?min(N,M)
B. Outage probability
The outage probability Pout is the probability that the
instantaneous e2e SNR γeqfalls below a threshold γth. Thus,
Poutcan be obtained as Pout= Pr(γeq≤ γth) = Fγeq(γth),
where Fγeq(γth) denotes the CDF of γeq(6) evaluated at γth.
C. Average symbol error rate
The average SER ¯Pe can be derived by averaging the
conditional error probability (CEP) Pe|γ(γ) over the PDF
of the e2e SNR; ¯Pe = Eγeq
wide range of modulation schemes is given by Pe|x(x) =
ηQ(√ζγ), where η and ζ are modulation-dependent con-
stants. By using integration by parts,¯Pe can be simplified
as¯Pe=
22π
exact lower bound for the average SER is derived by substi-
tuting (6) with β = 0 into the integral representation of¯Pe
and evaluating the integral by using [10, Eq. (6.621.3)] as
?
a,b,k,l,m
u=0
(αk)
(¯ γ1) (¯ γ2)
μ = m+b+3
1) Asymptotic outage probability: To obtain direct insights,
the asymptotic outage probability can be derived as
⎧
⎪
where Ω1=
Γn2(n2)
Γn2(m2+n2)(k2)NNr, and Ω3 =
Fγeq(x) = 1 −
a,b,k,l,m
u+v+m+1
2
m
?
x
2b−u−v+m+1
2
u=0
b
?
v=0
2αu?m
b! m!(a)
(αx + β)
(¯ γ2)
?
?(Ns+Nr)a−2a2
thecoefficients
?(N+M)i−2i2
u
??b
v
?d1(a,b)d2(k,l)
m−u+v+1
u+v−m−2b−1
2
m+2b+u−v+1
22
u+v+m+1
2
?
× e−x
a,b,k,l.m=?min(Ns,Nr)
m=0
a
¯ γ1+αk
¯ γ2
?
Ku+v−m+1
2
akx(αx+β)
¯ γ1¯ γ2
?
,
(6)
a=1
b=|Ns−Nr|
?min(Nr,Nd)
dl(i,j)=1 and can
k=1
dl(i,j)|2
?(Nr+Nd)k−2k2
readily be computed by using the efficient algorithm in [13].
?l
and
l=1
i=1
j=|N−M|
?Pe|γ(γ)?. The CEP for a
η
?
ζ
?∞
0x−1
2e−ζx
2 Fγeq(x)dx. An asymptotically
¯Plb
e=η
2−η
2
ζ
2
?
u+m+1
2
m+b
?
2?m+b
b! m!(a)
u
?d1(a,b)d2(k,l)
u−m−2b−1
2
×
2b−u+m+1
2
u+m+1
2
I(μ,ν,ψ,ω), where(7)
2, ν = u−m+1, ψ =ζ
2+a
¯ γ1+αk
¯ γ2, ω = 2
?
aαk
¯ γ1¯ γ2.
P∞
out=
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎨
Ω1
?
?
x
¯ γ
?NsNr+ o?¯ γ−(NsNr+1)?, Ns< Nd
?NdNr+ o?¯ γ−(NdNr+1)?, Ns> Nd,
Ω2
?
x
¯ γ
?NNr+ o?¯ γ−(NNr+1)?, Ns=Nd=N
Ω3
x
¯ γ
(8)
Γn1(n1)
Γn1(m1+n1)(k1)NsNr, Ω2=
Γn1(n1)
Γn1(m1+n1)(k1)NNr+
Γn2(n2)
Γn2(m2+n2)(k2)NdNr. Here
Γa(b) =?a
max(Nr,Nd), n1= min(Ns,Nr), and n2= min(Nr,Nd).
2) Asymptotic average SER: In the high SNR regime, the
average SER can be approximated as¯P∞
Gdis the diversity order, and Gais the array gain. By using
(8) and the integral expression of¯Pe in Section III-C, the
asymptotic average BER is derived as
⎧
⎪
Now, by using (8) and (9), the diversity order and array
gain are obtained as Gd = Nrmin(Ns,Nd) and Ga =
?Ωlη2Gd−1Γ(Gd+1
cases defined in (8).
