Wireless Information and Power Transfer in Relay Systems with Multiple Antennas and Interference

0.43 MB·English

by  Guangxu Zhu

1
Wireless Information and Power Transfer in Relay
Systems with Multiple Antennas and Interference
Guangxu Zhu, Student Member, IEEE, Caijun Zhong, Senior Member, IEEE, Himal A. Suraweera, Member, IEEE,
George K. Karagiannidis, Fellow, IEEE, Zhaoyang Zhang, Member, IEEE and Theodoros A. Tsiftsis, Senior
Member, IEEE
5 Abstract—In thispaper, an energy harvesting dual-hoprelay- energy transfer technique, first demonstratedby Nikola Tesla,
1 ing system without/with the presence of co-channel interference has rekindled its interest in the context of energy harvesting
0 (CCI) is investigated. Specifically, the energy constrained mutli- communication systems where radio-frequency (RF) signals
2 antenna relay node is powered by either the information signal
are used as an energy source [4–8]. Since RF signals can be
ofthesourceorviathesignalreceivingfromboththesourceand
n interferer.Inparticular,wefirststudytheoutageprobabilityand under control, it is much more reliable than external natural
a ergodic capacity of an interferencefree system, and then extend resources, hence, wireless energy harvestingusing RF signals
J
theanalysistoaninterferingenvironment.Toexploitthebenefit isapromisingtechniquetopowercommunicationdevices[9].
2 of multiple antennas, three different linear processing schemes
Since RF signals are capable of carrying both the infor-
2 are investigated, namely, 1) Maximum ratio combining/maximal
mation and energy, a new research area, namely simultane-
ratiotransmission(MRC/MRT),2)Zero-forcing/MRT(ZF/MRT)

] and 3) Minimum mean-square error/MRT (MMSE/MRT). For ous wireless information and power transfer (SWIPT), has
T
all schemes, both the systems outage probability and ergodic recentlyemerged.Theseminalworks[6,7]havecharacterized
I capacity are studied, and the achievable diversity order is the fundamental tradeoff between the harvested energy and
.
s also presented. In addition, the optimal power splitting ratio
informationcapacity. Nevertheless, it was assumed in [6] that
c minimizing the outage probability is characterized. Our results
[ the receiver can decode the information and harvest energy
show that the implementation of multiple antennas increases

the energy harvesting capability, hence, significantly improves from the same signal simultaneously, which is unfortunately
1
the systems performance. Moreover, it is demonstrated that the impossible due to practical circuit limitations. To this end,
v
6 CCI could be potentially exploited to substantially boost the the work in [10] proposed two practical receiver architec-
performance,whilethechoiceofalinearprocessingschemeplays
7 tures, namely, “time-switching”, where the receiver switches
a critical role in determininghow much gain could beextracted
3 between decoding information and harvesting energy, and
from the CCI.
5
“power-splitting”,wherethereceiversplitsthesignalintotwo
0 Index Terms—Dual-hop relay channel, wireless power trans-
streams,oneforinformationdecodingandtheotherforenergy
1. fer, co-channel interference, linear multiple-antenna processing, harvesting.Sincethen,anumberofworkshaveappearedinthe
0 performance analysis.
literatureinvestigatingdifferentaspectsofsimultaneousinfor-
5
mation and energy transfer with practical receivers [11,12].
1
: I. INTRODUCTION Specifically, in [11], an opportunistic RF energy harvesting
v
schemeforsingle-input-single-outputsystemswithco-channel
i Energy harvesting technique, as an emerging solution for
X interference (CCI) was investigated, where it was shown that
prolonging the lifetime of the energy constrained wireless
r devices, has gained significant interests in recent years. The the CCI can be identified as a potential energy source. Later
a
on, an improved receiver, i.e., the dynamic power splitting
conventionalenergyharvestingtechniquesrelyontheexternal
receiver was studied in [12]. The extension of [10] to the
natural resources, such as solar power, wind energy or ther-
scenario with imperfect channel state information (CSI) at
moelectriceffects[1–3].However,dueto the randomnessand
the transmitter was studied in [13]. For multiple-inputsingle-
intermittent property of external natural resources, communi-
output (MISO) channels, the optimal beamforming designs
cation systems employing the conventional energy harvesting
forSWIPT systemswith/withoutsecrecyconstrainthavebeen
technique can not guarantee the delivery of reliable and
investigated in [14,15], and the optimal transmission strat-
uninterrupted communication services. Recently, the wireless
egy maximizing the system throughput of MISO interference
G. Zhu, C. Zhong and Z. Zhang are with the Institute of Information channelhasbeenstudiedin[16].Moreover,theapplicationof
andCommunicationEngineering,ZhejiangUniversity,China.(email:caijun- RFenergytransfertechniqueincognitiveradionetworkswith
[e-mail]).
multipleantennasat the secondarytransmitterwas considered
H. A. Suraweera is with the Department of Electrical & Electronic
Engineering, University ofPeradeniya, Peradeniya 20400, SriLanka (email: in[17].Finally,cellularnetworkswithRFenergytransferwere
[e-mail]). considered in [8,18]. It is worth noting that all these prior
G. K. Karagiannidis is with the Department of Electrical and Computer
works focus on the point-to-point communication systems.
Engineering,KhalifaUniversity,POBox127788,AbuDhabi,UAEandwith
theDepartmentofElectrical andComputerEngineering,AristotleUniversity TheRFenergyharvestingtechniquealsofindsimportantap-
ofThessaloniki, 54124,Thessaloniki, Greece (e-mail:[e-mail]). plications in cooperativerelaying networks, where an energy-
T. A. Tsiftsis is with the Department of Electrical Engineering, Techno-
constrained relay with limited battery reserves relies on some
logicalEducationalInstituteofCentralGreece,35100Lamia,Greece(email:
[e-mail]). external charging mechanism to assist the transmission of
2
source information to the destination [2]. As such, a number means to evaluate key system performancemetrics, such
of works have exploited the idea of achieving simultaneous as the outage probability and ergodic capacity, without
information and energy transfer in cooperative relaying sys- resorting to time-consuming Monte Carlo simulations.
tems [4,9,19–21]. Specifically, [20] studied the throughput Therefore, a fast assessment of the impact of various
performanceof an amplify-and-forward(AF) relaying system key system parameters such as the energy harvesting
forbothtime-switchingandpower-splittingprotocolsand[21] efficiency η, the number of antennas N, the source
considered the power allocation strategies for decode-and- transmitting power ρ and the interference power ρ on
1 I
forward(DF)relayingsystemwithmultiplesource-destination the optimal power splitting ratio is enabled.
