Low-complexity End-to-End Performance Optimization in MIMO Full-Duplex Relay Systems
Himal A. Suraweera, Ioannis Krikidis, Gan Zheng, Chau Yuen, Peter J. Smith
aa r X i v : . [ c s . I T ] N ov Low-complexity End-to-End PerformanceOptimization in MIMO Full-Duplex RelaySystems
Himal A. Suraweera, Ioannis Krikidis, Gan Zheng, Chau Yuen, and Peter J. Smith
Abstract
In this paper, we deal with the deployment of full-duplex relaying in amplify-and-forward (AF)cooperative networks with multiple-antenna terminals. In contrast to previous studies, which focus onthe spatial mitigation of the loopback interference (LI) at the relay node, a joint precoding/decoding designthat maximizes the end-to-end (e2e) performance is investigated. The proposed precoding incorporatesrank-1 zero-forcing (ZF) LI suppression at the relay node and is derived in closed-form by solvingappropriate optimization problems. In order to further reduce system complexity, the antenna selection(AS) problem for full-duplex AF cooperative systems is discussed. We investigate different AS schemesto select a single transmit antenna at both the source and the relay, as well as a single receive antennaat both the relay and the destination. To facilitate comparison, exact outage probability expressionsand asymptotic approximations of the proposed AS schemes are provided. In order to overcome zero-diversity effects associated with the AS operation, a simple power allocation scheme at the relay node isalso investigated and its optimal value is analytically derived. Numerical and simulation results show thatthe joint ZF-based precoding significantly improves e2e performance, while AS schemes are efficientsolutions for scenarios with strict computational constraints.
Index Terms
MIMO relay networks, full-duplex relaying, precoding, antenna selection, outage probability.
H. A. Suraweera is with the Department of Electrical and Electronic Engineering, University of Peradeniya, Peradeniya 20400, Sri Lanka(e-mail: [email protected])H. A. Suraweera and C. Yuen are with the Singapore University of Technology and Design, 20 Dover Drive, Singapore 138682 (e-mail:{himalsuraweera, yuenchau}@sutd.edu.sg)I. Krikidis is with the Department of Electrical & Computer Engineering, University of Cyprus, Nicosia 1678, Cyprus (e-mail:[email protected])G. Zheng is with the Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, 4 rue Alphonse Weicker,L-2721 Luxembourg (e-mail: [email protected])P. J. Smith is with the Department of Electrical and Computer Engineering, The University of Canterbury, Private Bag 4800, Christchurch,New Zealand (email: [email protected])This work was presented in part at the IEEE Intl. Conf. Commun. (ICC 2013), Budapest, Hungry, June 2013.April 20, 2018 DRAFT
I. I
NTRODUCTION
Cooperative communications with relaying is a promising solution to extend the network coverage andensure higher throughputs and quality-of-service (QoS). Relaying techniques can be classified as eitherhalf-duplex or full-duplex [1]. In order to complete the relaying operation, half-duplex relaying requirestwo orthogonal channels and the associated bandwidth loss recovery has been an active research areafor several years. With full-duplex relaying, the relay node receives and transmits simultaneously on thesame channel and therefore utilizes the spectrum resources more efficiently [2], [3]. However, the mainlimitation in full-duplex operation is the loopback interference (LI) (also known in the literature as theloopback self-interference) due to signal leakage from the relay’s output to the input at the receptionside [4]–[7]. Specifically, the main drawback of full-duplex operation is the large power differentialbetween the LI generated by the full-duplex terminal and the received signal of interest coming froma distant source. The large LI spans most of the dynamic range of the analog-to-digital converter atthe receiver side and thus its mitigation is critical for the implementation of full-duplex operation. Inmodern communication systems such as WiFi, Bluetooth, and Femtocells, the transmission power and thedistance between communicating devices has been decreased. This important architectural modificationdecreases the power differential between the two received signals. This attribute, combined with thehigh computation capabilities of modern terminals, significantly facilitates the implementation of thefull-duplex radio technology [8]–[10].In the literature, the combination of multiple-input multiple-output (MIMO) techniques with relayinghas been invoked to further enhance the communication performance [11], [12]. While most work hasfocused on MIMO half-duplex relaying, recent work has also considered MIMO full-duplex relaying.MIMO provides an effective means to suppress the LI in the spatial domain [6], [13], [14]. With multipletransmit or receive antennas at the full-duplex relay, precoding at the transmitter and decoding at thereceiver can be jointly optimized to mitigate the LI effects. Zero forcing (ZF) and minimum mean squareerror (MMSE) are two widely adopted criteria in the literature for the precoding and decoding design[15]. ZF aims to completely null out undesired interference and provides an interference-free channel.Although ZF normally results in sub-optimal solutions, its performance is nearly optimal in the highsignal-to-noise ratio (SNR) regime. MMSE is an improved precoder/decoder design compared to ZF,which takes into account the background noise. The MMSE-based precoder has a more complicatedstructure but it can improve the achievable QoS. Due to the implementation simplicity and the efficiencyin the high SNR regime, ZF becomes a useful design criterion to completely cancel the LI and break theclosed-loop between the relay input and output.
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Assuming there is no closed-loop processing delay, the optimal precoding matrix for a full-duplexamplify-and-forward (AF) relay that maximizes the mutual information under an average power constraintis studied in [16]. In this case, the design approach and the resulting precoding solution are similar tothe half-duplex case. The joint precoding and decoding design for a full-duplex relay is studied in [6],[17], where both ZF solutions and MMSE solutions are discussed. Notice that the ZF solution used in[6], [17] and most early works uses a conventional approach based on the singular value decompositionof the loopback self-interference channel. The main drawback of this approach is that the ZF solutiononly exists given that the numbers of antennas at the source, full-duplex relay and the receiver satisfy acertain condition. In order to overcome this limitation, [13] adopts an alternative criterion and proposesto maximize the signal-to-interference ratios between the power of the useful signal to the power of LIat the relay input and output, respectively. Conventional ZF precoding and decoding are chosen via thesingular vectors of the LI channels, however, this design does not take into account the other channelsand the end-to-end (e2e) performance. In [18], a joint design of ZF precoding and decoding is proposedto fully null out the LI at the relay, taking into account the source-relay and relay-destination channels.A simple approach is studied in [19], where an iterative algorithm that jointly optimizes the precodingand decoding vectors in respect of the e2e performance, is investigated.Most of the work in the literature does not deal with the joint optimization of the precoding anddecoding process, even for scenarios with multiple antennas at the terminals. Hence, the focus hasbeen restricted to full duplex relay processing which has led to strictly suboptimal e2e performance.Furthermore, the available ZF-based solutions which do aim to optimize e2e performance are not givenin closed-form. Hence, in this paper, we consider a general case where each terminal can have arbitrarymultiple antennas and we jointly design precoding and decoding at the source, the relay and the destinationin order to maximize the achievable rate. For simplicity, a single data stream is transmitted and ZF criteriaare used by the full-duplex relay to handle the LI. We give the closed-form precoder/decoder solutions fortransmit and receive ZF schemes. Furthermore, the diversity orders are derived for the different schemes.In addition, we also propose several low-complexity antenna selection (AS) schemes for MIMO full-duplex relaying and analyze the outage probability of each scheme. The complexity of implementingMIMO systems can be significantly decreased with AS, which employs fewer radio frequency chainsthan antenna elements and then connects the chains to the best available antenna element [20]. Some We follow the footsteps of recent work such as [6] and investigate the performance of AS since both precoding/decoding andAS schemes belong to the general category of MIMO spatial suppression techniques. ZF precoding/decoding designs and differentAS schemes studied in this paper eliminate/mitigate the effect of LI respectively, and hence offer different performance/complexitytradeoff choices to a system designer. Moreover, AS can be viewed as a special case of precoding where the beamforming vectoronly contains a single non zero unit element whose entry depends on the selected transmit/receive antenna.