D. Impact of correlated fading among antenna elements
In this section, the impact of correlated fading among the
antenna elements of S, R and D on the system performance
is investigated. Assume that H1and H2undergo flat spatially
arbitrary-correlated Rayleigh fading. Then H1and H2can be
decomposed according to the Kronecker correlation structure
as follows: H1 = Υ
Υ1and Υ2are the transmit correlation matrices at S and R,
and Φ1and Φ2are the receive correlation matrices at R and
D, respectively. Further,˜H1∼ CN(0Nr×Nr,INr⊗ INs) and
˜H2∼ CN(0Nd×NrINd⊗ INr).
1) CDF of the e2e SNR: The exact CDF of the e2e
SNR is mathematically intractable. However, the CDF of an
asymptotically exact upper bound of the e2e SNR can be
derived. By using the SNR upper bound, γeq ≤ γub
min(γ1,γ2), the CDF of γub
1−(1 − Fγ1(x))(1 − Fγ2(x)), where Fγ1(x) and Fγ2(x) are
the CDFs of the first and second hop SNRs and given by [14]
(−1)nlΓnl(nl)det(Υl)n1−1det(Φl)m1−1det
Δnl(Υl)Δml(Φl)(x/γl)nl(nl−1)/2
where Δk(·) is a Vandermonde determinant in the eigenval-
ues of the k-dimensional matrix argument, and the (i,j)-th
element of Ψl(x) is given by [14, Eq. (1)]. By evaluating
Fγub
readily be derived.
2) High SNR analysis: In order to quantify the amount of
performance degradation due to spatial correlated fading, the
asymptotic outage probability and average SER are derived.
The asymptotic outage probability, when each channel under-
goes correlated fading, can be derived by replacing Ωl|3
(8) with
⎧
⎨
⎪
i=1Γ(b − i + 1) is the normalized complex multi-
variate gamma function. Further, m1= max(Ns,Nr), m2=
e
= (¯ γGa)−Gd, where
¯P∞
e=
⎪
⎪
⎪
⎩
⎨
Ω1η2NsNr−1Γ(NsNr+1
√π(ζ¯ γ)NsNr
Ω2η2NNr−1Γ(NNr+1
√π(ζ¯ γ)NNr
Ω3η2NrNd−1Γ(NrNd+1
√π(ζ¯ γ)NrNd
2)
+ o?¯ γ−(NNr+1)?,Ns=Nd=N
+o?¯ γ−(NsNr+1)?, Ns< Nd
+o?¯ γ−(NrNd+1)?, Ns> Nd.
2)
2)
(9)
2)
√π(ζ)Gd
?−
1
Gd, where Ωl|3
l=1stands for the three
1
2
1˜H1Φ
1
2
1and H2 = Υ
1
2
2˜H2Φ
1
2
2, where
eq =
eq(x) =
eqcan be derived as Fγub
Fγl(x)|2
l=1=
?
Ψl
?
x
γl
? ?
??????
2
l=1
(10)
eq(x) at γth, a tight outage probability lower bound can
l=1in
Ω?=
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Ω?
Ω?
+
Ω?
1=
2=
det(Υ2)m2det(Ψ2)n2Γn2(m2+n2)(k2)NNr, Ns=Nd=N
3=
Γn1(n1)
det(Υ1)m1det(Ψ1)n1Γn1(m1+n1)(k1)NsNr, Ns< Nd
Γn1(n1)
det(Υ1)m1det(Ψ1)n1Γn1(m1+n1)(k1)NNr
Γn2(n2)
Γn2(n2)
det(Υ2)m2v det(Ψ2)n2Γn2(m2+n2)(k2)NdNr,Ns> Nd.
(11)
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−10−505101520 25
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Average First Hop SNR
Average BER


Analysis
Asymptotic
Simulation
Ns=2, Nr=2, Nd=4
Ns=2, Nr=2, Nd=2
Ns=2, Nr=1, Nd=3
Ns=1, Nr=1, Nd=3
Ns=3, Nr=2, Nd=4
Fig. 1. The average BER of BPSK of MIMO beamforming AF relay networks.
The hop distances are L1=2L2and the path-loss exponent is ?=2.5.