pairs. More recently, the performance of energy harvesting • OurresultsdemonstratethattheCCIcouldbepotentially
cooperative networks with randomly distributed users was exploited to significantly improve the system’s perfor-
studied in [4,9]. However, all these works are limited to mance. However, the actual performance gain due to
the single antenna setup and all assume an interference free CCI depends heavily on the choice of linear processing
environment. schemes. It is shown that the MMSE/MRT scheme is
Motivated by this, we consider a dual-hop AF relaying always capable of turning the CCI as a desired factor,
system where the source and destination are equipped with and can achieve higher performance gain when the CCI
a single antenna while the relay is equipped with multiple isstrong.Ontheotherhand,CCIisnotalwaysbeneficial
antennas.1 The energy constrained relay collects energy from when the MRC/MRT and ZF/MRT schemes are used.
ambient RF signals and uses the harvested energy to forward The performance degrades significantly in the strong
the information to the destination node. The power-splitting CCI scenario if the MRC/MRT scheme is applied. In
receiverarchitectureproposedin [10] is adopted.Specifically, contrast, a weak interferer degrades the performance of
we first study the performance of the multiple antenna relay the ZF/MRT scheme, which on the other hand achieves
system without CCI, which serves as a benchmark for the almostthesameperformanceastheMMSE/MRTscheme
performance in the presence of CCI. Then, we present a in the presence of strong CCI.
detailedperformanceanalysisforthesystemassumingasingle Theremainderofthepaperisorganizedasfollows:Section
dominant interferer at the relay. It is worth pointing out II introducesthe system model. Section III investigatesof the
that, in the energy harvesting relaying system, while CCI performanceofthe system withoutCCI. Section IV addresses
provides additional energy, it corrupts the desired signal. In the scenario with CCI. Numerical results and discussions are
order to exploit CCI as a beneficial prospect, three differ- providedinSectionV.Finally,SectionVIconcludesthepaper
ent linear processing schemes, namely, 1) Maximum ratio and summarizes the key findings.
combining/maximalratio transmission (MRC/MRT), 2) Zero- Notation:Weusebolduppercaseletterstodenotematrices,
forcing/MRT(ZF/MRT),3)Minimummean-squareerror/MRT boldlowercaseletterstodenotevectorsandlowercaseletters
(MMSE/MRT) are investigated. to denote scalars. h denotes the Frobenius norm; E x
The main contributions of this paper are summarized as standsfortheexpecktatkioFnoftherandomvariablex; deno{te}s
follows: ∗
theconjugateoperator,whileT denotesthetransposeoperator
• For the scenario without CCI, we derive an exact outage and denotes the conjugate transpose operator; (0,1)
† CN
expressioninvolvinga single integral,anda tightclosed- denotesascalarcomplexGaussiandistributionwithzeromean
form outage probability lower bound. In addition, we and unit variance; Γ(x) is the gamma function; Ψ(a,b;z)
present a simple high signal-to-noise ratio (SNR) ap- is the confluent hypergeometric function [24, Eq. (9.210.2)];
proximation, which reveals that the system achieves a K (x)isthev-thordermodifiedBesselfunctionofthesecond
v
diversity order of N, where N is the number of relay kind [24, Eq. (8.407.1)]; Ei(x) is the exponential integral
antennas. A new tight closed-form upper bound for the function [24, Eq. (8.211.1)]; Γ(α,x) is the upper incomplete
ergodic capacity is also derived. Finally, the optimal gamma function [24, Eq. (8.350.2)]; F (a,b;c;z) is the
2 1
power splitting ratio minimizing the outage probability GaussHypergeometricfunction[24,Eq.(9.100)];ψ(x)isthe
is characterized. Digamma function [24, Eq. (8.360.1)];Gm,n() is the Meijer
p,q ·
• For the scenario with CCI, we present tight closed- G-function [24, Eq. (9.301)] and G1,1,1,1,1 () denotes the
formoutageprobabilitylowerboundsandcapacityupper generalizedMeijerG-functionoftw1o,[v1:a1r]i,a0,b[1le:1s][2·5]whichcan
bounds for all three schemes. In addition, we also char- be computed by the algorithm presented in [26, Table II].
acterize the high SNR outage behavior and show that
both the MRC/MRT and MMSE/MRT schemes achieve
II. SYSTEMMODEL
a diversity order of N, while the ZF/MRT only achieves
a diversity order of N 1. Moreover,the optimal power We consider a dual-hop multiple antenna AF energy har-
−
splitting ratio minimizing the outage probability is stud- vesting relaying system as shown in Fig. 1(a), where both
ied. the source and the destination are equipped with a single
• The presented analytical expressions provide an efficient antenna,whiletherelayisequippedwithN antennas[5].The
sourcesends informationto the destinationthroughan energy
1This particular system setup is applicable in several practical scenarios constrained relay node. Throughout this paper, the following
wheretwonodes(e.g.,machine-to-machine typelowcostdevices)exchange
assumptions are adopted: 1) It is assumed that direct link
information with the assistance of an advanced terminal such as a cellular
base-station/clusterhead sensor.[5,22,23] between the source and the destination does not exist due
3
to obstacles and/or severe fading. 2) The channel remains where P denotes the source power, h is an N 1 vec-
s 1
×
constant over the block time T and varies independently and tor with entries following identically and independently dis-
identically from one block to the other, and has a Rayleigh tributed (i.i.d.) (0,1), d denotes the distance between
1
CN
distributed magnitude. 3) As in [27–29], no CSI is assumed the source and the relay, τ is the path loss exponent, x is
at the source, full CSI is assumed at the relay, and local CSI the source message with unit power, n is an N 1 vector
r
×
is assumed at the destination. and denotes the additive white Gaussian noise (AWGN) with
E n n† =N I.
{ r r} 0
At the end of the first phase, the overall energy harvested
(cid:44) h
I during half of the block time T, can be expressed as
2
ηθP T
E = s h 2 , (2)
h dτ k 1kF 2
1
h h
1 2 where η denotes the RF-to-DC conversion efficiency.
B. Interference plus Noise Case
(cid:54) (cid:53) (cid:39)
We assume that the relay is subjected to a single dominant
(a)
interfererandAWGNwhilethedestinationisstillcorruptedby
theAWGN only.3 Itisworthpointingoutthesingledominant
Power interferer assumption has been widely adopt in the literature,
Antenna 1 Splitter (cid:84) see [31,32] and references therein. Moreover, such a system
Energy
nr1 1(cid:16)(cid:84) Receiver model enables us to gain key insights on the joint effect of
CCI and multiple antennas in an energy harvesting relaying
(cid:17)(cid:17)(cid:17) (cid:84) system.