April 20, 2018 DRAFT limited work on AS in full-duplex relay systems can be found in [6], [14]. In [6], several spatial LIsuppression techniques based on antenna sub-set selection and joint transmit/receive beam selection havebeen investigated. In [14], several low complexity antenna sub-set selection schemes have been proposedwith the objective to suppress LI at the relay’s transmit side. However, a basic limitation of the currentwork is that AS is used only to achieve LI suppression. On the other hand, from a system performancestandpoint, it is important to deploy MIMO AS techniques such that the e2e signal-to-interference noiseratio (SINR) at the destination is maximized.The performance of AS in half-duplex relay systems is a mature topic and well studied, see fore.g., [21]–[24]. On the other hand, to the best of authors’ knowledge, the current paper is the first toanalytically investigate the AS performance for full-duplex relay systems. Moreover, our analysis presentsnew results in addition to earlier work such as [25], [26] where the outage probability of single antennafull-duplex systems have been studied. Specifically, we select single transmit antennas at the source andthe relay, respectively, and single receive antennas at the relay and the destination, respectively. Theperformance of the aforementioned system set-up with different AS schemes is quantified by derivingexact, and asymptotic outage probability expressions. The asymptotic expressions illuminate the networkperformance by revealing the comparative performances of the AS schemes in terms of the system andchannel parameters. Furthermore, in order to eliminate the zero-diversity behavior of the full-duplexrelaying due to the LI, we propose a new simple power allocation scheme at the relay, which onlyinvolves a single parameter optimization. We also present optimal values for this parameter to minimizethe outage probability from a diversity perspective. These closed-form expressions are in the form offractions of the number of source/relay/destination antennas and reveal the spatial degrees of freedomoffered by each AS scheme. Moreover, these values can be calculated directly once a particular systemconfiguration is decided.The main contributions of this paper are twofold. • A low complexity joint precoding/decoding design for e2e SNR maximization is proposed. Specifi-cally, based on ZF loopback self-interference suppression, receive/transmit beamforming vectors atthe relay are designed. Closed-form solutions for the scheme’s outage probability as well as highSNR simple expressions are derived. Our analysis clearly reveals insights on system performanceand shows the impact on the achieved diversity order. • Several AS schemes are proposed including the optimal AS scheme that maximizes the e2e SNR atthe destination and various sub-optimal AS schemes. In order to eliminate the zero diversity behavior It should be noted that due to the influence of LI, power adaptation (or “gain control” [26]) is an important issue forfull-duplex AF relaying.
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(cid:10)(cid:11) (cid:12) (cid:13) (cid:14)(cid:15)(cid:16)(cid:17) (cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)
Fig. 1. Full-duplex MIMO relaying with multi-antenna source and destination nodes. The dashed line denotes the loopbackself-interference. in such full-duplex MIMO systems, we propose a simple power allocation method at the relay. Theoutage performance of the AS schemes are analytically investigated. Using the derived high SINRoutage approximations, we also investigate the optimal power allocation coefficient values.The rest of the paper is organized as follows. Section II presents the overall MIMO system model.Sections III and IV present the joint precoding/decoding designs and AS schemes, respectively. Theoutage probability of the precoding and AS schemes is analyzed in Section V and numerical results aregiven in Section VI. Finally, Section VII concludes the paper and summarizes several key findings.
Notation : The lowercase and uppercase boldface letters (e.g., x and X ) indicate column vectors andmatrices, respectively. I is the identity matrix and diag ( a , a , . . . , a n ) denotes a diagonal matrix withelements l = { a , a , . . . , a n } . We use ( · ) † to denote the conjugate transpose, k · k is the Frobenius normand Tr ( · ) is the trace operation. λ max ( X ) denotes the maximum eigenvalue of a matrix X and u max ( X ) represents the eigenvector associated with λ max ( X ) . The expectation operator is denoted by E ( · ) and Pr {·} is probability. K ν ( z ) is the modified Bessel function of the second kind of order ν .II. S YSTEM M ODEL
We consider a basic three-node MIMO relay network consisting of one source S , one relay R , andone destination, D as shown in Fig. 1. We use N T and N R to denote the number of transmit and receiveantennas at S and D , respectively. The relay is equipped with two groups of antennas; M R receive and M T transmit antennas for full-duplex operation. S has no direct link to D , which may result from heavypath loss and high shadowing between S and D . April 20, 2018 DRAFT
A. Channel Model
All wireless links in the network are subject to non-selective independent Rayleigh block fadingand additive white Gaussian noise (AWGN); H SR and H RD denote the S − R and R − D channels,respectively, while H RR denotes the LI channel. In order to reduce the effects of self-interference onsystem performance, an imperfect interference cancellation scheme (i.e. analog/digital cancellation) isused at R and we model the residual LI channel as a fading feedback channel [4], [27], [28]. Moreover,the noise at the nodes is modeled as complex AWGN with zero mean and normalized variance. Inaddition, the single-input single-output channel corresponding to the i -th receive and the j -th transmitantenna from terminal X to terminal Y , is denoted by h i,jXY where X ∈ { S, R } and Y ∈ { R, D } . As forthe average S − R and R − D channel statistics; we assume E{| h i,jSR | } = c SR and E{| h i,jRD | } = c RD .The experimental-based study in [8] has demonstrated that the amount of LI suppression achieved by ananalog/digital cancellation technique is influenced by several system and hardware parameters. Since eachimplementation of a particular analog/digital LI cancellation scheme can be characterized by a specificresidual power, a parameterization by H RR with elements satisfying E{| h i,jRR | } = c RR allows theseeffects to be studied in a generic way [5]. We assume that the channel coefficients remain approximatelystationary for a long observation time (time slot), but change independently from one slot to anotheraccording to a Rayleigh distribution. The channel coherence time is equal to one time slot. This assumptionapplies to networks with a low mobility and corresponds to slow fading (block) channels where codingis performed over one block. B. System Model
This work studies full-duplex operation at a system level using some well-known models for thecharacterization of the residual loop interference [6]. We note that the developed schemes do not referto a specific analogue or baseband implementation and can be applied to both by taking into accountrelated practical aspects (i.e., training sequence, antenna impedance mismatch, dynamic range etc). Furtherimplementation issues as well as more realistic radio environments (i.e., frequency selectivity) are beyondthe scope of this paper.In order to keep the complexity low, we assume that a single data stream is transmitted and eachnode employs only linear processing, i.e., S applies a precoding vector t on the data stream, while D Single stream beamforming delivers maximum diversity/array gains and is suitable in a slow fading environment. Also,with multiple antennas at all three nodes, multiple independent data streams can be simultaneously sent, namely, multi-stream beamforming systems. In such systems, although full-duplex operation mode aimed at utilizing spectrum resourcesmore efficiently can promise rate gains due to spatial multiplexing [30], in general they experience poor error performance [31].
April 20, 2018 DRAFT uses a linear receive vector r to decode the signal, where k t k = k r k = 1 . With the recent trend toincrease the number of antennas at the terminals (e.g., massive MIMO), linear processing solutions offeran attractive solution for low complexity implementation. In contrast, the complexity of the optimal nonlinear signal detection approach grows exponentially with the number of transmit antennas [29]. Therelaying operation is based on the AF policy with an amplification matrix W that keeps the transmittedpower at the relay node below the threshold P R . We jointly optimize t , r and W to maximize the e2esystem performance.
1) Joint Precoding/Decoding Design:
ZF is chosen as the design criterion for the relay amplificationmatrix W , such that there is no loopback self-interference from the relay output to relay input. To simplifythe problem, we further decouple W as W = w t w † r , where w r is the receive beamforming vector and w t is the transmit beamforming vector both at the relay node. By fixing w r (or w t ), w t (or w r ) can bejointly optimized with t at S and r at R to realize the overall zero loopback self-interference at R andmaximize the e2e SNR.
