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The University of Manchester Research
On the Outage Probability of MIMO Full-Duplex Relaying :Impact of Antenna Correlation and Imperfect CSIDOI:10.1109/TVT.2016.2602238
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Citation for published version (APA):Almradi, A., & Hamdi, K. (2016). On the Outage Probability of MIMO Full-Duplex Relaying : Impact of AntennaCorrelation and Imperfect CSI. IEEE Transactions on Vehicular Technology, PP(99).https://doi.org/10.1109/TVT.2016.2602238
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Download date:16. May. 2020
On the Outage Probability of MIMO Full-DuplexRelaying : Impact of Antenna Correlation and
Imperfect CSIAhmed Almradi, Student Member, IEEE, and Khairi Ashour Hamdi, Senior Member, IEEE
Abstract—This paper analyzes the performance of multiple-input multiple-output (MIMO) full-duplex (FD) relaying systems,where the source and destination nodes are equipped withsingle antenna and communicating via a dual-hop amplify-and-forward (AF) relay with multiple receive and transmit antennas.The system performance due to practical wireless transmissionimpairments of spatial fading correlation and imperfect channelstate information (CSI) is investigated. At the relay, the loopbackself-interference (LI) is mitigated by using receive zero-forcing(ZF) precoding scheme, then steering the signal to the destinationby using maximum-ratio transmission (MRT) technique. To thisend, new exact closed-form expressions for the outage probabilityare derived, where the case of arbitrary, exponential, and nocorrelations are considered. Meanwhile, for a better systemperformance insights, simpler outage probability lower-boundexpressions are also included, through which the acheiveablediversity order of the receive ZF/MRT scheme is shown to bemin (NR − 1, NT ), where NR and NT are the number of relayreceive and transmit antennas, respectively. Numerical resultssustained by Monte Carlo simulations show the exactness andtightness of the proposed closed-form exact and lower-boundexpressions, respectively. In addition, it is seen that the outageprobability performance of FD relaying outperforms that of theconventional half-duplex (HD) relaying at low to medium signal-to-noise ratio (SNR). However, at high SNR, the performance ofHD relaying outperforms that of the FD relaying. Furthermore, inthe presence of channel estimation errors, an outage probabilityerror floor is seen at high SNR. Therefore, for optimum outageperformance, hybrid relaying modes is proposed which switchesbetween HD and FD relaying modes.
Index Terms—MIMO relaying, full-duplex relaying, half-duplex relaying, maximum-ratio combining (MRC), maximumratio transmission (MRT), zero-forcing (ZF), outage probability.
I. INTRODUCTION
COOPERATIVE relaying techniques have gained a greatdeal of attention due to their ability to extend network
coverage, connectivity and attain higher capacity without sac-rificing extra power resources. In a dual-hop relaying systems,
Copyright (c) 2015 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].
A. Almradi is with the Department of Electrical and Electronic Engineering,Azzaytuna University, Tarhuna, Libya. He is also with the school of Electricaland Electronic Engineering, The University of Manchester, Manchester, M139PL, UK. (e-mail: [email protected]).
K. A. Hamdi is with the school of Electrical and Electronic Engineer-ing, The University of Manchester, Manchester, M13 9PL, UK. (e-mail:[email protected]).
Part of this paper was presented at the IEEE Wireless Communications andNetworking Conference (WCNC), Doha, Qatar, April 2016.
an intermediate idle node operates as a relay between thesource and destination nodes when the direct link betweenthe source and destination is in deep fade. Two orthogonalchannels are required for communications to take place inthe conventional dual-hop one-way relaying networks. As aresult, a significant loss of spectrum efficincy is incurredbecause of the inherent half-duplex (HD) relaying transmissionconstraint at the relay (in the HD mode, time-division duplexor frequency-division duplex is utilized to assure orthogo-nal transmission (also known as out-of-band full-duplex)).Recently, full-duplex (FD) relaying, also known as in-bandfull-duplex, has received a lot of research interest becauseof its potential to double spectral efficiency (see e.g., [1]–[13]). This is due to the fact that full-duplex relays receiveand re-transmit its information at the same time over the samefrequency. Hence, efficiently utilizing the spectrum resourcesof the system. However, the main limitation of FD relayingis the loopback self-interference (LI) caused by the signalleakage from the relay’s transmission to its own reception,namely, the large power differences between the LI (powertransmitted from the FD relay) and the FD relay receivedsignal (which is much weaker than the transmitted signaldue to path loss and fading) exceeds the dynamic range ofthe analoge-to-digital converter. Therefore, LI mitigation andcancellation is vital in the implementation of FD relayingoperation [1]–[6], [14].