Information
In such case, the signal at the input of the information
Receiver
Power 1(cid:16)(cid:84) receiver at the relay is given by
Antenna N Splitter
nrN yr = (1−θ)Ps/dτ1h1x+ (1−θ)PI/dτIhIsI +nr,
q q (3)
(b)
where P is the interference power, d denotes the distance
I I
Fig. 1: (a) System model: S, R and D denote the source, between the interferer and the relay, s is the interference
I
relay and destination node, respectively. (b) Block diagram symbol with unit power, and h is an N 1 vector with
I
×
of the relay receiver with the power splitting protocol. entries following i.i.d. (0,1).
CN
Also, according to [11], at the end of the first phase, the
overall energy harvested during half of the block time T is
We focus on the power splitting protocol proposed in [10]. 2
given by
Specifically, the entire communication consists of two time
slots with duration of T each. At the end of the first phase, P P T
2 E =ηθ s h 2 + I h 2 . (4)
eachantennaattherelaynodesplitsthereceivedsourcesignal h dτ k 1kF dτ k IkF 2
(cid:18) 1 I (cid:19)
into two streams, one for energy harvesting and the other for
Forbothcases,duringthesecondphase,therelaytransmits
informationprocessingasdepictedinFig.1(b).Asin[12,21],
a transformedversionof the receivedsignal to the destination
we consider the pessimistic case where power splitting only
usingtheharvestedpower.Hence,thesignalatthedestination
reducesthe signalpower, butnot the noise power. Hence, our
can be expressed as
resultsprovidealowerboundontheperformanceforpractical
systems. We now consider two separate cases depending on
y = 1/dτh Wy +n , (5)
whether the relay is subject to CCI or not. d 2 2 r d
q
where h is a 1 N vector and denotes the relay-destination
2
×
channel and its entries follow i.i.d. (0,1), d denotes the
2
A. Noise-limited Case CN
distance between the relay and the destination, n is the
d
Let θ denote the power splitting ratio2, then the signal AWGN at the destination with E{n∗dnd} = N0, W is the
component at the input of the information receiver is given transformation matrix applied at the information receiver at
by therelaywithE{kWyrk2F}=Pr.Obviously,theperformance
of the system depends on the choice of W, which will be
y = (1 θ)P /dτh x+n , (1) elaborated in the ensuing sections.
r − s 1 1 r
q
2Theoptimality ofuniformθcanbeestablishedbyusingsimilarmethods 3The scenario where the relay and the destination experience different
asin[12]. interference patterns willoccurinfrequency-division relaying systems[30].
4
III. THENOISE-LIMITEDSCENARIO which can be evaluated as
d/c
In this section, we consider the scenario where the relay
P = f (x)dx
is corrupted by AWGN only. In such case, it can be shown out Z0 kh1k2F
that the optimal transformation matrix W has the following ∞ ax+b
+ f (x)F dx. (12)
structure: Zd/c kh1k2F kh2k2F (cid:18)cx2−dx(cid:19)
h†h† Since the squared Frobenius norm of a complex Gaussian
W=ω 2 1 , (6)
h h vector is Chi-square distributed, h 2 and h 2 are i.i.d.
k 2kFk 1kF k 1kF k 2kF
gammarandomvariables.Aftersomesimplealgebraicmanip-
where ω is the power constraint factor, i.e., the information
ulations (10) is obtained. (cid:3)
receiver first applies the MRC principle to combine all the
Theorem1presentstheexactoutageprobabilityoftheofthe
signals from N antennas, and then forward the signal to the
systemwitharbitrarynumberofantennas.Forthespecialcase
destination by using the MRT principle. To guarantee the
with a single antenna at the relay, Theorem 1 reduces to the
transmit power constraint at the relay, ω can be computed
resultderivedin[20,Proposition3].Tothebestoftheauthors’
as
knowledge, the integral in (10) does not admit a closed-
P
ω2 = r , (7) form expression. However, this single integral expression can
(1−dθτ1)Ps kh1k2F +N0 be efficiently evaluated numerically using software such as
Matlab or MATHEMATICA. Alternatively, we can use the
whereP istheavailablerelaypower.Sincetherelaycommu-
r
nicates with the destination for half of the block time T, we followingclosed-formlowerboundforthe outageprobability,
have P = Eh = ηθPs h 2 . Hence, the end-to-end2SNR which will be shown to be tight across the entire SNR range
of the srystemT/c2an bedeτ1xpkres1skeFd as in the Section V.
Corollary 1: Theoutageprobabilityofthemultipleantenna
ω2 h 2 h 2 (1−θ)Ps energy harvesting relaying system can be lower bounded as
γ = k 2kF k 1kF dτ1dτ2
= ω2kh2ηkθd(2F121−τNddθτ2τ20)ρ+21 kNh02k2F kh1k4F , (8) Polouwt =1− 2Γe(−Nd/)c NXi=−01i1!NXj=−01(cid:18)Nj−1(cid:19)(cid:18)dc(cid:19)N−j−1×
ηθρ1 h 2 h 2 + (1−θ)ρ1 h 2 +1 a i+j+1 a
dτ1dτ2 k 2kF k 1kF dτ1 k 1kF 2 Ki−j−1 2 . (13)
c c
where ρ1 is defined as ρ1 =Ps/N0. (cid:16) (cid:17) (cid:18) r (cid:19)
In the following, we give a detailed performance analysis Proof: See Appendix I. (cid:3)
in terms of the outage probability and ergodic capacity. In While Theorem 1 and Corollary 1 are useful to study the
addition, the optimal θ minimizing the outage probability is system’soutageprobability,the expressionsare ingeneraltoo
investigated. complex to gain insight. Motivated by this, we now look into
the high SNR regime, and derive a simple approximation for
the outage probability, which enables the characterization of
the achievable diversity order.
A. Outage Probability
Theorem 2: In the high SNR regime, i.e., ρ , the
1
→ ∞
The outageprobabilityis an importantperformancemetric, outage probability of the multiple antenna energy harvesting
which is defined as the instantaneous SNR falls below a pre- relaying system can be approximated as
defined threshold γ . Mathematically, outage probability can
th
be expressed as P∞ dτ1ργ1th N 1 +
Pout =Prob(γ <γth). (9) out ≈ Γ(cid:16)(N +(cid:17)1) (1 θ)N
−
ln((1 θ)ρ ) ln(dτγ ) C dτ N
− 1 − 1 th − 2 , (14)
Theorem 1: The outage probabilityof the multiple antenna Γ(N) ηθ
(cid:18) (cid:19) !
energy harvesting relaying system can be expressed as
where C is the Euler-Mascheroniconstant [24, Eq. (9.73)].