2) Antenna Selection:
AS schemes can be considered as a special case of our system model withone element of r and t being unity and the rest zero. Hence, only one element of W is non-zeroand this entry depends on the selected transmit and receive antennas at the relay. Specifically, in thecase of AS, we assume that at each terminal, a single antenna is selected either to maximize the e2eSINR at D (with optimal AS) or to maximize SNRs/SINRs associated with S − R , R − R and R − D links (with sub-optimal AS). The search complexity of the optimal scheme is high especially with alarge number of antennas at each terminal, therefore, the sub-optimal schemes provide a better trade-offbetween implementation complexity and e2e system performance. Moreover, if S transmits with a power P S , we model the transmit power at R , as P αS where < α ≤ . The parameter α provides a dB scalingof the relay transmit power which is necessary in the presence of residual LI. Hence, α captures theeffects of power control on the achieved performance as it allows the analysis of different relative powergains between the SINR and the SNR of the S − R and R − D hops, respectively. Although the proposedZF precoding design operation is optimal with the use of full power at the relay ( α = 1) , as we showlater (in Section V), when AS schemes are implemented, an appropriate α can protect the MIMO relaysystem from error floor effects and thus a zeroth-order diversity.III. J OINT P RECODING /D ECODING D ESIGN
Based on the above system model, the equivalent S − R and R − D channels become h SR , H SR t , and h RD = H † RD r . (1) April 20, 2018 DRAFT
We first assume t , r are fixed and study their optimal design together with w r and w † t according todifferent criteria.By assuming a processing delay at R , given by τ [5], [6], the input and the output at R can be writtenas r [ n ] = h SR x S [ n ] + H RR x R [ n ] + n R [ n ] , (2)and x R [ n ] = Wr [ n − τ ] , (3)respectively, where x S [ n ] is the transmitted symbol at S with zero mean, average power P S and n R isthe M R × AWGN vector with zero mean and identity covariance matrix.Using (2) and (3) recursively, the relay output can be rewritten as x R [ n ] = Wr [ n − τ ] = Wh SR x S [ n − τ ] + WH RR x R [ n − τ ] + Wn R [ n ]= W ∞ X j =0 ( H RR W ) j ( h SR x S [ n − jτ − τ ] + n R [ n − jτ ]) . (4) Note that we aim to maximize the e2e SNR, and the optimal W should possess a minimum meansquare error (MMSE) structure, which is nontrivial to solve. To simplify the signal model, and find low-complexity closed-form rather than optimal solutions, we add the additional ZF constraint that the designof W ensures no loopback self-interference for the full-duplex operation. To realize this, it is easy tocheck from (4) that the following condition is sufficient, WH RR W = . (5)As a result, (4) becomes x R [ n ] = W ( h SR x S [ n − τ ] + n R [ n ]) , (6)with the covariance matrix E [ x R x † R ] = P S Wh SR h † SR W † + WW † . (7)The relay output power is P R = Tr ( E [ x R x † R ]) = k Wh SR k P S + k W k . (8) April 20, 2018 DRAFT
The received signal at D can be written as r D [ n ] = h † RD x R [ n ] + n D [ n ]= h † RD Wh SR x S [ n − τ ] + h † RD Wn R [ n ] + n D [ n ] . (9)The e2e SINR, denoted as γ , is expressed as γ = P S | h † RD Wh SR | k h † RD W k + 1 . (10) We aim to optimize the relay processing matrices W in order to maximize the e2e SINR. Mathematically,the optimization problem is formulated as max W γ ( in Eq. (11)s.t. P S k Wh SR k + k W k ≤ P R , WHW = . To further simply the problem, we assume W = w t w † r , where w r is the receive beamforming vectorand w t is the transmit beamforming vector. It is noted that W is of rank-1 and this is reasonable sincethere is only a single data stream. Then the ZF condition is simplified to w † r H RR w t = 0 . To achievethis requirement, we can design w r or w t jointly with t and r , as described below. A. Receive ZF with M R > We assume maximum ratio transmission (MRT) with w t = h RD and optimize w r based on the ZFcriterion. Consequently, problem (11) reduces to max w r P S k h RD k | w † r h SR | k h RD k k w r k + 1 (12)s.t. P S k h RD k | w † r h SR | + k h RD k k w r k ≤ P R , w † r H RR h RD = . Note that the first power constraint needs to be satisfied with equality, otherwise, k w r k can be increasedwithout violating any constraint and this leads to a higher objective value. Hence, the objective function(12) can be written as P S k h RD k | w † r h SR | k h RD k ( P R − P S k h RD k | w † r h SR | ) + 1 , April 20, 2018 DRAFT0 which is monotonically increasing in | w † r h SR | . As a result, (12) is equivalent to max w r | w † r h SR | (13)s.t. P S k h RD k | w † r h SR | + k h RD k k w r k ≤ P R , w † r H RR h RD = . Let E , I + P S h SR h † SR and E / w r = v r . With this definition, we can formulate a simple optimizationproblem for v r as follows: max v r | v † r E − / h SR | (14)s.t. k v r k ≤ P R k h RD k , v † r E − / H RR h RD = . From the ZF constraint, we know that v r lies in the null space of E − / H RR h RD . Hence, v r = Du r ,where D , I − E − / H RR h RD h † RD H † RR E − / k E − / H RR h RD k . The objective function in (14) then becomes | u † r DE − / h SR | and the optimal u r should align with DE − / h SR . Using the facts that the first power constraint shouldbe met with equality and D is idempotent, we can express the optimal solutions of (14) and (13) as v r = DE − / h SR k DE − / h SR k s P R k h RD k , and (15) w r = E − / DE − / h SR k DE − / h SR k s P R k h RD k . The objective value in (12) involves | w † r h SR | and k w r k which, from (15), are given by | w † r h SR | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h † SR E − / DE − / h SR k DE − / h SR k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P R k h RD k = P R k h RD k h † SR E − / DE − / h SR = P R k h RD k k h † SR E − / k − k h † SR E − H RR h RD k k E − / H RR h RD k ! = P R k h RD k k H RR h RD k k h SR k − | h † SR H RR h RD | k H RR h RD k + P S ( k H RR h RD k k h SR k − | h † SR H RR h RD | ) , (16) April 20, 2018 DRAFT1 and k w r k = P R k h RD k − P S | w † r h SR | (17) = P R k h RD k − P S P R k h RD k k H RR h RD k k h SR k − | h † SR H RR h RD | k H RR h RD k + P S ( k H RR h RD k k h SR k − | h † SR H RR h RD | ) (18) = P R k h RD k k H RR h RD k k H RR h RD k + P S ( k H RR h RD k k h SR k − | h † SR H RR h RD | ) . (19) Using (16) and (17) in (12), the achievable e2e SNR can be derived as γ = P S k b Dh SR k P R k h RD k P S k b Dh SR k + P R k h RD k + 1 , (20)where b D , I − H RR h RD h † RD H † RR k H RR h RD k .Next, we can address the design of t and r . Notice from (1) that t and r are embedded in k b Dh SR k and k h RD k , respectively, so we propose the following solution to separately optimize t and r : t ∗ = arg max k t k =1 k b Dh SR k = arg max k t k =1 k b DH SR t k (21) = u max ( H † SR b DH SR ) , and r ∗ = arg max k r k =1 k h RD k = arg max k r k =1 k H † RD r k (22) = u max ( H † RD H RD ) , where we have used the fact that b D is idempotent. Note that b D also depends on r via h RD , so the abovesolutions may not be optimal. Nevertheless, the choice of r ∗ in (22) uniquely maximizes k h RD k andgiven this choice of r , t ∗ in (21) uniquely maximizes k b Dh SR k . Hence, this approach is very appealingand these simple closed-form solutions facilitate both the precoder/receive vector design and performanceanalysis.Substituting t ∗ and r ∗ back into (20), the e2e SNR can be expressed as γ = P S k b DH SR k P R k H RD k P S k b DH SR k + P R k H RD k + 1 , (23)where k X k = λ max ( XX † ) . April 20, 2018 DRAFT2