The ever increasing demand of wireless communicationdevices with higher throughput led to the deployments ofshort range systems, such as small-cell systems, WiFi, andFemtocells, where the cell-edge path loss is much less than thatin the traditional cellular systems. Therefore, the transmissionpower and distance between devices has been decreased. Thissignificant constructural modification alongside with the recentadvances in antennas and radio-frequency (RF) circuit designis the breakthrough in making full duplex communicationsfeasible as the LI mitigation problem becomes viable. SeveralLI mitigation techniques have been proposed in the literaturewhich may be categorized into three stages: propagation(passive) domain LI cancellation, analog and digital (active)domain LI cancellation, spatial domain LI cancellation in thepresence of multiple receive and/or multiple transmit antennasat the FD relay [3], [5], [6], [10]. Meanwhile, simultaneousreception and transmission in the FD mode requires the relayto have a separate transmit and receive RF chains. In addition,theoretically, FD systems can be equipped with either sharedantenna or separate antennas. In shared antenna FD systems,
2
only one antenna is used for both transmission and receptionsimultaneously, where a circulator is used to separate thereceived and transmitted signals. In separate antennas FDsystems, two antennas are required, one for reception and theother for transmission, where natural isolation is used as a firststep LI reduction between the transmit and receive antennasby using path loss. It is to be emphasized that in the case ofmultiple antennas FD systems, it is challenging to use sharedantennas in multiple antennas FD systems as it results in severinterefernce between the shared antennas [3], [5].
Linking multiple-input-multiple-output (MIMO) with re-laying systems substantially increase capacity and improvereliability. While most previously published work has focusedon MIMO HD relaying transmission (see e.g., [15]–[18]),recent work has also considered the combination of MIMOtechniques with FD relaying transmission mode. MIMO FD re-laying transmission offer a powerful technique of suppressingthe LI in the spatial domain and can offer higher capacity thanthe conventional MIMO HD relaying mode (e.g., [10], [11]).Therefore, in the presence of MIMO FD relaying systems,joint optimization by precoding and decoding at the transmitterand receiver, respectively, can be used to mitigate the LIeffects. Zero-forcing (ZF) precoding and decoding vectorsbased on the conventional singular value decomposition (SVD)of the LI channel is proposed in [10] to null out the LIat the relay. In addition, for simplicity and mathematicaltractability, a low complexity joint precoding/decoding designfor maximizing the overall signal-to-noise ratio (SNR) is in-vestigated in [11], where a closed-form overall SNR is derived.More specifically, both receive ZF precoding scheme withmaximum-ratio transmission (MRT) scheme, and maximal-ratio combining (MRC) with transmit ZF scheme have beenpresented.
The performance analysis of the classical MIMO HD re-laying systems, with multiple antennas at the source anddestination, and/or the relay, have been studied extensivelyin the literature (see e.g., [15]–[18]). A new exact closed-form expression for the outage probability of spatially cor-related MIMO HD relaying systems, with multiple antennasat the source and destination applying the MRT/MRC schemeassuming perfect channel state information (CSI), has beenderived in [16]. Moreover, in [15], the performance analysisof MIMO HD relaying beamforming is addressed, wherean exact outage probability, approximate and upper-boundergodic capacity expressions are derived. In [10], differentspatial LI suppression techniques through MIMO FD relayingare presented, namely, antenna selection, beam selection, andnull space projection.
More relevant to out work are [11], [19], where the outageprobability and/or ergodic capacity expressions are derived. In[19], MIMO FD relaying systems and MIMO HD relayingsystems’ outage probability with antenna subset selection isconsidered. In [11], an exact closed-form and asymptoticoutage probability expressions are addressed, where receiveZF/MRT scheme, MRC/transmit ZF scheme, and antennaselection are considered for MIMO FD relaying systems,with multiple antennas at the source and destination, spatiallyuncorrelated fading and perfect CSI. It is to be emphasized
that all previously published work on MIMO FD relayingsystems address the outage probability performance with spa-tially independent antennas and perfect CSI (see e.g., [7], [8],[11], [12], [19]–[23]). However, the joint impact of antennacorrelation and imperfect CSI on the performance of MIMOFD relaying systems is of practical importance and has notbeen investigated yet.
Motivated by the above mentioned limitations and owingto the practical MIMO wireless transmission impairments, aspatial fading correlation exists due to the use of multipletransmit and receive antennas in a limited space, and alsodue to poor scattering conditions and limited angular spread.Meanwhile, perfect CSI is not available as channel estimationis performed in practice, hence, a residual desired and LIchannels error exist due to imperfect CSI. Therefore, thispaper investigates the impact of spatial fading correlation,and channel estimation errors for both the desired and LIchannels, on the performance of MIMO FD relaying systemswith receive ZF/MRT scheme.
The main contributions of this paper are summarized asfollows:
1) An exact closed-form outage probability expression isderived for spatially correlated MIMO FD relayingsystems with arbitrary correlation matrix and in thepresence of channel estimation errors, where receiveZF/MRT scheme is applied to maximize the overall SNRat the FD relay.
2) A simpler special cases of the outage probability expres-sion with exponential correlation matrix and independentantennas are also presented. In addition, a simpler tightclosed-form lower-bound outage probability expressionsare also included. Besides, the characterization of highSNR outage probability show that the achievable di-versity order is min (NR − 1, NT ), where NR and NTare the number of relay receive and transmit antennas,respectively.
3) The presented analytical expressions efficiently evaluatethe outage probability of MIMO FD relaying with re-ceive ZF/MRT scheme. Therefore, without resorting tothe time consuming Monte Carlo simulations, the impactof some system key parameters such as the number ofrelay receive and transmit antennas, spatial correlationstructure and coefficient, diversity order, estimation errorvariance, and the source transmit power on the system’soutage probability are investigated.
The structure of the rest of the paper is as follows. Insection II, we introduce the system model. In section III,the instantaneous overall SNR is addressed. In section IV,outage probability analysis is considered. Numerical resultsare provided in section V. Finally, section VI concludes thepaper.
II. THE SYSTEM MODEL
Fig. 1 depicts the considered three node MIMO FD relayingnetwork, where the source node S is equipped with singleantenna, and intends to transmit data to the destination nodeD which is also equipped with single antenna via a MIMO
3
D
R
S
Figure 1: The MIMO FD relaying system model.
full-duplex AF relay R. The relay is equipped with NRantennas for reception and NT antennas for transmission. It isassumed here that there is no direct link between the sourceand destination due to high path loss and heavy shadowing.Meanwhile, for comparison purposes, the case of MIMO HDrelaying is also included with the constraint that the totalnumber of antennas at the HD relay is N = NR +NT .
A. Channel Model
The statistical spatial correlation among the receiving andtransmitting antennas, respectively, at the MIMO FD relay hasto be characterized in practice due to space and/or scatteringlimitations. Therefore, the S → R channel, h1 and theR → D channel, h2 are subject to frequency flat corre-lated Rayleigh fading channel distributed according to h1 ∼CN (0, Ω) with Ω = E
[h1h
†1
]and h2 ∼ CN (0, Λ) with
Λ = E[h2h
†2
], respectively, where E (·) is the expectation
operator, and † denotes the conjugate transpose, assuming thatthe channel coherence time equals at least two time slots (thiscondition is necessary for the MIMO HD relaying systemscase as MIMO FD relaying systems use only one time slotto transmit its information from the S → D). The noise atthe relay and destination are modeled by complex additivewhite Gaussian noise (AWGN) with zero mean and covarianceof σ2
nI . In addition, HR denotes the residual LI R → Rchannel, which is the resultant error due to imperfect LImitigation performed by antennas isolation, analog and digitalcancellation at the FD relay. Here, we model the residualLI channel HR via independent Rayleigh fading channeldistributed according to HR ∼ CN (0, I). In order to furtherreduce the effect of residual LI at the MIMO FD relaying,receive ZF precoding scheme with MRT are performed. Notethat this interference suppression and beamforming requirean estimate of h1, h2, and HR where channel estimation isrequired.
B. Imperfect Channel Estimation
In practice, a training sequence is used to estimate thechannel. Let x and x denote the actual channel and theestimated channel, respectively. It is assumed that x and x arejointly ergodic and stationary Gaussian processes. In addition,it is also assumed that the channel estimate x and estimation
error ε are orthogonal. Therefore, in the case of imperfectchannel estimation, the channel can be decomposed as
x = x + ε (1)
where x ∈ h1, h2, HR denotes the true parameter value,x ∈
h1, h2, HR
denotes the estimated parameter value,
and ε ∈ ε1, ε2, ε3 is the estimation error, for the S →R channel, R → D channel, and R → R self-interferencechannel, respectively, where ε is the channel estimation errorvector (matrix in the LI channel case), distributed accordingto ε ∼ CN
(0, σ2
εΥ), where Υ ∈ Ω, Λ, I. It is to be
emphasized that the parameter σ2ε captures the quality of the
channel estimation, which is chosen depending on the usedmethod of estimation, and depends on the training sequencelength as well as the pilot power.
III. THE INSTANTANEOUS OVERALL SNR
Following similar derivations to [11, Eq. (20)], and with thehelp of [24, Proposition 1], we arrive at the following overallSNR of MIMO full duplex relaying [25]
γ =γ1γ2
γ1 + γ2 + c(2)
where γ1 =ρ1‖P h1‖2(
ρ1σ2ε1
(P h1)†Σ(P h1)
‖P h1‖2+ρ2σ2
ε3+1
) , with ρ1 = PSσ2n
denotes the first-hop SNR and PS is the source average trans-
mit power, γ2 =ρ2‖h2‖2(
ρ2σ2ε2
h†2Λh2
‖h2‖2+1
) , with ρ2 = PRσ2n
denotes the
second-hop SNR and PR is the relay average transmit power,
and c =ρ2σ
2ε2
h†2Λh2
‖h2‖2(ρ1σ
2ε1
(P h1)†Σ(P h1)
‖P h1‖2+ρ2σ
2ε3
+1
)+1(
ρ1σ2ε1
(P h1)†Σ(P h1)
‖P h1‖2+ρ2σ2
ε3+1
)(ρ2σ2
ε2
h†2Λh2
‖h2‖2+1
) . The
projection matrix is given as P = I − HRh2h†2H†R
‖HRh2‖2, with ‖·‖
being the Frobenius norm.As far as a simple closed-form outage probability expression
is concern, equation (2) is mathematically intractable and doesnot lend itself to a closed-form outage probability expression.Therefore, in this paper, for simplicity and mathematicaltractability, the estimation error covariance matrix is assumedto be distributed according to ε ∼ CN
(0, σ2
εI)
1. Hence, theresultant overall SNR is simplified to [26]
γ =γ1γ2
γ1 + γ2 + c(3)
where γ1 =ρ1‖P h1‖2
(ρ1σ2ε1
+ρ2σ2ε3
+1), γ2 =
ρ2‖h2‖2(ρ2σ2
ε2+1)
, and c =
ρ2σ2ε2
(ρ1σ2ε1
+ρ2σ2ε3
+1)+1
(ρ1σ2ε1
+ρ2σ2ε3
+1)(ρ2σ2ε2
+1).
It is to be emphasized that in the case of perfect CSI (i.e.,σ2ε = 0) and/or independent fading channels (i.e., Ω = Λ =
I), Eq. (2) and Eq. (3) becomes identical and hence result inthe same exact outage probability analysis.
The matrix P is an idempotent orthogonal projection (null-space projection) matrix, which is used to eliminate theloopback self-interference channel (i.e. the relay receives
1Note that this assumption is shown to be a tight lower-bound to the exactoutage analysis of (2) in Fig. 3.
4
and transmits on orthogonal subspaces). The vector h1 =P h1 has the same statistics as h1 with dimensionality re-duced by one [24], hence h1 ∼ CN (0, Σ), where Σ =Ω (1 : NR − 1, 1 : NR − 1). For the ZF to be applicable inspatially correlated MIMO systems, channels need not be fullycorrelated (i.e., ρ < 1). Note that for independent fadingchannels (i.e., Σ = Λ = I) with perfect CSI (i.e., σ2
εi = 0)2,equation (3) reduces to [11, Eq. (20)].
IV. OUTAGE PROBABILITY ANALYSIS
In this section, the information outage probability of theproposed overall SNR of MIMO FD relaying systems isinvestigated. An exact as well as lower-bound expressions forthe outage probability are derived for arbitrary correlation,where simpler correlation scenarios and lower bounds are alsoincluded. The outage probability is defined as the probabilitythat the instantaneous mutual information, I = log2 (1 + γ),falls below a target rate of R0 bits per channel use3. And canbe given as
Pout (R0) = Pr (log2 (1 + γ) < R0)
= Fγ (γT ) . (4)
where γT = 2R0 − 1, and Fγ (·) denotes the cumulativedistribution function (CDF) of the overall SNR.
A. Arbitrary Correlation Case
The CDF and the probability density function (PDF) of therandom variables (RVs) γ1 and γ2, respectively, with arbitrarycorrelation matrix, can be written as [27]
Fγ1 (x) = 1−%(Σ)∑i=1
τi(Σ)∑j=1
j−1∑k=0
χi, j (Σ)
k!
×(
x
γ1α〈i〉
)ke− xγ1α〈i〉 , x ≥ 0 (5)
and
fγ2 (y) =
%(Λ)∑l=1
τl(Λ)∑m=1
χl,m (Λ)
(γ2β〈l〉
)−mΓ (m)
× ym−1e− yγ2β〈l〉 , y ≥ 0 (6)
where Σ is the correlation matrix of h1 with eigenvaluesα1, α2, . . . , αNR−1 in any order, % (Σ) is the number ofdistinct eigenvalues of Σ, α〈1〉 > α〈2〉 > . . . > α〈%(Σ)〉are the distinct eigenvalues in decreasing order, τi (Σ) isthe multiplicity of α〈i〉, χi, j (Σ) is the (i, j)
th characteristicfunction of Σ [27, Eq. (129)]. Similarly, Λ is the correlationmatrix of h2 with eigenvalues β1, β2, . . . , βNT in any order,% (Λ) is the number of distinct eigenvalues of Λ, β〈1〉 >
2It is to be emphasized that c can be accurately approximated to 1, this hasbeen proved in [25].
3Note that in contrast to (4), the SNR outage probability can be defined asthe probability that the instantaneous overall γ, falls below a threshold γT ;Pr (γ < γT ) = Fγ (γT ). Note that according to (4), γT in the case of HDrelaying is given as γT = 22R0 − 1.
β〈2〉 > . . . > β〈%(Λ)〉 are the distinct eigenvalues in decreasingorder, τl (Λ) is the multiplicity of β〈l〉, and χl,m (Λ) is the(l, m)
th characteristic function of Λ.1) Exact Outage Probability: The CDF of the overall SNR
can be derived as [15, Appendix I]
Fγ (γT ) = Pr(
γ1γ2
γ1 + γ2 + c< γT
)= 1−
∞
0
F γ1
(γT (γT + w + c)
w
)fγ2 (γT + w) dw (7)
where F γ1 (·) is the complementary CDF of γ1.
Theorem 1. The exact outage probability of the overall SNRγ, with arbitrary correlation matrices, can be derived as givenin (8), shown at the top of next page.
where Kv (z) is the modified bessel function of the secondkind of order v.
Proof: The proof is given in Appendix A.2) Lower-Bound Outage Probability: The overall SNR can
be upper bounded by4
γ ≤ γup = min (γ1, γ2) . (9)
Note that the overall SNR upper-bound for AF relayingsystems (9) substitutes the exact overall SNR for the decode-and-forward (DF) relaying systems.
Therefore, the outage probability of the overall SNR γ, witharbitrary correlation matrices, is lower-bounded by
Fγup (γT ) = 1−%(Σ)∑i=1
τi(Σ)∑j=1
χi, j (Σ)Γ(j, γT
γ1α〈i〉
)Γ (j)
×%(Λ)∑k=1
τk(Λ)∑l=1
χk, l (Λ)Γ(l, yγ2β〈k〉
)Γ (l)
. (10)
Proof: The proof is given in Appendix B.
B. Exponential Correlation CaseIn the case of exponential correlation matrix, all eigenvalues
are distinct with multiplicity of one. Therefore, % (Σ) = NR−1, τi (Σ) = 1, % (Λ) = NT , τm (Λ) = 1. Hence, the simplifiedCDF and PDF of γ1 and γ2, respectively, given as
Fγ1 (x) = 1−NR−1∑i=1
χi (Σ) e− xγ1αi (11)
and
fγ2 (y) =
NT∑j=1
χj (Λ)
γ2βje− yγ2βj (12)
where χi (Σ) = αNR−2i
∏NR−1k=1, k 6=i (αi − αk)
−1, andχj (Λ) = βNT−1
j
∏NTn=1, k 6=j (βj − βn)
−1.
4It is well known in the conventional half-duplex relaying literature thatthe overall SNR γ = γ1γ2
γ1+γ2+1can be tightly upper-bounded by γ1γ2
γ1+γ2(see
e.g., [28, Eq. (6)]). In addition, the upper-bound γ1γ2γ1+γ2
can be further upper-bounded by min (γ1, γ2) (see e.g., [29, Eq. (8)]). It is to be emphasized thatthe asymptotic results of these bounds are exact.
5
Fγ (γT ) = 1− 2
%(Σ)∑i=1
τi(Σ)∑j=1
j−1∑k=0
χi, j (Σ)
k!
(γT
γ1α〈i〉
)ke− γTγ1α〈i〉
k∑l=0
(kl
)(γT + c)
l%(Λ)∑m=1
τm(Λ)∑n=1
χm,n (Λ)1
Γ (n)e− γTγ2β〈m〉
×n−1∑p=0
(n− 1p
)γn−p−1T
(γ2β〈m〉
)p−n−l+1(
γ2T + γT c
γ1γ2α〈i〉β〈m〉
) p−l+12
Kp−l+1
(2
√γ2T + γT c
γ1γ2α〈i〉β〈m〉
)(8)
1) Exact Outage Probability: The exact outage probabilityof the overall SNR γ, with exponential correlation matrices,can be written as
Fγ (γT ) = 1− 2
NR−1∑i=1
χi (Σ) e− γTγ1αi
NT∑j=1
χj (Λ) e− γTγ2βj
×
√γ2T + γT c
γ1γ2αiβjK1
(2
√γ2T + γT c
γ1γ2αiβj
). (13)
Proof: The proof is given in Appendix C.2) Lower-Bound Outage Probability: The outage probabil-
ity of the overall SNR γ, with exponential correlation matrices,is lower-bounded by
Fγup(γT ) = 1−
NR−1∑i=1
χi (Σ) e− γTγ1αi
NT∑j=1
χj (Λ) e− γthγ2βj .
(14)Proof: Following similar steps to the derivations of (10),
the outage probability of the upper-bound overall SNR can bederived by substituting the CDF of γ1 and γ2 into (25) whichupon tedious simplification reduces to (14).
3) Asymptotic Analysis: In the asymptotic high SNRregime, the asymptotic outage probability of the overall SNRγ can be expressed as
F∞γ (γT ) =
(γTγ1
)NR−1
∏NR−1i=0 αiΓ (NR)
+
(γTγ1
)NT (1κ
)NT∏NTi=0 βiΓ (NT + 1)
+O
((γTγ1
)min(NR, NT+1)). (15)
Proof: The proof follow similar steps to the derivation of[17, Eq. (17)].
It is easily seen from the above asymptotic expressionthat the achievable diversity order of correlated MIMO FDrelaying systems with ZF/MRT scheme is min (NR − 1, NT ).Furthermore, the negative impact of correlation on the outageprobability is clearly seen here, where the outage probabilityincrease is quantified by the factors 1∏NR−1
i=0 αiand 1∏NT
i=0 βi.
Meanwhile, the outage probability increase due to estimation
error is determined by the factors(ρ1σ
2ε1
+ρ2σ2ε3
+1
1−σ2ε1
)NR−1
and(ρ2σ
2ε2
+1
1−σ2ε2
)NT.
C. Independent Case
In the case of independent channels, the CDF and PDF ofthe RVs γ1 and γ2 are (respectively) given as
Fγ1 (x) = 1− e−xγ1
NR−2∑k=0
1
k!
(x
γ1
)k, x ≥ 0 (16)
and
fγ2 (y) =γ2−NT
(NT − 1)!yNT−1e−
yγ2 , y ≥ 0. (17)
1) Exact Outage Probability: The exact outage probabilityof the overall SNR γ, with independent channels, can bederived as in (18), shown at the top of next page.
Proof: The proof follow similar steps to the derivation of(8).
2) Lower-Bound Outage Probability: The outage probabil-ity of the overall SNR γ, with independent channels, is lower-bounded by
Fγup (γT ) = 1−Γ(NR − 1, γTγ1
)Γ (NR − 1)
Γ(NT ,
γTγ2
)Γ (NT )
. (19)
Proof: The proof follow similar steps to the derivation of(10).
3) Asymptotic Analysis: To validate and characterize theachievable diversity order of the MIMO full-duplex relayingsystem receive ZF/MRT scheme at the AF relay, we approxi-mate (19) in the asymptotic high SNR regime where γ2 = κγ1,with κ denotes a finite constant number and γ1 →∞. Hence,the asymptotic outage probability of the overall SNR γ canbe expressed as
F∞γ (γT ) =
(γTγ1
)NR−1
Γ (NR)+
(γTγ1
)NTΓ (NT + 1)
(1
κ
)NT+O
((γTγ1
)min(NR, NT+1)). (20)
Proof: The proof is given in Appendix D.Equation (20) can be further simplified to
F∞γ (γT ) =
(γTγ1
)NR−1
Γ(NR) , NR − 1 < NT(γTγ1
)NR−1
Γ(NR) +
(γTγ2
)NTΓ(NT+1) , NR − 1 = NT(
γTγ2
)NTΓ(NT+1) , NR − 1 > NT
.
(21)It is straight forward to show from (21) that the achievable
diversity order of the ZF/MRT MIMO FD relaying systems is
6
Fγ (γT ) = 1− 2γNT− 1
2
T
√γT + c
(NT − 1)!γX2
NXT
√γ2
γ1e−
γTγ1− γTγ2
NR−2∑k=0
1
k!
(γTγ1
)k k∑l=0
(kl
)
×(γ1 (γT + c)
γ2γT
) l2NT−1∑m=0
(NT − 1m
)(γT + c
γ1γTγ2
)m2
Km−l+1
2
√γ2T + γT c
γ1γ2
. (18)
min (NR − 1, NT ). Therefore, correlation has no impact onthe diversity order of the system.
D. Hybrid Relaying Modes
The case of MIMO half-duplex relaying with maximum-ratio combining (MRC)/MRT scheme [25, Eq. (5)] is consid-ered with the constraint that the total number of antennas at theHD relay is N = NR +NT . The need for proper mode selec-tion is motivitaed by Figs. 4-5. The outperform region of FDrelaying can be derived by solving PFD
out (R0)−PHDout (R0) ≤ 0.
Note that the outage probability expression for a hybridrelaying mode that switches to the appropriate mode accordingto the instantaneous CSI is given by5
PHybridout (R0) = Pr
(max
(CFD
Inst, CHDInst
)< R0
)(22)
where CFDInst = log2
(1 + γFD
), CHD
Inst = log2
(√1 + γHD
),
γFD is given in (3), and γHD is given in [25, Eq. (5)].
V. NUMERICAL RESULTS
In this section, we analyze and validate the presentedtheoretical results with Monte Carlo simulations. Without lossof generality, two common correlation scenarios are consid-ered here; the uniform correlation case where Σ (i, j) =
1 i = j
ρ i 6= j, and the exponential correlation case where
Σ (i, j) = ρ|i−j|, with ρ ∈ [0, 1) being the correlationcoefficient. For simplicity, symmetric settings (ρ1 = ρ2)are assumed for both links, with a target rate of R0 = 2bits per channel use. The Cholesky decomposition methodis used to transform a set of uncorrelated Gaussian randomvector to a set of correlated random vector of a predefinedcorrelation matrix6. The correlation coefficient is defined usingthe practical channel model given in [30], assuming that thereis a uniform linear antenna arrays at the relay. Let d bethe equidistant antenna spacing (spacing between adjacentantennas), normalized by the carrier wavelength, denoted as drat the receiving antennas, and dt at the transmitting antennas
5Note that as far as a closed-form outage probability expression is concern,the analysis of such hybrid relaying mode is a challenging mathematicalproblem due to the presence of high correlation between γHD and γFD.
6The Cholesky decomposition of the predefined correlation matrix Υ
may be derived as: Υ = UΨU† =(U√
Ψ)(
U√
Ψ)†
, where U is aunitry eigenvector matrix and Ψ is a diagonal matrix whose entries are theeigenvalues of Υ. Therefore, a spatially correlated channel random vectory is generated by y = Lx, where x is a vector of independent randomvariables distributed according to x ∼ CN (0, I) and L = U
√Ψ. Therefore,
y ∼ CN(0, LL†), note that Υ = LL†.
0 5 10 15 20 25 30
10−4
10−3
10−2
10−1
100
Average per hop SNR in dBs
Out
age
Pro
babi
lity
Theory (Exp. Corr. Exact)
Theory (Exp. Corr. Lower−Bound)
Theory (Exp. Corr. Asymptotic)
Theory (Unif. Corr. Exact)
Theory (Unif. Corr. Lower−Bound)
Theory (Indep. Exact)
Theory (Indep. Lower−Bound)
Theory (Indep. Asymptotic)
Simulation
Figure 2: MIMO FD relaying outage probability with perfectCSI (σ2
ε = 0) against the first hop SNR (ρ1) where ρ1 = ρ2,for the case of high (σ2
θr= σ2
θt= π
64 , dr = dt = 13 ) uniform,
exponential and no correlations, and NR = NT = 3.
(dr = dt in the MIMO HD relaying case). Two cases for d areassumed here; the fixed antenna spacing case where d is fixedregardless of the chosen number of antennas, and the variable dcase where the inter-antenna spacing is changing according tothe limited available space and the chosen number of antennasput in that space, hence d = D
N−1 , where D is the limitedmulti-antenna space at the relay, normalized by the carrierwavelength. Also, let θr and θt be the mean angle or arrival(AoA) and mean angle of departure (AoD), respectively. In ad-dition, let σ2
θrand σ2
θtas the receive and transmit cluster angle
spread, respectively. Assuming that the actual AoA and AoDmay be given by θr = θr + θr and θt = θt + θt, respectively,with θr ∼ CN
(0, σ2
θr
)and θt ∼ CN
(0, σ2
θt
). Moreover, The
receive and transmit antennas correlation matrices are definedas Σ = f
(ρ(dr, θr, σ
2θr
))and Λ = f
(ρ(dt, θt, σ
2θt
)),
respectively, where the function f (·) used to differentiatebetween different correlation scenarios (i.e., the defined uni-form and exponential correlation). With these assumptions, thecorrelation coefficient ρ can be estimated as
ρ(d, θ, σ2
θ
)= e−j2πd cos(θ)e−
12 (2πd sin(θ)σθ)
2
. (23)
Which implies that large antenna spacing and/or largecluster angle spread results in small spatial correlation andvice versa. For all results in this section, it is assumed thatθr = θt = π
2 .
7
0 5 10 15 20 25 3010
−5
10−4
10−3
10−2
10−1
100
Average per hop SNR in dBs
Out
age
Prob
abili
ty
Theory (Eq. (13))
Simulation (Eq. (4))
Simulation (Eq. (2))
σ2ε = 0.05
σ2ε = 0.025
σ2ε = 0.005
σ2ε = 0
Figure 3: MIMO FD relaying outage probability with imper-fect CSI against the first hop SNR (ρ1) where ρ1 = ρ2, forthe case of high (σ2
θr= σ2
θt= π
64 , dr = dt = 13 ) exponential
and no correlations, and NR = NT = 3.
0 5 10 15 20 25 3010
−6
10−5
10−4
10−3
10−2
10−1
100
Average per hop SNR in dBs
Out
age
Prob
abili
ty
FDR (3,3)
FDR (3,1)
FDR (9,3)
FDR (2,1)
HDR (3)
HDR (6)
Figure 4: MIMO relaying outage with perfect CSI probabilityagainst the first hop SNR (ρ1) where ρ1 = ρ2, for highexponential correlation (σ2
θr= σ2
θt= π
64 , dr = dt = 13 ).
In Fig. 2, the outage probability of MIMO FD relayingagainst the first-hop SNR is presented with different cor-relation matrix scenarios, where Monte Carlo simulationsof (4) is used to validate the new closed-form analyticalexpressions (8), (13), and (18). It is seen that the simulationand proposed analytical expression provide a perfect matchwhich corroborate the exactness of the proposed exact closed-form analytical expressions. In addition, the tightness of theproposed lower-bound and asymptotic expressions are alsoverified. The chosen values of the number of relay receiveand transmit antennas in the FD relaying case are shown as(NR, NT ), where in the HD relaying is shown as (N). Fig. 3
0 5 10 15 20 25 3010
−4
10−3
10−2
10−1
100
Average per hop SNR in dBs
Out
age
Pro
babi
lity
Full−Duplex
Half−Duplex
Hybrid Relaying
NR = N
T = 3 & σ2
ε = 0.05
NR = N
T = 3 & σ2
ε = 0.005
Figure 5: MIMO FD, HD, and hybrid relaying outage probabil-ity with imperfect CSI against the first hop SNR (ρ1) whereρ1 = ρ2,for exponential correlation with σ2
θr= σ2
θt= π
64 ,dr = dt = 2
NX−1 , where NX = NR = NT in the FD relayingcase, and NX = NR+NT = N in the HD relaying case, withdifferent σ2
ε .
shows the effect of imperfect CSI on the MIMO FD relayingoutage probability in the case of fixed antennas spacing andexponential correlation matrix. The tightness of the approxi-mation of the estimation error covariance is also verified. Ahigher outage probability degradation is seen as the estimationerror variance σ2
ε increases, where in the presence of imperfectCSI an error floor is seen at high SNR. In Fig. 4, the outageprobability of MIMO FD relaying with receive ZF/MRTscheme is shown with different antenna configurations, whereMIMO HD relaying with MRC/MRT is also presented. Forinstance, once comparing the FD relaying settings (3, 3) withthe HD relaying of (6) and (3) for the separate and sharedFD relaying antenna system configurations. It is seen that FDrelaying outperforms the conventional HD relaying in the prac-tical SNR range. Furthermore, a performance improvement isseen as a result of higher diversity order and/or array gain. Forsimplicity, it is assumed that σ2
ε1 = σ2ε2 = σ2
ε3 = σ2ε . Fig. 5
shows the impact of imperfect CSI on MIMO FD relaying,HD relaying, and hybrid relaying outage probability7 withvariable antennas spacing. It is to be emphasized here that thedegradation in outage probability due to spatial correlation ishigher in MIMO HD relaying compared to MIMO FD relayingsystems since higher number of antennas are planted on thesame limited space. Meanwhile, the degradation in outageprobability due to estimation error is higher in the MIMOFD relaying systems owing to the addition of the estimationerror of the loopback self-interference channel. Therefore,for optimum performance, hybrid relaying modes is proposed
7Note that a lower bound outage probability for the hybrid relaying caseis shown in Fig. 5 where γHD and γFD are assumed independent.
8
which switches between HD and FD relaying modes.
VI. CONCLUSIONS
In this paper, new closed-form expressions for the outageprobability of spatially correlated MIMO FD relaying withreceive ZF/MRT are presented, where simpler lower boundsare also included. The case of imperfect CSI is also addressed.Numerical results demonstrated that FD relaying systemsoutperforms HD relaying systems at low to medium SNR.In addition, in the presence of channel estimation errors,an outage probability error floor is seen as SNR increases.Besides, The performance degradation due to imperfect CSIis higher in the FD relaying case because of the extra LIchannel estimation errors. Furthermore, although the sourceand destination are equipped with one antenna, the systemperformance can be improved by appropriately choosing theprecoding scheme (receive ZF/MRT or MRC/transmit ZF). Forsuperior system performance, receive ZF is performed whenNR > NT , and transmit ZF is performed when NR < NT .However, when NR = NT , both receive ZF and transmitZF have the same performance assuming symmetric settings(ρ1 = ρ2) case.
APPENDIX APROOF OF THEOREM 1
From (7), it can be shown that the CDF of the overall SNRis given as (24), shown at the top of next page.
Where with the aid of [31, Eq. (3.471.9)], we arrive at (8),which concludes the proof.
APPENDIX BPROOF OF EQUATION (10)
From the overall SNR upper-bound (9), we have
Fγup(γT ) = Pr (min (γ1, γ2) < γT )
= 1− (1− Fγ1 (γT )) (1− Fγ2 (γT )) (25)
which upon substituting the CDF of γ1 and γ2 reduces to(10), that concludes the proof.
APPENDIX CPROOF OF EQUATION (13)
From (7), the CDF of the overall SNR can be derived as
Fγ (γT ) = 1−NR−1∑i=1
χi (Σ)
NT∑j=1
χj (Λ)
βje− γTγ1αi
− γTγ2βj
×∞
0
e− γ
2T+γT c
wγ1αi e− wγ2βj dw (26)
Which with the aid of [31, Eq. (3.471.9)], reduces to (13),that concludes the proof.
APPENDIX DPROOF OF EQUATION (20)
The asymptotic results can be easily obtained once invokingthe asymptotic expansion of the incomplete gamma function[31, Eq. (8.354.1)]. Hence, in the high SNR regime whereγ2 = κγ1 and γ1 →∞, we have
γ
(NR − 1,
γTγ1
)=
(γTγ1
)NR−1
NR − 1+O
((γTγ1
)NR). (27)
where γ(NR − 1, γTγ1
)+ Γ
(NR − 1, γTγ1
)= Γ (NR − 1).
Therefore,
Γ(NR − 1, γTγ1
)Γ (NR − 1)
= 1−
(γTγ1
)NR−1
Γ (NR)+O
((γTγ1
)NR).
(28)Similarly, we have
Γ(NT ,
γTγ2
)Γ (NT )
= 1−
(γTγ1
)NTΓ (NT + 1)
(1
κ
)NT+O
((γTγ2
)NT+1).
(29)Substituting (28) and (29) into (19) yields (20), that con-
cludes the proof.
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9
Fγ (γT ) = 1−%(Σ)∑i=1
τi(Σ)∑j=1
j−1∑k=0
χi, j (Σ)
k!
(γT
γ1α〈i〉
)ke− γTγ1α〈i〉
k∑l=0
(kl
)(γT + c)
l%(Λ)∑m=1
τm(Λ)∑n=1
× χm,n (Λ)
(γ2β〈m〉
)−nΓ (n)
e− γTγ2β〈m〉
n−1∑p=0
(n− 1p
)γn−p−1T
∞
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wp−le− γ
2T+γT c
wγ1α〈i〉 e− wγ2β〈m〉 dw. (24)
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Ahmed Almradi (S’14) received the B.Sc. degreein Electrical and Electronic Engineering in 2004from the University of Tripoli, Tripoli, Libya, thenreceived the M.Sc. degree in Electrical Engineeringin 2012 from Rochester Institute of Technology,Rochester, NY, USA. He is on study leave from Az-zaytuna University, Tarhuna, Libya. He is currentlyworking towards the Ph.D degree at the University ofManchester, Manchester, U.K. His research interestsare in the modeling, design, and performance analy-sis of wireless communication systems with special
emphasis on MIMO half-duplex and full-duplex relaying systems, wirelessinformation and power transfer (energy harvesting) systems, diversity andbeamforming, and OFDM systems.
Khairi Ashour Hamdi (M’99-SM’02) received theB.Sc. degree in Electrical Engineering from theUniversity of Tripoli, Tripoli, Libya, in 1981; theM.Sc degree (with distinction) from the Techni-cal University of Budapest, Budapest, Hungary, in1998; and the Ph.D. degree in TelecommunicationEngineering from Hungarian Academy of Sciences,Budapest, in 1993. He was with the University ofEssex, Colchester, U.K. He is currently with theSchool of Electrical and Electronic Engineering,The University of Manchester, Manchester, U.K. His
current research interests include modelling and performance analysis ofwireless communication systems and networks, green communication systems,and heterogeneous mobile networks.