P =1 ∞ Γ N,cxa2x−+dbx xN−1e−xdx, (10) Proof: See Appendix II. (cid:3)
out − (cid:16) Γ(N) (cid:17)Γ(N) We observe that the system achieves a diversity order of
Zd/c N, which is the same as the conventional case with constant
where a= (1−θd)τ1ρ1γth, b=γth, c= ηθd(121−τdθτ2)ρ21, d= ηdθρτ11dγτ2th. pasowρe−rNrellnayρ noradtehe[2r2t]h.aHnoρw−eNvera,swienntohteicecothnavtenPtiooutnadleccaayses
Proof: Substituting(8) into (9), the outage probabilityof 1 1 1
[22]. This important observation implies that, in the energy
the system can be expressed as
harvesting case, the slope of P converges much slower
out
P = (11) compared with that in the constant power case. Please note
out
Prob h 2 c h 4 d h 2 < a h 2 +b , that similar observations have been made in prior work [21].
k 2kF k 1kF − k 1kF k 1kF The possible reason is that, in SWIPT systems, the available
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
5
transmitpowerattherelayisarandomvariable,whichresults moretransmissionpowerattherelay,whichbenefitstherelay-
in higher outage probability compared to the conventional destination transmission. Nevertheless, a large θ also dete-
constant relay power case. riorates the quality of the source-relay transmission. Hence,
there exists a delicate balance, which we now investigate.For
B. Ergodic Capacity tractability, we only focus on the outage performance in the
highSNR region,and the impactof θ on the ergodiccapacity
Noticing that the end-to-end SNR given in (8) can be
will be numerically illustrated in the Section V.
alternatively expressed as
Starting from the high SNR approximation of P in (14),
γ γ out
γ = 1 2 , (15) the optimal θ, which is the solution of the optimization
γ +γ +1
1 2 problem min P ,canbeobtainedbysolvingtheequivalent
out
whereγ = (1−θ)ρ1 h 2 andγ = ηθρ1 h 2 h 2.The problem0in<θ(<210) shown on the top of the next page.
ergodic1capacitdyτ1is gkive1nkFby 2 dτ1dτ2 k 2kF k 1kF Proposition 1: The optimal θ is the root of the following
polynomial
1 γ γ
C = E log 1+ 1 2 . (16)
2 (cid:20) 2(cid:18) γ1+γ2+1(cid:19)(cid:21) a1θN+1 b1(1 θ)N+1 c1θ(1 θ)N
− − − −
Unfortunately, an exact evaluation of the ergodic capacity is d (1 θ)N+1ln(1 θ)=0, (21)
1
generallyintractable,sincethecumulativedistributionfunction − − −
(c.d.f.) of (8) can not be given in closed-form. Motivated by where a = N, b = dN2τN(lnρ1−lndτ1γth−C), c = dN2τ ,
1 1 ηNΓ(N) 1 ηNΓ(N)
this,we hereafterseektodeduceatightboundfortheergodic
d = NdN2τ and 0<θ <1 .
capacity. 1 ηNΓ(N)
Proof: It is easily to prove that, when ρ , there is
Startingfrom(16),theergodiccapacitycanbealternatively 1
→∞
only one root (denotedby θ∗) on the intervalof (0,1) for the
expressed as
equation f′(θ) = 0, and we can also note that f′(0) =
1 (1+γ1)(1+γ2) and f′(1) = + . Due to the continuity of f′(θ), we h−a∞ve
C = 2E(cid:20)log2(cid:18) 1+γ1+γ2 (cid:19)(cid:21)=Cγ1 +Cγ2 −Cγ(1T7,) fm′e(aθn)s<tha0t, fθ(θ∈)∞(fi0r,sθt∗d)ecarnedasfes′(θas) θ>fr0o,mθ 0∈to(θθ∗∗,1a)n,dwthhiecnh
wCloγhgTer(e=1+C21Eγeix[lo+=g2ey(1)12+iEs[γalo1cg+o2n(γv12ex)+].fγuUin)sci]tn,igofnothrweifithacrt∈etshpae{tc1ft,t2(ox},,xya)ann=dd ionfcDfreu(aθes)etsocaantshebθepforreobsmteaninθce∗edtoobfy1lo.sogTlavhriientrhgemffoirc′e(,fθut)hn=ectig0ol.noblanl(m1inimθ)u,(cid:3)ma
2 −
closed-formexpressionfortherootof(21)cannotbeobtained.
y, we have
However, it can be efficiently solved numerically.
1
C log 1+eE(lnγ1)+eE(lnγ2) . (18)
γT ≥ 2 2
(cid:16) (cid:17) IV. THE INTERFERENCEPLUSNOISE SCENARIO
We now establish the ergodic capacity upper bound of the
system using the following theorem: We now assume that the relay is subject to the influence
Theorem 3: The ergodic capacity of the multiple antenna of a single dominant interferer. In the presence of CCI, the
energy harvesting relaying system is upper bounded by optimalrelayprocessingmatrixWmaximizingtheend-to-end
signal-to-interference-and-noiseratio (SINR) of the system is
C = e(1−dθτ1)ρ1 N−1 dτ1 kΓ k, dτ1 + the solution of the following optimization problem:
up
2ln2 kX=0 (cid:18)(1−θ)ρ1(cid:19) (cid:18)− (1−θ)ρ1(cid:19) max γ = (1d−τ1θd)τ2Ps|h2Wh1|2
2l1n2Γ(1N)N−1m1! dητ1θdρτ2 mG31,,13 dητ1θdρτ2 −−mm,N−m,0 − W (1d−τ1θd)τ2PI|h2WhI|2+ kh2dWτ2k2FN0+N0
mX=0 (cid:18) 1(cid:19) (cid:18) 1(cid:12)(cid:12) (cid:19) s.t. E Wy 2 =P =ηθ Ps h 2 + PI h 2 .
21log2 1+ (1−dτθ)ρ1eψ(N)+ dητθdρτ1(cid:12)(cid:12)e2ψ(N) . (19) {k rkF} r (cid:18)dτ1 k 1kF dτI k IkF(cid:19)(22)
(cid:18) 1 1 2 (cid:19)
Proof: See Appendix III. (cid:3) Due to the non-convex nature of the problem, a closed-form
Theorem 3 presents a new upper bound for the ergodic solution for W is hard to find. Hence, in the following, we
capacity of the system, which is quite tight across the entire consider three heuristic two-stage relay processing strategies
SNR range as shown in the section V, hence, providing proposedin[23],i.e.,thematrixWadmitstherank-1structure
an efficient means to evaluate the ergodic capacity without h†
W=ω 2 w ,wherew isa1 N linearcombiningvec-
resortingto MonteCarlo simulations.In addition,aswe show tor,whickhh2dkeFpen1dsonthe li1nearco×mbiningschemeemployed
in the next subsection, it enables the study of the optimal
at the relay and will be specified in the following subsection.
power splitting ratio.