B. Transmit ZF with M T > We assume that w r = h SR , i.e., the relay employs a maximal-ratio combining (MRC) receivebeamforming vector, and optimizes the transmit ZF vector w t . In this case, we can simplify problem(11) as: max w t P S | h † RD w t | k h SR k | h † RD w t | k h SR k + 1 (24)s.t. k w t k ≤ P R k h SR k P S + k h SR k , h † SR H RR w t = 0 , or equivalently using monotonicity, max w t | h † RD w t | (25)s.t. k w t k ≤ P R k h SR k P S + k h SR k , h † SR H RR w t = 0 . Following the same procedure employed to obtain (15), the solution of (25) is given by w ∗ t = s P R k h SR k P S + k h SR k Bh RD k Bh RD k , (26)where we have defined B , I − H † RR h SR h † SR H RR k h † SR H RR k . With w ∗ t , the optimized e2e SNR can be expressed as γ = P S k h SR k P R k Bh RD k P S k h SR k + P R k Bh RD k + 1 . (27)Similar to the receive ZF scheme, we propose the following solutions for t and r (which may not beoptimal) t ∗ = arg max k t k =1 k h SR k = arg max k t k =1 k H SR t k (28) = u max ( H † SR H SR ) , and r ∗ = arg max k r k =1 k Bh RD k = arg max k r k =1 k BH RD r k (29) = u max ( H † RD BH RD ) , April 20, 2018 DRAFT3 respectively. Finally, substituting t ∗ and r ∗ into (27), the e2e SNR can be expressed as γ = P S k H SR k P R k BH † RD k P S k H SR k + P R k BH † RD k + 1 . (30)IV. A NTENNA S ELECTION
This section deals with the problem of AS for the full-duplex MIMO relay channel considered. ASis proposed as an alternative to the e2e optimization and is particularly relevant to systems with strictercomputational/energy constraints. Full-duplex relay AS introduces new design challenges due to thepresence of LI and differs from the existing body of AS literature in several ways. As explained below,with full-duplex operation, several AS choices that provide different performance/complexity tradeoffexist while a straightforward AS strategy (see for e.g. [21]) can be used to maximize the performance inhalf-duplex AS systems. Moreover, power allocation is an important issue with different full duplex ASschemes while half-duplex AS schemes can use full power at the relay (in the absence of LI).The AF process at R employs the conventional amplification factor [5, Eq. (4)] which guarantees thestability of the relay and prevents oscillation. This particular choice of amplification process is also simpleto use since R can adaptively adjust its transmit power to a constant level. In this case, the instantaneouse2e SINR is expressed as [4], [5] γ i,j,k,l = γ i,jSR γ i,lRR +1 γ k,lRDγ i,jSR γ i,lRR +1 + γ k,lRD + 1 , (31)where γ i,jSR = P S | h i,jSR | , and γ k,lRD = P αS | h k,lRD | are the instantaneous SNRs of the S − R and the R − D links while γ i,lRR = P αS | h i,lRR | is the instantaneous interference-to-noise ratio (INR) of the R − R link.In order to facilitate the analysis of the outage probability in Section V-B, we also restate the averageSNRs of the S − R and the R − D links as ¯ γ SR , P S c SR and ¯ γ RD , P αS c RD , respectively. Moreover, ¯ γ RR , P αS c RR is the average INR of the R − R link. A. Optimal Antenna Selection
Denote the selected receive and transmit antenna indexes at R and S , and the receive and transmitantenna indexes at D and R are by I, J, K, L , respectively. The optimal AS (OP AS) scheme can beexpressed as { I, J, K, L } = argmax ≤ i ≤ M R , ≤ j ≤ N T ≤ k ≤ N R , ≤ l ≤ M T (cid:16) γ i,j,k,l (cid:17) . (32) April 20, 2018 DRAFT4
The OP AS scheme maximizes the e2e SINR, however it has a high computation and implementationcomplexity. In a centralized architecture, a central unit requires the knowledge of all links ( S − R , R − R and R − D ) in order to decide on the selected antennas. B. max - max Antenna Selection
The max − max AS (MM AS) scheme selects the best S − R and R − D links without consideringthe LI and can be expressed as { I, J } = argmax ≤ i ≤ M R , ≤ j ≤ N T (cid:16) γ i,jSR (cid:17) , { K, L } = argmax ≤ k ≤ N R , ≤ l ≤ M T (cid:16) γ k,lRD (cid:17) . (33)Note that the MM AS scheme, which is SNR optimal in conventional half-duplex relaying [21], becomesstrictly sub-optimal in full-duplex relaying since it does not take into account the effect of LI. However,the MM AS scheme can be easily implemented by estimating the S − R channels at R and using channelfeedback (on the R − D link) from D to R , related to the selected antenna index K . C. Partial Antenna Selection
The partial AS (PR AS) scheme simplifies the selection problem by decoupling the two relaying hopsaccording to the following rule { I, J, L } = argmax ≤ i ≤ M R , ≤ j ≤ N T , ≤ l ≤ M T γ i,jSR γ i,lRR + 1 ! , { K } = argmax ≤ k ≤ N R (cid:16) γ k,LRD (cid:17) . (34)The PR AS scheme provides a good performance/implementation complexity trade-off since it reducesthe searching set of the optimal solution while it also takes into account the LI. It is worth noting thatchannel feedback from D to R is not required since the relay transmit antenna is selected independentlyof the second hop. D. Loop Interference Antenna Selection
The loop interference AS (LI AS) scheme selects the receive/transmit antennas in order to minimizethe effects of LI according to { I, L } = argmin ≤ i ≤ M R , ≤ l ≤ M T (cid:16) γ i,lRR (cid:17) , { J } = argmax ≤ j ≤ N T (cid:16) γ I,jSR (cid:17) , { K } = argmax ≤ k ≤ N R (cid:16) γ k,LRD (cid:17) . (35) The name for this AS scheme was adopted in the same spirit where selection schemes based on the first-hop CSI are identifiedas partial relay selection in the literature [28].
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This scheme is analogous to the LI suppression policies with relay precoders proposed in [6], [14]. TheLI AS aims to minimize the deleterious effects of LI, while some improvement in the S − R , R − D channels is also extracted by selecting antennas at S and D .V. O UTAGE P ROBABILITY A NALYSIS
In this section, we study the outage probability of the precoding/decoding designs as well as theAS schemes presented in Sections III and IV, respectively. We derive exact expressions for the outageprobability and based on these results, the asymptotic behavior is also studied to reveal important insightssuch as the diversity order.
A. Joint Precoding/Decoding Designs
The rate outage probability, P out , is defined as the probability that the instantaneous mutual information, I = log (1 + γ ) , falls below a target rate of R bits per channel use (BPCU). Hence, P out = Pr (log (1 + γ ) ≤ R ) = F γ ( γ T ) , (36)where γ T = 2 R − and F γ ( · ) is the cumulative distribution function (cdf) of the e2e SNR.
1) Receive ZF:
From (23), we can now derive the outage probability of the system. To this end, wefirst note that k b DH SR k = λ max (cid:16) H † SR b D † b DH SR (cid:17) can be written as k b DH SR k = λ max H † SR I − H RR h RD h † RD H † RR k H RR h RD k ! H SR ! = λ max (cid:16) H † SR Φ † ( I − diag (1 , , . . . , ΦH SR (cid:17) = λ max (cid:16) b H † SR diag (0 , , . . . , b H SR (cid:17) = λ max (cid:16) ˘ H † SR ˘ H SR (cid:17) , (37) where Φ is a unitary matrix, b H SR = ΦH SR and ˘ H SR is a ( M R − × N T matrix. In (37), the firstequality follows from the fact that b D = b D † b D . The second equality is due to the eigen decomposition( H RR h RD k H RR h RD k is a M R × normalized column vector and has rank 1). Hence, k b DH SR k is the maximumeigenvalue of a Wishart matrix (cid:16) ˘ H † SR ˘ H SR (cid:17) with dimensions ( M R − × N T .We now derive the exact outage probability with receive ZF using the result for k b DH SR k in (23) inconjunction with k H RD k . The required cdf of the e2e SNR can be derived by adopting a similar approachas in [33, Appendix I]. Specifically, we can express the cdf of γ as F γ ( γ T ) = Pr (cid:16) γ SR γ RD γ SR + γ RD +1 < γ T (cid:17) =1 − R ∞ ¯ F γ RD (cid:16) ( γ T + y +1) γ T y (cid:17) f γ SR ( γ T + y ) dy , where ¯ F γ RD ( x ) is the complementary cdf of γ RD , and f γ SR ( x ) April 20, 2018 DRAFT6 is the probability density function (pdf) of γ SR , with γ SR = P S k b DH SR k and γ RD = P R k H SR k . Byusing [32, Eq. (23)], we can obtain the pdf of γ SR and the cdf of γ RD as f γ SR ( x ) = min( N T ,M R − X a =1 ( N T + M R − a − a X b = | N T − M R +1 | a b +1 d ( a, b )( b )!¯ γ b +1 SR x b e − ax ¯ γSR , and F γ RD ( x ) = 1 − min( M T ,N R ) X k =1 ( M T + N R ) k − k X l = | M T − N R | l X m =0 k m d ( k, l )( m )!¯ γ mRD x m e − kx ¯ γRD , respectively, where the average SNR of the S − R and R − D links are given by ¯ γ SR = P S c SR and ¯ γ RD = P R c RD . The coefficients, d l ( i, j ) , l = 1 , are given in [32] for some system configurations andcan be efficiently computed using the algorithm in [34]. We now substitute the above pdf and cdf intothe integral representation of F γ ( γ T ) and solve it in closed-form using [35, Eq. (3.471.9)] to yield F γ ( γ T ) = 1 − s X a =1 ( N T + M R − a − a X b = | N T − M R +1 | s X k =1 ( M T + N R ) k − k X l = | M T − N R | l X m =0 (38) × m X u =0 b X v =0 (cid:0) mu (cid:1)(cid:0) bv (cid:1) d ( a, b ) d ( k, l ) k u + v + m +12 γ m +2 b + u − v +12 T (1 + γ T ) m − u + v +12 b ! m ! a u + v − m − b − ¯ γ b − u − v + m +12 SR ¯ γ u + v + m +12 RD × e − (cid:16) a ¯ γSR + k ¯ γRD (cid:17) γ T K u + v − m +1 s ak (1 + γ T ) γ T ¯ γ SR ¯ γ RD ! , where s = min ( N T , M R − and s = min ( M T , N R ) .In order to further obtain insights, such as diversity order, we now present a simplified asymptoticoutage probability. Specifically, we adopt the upper bound, γ ≤ min ( γ SR , γ RD ) , to γ . This bound istight for medium-to-high SNR values and in [36] it was shown that it is also asymptotically-exact in thehigh SNR regime [36]. Therefore, using simple order statistics we can express the asymptotic cdf of thee2e SNR as F ∞ γ ( x ) = F γ ∞ SR ( x ) + F γ ∞ RD ( x ) − F γ ∞ SR ( x ) F γ ∞ RD ( x ) .It can be easily shown that at high SNRs, F ∞ γ ( x ) can be approximated by a single term polynomialapproximation. To see this, we first need polynomial approximations for γ SR and γ RD . These results canbe borrowed from [37, Eq. (7)] and with the aid of F ∞ γ ( x ) we can show that P ∞ out = Q s − k =0 k ! Q s − k =0 ( t + k )! (cid:16) γ T ¯ γ SR (cid:17) N T ( M R − N T ( M R − < M T N R , Q s − k =0 k ! Q s − k =0 ( t + k )! (cid:16) γ T ¯ γ SR (cid:17) N E + Q s − k =0 k ! Q s − k =0 ( t + k )! (cid:16) γ T ¯ γ RD (cid:17) N E N T ( M R −
1) = M T N R = N E , Q s − k =0 k ! Q s − k =0 ( t + k )! (cid:16) γ T ¯ γ RD (cid:17) M T N R N T ( M R − > M T N R , (39) April 20, 2018 DRAFT7 where t = max ( N T , M R − and t = max ( M T , N R ) . By inspecting (39), we see that our full-duplexreceive ZF design achieves a diversity order of min ( N T ( M R − , M T N R ) .
2) Transmit ZF:
Using an equivalent approach to that used for the receive ZF scheme and omittingdetails for conciseness, the exact outage probability can be expressed as F γ ( γ T ) = 1 − s X a =1 ( N T + M R ) a − a X b = | N T − M R | s X k =1 ( M T + N R − k − k X l = | M T − N R − | l X m =0 (40) × m X u =0 b X v =0 (cid:0) mu (cid:1)(cid:0) bv (cid:1) d ( a, b ) d ( k, l ) k u + v + m +12 γ m +2 b + u − v +12 T (1 + γ T ) m − u + v +12 b ! m ! a u + v − m − b − ¯ γ b − u − v + m +12 SR ¯ γ u + v + m +12 RD × e − (cid:16) a ¯ γSR + k ¯ γRD (cid:17) γ T K u + v − m +1 s ak (1 + γ T ) γ T ¯ γ SR ¯ γ RD ! , where s = min ( N T , M R ) and s = min ( M T − , N R ) .Furthermore, we can express the asymptotic outage probability of transmit ZF as P ∞ out = Q s − k =0 k ! Q s − k =0 ( t + k )! (cid:16) γ T ¯ γ SR (cid:17) N T M R N T M R < ( M T − N R , Q s − k =0 k ! Q s − k =0 ( t + k )! (cid:16) γ T ¯ γ SR (cid:17) M E + Q s − k =0 k ! Q s − k =0 ( t + k )! (cid:16) γ T ¯ γ RD (cid:17) M E N T M R = ( M T − N R = M E , Q s − k =0 k ! Q s − k =0 ( t + k )! (cid:16) γ T ¯ γ RD (cid:17) ( M T − N R N T M R > ( M T − N R , (41)where t = max ( N T , M R ) and t = max ( M T − , N R ) . From Eq. (41) we see that with transmit ZF,a diversity order of min ( N T M R , ( M T − N R ) can be achieved.On the other hand, half-duplex MIMO hop-by-hop (MRT/MRC) beamforming exhibits a diversity orderof min ( N T M R , M T N R ) . As a result, although half-duplex hop-by-hop beamforming delivers a superiordiversity performance in general, in certain antenna configurations, half-duplex hop-by-hop beamformingand full-duplex ZF designs offer the same diversity. B. Antenna Selection
In this subsection, we investigate the outage probability of the proposed full-duplex based AS schemes.We derive exact as well as approximate outage expressions when P S → ∞ for comparison of the proposedAS schemes. By considering the definition of the outage probability, we can write P ⋆ = Pr log γ I,JSR γ I,LRR +1 γ K,LRDγ
I,JSR γ I,LRR +1 + γ K,LRD + 1 < R . (42) In the following subsections, the statistical distributions of γ I,JSR , γ I,LRR and γ K,LRD may differ depending on the AS scheme.Any remark concerning the distributions of these RVs is strictly limited to the particular AS scheme.
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For “optimal”, “ max − max ”, “partial” and “loop interference” AS schemes, the subscript ⋆ in (42) refersto OP, MM, PR and LI, respectively.