The asymptotic average SER can readily be derived again by
substituting the above Ω?
order of the system remains the same regardless of the amount
of correlation, the array gain significantly reduces with respect
to the uncorrelated fading by factors of Ω1/Ω?
Ω3/Ω?
respectively (see Section IV).
IV. NUMERICAL RESULTS
l|3
l=1into (9). Although the diversity
1, Ω2/Ω?
2, and
3for the cases Ns ≤ Nd, Ns = Nd, and Ns ≥ Nd,
This section presents the numerical results and verifies
our analysis through Monte-Carlo simulations. To capture the
effect of the network geometry, the average SNR of the i-th
hop is modeled as ¯ γi|2
transmit SNR, L0is the reference distance, and ? is the path-
loss exponent. The distances between the terminals S → R,
and R → D are denoted by L1and L2, respectively.
1) Diversity order and array gain: In Fig. 1, the average
bit error rate (BER) of binary phase shift keying (BPSK) is
plotted for the perfect CSI case (ρl = 1 and σ2
ideal CA-AF relay (α = 1,β = 0 in (5)) is treated and
the BER curves are plotted by using (7). The asymptotic
average BER curves are also plotted to depict the valuable
system-design parameters such as the diversity order and array
gain. The asymptotic BER curves clearly show the diversity
gains obtained by different antenna set-ups at each terminal.
Specifically, the asymptotic BER curves corresponding to
Ns= 2,Nr= 2,Nd= 2, and Ns= 2,Nr= 2,Nd= 4 reveal
that the diversity order of both set-ups is the same (Gd= 4),
and thus verify our diversity analysis Gd= Nrmin(Ns,Nd).
However, the latter network set-up experiences a higher array
gain than that of the former because of the two additional
antennas at D.
2) Impact of correlated fading on the outage probability:
Fig. 2 shows the effect of correlated fading among the antennas
at S, R and D on the outage probability. The transmit and
receive arbitrary correlation matrices Υ1, Υ2, Ψ1and Ψ2for
uniform linear antenna arrays at S, R and D are constructed by
using [15, Eq. (4)]. The amount of spatial correlation between
i=1= ¯ γ
?
L0
Li
??
, where ¯ γ is the average
e,l= 0). An
−10−505
Average First Hop SNR
101520 2530
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
10
1
Outage Probability


Exact (Simulation)
Lower Bound (Analytical)
Asymptotic
High Correlation
(l1 = l2 =0.2), (σ2
as,1=σ2
as,2=π/24)
Low Correlation
(l1 = l2 =0.8), (σ2
as,1=σ2
as,2=π/12)
Uncorrelated
Fading
Medium Correlation
(l1 = l2 =0.5), (σ2
as,1=σ2
as,2=π/18)
Ns=2, Nr=2, Nd=3
Fig. 2.
distances are L1=L2, and the path-loss exponent is ?=2.5.
The impact of spatial correlation on the outage probability. The hop
−10 −50510 1520 25
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
10
1
Average First Hop SNR
Outage Probability


Analysis
Simulation
τ1=0, f2τ2=0.075
(ρ1=1, ρ2=0.945)
Ns = 3, Nr = 2, Nd = 3
τ1=0.175, f2τ2=0.17
(ρ1=0.72, ρ2=0.735)
f1τ1=0.125, τ2=0
(ρ1=0.852, ρ2=1)
τ1=0, f2τ2=0.17
(ρ1=1, ρ2=0.735)
f1τ1=0.125, f2τ2=0.13
(ρ1=0.852, ρ2=0.84)
Asymptotic Outage
for Perfect CSI
τ1=τ2=0, ρ1=ρ2=1
Perfect CSI
f1τ1=0.08, f2τ2=0.075
(ρ1=0.938, ρ2=0.945)
Fig. 3.
distances are L1= 2L2and the path-loss exponent is ? = 2.5.
The impact of outdated CSI on the outage probability. The hop
adjacent antenna elements then can be quantified by using their
relative antenna spacing (l1,l2), angular spreads (σ2
and the angle of arrival or departure (θ1,θ2). Three different
correlation scenarios are obtained as (a) high correlation, (b)
medium correlation, and (c) low correlation. Since l1 and
l2 are the relative antenna spacing, and σ2
the angular spreads, smaller values of l1, l2, σ2
result in higher spatial correlation [15]. The asymptotic outage
curves reveal that the amount of correlation does not affect the
attainable diversity order, but the array gain is severely affected
by higher correlation. Our outage lower bounds are tighter in
moderate-to-high SNR regimes, asymptotically exact at high
SNRs, and provide valuable system-design insights.