C. Optimization of the Parameter θ in High SNR Value A. MRC/MRT Scheme
The right selection of the power splitting ratio θ is crucial FortheMRC/MRT scheme,w issettomatchthefirsthop
1
forthesystem’sperformance.Ahighvalueofθ couldprovide channelgivenin (6). To meetthe transmitpowerconstraintat
6
1 ln((1 θ)ρ ) lndτγ C dτ N
min f(θ)= + − 1 − 1 th− 2 . (20)
0<θ<1 (1 θ)N Γ(N) ηθ
− (cid:18) (cid:19)
the relay, the power constraint factor ω2 should be given by Theorem 5: InthehighSNRregion,i.e.,ρ ,theout-
1
→∞
ageprobabilityoftheMRC/MRTschemecanbeapproximated
P
ω2 = r , (23) as5
(1−dθτ1)Ps kh1k2F + (1−dθτI)PI |khh†11hkI2F|2 +N0 PMRC dτ1γth N 1 N N ((1−θ)ρI)n +
wtphrheeesrseeenddP-atros-=endTE/hS2IN=Rηθof(cid:16)tPdhτ1sekMh1RkC2F/M+RPdTτII kshchIekm2F(cid:17)e.cTanhebreefoerxe-, dINouτtN≈−1(cid:18)Nρ−11 ((cid:19)1) N(cid:18)−i1−−12θF1(cid:19)(cid:18)N,nX2=N0−di−nI1τ;(2NN−−i;1n−)dd!τIτ1ρρ1I(cid:19)
γMRC = γIM1RCγIM2RC , (24) 2 iP=0 (cid:0) i (cid:1) −(ηθ)NΓ(N +1)Γ(N2)N−i−1 .
I γMRC+γMRC+1 
I1 I2 

where γMRC = (1−dθτ1)ρ1kh1k2F , γMRC = (29)
I1 (1−dθτI)ρI |khh†11hkI2F|2+1 I2 Proof: See Appendix V. (cid:3)
ηθ ρ1 h 2 + ρI h 2 h 2 and ρ is defined as For the special case where the relay is equippedwith a single
dτ2 dτ1 k 1kF dτI k IkF k 2kF I antenna,i.e.,N =1,with thehelpof[24,Eq.(9.121.6)],(29)
ρ =(cid:16) PI. (cid:17)
I1) ONu0tage Probability: Since the exact analysis appears to reduces to
be difficult, in the following we focus on deriving an outage PMRC 1 + ρI + dτ2(lndρτ11 −lndρτII) dτ1γth. (30)
lower bound and a simple high SNR outage approximation. Iout ≈ 1−θ dτI ηθ ! ρ1
According to [22,23], the end-to-end SINR in (24) can be
tightly upper bounded by Theorem 5 indicates that a full diversity order of N is
still achievable in the presence of CCI for the MRC/MRT
γMRC γup =min γMRC , γMRC , (25)
I ≤ I I1 I2 scheme. Moreover, from (30), we see that the effect of CCI
could be either beneficial or detrimental, depending on the
the outage probability of the M(cid:0)RC/MRT sch(cid:1)eme is lower
bounded by PLMRC =Prob(γup <γ ). (26) rdeτ2l(altnioρηInθ−shlnipdτI)beitswpeoesnitiρvIe,, tdhτIe,CdCτ2Iηisadnedtriθm,ein.eta.,l,wwhheilne wdρτIIhe−n
Iout I th ρI dτ2(lnρI−lndτI) is negative, the CCI becomes beneficial,
We have the following key result: wdτIhi−ch suggηeθsts that, in wireless powered relaying systems,
Theorem 4: If ρ = ρ ,4 the outage probability of the
1 I CCI could be potentially exploited to improve the perfor-
6
MRC/MRT scheme can be lower bounded as
mance.
PLMRC =1 FMRCFMRC, (27) 2) Ergodic Capacity: Utilizing similar techniques as in
Iout − 1 2
Section III-B, we establish the following ergodic capacity
with
upper bound:
FMRC = dτIe−(1d−τ1θγ)tρh1 N−1 dτ1γth m Theorem 6: If ρ1 6= ρI, the ergodic capacity of the
1 (1 θ)ρ (1 θ)ρ × MRC/MRT scheme is upper bounded by
− I m=0(cid:18) − 1(cid:19)
X
m 1 (1 θ)ρ ρ n+1 CMRC =C +C
− 1 I , Iup γIM1RC γIM2RC−
nX=0(m−n)!(cid:18)dτIρ1+dτ1ρIγth(cid:19) 12log2 1+eE(lnγIM1RC)+eE(lnγIM2RC) , (31)
athnednFe2xMtRpCagcea.n be expressed as in (28) shown on the top of where CγMRC, CγMRC, E(cid:16)lnγIM1RC and E lnγIM2RC a(cid:17)re given
I1 I2
Proof: See Appendix IV. (cid:3) by (32) - (35) shown on the next page.
(cid:0) (cid:1) (cid:0) (cid:1)
WhileTheorem4isusefulfortheevaluationofthesystem’s Proof: See Appendix VI. (cid:3)
outage probability, the expression is too complex to yield 3) Optimalθ Analysis: We nowstudy the optimalvalue of
much useful insights. Motivated by this, we now look into θ minimizing the outage probability. Based on the high SNR
the high SNR region, and derive a simple approximation for approximation for PMRC in (29), the optimal θ can be found
Iout
the outage probability, which enables the characterization of as:
the achievable diversity order of the system.
Proposition 2: The optimal θ is a root of the following
4For mathematical tractability, we only provide the result for the general
casewherethesignalfromthesourceandtheCCIhavedifferentpower,i.e.,
ρ1 6= ρI. But the result for the special case ρ1 = ρI is much more easier 5It is worth pointing out that the result in Theorem 5 holds for all cases
andcanbeobtained inasimilarway. whetherthesignalpowerandtheCCIpowerisequal ornot.
7
2dNτdNτ N s−1(1 N j) dτ dτ 1−N−sN−1 1 dτγ N+1−s
FMRC = 1 I j=1 − − I 1 2 th
2 ρNρN (N s)!(s 1)! ρ − ρ m! ηθ ×
1 I s=1 Q − − (cid:18) I 1(cid:19) m=0 (cid:18) (cid:19)
X X
dτ1dτ2γth m+s−2N−1K 2 dτ1dτ2γth + 2dN1 τdNI τ N js=−11(1−N −j)
(cid:18) ηθρ1 (cid:19) m+s−N−1 s ηθρ1 ! ρN1 ρNI s=1 Q(N −s)!(s−1)! ×
X
dτ1 dτI 1−N−sN−1 1 dτ2γth N+1−s dτ2dτIγth m+s−2N−1K 2 dτ2dτIγth . (28)
m+s−N−1
(cid:18)ρ1 − ρI(cid:19) m=0m!(cid:18) ηθ (cid:19) (cid:18) ηθρI (cid:19) s ηθρI !