1) Optimal Antenna Selection:
Let γ OP denote the e2e SINR at D for the OP AS scheme. The outageprobability of the OP AS scheme can be written as P OP = F γ OP ( γ T ) , (43)where F X ( · ) denotes the cdf of the random variable (RV), X . Obtaining an analytical expression for P OP appears to be a cumbersome problem due to the dependencies between the SINR variables beingmaximized. Therefore, we have performed simulations to evaluate the outage performance of the OP ASscheme in Section V. Further, under some special antenna configurations, for example with M R = M T = 1 ,the OP AS scheme is equivalent to the MM AS scheme for which an analytical expression is presentedbelow.We now state the asymptotic behavior of the OP AS scheme in Proposition 1 . Proposition 1:
The outage probability of the OP AS scheme as P S → ∞ can be approximated by P OP ≈ C (cid:18) ¯ γ RR γ T ¯ γ SR (cid:19) N T M R + C (cid:18) γ T ¯ γ RD (cid:19) M T N R , (44)where C > and C > are two positive constants. Proof:
We first lower bound γ OP by γ MM , where γ MM is the SINR of the suboptimal MM AS scheme.In the following subsection, we show that as P S tends to infinity, the corresponding upper bound, P OP ≤ P MM , can be approximated by P MM ≈ ( N T M R )! (cid:16) ¯ γ RR γ T ¯ γ SR (cid:17) N T M R + (cid:16) γ T ¯ γ RD (cid:17) M T N R . Next we upper bound γ OP by γ UB , defined as γ UB , X Y X X Y + X +1 where X and X are the maximum of N T M R and M T N R exponential RVs with parameters, ¯ γ SR and ¯ γ RD , respectively, while Y is a RV chosen as the minimum of M T M R exponential RVs with parameter ¯ γ RR . As P S tends to infinity, we can show that the correspondinglower bound, P OP > P LB can be approximated by P LB ≈ ( N T M R )!( M T M R ) NT MR (cid:16) ¯ γ RR γ T ¯ γ SR (cid:17) N T M R + (cid:16) γ T ¯ γ RD (cid:17) M T N R .Since the upper and lower bounds of P OP have the same diversity order, (44) follows and the proof iscompleted.Using the above asymptotic result, we now derive the optimal α to yield the power allocation solution atthe relay. Following the respective definitions and expressing ¯ γ SR , ¯ γ RR and ¯ γ RD explicitly in terms of P S ,we see that the first term in (44) decays as P − (1 − α ) N T M R S while the second term decays as P − αM T N R S . The SINR upper bound, γ UB corresponds to a “virtual” system in which transmit/receive AS is decoupled to consider thebest S − R and R − D links and the weakest LI ( R − R ) link, respectively, since such a strategy will maximize the e2e SINRin (31). However, clearly such a AS scheme is not possible in our system, since selecting a particular transmit/receive antennapair at R will automatically fix the LI link, i.e., AS for the links can not be performed independently. April 20, 2018 DRAFT9
Therefore, depending on the value of α , the first or the second term in (44) becomes dominant anddetermines the total asymptotic outage probability. Outage minimization from a diversity perspectiveoccurs when (1 − α ) N T M R = αM T N R and we have α OP opt = N T M R N T M R + M T N R , (45)with P α OS opt S as the optimal power allocation solution at the relay. Moreover, the highest diversity order, d max,OP , achieved with the OP AS scheme is given by d max,OP = 1( M T N R ) − + ( N T M R ) − . (46) max − max Antenna Selection:
With this scheme, γ I,JSR is simply the largest of N T M R exponentialRVs with parameter ¯ γ SR , γ K,LRD is simply the largest of M T N R exponential RVs with parameter ¯ γ RD , and,since the R − R link is ignored, γ I,LR,R is an exponential RV with parameter ¯ γ RR . The outage probabilityof MM AS can be written as P MM = 1 − Z ∞ F X (cid:18) ( y + γ T + 1) γ T y (cid:19) f Y ( y + γ T ) dy, (47)where X = γ I,JSR γ I,LRR +1 , Y = γ K,LRD and F X ( · ) denotes the complementary cdf of the RV, X . Clearly, in orderto evaluate (47) we first need to find the cdf and the pdf of X and Y , respectively. The cdf of X can beexpressed as F X ( x ) = 1¯ γ RR Z ∞ F γ I,JSR (( y + 1) x ) e − y ¯ γRR dy (48) = 1 − N T M R N T M R − X p =0 ( − p (cid:0) N T M R − p (cid:1) e − ( p +1) x ¯ γSR ( p + 1) (cid:16) ( p +1)¯ γ RR x ¯ γ SR (cid:17) . The second equality in (48) follows since the binomial expansion F γ I,JSR ( x ) = (cid:16) − e − x ¯ γSR (cid:17) N T M R canbe written as F γ I,JSR ( x ) = 1 − N T M R P N T M R − p =0 ( − p ( NT MR − p ) p +1 e − ( p +1) x ¯ γSR . We can now write (47) as P MM = 1 − N T M R M T N R N T M R − X p =0 ( − p (cid:0) N T M R − p (cid:1) p + 1 M T N R − X q =0 ( − q (cid:0) M T N R − q (cid:1) ¯ γ RD (49) × Z ∞ e − ( p +1)( y + γT +1) γT ¯ γSRy − ( q +1)( y + γT )¯ γRD (cid:16) ( p +1)( y + γ T +1)¯ γ RR γ T ¯ γ SR y (cid:17) dy. Eq. (49) does not admit a closed-form solution. However, it can be easily evaluated numerically usingstandard mathematical software tools.
April 20, 2018 DRAFT0
In order to derive an accurate closed-form outage expression applicable in the asymptotic regime ( P S → ∞ ) , we consider P MM ≥ Pr ( min γ I,JSR γ I,KRR + 1 , γ
K,LRD ! < γ T ) (50) → − N T M R M T N R N T M R − X p =0 ( − p (cid:0) N T M R − p (cid:1) ( p + 1) (cid:16) ( p +1)¯ γ RR γ T ¯ γ SR (cid:17) M T N R − X q =0 ( − q (cid:0) M T N R − q (cid:1) q + 1 e − (cid:16) p +1¯ γSR + q +1¯ γRD (cid:17) γ T , where F min( X,Y ) ( · ) = 1 − (1 − F X ( · )) (1 − F Y ( · )) has been used. Ignoring the product term F X ( · ) F Y ( · ) as it gives higher order terms, we observe that the asymptotic behavior of P MM can be further approximatedas P MM ≈ F X ( γ T ) + F Y ( γ T ) . Consider F X ( γ T ) as P S → ∞ ; for small x = γ T ¯ γ SR we can simplify F X ( x ) = e − x ¯ γSR ¯ γ RR R ∞ (cid:16) − e − x ¯ γSR (cid:17) N T M R e − y ¯ γRR dy as F X ( x ) ≈ x N T M R ¯ γ RR Z ∞ y N T M R e − y ¯ γRR dy (51) = ( N T M R )! (¯ γ RR x ) N T M R . Similarly, we can show that as P S → ∞ , F Y ( γ T ) ≈ (cid:16) γ T ¯ γ RD (cid:17) M T N R . Therefore, (50) can be simplified for < α < as P MM ≈ ( N T M R )! (cid:18) ¯ γ RR γ T ¯ γ SR (cid:19) N T M R + (cid:18) γ T ¯ γ RD (cid:19) M T N R . (52)As an immediate observation, from (44) and (52) we see that the OP AS scheme and the MM AS schemeachieve the same diversity performance. As a result, we have α MM opt = α OP opt with P α MM opt S as the optimalpower allocation solution at the relay and the highest diversity order, achieved with the MM AS schemeis also d max,MM = M T N R ) − +( N T M R ) − . However, compared to the MM AS scheme, the OP AS schemehas a higher array gain as verified in Section VI.
3) Partial Antenna Selection:
The outage probability of this scheme can be evaluated from P PR = 1 − Z ∞ F X (cid:18) ( y + γ T + 1) γ T y (cid:19) f Y ( y + γ T ) dy, (53)with X = γ I,JSR γ I,LRR +1 and Y = γ K,LRD . The required distributions of X and Y are different to the previous caseof max − max AS and in order to calculate P PR we need to evaluate them. For any i -th relay receiveantenna, the ratio γ i,jSR γ i,lRR +1 is maximized when the strongest S − R channel and the weakest R − R channelfrom the i th antenna ( i = 1 , . . . , M R ) are selected. Since there are M R antennas, the cdf of X can beevaluated as F X ( x ) = (cid:0)R ∞ F A (( y + 1) x ) f B ( y ) dy (cid:1) M R , where A is a RV defined as the largest among N T exponentially distributed RVs, while B is the smallest out of M T exponentially distributed RVs. April 20, 2018 DRAFT1
Substituting the required cdf and the pdf into F X ( x ) with simplifications yields F X ( x ) = − N T N T − X p =0 ( − p (cid:0) N T − p (cid:1) e − ( p +1) x ¯ γSR ( p + 1) (cid:16) ( p +1)¯ γ RR xM T ¯ γ SR (cid:17) M R . (54)Furthermore, we notice that the RV, Y = γ K,LRD , is simply the largest among N R exponential RVs withparameter ¯ γ RD . Therefore, the pdf of Y can be written as f Y ( y ) = N R ¯ γ RD P N R − q =0 ( − q (cid:0) N R − q (cid:1) e − ( q +1) y ¯ γRD .Combining these results, the exact outage probability of the PR AS scheme can be written as P PR = 1 − N R ¯ γ RD N R − X q =0 ( − q (cid:18) N R − q (cid:19) I q , (55)where the integral I q is defined as I q = Z ∞ − − N T N T − X p =0 ( − p (cid:0) N T − p (cid:1) e − ( p +1)( y + γT +1) γT ¯ γSRy ( p + 1) (cid:16) ( p +1)¯ γ RR ( y + γ T +1) γ T M T ¯ γ SR y (cid:17) M R e − ( q +1)( y + γT )¯ γRD dy. (56)In order to derive an accurate closed-form expression for the outage probability with P S → ∞ weconsider P PR ≥ Pr ( min γ I,JSR γ I,KRR + 1 , γ
K,LRD ! < γ T ) (57) ≈ N T ! M N T T ! M R (cid:18) ¯ γ RR γ T ¯ γ SR (cid:19) N T M R + (cid:18) γ T ¯ γ RD (cid:19) N R , for < α < . We see that the first term decays as P − (1 − α ) N T M R S while the second term decays as P − αN R S . Therefore, outage minimization occurs when (1 − α ) N T M R = αN R and we have α PR opt = N T M R N T M R + N R , (58)with P α PR opt S as the optimal power allocation solution at the relay. Therefore, the highest diversity order, d max,PR , achieved with the PR AS scheme can be expressed as d max,PR = 1 N − R + ( N T M R ) − . (59)
4) Loop Interference Antenna Selection:
In the case of the LI AS scheme, the outage probability canbe evaluated from P LI = 1 − Z ∞ F X (cid:18) ( y + γ T + 1) γ T y (cid:19) f Y ( y + γ T ) dy, (60) April 20, 2018 DRAFT2 where X = γ I,JSR γ I,LRR +1 and Y = γ K,LRD . Since receive/transmit antennas at R are selected to minimize theLI, with this scheme γ I,LRR is the minimum of M R M T exponential RVs with parameter ¯ γ RR , while γ I,JSR and γ K,LRD are the largest of N T and N R exponential RVs with parameters ¯ γ SR and ¯ γ RD , respectively.Therefore, the required cdf of X can be found using F X ( x ) = 1 − N T M R M T ¯ γ RR N T − X p =0 ( − p (cid:0) N T − p (cid:1) p + 1 Z ∞ e − ( p +1)( y +1) x ¯ γSR e − MRMT y ¯ γRR dy. (61)Simplifying the integral in (61) yields F X ( x ) = 1 − N T N T − X p =0 ( − p (cid:0) N T − p (cid:1) e − ( p +1) x ¯ γSR ( p + 1) (cid:16) ( p +1)¯ γ RR xM R M T ¯ γ SR (cid:17) . (62)Now, combining the pdf of Y and (62) we can express the exact outage probability as P LI = 1 − N T N R ¯ γ RD N T − X p =0 ( − p (cid:0) N T − p (cid:1) p + 1 N R − X q =0 ( − q (cid:18) N R − q (cid:19) Z ∞ e − ( p +1)( y + γT +1) γT ¯ γSRy e − ( q +1)( y + γT )¯ γRD (cid:16) ( p +1)¯ γ RR γ T ( y + γ T +1) M R M T ¯ γ SR y (cid:17) dy. (63)We now present an asymptotic approximation for the outage probability of the LI AS scheme. The outageprobability as P S → ∞ can be approximated by P LI ≥ Pr ( min γ I,JSR γ I,KRR + 1 , γ
K,LRD ! < γ T ) (64) → − N T N R N T − X p =0 ( − p (cid:0) N T − p (cid:1) e − ( p +1) x ¯ γSR ( p + 1) (cid:16) ( p +1)¯ γ RR xM R M T ¯ γ SR (cid:17) N R − X q =0 ( − q (cid:0) N R − q (cid:1) q + 1 e − ( q +1) y ¯ γRD . Eq. (64) can be simplified as P LI ≈ N T !( M R M T ) N T (cid:18) ¯ γ RR γ T ¯ γ SR (cid:19) N T + (cid:18) γ T ¯ γ RD (cid:19) N R , (65)for < α < . We see that the first term decays as P − (1 − α ) N T S while the second term decays as P − αN R S .As for the previous AS schemes, the optimum α value can be found from (1 − α ) N T = αN R and isgiven by α LI opt = N T N T + N R , (66)to yield P α LI opt S as the optimal power allocation solution at the relay. April 20, 2018 DRAFT3
TABLE ID
IVERSITY ORDER AND COMPLEXITY FOR THE PRECODING DESIGNS AND
AS S
CHEMES . Scheme Diversity Order Complexity
Receive ZF min ( N T ( M R − , M T N R ) high Transmit ZF min ( N T M R , ( M T − N R ) high OP AS M T N R ) − +( N T M R ) − N T M R M T N R MM AS M T N R ) − +( N T M R ) − N T M R + M T N R PR AS N − R +( N T M R ) − N T M R M T + N R LI AS N − T + N − R N T + M R M T + N R Further, the highest diversity order, d max,LI , achieved with the LI AS scheme can be expressed as d max,LI = 1 N − T + N − R . (67) C. Comparisons of the Schemes
Table 1 summarizes the diversity order achieved from the investigated schemes as well as theirassociated complexity. The first main observation is that the precoding designs outperform the ASschemes in terms of diversity gain. The utilization of all antenna elements mitigates the LI effectsand ensures a diversity order that is dominated by the weakest relaying branch. We note that due to thereceived/transmitted ZF operation, one antenna element is reserved for spatial cancellation at the relay’sinput/output, respectively. On the other hand, OP AS and MM AS schemes achieve similar diversityperformance and significantly outperform the PR AS and LI AS schemes. Another interesting observationis that the diversity order of the PR AS scheme does not depend on M T . Similarly, in the case of LI ASthe diversity order is independent of the number of relay antennas. By comparing the results in Column2 of Table 1, it is easy to see that with M R , M T > d precoding > d OP = d MM > d PR > d LI . (68)As for the complexity, the precoding schemes utilize all the antennas and require a radio frequency chainfor each antenna element. In addition, the computation of the beamforming vectors involves demandingmathematical operations such as matrix multiplication, matrix inversion and eigen-decomposition givinga general complexity of O ( n ) . Therefore, although ZF precoding designs achieve higher diversity April 20, 2018 DRAFT4 performance, they are characterized by a higher complexity in comparison to AS schemes. The proposedAS schemes also correspond to different complexities and are appropriate for networks with differentcomputational capabilities. In order to provide a simple comparison of their complexity, we use as ametric the number of channels that should be examined in order to apply each AS scheme. It is worthnoting that each channel in most of the cases is associated with a feedback channel (and a trainingprocess) in a centralized implementation. The OP AS examines all the possible combinations and thereforecorresponds to a high complexity equal to N T M R M T N R channels. The MM AS scheme decouplesthe AS selection into two independent groups and therefore has a complexity of ( N T M R + M T N R ) channels. The PR AS scheme decouples the R − D link in the selection process and gives a complexityof ( N T M R M T + N R ) channels. Finally, the LI AS scheme is based on the LI channel and thus has acomplexity of ( N T + M R M T + N R ) channels.VI. N UMERICAL R ESULTS
In this section, we give numerical examples for the outage probability of the proposed precoding andAS schemes. The simulation set-up follows the system model of Section II with R = 2 BPCU, and c SR = c RD = 1 . Although we have considered a symmetric setup, i.e., c SR = c RD , the main observationsshown for AS schemes in Figs. 4-6 are also valid for asymmetric setups, where c SR = c RD . A. Joint Precoding/Decoding Designs