3) Impact of outdated CSI on the outage probability: In
Fig. 3, the impact of outdated CSI due to feedback delay on
the outage probability is shown. The relay does not have the
knowledge of feedback channel statistics ρl. The beamforming
vectors at S, R and D are selected based on the outdated
CSI received via the local feedback channels R → S and
D → R with time delays τ1 and τ2. Several outage curves
as,1,σ2
as,2),
as,1and σ2
as,1and σ2
as,2are
as,2
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Page 5

are obtained by changing ρ1 and ρ2, where they are related
to the time delays by following Clarke’s fading model. Thus,
ρ1 = J0(2πf1τ1) and ρ2 = J0(2πf2τ2), where f1 and f2
are the Doppler fading frequencies. The outage performance
degrades significantly even when the feedback channels expe-
rience slight time delays. In fact, the achievable diversity gain
diminishes completely whenever there exists even a slight time
delay in either the R → S or D → R feedback channel.
4) Impact of channel estimation errors on the average BER:
Fig. 4 depicts the effect of channel estimation errors on the
average BER of BPSK. The channels S → R and R → D are
modeled by using (1). Even a slight estimation error in either
channel degrades the BER performance significantly. When
the estimation error is assumed to be fixed, the BER curves
exhibit error floors as the average first hop SNR increases.
However, in practice, the estimation errors are inversely propo-
sitional to the pilot symbol SNR, σ2
Ep is the pilot symbol power and σ2
Specifically, when the variance of the Gaussian estimation
errors decreases as the SNR of the data symbols increases (the
pilot symbols have the same energy as the data symbols), the
error floors do not occur and the achievable diversity order
is persevered; however, the array gain is severely degraded.
Two types of relay gains, G1and G2, are treated. Specifically,
G1does not have the knowledge of channel estimation error
variance σ2
having the knowledge of σ2
e,l∝
lis the noise variance.
?Ep
σ2
l
????
2
l=1, where
e,l, whereas G2does. Fig. 4 shows that the relays
e,lalways perform the best.
V. CONCLUSION
The performance of hop-by-hop beamforming for dual-hop
MIMO AF relay networks was studied. The effect of spatially-
correlated fading, feedback delays and channel estimation
errors on the system performance was derived, thereby quan-
tifying the amount of performance degradation. Our results
show that these practical transmission impairments result in
significant performance degradations. Valuable system-design
parameters such as the diversity order and array gains were de-
rived by using our asymptotic analysis of performance metrics.
Our analytical and simulation results provide valuable insights
for designing practical dual-hop MIMO relay networks with
beamforming.
VI. APPENDIX
The CDF of γeq in (5) can be derived by using Fγeq(x)=
1−?∞
derive Fγeq(x), one needs the closed-form statistics ofˆλ1and
ˆλ2, the largest eigenvalues of the central Wishart matrices. By
using [2], fγ1(x) can be obtained as
min(Ns,Nr)
?
Similarly,¯Fγ2(x) can readily be derived as
min(Nr,Nd)
?
0
¯Fγ2
?(α(x+z)+β)x
z
?
fγ1(z+x)dz, where fγ1(x) is the PDF
of γ1and¯Fγ2(x) is the complementary CDF of γ2. In order to
fγ2(x) =
a=1
(Ns+Nr)a−2a2
?
b=|Ns−Nr|
ab+1d1(a,b)
(¯ γ1)b+1(b)!xbe−ax
¯ γ1.(12)
¯Fγ1(x)=
k=1
(Nr+Nd)k−2k2
?
l=|Nr−Nd|
l?
m=0
kmd2(k,l)
(¯ γ2)m(m)!xme−kx
¯ γ2,(13)
−10 −505 10 1520
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10
0
Average First Hop SNR
Average BER


Lower Bound − Relay Gain G1 (Analytical)
Lower Bound − Relay Gain G2 (Analytical)
Exact − Relay Gain G1 (Simulation)
Exact − Relay Gain G2 (Simulation)
σe,1
2 =0.022, σe,2
2=0
σe,1
2 =0.1, σe,2
2=0.11
Ns = 2, Nr = 2, Nd = 2
Asymptotic BER
for Perfect CSI
σe,1
Perfect Channel Estimation
2 =0, σe,2
2=0
σe,1
2 =0.05, σe,2
2=0.049
σ2
e,1=
?