X
CγIM1RC = (12d−τ1θln)2ρ1 mNX=−01nXm=0d(nI(τ1(−mθ−)ρnI))!nn!G11,,1[1,:11,]1,,01,[1:1] ((11−−ddθθτ1τI))ρρI1(cid:12)(cid:12)0m0;−−+;0n1!, (32)
(cid:12)
(cid:12)
(cid:12)
dNτdNτ N s−1(1 N j) dτ dτ 1−N−sN−1 1 dτ N+1−s
C = 1 2 j=1 − − I 1 2
γIM2RC ρN1 ρNI 2ln2 s=1 Q(N −s)!(s−1)! (cid:18)ρI − ρ1(cid:19) m=0m!(cid:18)ηθ(cid:19) ×
X X
dτdτ dNτdNτ N s−1(1 N j) dτ dτ 1−N−s
G3,1 1 2 s−N−1 + 1 2 j=1 − − 1 I
1,3 ηθρ s−N−1,m+s−N−1,0 ρNρN2ln2 (N s)!(s 1)! ρ − ρ
(cid:18) 1(cid:12) (cid:19) 1 I s=1 Q − − (cid:18) 1 I(cid:19)
(cid:12) X
(cid:12) N−1 1 dτ N+1−s dτdτ
(cid:12) 2 G3,1 2 I s−N−1 , (33)
× m! ηθ 1,3 ηθρ s−N−1,m+s−N−1,0
m=0 (cid:18) (cid:19) (cid:18) I(cid:12) (cid:19)
X (cid:12)
(cid:12)
(cid:12)
E lnγIM1RC =ln((1−θ)ρ1)−lndτ1 +ψ(1)−e(1−dθτI)ρIG23,,30 (1 dτIθ)ρ 01,,01,1 +
(cid:18) − I(cid:12) (cid:19)
(cid:0) (cid:1) N−1 m ((1 θ)ρ(cid:12)(cid:12) )n−m dτ
− (cid:12)I Γ(m)Ψ m,m n; I , (34)
mX=1nX=0(m−n)!dI(n−m)τ (cid:18) − (1−θ)ρI(cid:19)
polynomial In the special case of N = 1, the optimal solution can be
given in closed-form as follows:
N−1
(n)(1 θ)n−N−1 B =0, (36)
A − − θN+1 dτIρ1(lnρI−lnρ1−lndτI+lndτ1)
nX=0 θopt = r η(dτ1ρI−dτIρ1) . (37)
where (n) = dρnInIτ , = MRC 1+ dτIρ1(lnρI−lnρ1−lndτI+lndτ1)
dN2τ N−A1 N−1 ( 1)N−i−12F1((cid:18)NN−,2nN−−1)i!−1;2N−Bi;1−ρρ1IddτIτ1(cid:19) This simple expressiorn is quitηe(diτ1nρfIo−rmdτIaρt1iv)e, and it can be
ηNΓ2(N) i=0 i − 2N−i−1 observed that the optimal θ in (37) is a decreasing function
and 0<θP<1(cid:0). (cid:1) of η and ρ , and an increasing function of ρ , which can be
I 1
explained as follows:
Proof: The result is derived by following the same steps • Asηincreases,moretransmissionpowercanbecollected
as in the proof of Proposition 1. (cid:3) at the relay, hence the bottleneck of the system perfor-
dNτ N s−1(1 N j) dτ dτ 1−N−s ρ 1−s
E lnγMRC =lnηθ lndτ +ψ(N)+ I j=1 − − I 1 1
I2 − 2 ρN (s 1)! ρ − ρ dτ ×
I s=1 Q − (cid:18) I 1(cid:19) (cid:18) 1(cid:19)
(cid:0) (cid:1) X
dNτ N s−1(1 N j) dτ dτ 1−N−s ρ 1−s
(ψ(N s+1)+lnρ lndτ)+ 1 j=1 − − 1 I I
− 1− 1 ρN (s 1)! ρ − ρ dτ ×
1 s=1 Q − (cid:18) 1 I(cid:19) (cid:18) I(cid:19)
X
(ψ(N s+1)+lnρ lndτ). (35)
− I − I
8
mance lies in the SINR of the signal at the input of the Then, utilizing (42) and following the similar lines as in the
information receiver. As a result, we should choose a proof of Theorem 5, we can obtain
smaller θ to improve the first hop performance.
• A largeρI providesmore energy,while at the same time PZF 1 dτ1γth N−1+
reduces the SINR of the first hop transmission. Hence, a Iout ≈ (N 1)! (1 θ)ρ
− (cid:18) − 1(cid:19)
sSmINalRle.rθ shouldbe chosentocompensatethe lossof the dτ1ηdθτ2ργ1th N N−1 N −1 ( 1)N−i−1
• For largeρ1, in generalthe first hoptransmission quality Γ((cid:16)N +1)Γ(cid:17)(N) i=0 (cid:18) i (cid:19) − ×
is quite good, hence,it is beneficialto have more energy X
at the relay, i.e., a larger θ is desirable. 2F1 N,2N −i−1;2N −i;1− ddτIτ1ρρI1 . (43)
(cid:16) 2N i 1 (cid:17)
− −
The desired result follows by noticing that the second term is
B. ZF/MRT Scheme negligible compared with the first term in (43). (cid:3)
For the ZF/MRT scheme, the relay utilizes the available Theorem 8 indicates that the achievable diversity order of
multiple antennas to completely eliminate the CCI. To ensure theZF/MRTschemeisN 1.ComparedwiththeMRC/MRT
−
this is possible, the number of the antennas equipped at the scheme, the ZF/MRT scheme incurs a diversity loss of one.
relay shouldbe greaterthan the numberof interferers.Hence, This is an intuitive and satisfying result since one degree of
fortheZF/MRTscheme,itisassumedthatN >1.According freedom is used for the elimination of the CCI.
to [23], the optimal combining vector w is given by w = 2) ErgodicCapacity: We nowlookintothe ergodiccapac-
1 1
h†1P , where P = I h h†h −1h†. Therefore, the ity of the system, and we can establish the following upper
√h†Ph N − I I I I bound of the ergodic capacity:
end1-to-e1ndSINR of the ZF/MRT(cid:16)schem(cid:17)e can be expressed as Theorem 9: Ifρ =ρ ,theergodiccapacityoftheZF/MRT
1 I
6
γZFγZF scheme is upper bounded by
γZF = I1 I2 , (38)
I γIZ1F+γIZ2F+1 CIZuFp =CγIZ1F +CγIZ2F − 21log2 1+eE(lnγIZ1F)+eE(lnγIZ2F) ,
where γIZ1F = h†1Ph1 (1−dθτ)ρ1, γIZ2F = (cid:16) ((cid:17)44)
1
dητ2θ1(cid:16))dρOτ11uktha1gke2FP+robdρaτIIbkilhitIyk:2FW(cid:17)(cid:12)(cid:12)(cid:12)ekhfi2rkst2Fp.(cid:12)(cid:12)(cid:12)resent the following out- where CγIZ1F = e(21−ldnθτ1)2ρ1 Nk=−02 (1−dθτ1)ρ1 kΓ −k,(1−dθτ1)ρ1 ,
ZagFeT/MhloeRwoTreermscbho7eu:mnIdef:cρa1n b6=e loρwI,erthbeouonudteadgeasprobability of the CEγ(cid:0)IMl2nRCPγraIZon1Fod(cid:1)f:E=Wlilntnhγ(tIZ(h2F1e−h=eθl)Epρo1l)fnP−tγhIMe(cid:16)2lRncCd.dτ1..f+. oψ(cid:17)f (γNZF(cid:16)−gi1v)e,nCinγIZ2F(4(cid:17)0=)
(cid:0) (cid:1) (cid:0) (cid:1) I1
PLZF =1 FZFFZF, (39) and following the similar lines as in the proof of Theorem 6
Iout − 1 2 yields the desired result. (cid:3)
where FZF = Γ(cid:16)N−1,(1d−τ1γθt)hρ1(cid:17) and FZF =FMRC. 3) Optimal θ Analysis: We now study the optimal θ
1 Γ(N−1) 2 2 minimizing the outage probability. Based on the high SNR
Proof: According to [23], the c.d.f. of γIZ1F is given by approximation for PZF in (43), the optimal θ can be found
Iout
Γ N 1, dτ1x as:
F (x)=1 − (1−θ)ρ1 . (40) Proposition 3: The optimal θ is a root of the following
γIZ1F − (cid:16) Γ(N 1) (cid:17) polynomial
−
Then, the desired result can be obtained by following the
similar lines as in the proof of Theorem 4. (cid:3) A1 B1 =0, (45)
(1 θ)N − θN+1
To gain further insights, we now look into the high SNR −
region, and present a simple and informative approximation where A1 = (N−12)!, B1 =
forTthheeoroeumtag8e: pInrotbhaebhiliigtyh.SNRregion,i.e.,ρ1 →∞, theout- ηdNN2Γτ2d(Nτ1γ)tρh1 Ni=−01 N−i 1 (−1)N−i−12F1(cid:18)N,2N−2iN−1−;2i−N1−i;1−ddτIτ1ρρ1I(cid:19)
age probability of the ZF/MRT scheme can be approximated and 0<θ <P1.(cid:0) (cid:1)
as Proof: The result is derived by following the same steps
1 dτγ N−1 as in the proof of Proposition 1. (cid:3)
PZF 1 th . (41)
Iout ≈ (N 1)! (1 θ)ρ
− (cid:18) − 1(cid:19)
C. MMSE/MRT Scheme
Proof: With the help of the asymptotic expansion of
incomplete gamma function given in [24, Eq. (8.354.2)], it The ZF scheme completely eliminates the CCI at the
is easy to note that the c.d.f. of γZF can be approximated as relay, which however may cause an elevated noise level. In
I1
contrast,the MMSE schemedoesnotfullyeliminate the CCI,
1 dτx N−1
F (x) 1 . (42) instead,itprovidestheoptimumtrade-offbetweeninterference
γIZ1F ≈ (N −1)!(cid:18)(1−θ)ρ1(cid:19) suppressionand noise enhancement.Accordingto [23],w1 is
9
given by scheme.
A close observation of (29), (43) and (50) reveals that the
dτ −1
w =h† h h† +h h† + I I . (46) difference among all three schemes only lies in their first
1 1(cid:18) 1 1 I I (1−θ)ρI (cid:19) terms, which can be expressed as follows:
Therefore, the end-to-end SINR of the MMSE/MRT scheme
N ((1 θ)ρ )n
can be expressed as a = − I ,
MRC dnτ(N n)!
γMMSE = γIM1MSEγIM2MSE , (47) nX=10 I −
I γMMSE+γMMSE+1 a = ,
I1 I2 ZF (N 1)!
wγIMh2eMrSeEγ=IM1Mdητ2θSE =dρτ11 kddτIτ1hρρ1I1kh2F†1R+−dρ1τIIhk1h,IRk2F=khhI2hk†I2F+. (1−dθτI)ρII and aMMSE = N−1! + d(τI1(N−θ−)ρ1I)!. (52)
1) Outage(cid:16)Probability: (cid:17)
It can be easily observed that a is strictly smaller
MMSE
Theorem 10: If ρ = ρ , the outage probability of the
1 6 I than aMRC, since aMMSE only includes the first two terms of
MMSE/MRT scheme can be lower bounded as
a . As such, we conclude that the MMSE/MRT scheme
MRC
PLMMSE =1 FMMSEFMMSE, (48) always achieves a strictly better outage performance than
Iout − 1 2 the MRC/MRT scheme due to the higher array gain. For
Γ N, dτ1γth the ZF/MRT scheme, although a diversity loss leads to its
where FMMSE = (cid:16) (1−θ)ρ1(cid:17)
1 Γ(N) − inferior performance in the high SNR region, it should be
e−(1d−τ1dθγτI)tΓρh1(N(1)−θ)ρI (1d−τ1θγ)thρ1 N2F1 2,1;2;−ddτIτ1ρρI1γth and nthoattedthtehaZtFa/ZMFRisCgsecnheeramlleyhsamsalalerlatrhgaenr aarMraRyC,gwaihnicthhamneathnes
FMMSE =FMRC.(cid:16) (cid:17) (cid:16) (cid:17)
2 2 MRC/MRT scheme. Therefore, in the low SNR region, the
Proof: According to [23] we know that the c.d.f. of
ZF/MRT scheme may achievebetter outageperformancethan
γMMSE is given by
I1 the MRC/MRT scheme.
2) Ergodic Capacity:
F (x)=1 Γ N,(1−dτ1θx)ρ1 + e−(1−dτ1θx)ρ1 (1−θ)ρI Theorem 12: If ρ1 6= ρI, the ergodic capacity of the
γIM1MSE − (cid:16) Γ(N) (cid:17) dτΓ(N) MMSE/MRT scheme is upper bounded by
I
× (1 dτ1θx)ρ N2F1 2,1;2;−ddττ1ρρIx . (49) CIMuMpSE =CγIM1MSE +CγIM2MSE−
(cid:18) − 1(cid:19) (cid:18) I 1 (cid:19) 1log 1+eE(lnγIM1MSE)+eE(lnγIM2MSE) , (53)
Then, following the similar lines as in the proof of Theorem 2 2
4, we can obtain the desired result. (cid:3) where (cid:16) (cid:17)
To gain further insights, we now look into the high SNR
rpergoiboanb,ilaitnyd. present a simple approximation for the outage CγIM1MSE = e2(1−ldnθτ1)2ρ1 N−1dk1τ(Γ(1(cid:16)−kθ,)(ρ1−1d)θτ1k)ρ1(cid:17)−
Theorem 11: In the high SNR region, i.e., ρ1 , Xk=0 −
the outage probability of the MMSE/MRT scheme →can∞be (1−θ)3ρ2Iρ1 G1,1,2,1,1 (1−dθτ1)ρ1 0;(N−2+,−21) , (54)
approximated as 2ln2Γ(N)d2Iτdτ1 1,[1:2],0,[1:2] (1−dθτI)ρI(cid:12)(cid:12)0;(−−1,−2)!
(cid:12)
dτγ N 1 (1 θ)ρ 1 (cid:12)
PIMouMtSE ≈(cid:18)dητ2θ1ρ1Nth(cid:19) N(cid:18)−1N!N+−d1τI−Γ((N1))IN(cid:19)−(i−11−θ)N + E(cid:0)lnγIM1M((S1dE2I(cid:1)−τ=Γθ()ψNρ(I)N)2)G+31,,23ln(cid:18)(((11−−dτIθθ))ρρ1I)(cid:12)(cid:12)−−−1N,l−,n−2d2,τ1−−1(cid:19), (55)
Γ(N2F(cid:16)+1 1(cid:17)N)Γ,2(NN)−Xii=−0 (cid:18)1;2Ni −(cid:19)i;−1− ddτIτ1ρρ1I ×. (50) aanndd CfPoγrlIMol2oMowSfE:s=WthiCtehγsIMti2hRmCeihalaserlwplieonlfelstahsaesEc.i(cid:0)dnl.nft.hγoeIMf2Mpγ(cid:12)(cid:12)(cid:12)rSIMoE1oM(cid:1)fS=EofgEiTv(cid:0)ehlnneoγirnIMe2Rm(4C9(cid:1)6).
(cid:16) 2N i 1 (cid:17) yields the desired result. (cid:3)
− −
3) Optimal θ Analysis: We now study the optimal θ

Proof: After some simple manipulation the c.d.f. of
minimizing the outage probability. Based on the high SNR
γIM1MSE can be approximated as approximationforPMMSE in (50), the optimalθ canbe found
Iout
1 (1 θ)ρ dτγ N as:
FγIM1MSE(x)≈(cid:18)N! + dτI−Γ(N)I(cid:19)(cid:18)(1−1θt)hρ1(cid:19) . (51) equPartoipoonsition 4: The optimal θ is a root of the following
Then, following the similar lines as in the proof of Theorem
1 (N 1)ρ
5, we can obtain the desired result. (cid:3) + − I B =0, (56)
(1 θ)N+1Γ(N) dτ(1 θ)NΓ(N) − θN+1
Theorem 11 indicates that the MMSE/MRT scheme − I −
achieves a diversity order of N, the same as the MRC/MRT where have been defined in (36) and 0<θ <1.
B
10
Proof: The result is derived by following the same steps the ZF/MRT scheme outperforms the MRC/MRT scheme in
as in the proof of Proposition 1. (cid:3) thelowSNRregion,whiletheoppositeholdsinthehighSNR
region.
V. NUMERICAL RESULTSAND DISCUSSION From Fig. 2(b), we see that, for all three schemes, the
proposedergodiccapacityupperboundsin(31),(44)and(53)
In this section, we present numerical results to validate the
are sufficiently tight across the entire SNR range of interest.
analyticalexpressionspresentedin Section IV, andinvestigate
In addition, we observe the intuitive result that increasing N
the impact of various key system parameters on the system’s
results in an improvement of the ergodic capacity. Moreover,
performance. Unless otherwise specified, we set γ = 0 dB,
th
the MMSE/MRT scheme always has the best performance,
η =0.8, θ =0.5, ρ =9.5dB, τ =2 and d =d =d =1.
I 1 2 I
while the ZF/MRT scheme is slightly inferior,and the perfor-
mance gap between them disappears as N increases. On the
A. Effect of Multiple Antennas
other hand, the MRC/MRT scheme always yields the lowest
ergodic capacity, and as N increases, the performance gap
becomes more pronounced.
100
Monte Carlo Simulation
Lower Bound
High SNR Approximation B. Effect of CCI
10−1 MRC/MRT
ZF/MRT
Outage Probability10−2 N=2 MMSE/MRT 100 NMZMFoRM/nMCS−/REiMn/TtMRerTRfeTrence Case
N=3
10−3 Probability10−1
10−40 5 10 15 2ρ0 (dB)25 30 35 40 Outage
1
(a) Outageprobability
5 10−−21 0 −5 0 5 10 15 20
4.5 2.9 ρ (dB)
N=4 I
4 2.8
(a) Outageprobability
Hz)3.5 2.7
Ergodic Capacity (bits/s/012...01235550 15.55 16 16.51017 ρ1 1(5dB) 2MU0popneter BCoaurlNnoMZM2d =FS5RM2/iMCmS/REuM/lTaMRtiTRoTn30 Ergodic Capacity (bits/s/Hz)0000000.......13456789
(b) Ergodiccapacity 0.2 Non−interference Case
MRC/MRT
0.1 ZF/MRT
Fig. 2: Impact of N on the system performance. MMSE/MRT
0
−10 −5 0 5 10 15 20
ρ (dB)
I
Fig. 2 illustrates the impact of antenna number N on the
(b) Ergodiccapacity
outage probability and ergodic capacity. It can be readily
observed from Fig. 2(a) that for all the three considered Fig. 3: Impact of CCI on the system performance.
schemes, the proposed lower bounds in (27), (39) and (48)
are sufficiently tight across the entire SNR range of interest, Fig. 3 investigates the impact CCI on the system perfor-
especially when N is large, and become almost exact in the mance. The scenario without CCI is also plotted for compari-
highSNRregion,whilethehighSNRapproximationsin(29), son. It can be readily observed from Fig. 3(a) that the outage
(41) and (50) work quite well even at moderate SNR values probability of the MRC/MRT scheme decreases slightly for
(i.e.,ρ =20dB).Inaddition,weseethatboththeMRC/MRT smaller ρ (i.e., ρ < 0 dB), and then increases as the inter-
1 I I
and MMSE/MRT schemes achieve the full diversity order of ference becomes stronger. This phenomenonclearly indicates
N, while the ZF/MRT scheme only achievesa diversity order thattheCCIcancauseeitherbeneficialorharmfuleffectonthe
of N 1, which is consistent with our analytical results. system’sperformance.ThisisbecausethatCCIprovidesaddi-
−
Moreover, the MMSE/MRT scheme always attains the best tionalenergybutat the same time corruptsthe desired signal.
outage performance among all three proposed schemes, and For the MRC/MRT scheme, when the CCI is too strong, the