Fig. 2 shows the results for the receive ZF based precoding design with different antenna configu-rations. The specific values of N T , M R , M T , N R for each antenna configuration are shown inside thefigure labels as ( N T , M R , M T , N R ) respectively. These results reveal several interesting observationsuseful for system designers. The achievable diversity orders of the considered configurations, given by min ( N T ( M R − , M T N R ) , are , and , respectively. Therefore, although only one receive antenna isused at D , the performance can be improved by selecting appropriate design parameters at S and R . Thisattribute of the system is useful under different conditions; e.g., when fixed infrastructure based relays areemployed, they can be equipped with many antennas while user terminals that act as relays have spaceconstraints, and here the source can be equipped with many antennas. We also observe that although (2 , , , and (2 , , , enjoys a diversity order of two, the latter has a superior performance as a resultof higher array gain. The same observation can be seen when (3 , , , and (2 , , , are compared.In the first case, additional performance gain is obtained via increasing M R (also (2 , , , has onemore total number of antennas compared to (2 , , , ). However, in the second case, while (3 , , , and (2 , , , have the same number of total antennas, swapping N S with M R improves the outage April 20, 2018 DRAFT5 −6 −5 −4 −3 −2 −1 Average per hop SNR in dB O u t age P r obab ili t y SimulationTheoryAsymptotic (2,2,1,1)(2,3,2,1)(2,2,2,1)(3,2,3,1) (2,3,3,1) (2,2,1,1) half−duplex(2,3,3,1) half−duplex
Fig. 2. Outage probability versus per hop average SNR for the receive ZF based precoding design with different antennaconfigurations. −6 −5 −4 −3 −2 −1 Average per hop SNR in dB O u t age P r obab ili t y SimulationTheoryAsymptotic
Transmit (2,1,2,1)Receive (2,3,2,3) Transmit (2,3,2,3)Transmit (2,2,2,2)
Fig. 3. Outage probability versus per hop average SNR of precoding designs with different antenna configurations. probability. For comparison, we have included results for half-duplex hop-by-hop beamforming [33] withtwo configurations, namely (2 , , , and (2 , , , and γ T = 2 R − . These results can be comparedfor example with (2 , , , full-duplex operation and refer to the so called “RF chain preserved” conditionand the “number of antenna preserved” (at the relay) condition. April 20, 2018 DRAFT6 −6 −5 −4 −3 −2 −1 P S in dB O u t age P r obab ili t y LI AShalf duplex (n v = 2)OP ASPR AShalf duplex (n v = 4) MM AS Fig. 4. Outage probability versus P S ; c RR = 0 . and α = 1 . The results for OP, MM, PR and LI AS schemes are computedvia simulations and (49), (55), (63) respectively. We show results for transmit ZF based precoding design with different antenna configurations in Fig. 3.The achievable diversity orders of the considered configurations, given by min ( N T M R , ( M T − N R ) ,are again , and , respectively. We also compare the performance of the (2 , , , configurationunder receive and transmit ZF designs, and the achievable diversity order of the former design given by min ( N T ( M R − , M T N R ) is four. Interestingly, receive ZF design exhibits a superior performance totransmit ZF since the former enjoys fourth order diversity order while the latter only has a diversity orderof three. Clearly, this observation demonstrates that while under some configurations ( M T = 1) or ( M R =1) only one form (receive or transmit) of precoding design can be deployed, in other configurations, whenboth designs can be applied, the system designer has to carefully decide on the configuration as well asthe precoding design. B. Antenna Selection
In Figs. 4-6, we have set N T = M R = M T = N R = 2 . Fig. 4 shows the outage probability as afunction of P S for the considered AS schemes. No power control at R is adopted and thus we adopt α = 1 .Clearly, we see that all full-duplex schemes suffer from a zero-diversity order. Among the full-duplexAS schemes, the OP AS scheme provides the best performance. The PR AS scheme exhibits the nextbest performance and converges to the same error floor as the OP AS scheme. With low P S , the MM ASperforms better than both PR AS and LI AS schemes. Furthermore, for comparison with full-duplex, we April 20, 2018 DRAFT7 −6 −5 −4 −3 −2 −1 P S in dB O u t age P r obab ili t y OP AS (c RR = 0.1)OP AS (c RR = 0.5)MM AS (c RR = 0.1)MM AS (c RR = 0.5)PR AS (c RR = 0.1)LI AS (c RR = 0.1)Asymptotic Fig. 5. Outage probability versus P S with optimum α . The results for OP AS scheme are computed from simulations whileasymptotic results for MM, PR, and LI AS schemes are due to (52), (57), (65) respectively. have also plotted results for half-duplex operation with two cases; namely, the total number of antennasat the relay ( n V ) is and , respectively. With half-duplex transmission, the AS principle is simple; i.e.,antennas are selected at each node to maximize the SNRs of the S − R and R − D links, respectively.The half-duplex results were plotted using [21, Eq. (9)] with γ T = 2 R − due to the two time slotoperation. The full-duplex AS schemes shows a favorable outage performance at a low-to-medium rangeof P S , while the superiority of half-duplex transmission at high P S is clearly evident since it avoids LIand enjoys the benefits of diversity.Fig. 5 shows the outage probability of the AS schemes with optimal α . In contrast to the results inFig. 4, where outage probability exhibits a saturated behavior at high P S (zero diversity), all AS schemesare now able to provide some diversity and outage decays as P S increases. For the considered systemset up, α OS opt = 0 . , α MM opt = 0 . , α PR opt = 0 . and α LI opt = 0 . , and the achieved diversity orders of theOP, MM, PR and LI AS schemes are respectively, , , . and . Moreover, as expected, the OP ASscheme is able to provide the best performance among all the considered AS schemes in the work. When c RR is high (0 . , a performance gap between OP AS and MM AS is observed (although both OP ASand MM AS provides the same diversity, the former has a higher array gain). However, we see that theperformance difference between MM AS and OP AS schemes are almost negligible at c RR = 0 . . Theusefulness of our asymptotic results can also be appreciated from Fig. 5. With increasing P S , we see thatthe asymptotic plots match the exact results very well. April 20, 2018 DRAFT8 −4 −3 −2 −1 P S in dB O u t age P r obab ili t y α = 0.5 α = 0.3 α = 0.667 α = 0.99 α = 0.9 Fig. 6. Outage probability versus P S for the PR AS scheme and different α ; c RR = 0 . . In Fig. 6, the outage behavior of the PR AS scheme with several values of α is illustrated. For α values close to one, the outage begins to suffer from low diversity (e.g., the curve corresponding to α = 0 . almost converge to an error floor and exhibit a near zero diversity behavior). Clearly, the valueof α PR opt = 0 . yields the best performance in the asymptotic regime. Interestingly, for P S < dB, α = 0 . and . are able to provide a better performance than the optimal case before they begin toexperience the decremental effects of low diversity. Therefore, depending on the operating region, anappropriate value for α can be selected. In the cases of OP AS, MM AS and LI AS, similar outagebehavior with different α values can be observed as well.VII. C ONCLUSION
In this paper, we considered full-duplex MIMO relaying with multi-antenna source and destina-tion nodes. We introduced joint precoding/decoding designs which incorporate rank-1 zero-forcing self-interference suppression at the relay node. Our analysis delivered closed-form results which were furtheranalyzed to reveal several interesting observations. Exact as well as asymptotic expressions for the outageprobability were derived to explicitly reveal insights such as the achievable diversity order and the arraygain. These results were also verified from simulations to confirm their correctness. The outage probabilityis influenced by the number of antennas deployed at each node as well as the adopted precoding (receiveZF or transmit ZF) design. In order to further reduce system complexity, we also presented several ASschemes. The investigated AS schemes have been analyzed in terms of the outage probability and exact
April 20, 2018 DRAFT9 expressions as well as asymptotic approximations have been derived. A simple power allocation schemeat the relay was proposed to overcome the zero-diversity limitation. A single parameter in the powerallocation scheme can be set to obtain the desired outage performance while optimum values of thisparameter were presented for diversity maximization of the investigated AS schemes.R
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