EP
σ2
1
?−1
,σ2
e,2=
?Ep
σ2
2
?−1
Fig. 4. The impact of channel estimation errors on the average BER of BPSK.
The hop distances are L1= 0.75L2and the path-loss exponent is ? = 2.5.
where d1(a,b) and d2(k,l) are defined in (6). Now, by substi-
tuting (12) and (13) into the integral representation of Fγeq(x)
and evaluating the integral by using [10, Eq. (3.471.9)], the
desired result in (6) can be derived.
REFERENCES
[1] Y. Yang et al., “Relay technologies for WiMAX and LTE-Advanced
mobile systems,” IEEE Commun. Mag., vol. 47, pp. 100–105, Oct. 2009.
[2] P. Dighe et al., “Analysis of transmit-receive diversity in Rayleigh
fading,” IEEE Trans. Commun., vol. 51, no. 4, pp. 694–703, Apr. 2003.
[3] R. H. Y. Louie et al., “Performance analysis of beamforming in two
hop amplify and forward relay networks,” in Proc. IEEE Int. Conf. on
Commun., May 19–23, 2008, pp. 4311–4315.
[4] ——, “Performance analysis of beamforming in two hop amplify and
forward relay networks with antenna correlation,” IEEE Trans. Wireless
Commun., vol. 8, no. 6, pp. 3132–3141, Jun. 2009.
[5] T. Duong et al., “Symbol error probability of hop-by-hop beamforming
in Nakagami-m fading,” Electronics Letters, vol. 45, no. 20, pp. 1042–
1044, Sep. 2009.
[6] D. da Costa and S. Aissa, “Cooperative dual-hop relaying systems with
beamforming over Nakagami-m fading channels,” IEEE Trans. Wireless
Commun., vol. 8, no. 8, pp. 3950–3954, Aug. 2009.
[7] J.-B. Kim and D. Kim, “Performance of dual-hop amplify-and-forward
beamforming and its equivalent systems in Rayleigh fading channels,”
IEEE Trans. Commun., vol. 58, no. 3, pp. 729–732, Mar. 2010.
[8] H. Min et al., “Effect of multiple antennas at the source on outage
probability for amplify-and-forward relaying systems,” IEEE Trans.
Wireless Commun., vol. 8, no. 2, pp. 633–637, Feb. 2009.
[9] H. Suraweera et al., “Effect of feedback delay on downlink amplify-
and-forward relaying with beamforming,” in IEEE GLOBECOM 2009.,
Nov. 2009.
[10] I. Gradshteyn and I. Ryzhik, Table of integrals, Series, and Products,
7th ed. Academic Press, 2007.
[11] S. Zhou and G. Giannakis, “Adaptive modulation for multi-antenna
transmissions with channel mean feedback,” IEEE Trans. Wireless
Commun., vol. 3, no. 5, pp. 1626–1636, Sep. 2004.
[12] Y. Chen and C. Tellambura, “Performance analysis of maximum ratio
transmission with imperfect channel estimation,” IEEE Commun. Lett.,
vol. 9, no. 4, pp. 322 – 324, Apr. 2005.
[13] A. Maaref and S. Aissa, “Closed-form expressions for the outage
and ergodic Shannon capacity of MIMO MRC systems,” IEEE Trans.
Commun., vol. 53, no. 7, pp. 1092–1095, Jul. 2005.
[14] M. McKay et al., “Performance analysis of MIMO-MRC in double-
correlated Rayleigh environments,” IEEE Trans. Commun., vol. 55,
no. 3, pp. 497–507, Mar. 2007.
[15] H. Bolcskei et al., “Impact of the propagation environment on the
performance of space-frequency coded MIMO-OFDM,” IEEE J. Sel.
Areas Commun., vol. 21, pp. 427–439, Apr. 2003.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings