Source Optimization in MISO Relaying with Channel Mean Feedback: A Stochastic Ordering Approach

0.12 MB·English

by  Minhua Ding

Source Optimization in MISO Relaying
with Channel Mean Feedback:
A Stochastic Ordering Approach
Minhua Ding, Member, IEEE, and Q. T. Zhang, Fellow, IEEE
Department of Electronic Engineering, City University of Hong Kong
Email: [email protected], [email protected]
1
1
Abstract—This paper investigates the optimum source trans- whereas the relay and the source are only aware of the long-
0
2 mission strategy to maximize the capacity of a multiple-input term statistics of the source-relay channel. Relaying without
single-output (MISO) amplify-and-forward relay channel, as-
instantaneous CSI is commonly referred to as noncoherent
n suming source-relay channel mean feedback at the source. The
relaying.
a challenge here is that relaying introduces a nonconvex structure
J in the objective function, thereby excluding the possible use of The above scenario has been considered in [5], where the

5 previousmethodsdealingwithmeanfeedbackthatgenerallyrely sourceandtherelayareassumedtohavethelong-termcovari-
2 on the concavity of the objective function. A novel method is ance information of a rapidly changing source-relay channel,
employed, which divides the feasible set into two subsets and and the optimum source covariance to maximize the ergodic
] establishes the optimum from one of them by comparison. As
T capacityisdetermined.However,whenthemeaninformation1
such,theoptimizationistransformedintothecomparisonoftwo
I nonnegative random variables in the Laplace transform order, of a slowly varying source-relay channel is available at the
.
s whichisoneoftheimportantstochasticorders.Itturnsoutthat source, the capacity-achieving transmission strategy remains
c theoptimumtransmissionstrategyistotransmitalongtheknown asanopenproblemandistackledinthiswork.Meanfeedback
[ channel mean and its orthogonal eigenchannels. The condition modelstheuncertaintyinCSIduetochannelestimationerrors,
forrank-oneprecoding(beamforming)toachievecapacityisalso
1 quantizationerrors or imperfectnessof the feedback link, and
determined. Our results subsume those for traditional MISO
v precoding with mean feedback. is feasible in slowly varying fading channels [8].
5
Channel mean and covariance feedback generally require
1
differenttreatments, as they constrain the solution differently.
8 I. INTRODUCTION
4 Regarding utilizing mean feedback for maximum ergodic ca-
. Wireless relay systems have been a recent subject of in- pacityintraditionalMISO,theoptimizationmethodin[9][10]
1
0 tensive research due to their potential of providing increased isbasedoncalculusofvariationsusingtheFre´chetdifferential,
1 diversity, extended coverage, or flexibility in compromising whereas in [11], the expression for ergodic capacity is first
1 systemperformanceandcomplexity/powerconsumption.Effi- obtained and then ordinary calculus is used for optimization.
v: cient relaying protocols have been developed, among which Bothmethodsrelyontheconcavityoftheobjectivefunctionin
i the most popular are the amplify-and-forward (AF) and transmit covariance [9]-[11]. However, such concavity cannot
X
decode-and-forwardprotocols [1]. be established in our case, and these methods do not apply.
r
To further enhance system performance, multiple antennas We, therefore, take a nonconventional powerful approach
a
are deployed on one or more nodes of cooperative relay based on one of the stochastic orders, the Laplace transform
networks[2]-[6].However,inmulti-antennasystems,theopti- order [17], which circumvents the requirement of concavity.
mumsystemstructureandtheresultantperformance(capacity, Optimumsourceprecodingmatrix(orequivalently,thecovari-
error rate, mean-square error, etc.) depend heavily on the na- ance matrix) is determined. A special case of precoding, i.e.,
tureofchannelsandthechannelstateinformation(CSI)avail- beamforming, occurs when the covariance matrix is of rank
able at the transceiver (see [7]-[13] and references therein). one, which has appealing reduced complexity but may not
Therefore, research on multi-antenna relay communications always be capacity-optimum. Here we derive the necessary
has progressed from early work assuming perfect CSI [3][4] and sufficient condition for it to achieve capacity. All our
to recent studies assuming more realistic partial CSI [5]. results subsume as a special case those for traditional MISO
Consider a half-duplex AF relay link with M antennas precoding with channel mean feedback.
(M 2) at the source and a single antenna at both the
≥ II. SYSTEMMODELAND PROBLEM STATEMENT
relay and the destination. Thisscenario typically occurswhen
We focus on a half-duplexAF link with M antennas at the
atraditionalmultiple-inputsingle-output(MISO)link(without
source and with a single antenna at both the relay and the
relaying) is obstructed, and then a relay is used to maintain
the link andcoverage.We assume thatthe destinationhasfull
1Channel covariance (mean)information atthetransmitter isalsoreferred
knowledgeof the source-relayand relay-destinationchannels, toaschannel covariance (mean)feedback [7].
destination,2 resulting in a MISO source-relaylink (backward Our goal is to find the optimumQ for the problem defined
channel)andasingle-inputsingle-outputrelay-destinationlink by (4)-(5). The main challenge here is that in (5), the log
(forward channel). No direct source-destination link exists. function inside the expectation operator is nonconvex in Q.
The backward channel h is modeled as [9]: Previous methods for MISO precoding with channel mean
B
feedback [9]-[11], which generally utilize the concavity of
h =µ+√αh , (1)
B w the objective function in the transmit covariance matrix, are
where h represents the scattering and is distributed as not applicable here. Below we employ a new method based
w
(0,I )(circularlysymmetriccomplexGaussian),andα is on the Laplace transform order [17]. Some mathematical
c M
N
a nonnegative scaling constant (α 0). Only the knowledge preliminaries pertaining to the Laplace transform order are
≥
of the source-relay channel mean µ and α is provided to the given in Appendix I.
source, which we refer to as channel mean feedback (at the
source).Inthefirsttimeslot,thereceivedsignalinthesource- III. OPTIMUMSOURCE COVARIANCE MATRIX
relay link is given by Theorem 1 The optimum Q for the problem(4) is given
by Q =VΦV , where
r1 =h†BFx+n1, opt †
where x is the source signal with E{xx†}= Mγ IM and n1 is V= [µ/kµk v2 ... vM], (6)
the noise at the relay distributed as (0,1). The precoding 1 φ 1 φ
Nc Φ=diag φ, − ,..., − , (7)
(shaping)matrixFisrelatedtothetransmitcovariancematrix M 1 M 1
Q through Q , 1 FF . A transmit power constraint is (cid:26) − − (cid:27)
M † andv ...v arearbitraryorthonormalvectorsorthogonalto
imposed on the source: E tr(Fx(Fx) ) = γtr(Q) γ, 2 M
{ † } ≤ µ in CM.
which yields the constrainton the transmitcovariancematrix:
Proof: Due to space constraint, we only provide a detailed
tr(Q) 1.Denotetheamplifyingfactoroftherelayasη.Due
≤ outline here. A complete proof can be found in [14].
to noncoherent relaying, a long-term power constraint, G, is
To ease the presentation, we first sketch the basic idea. A
imposed on the relay, i.e.,
globally optimum Q exists for (4) due to the Weierstrass’
G=E ηr1 2 =η2[㵆Qµ+γαtr(Q)+1]. (2) Theorem [19]. We divide the feasible set into two subsets,
{| | }
represented by Q and Q . Specifically, denote as Q a
Thus, η = G/ 1+γ[µ Qµ+αtr(Q)] . In (2), we have 1 2 1
used: E h Qh { = tr(Q†), since h is} (0,I ). In the feasible but otherwise arbitrary covariance matrix, which has
second t{im†wepslowt,}the received signalwin theNrcelay-Mdestination noneofitseigenvectorsalignedwithµ.LetQ2 havethesame
eigenvectors as Q [see (6)]. Here Q is customized for
link is given by opt 2
the proof. Our goal is to show that given any Q , by proper
1
r2 =η hF r1+n2, powerallocationinQ2,wecanalwayshaveC(Q2) C(Q1).
≥
We will show that to maximize C(Q ), equal power must be
where h is the forward channel coefficient, and n is the 2
F 2 allocated to v ...v as in (7). We will also show that Q
noise at the destination with distribution (0,1). 2 M 2
c
N with its optimum power allocation achieves at least the same
Perfect knowledge of h and h is assumed at the desti-
B F
ergodiccapacityasQ does. Theoptimalityof Q canthen
nation. No knowledge about h is available at the source or 1 opt
F
be established. Below the proof starts.
relay.Theergodiccapacityoftheaboverelaychannelisgiven
To facilitate subsequent comparison, our first step is to
below [5]
exploit the eigen-structures of covariance matrices. Let Q
1
max 1E log 1+ η2γ|hF|2h†BQhB , (3) be eigen-decomposed as Q1 =UΛU† with
Q 0 2 hB,hF ( " η2 hF 2+1 #)
tr(Q(cid:23)) 1 | | U=[u1...uM],Λ=diag λ1,...,λM . (8)
≤ { }
where 21 isduetothehalf-duplexassumption.Thelogfunction Here u1...uM is an arbitrary orthonormal basis in CM,
here denotes natural logarithm, and thus the unit is nats per among{which non}e is aligned with µ, and M λ = 1. Let
i=1 i
channel use. After substituting η into (3) and applying a Q have the same eigenvectors as given in (6), i.e.,
2
technique used in [21, Appendix A], the ergodic capacity P
under mean feedback can be further shown equivalent to: M
Q =VΦˆV , Φˆ =diag φˆ ,φˆ ,...,φˆ , φˆ =1. (9)
2 † 1 2 M i
max C(Q) (4) { }
i=1
Q 0,tr(Q)=1 X
where (cid:23) Also define β , (β1...βM)T , U†µ. Clearly, µ = β .
From (8) and (1), we obtain µ Q µ= M λ kβ k2, ankd k
C(Q)= 1E log 1+ G|hF|2h†BQhB . † 1 i=1 i| i|
2 hB,hF ( " µ†Qµ+α+ G|hFγ|2+1#) h†BQ1hB =[β+√αh˘w]†Λ[β+√Pαh˘w]= αWQ1,
(5) where h˘ ,(h˘ ...h˘ )T ,U h has the same distribu-
w w1 wM † w
tion as h , h˘ + βi 2 is a noncentral chi-square random
2Theanalysisinthispapercanbeextendedtothecasewhentherelayhas w | wi √α|
multiple antennas. Details arenotdiscussedduetospaceconstraint. variable of two degrees of freedom with the noncentrality
parameter given by β 2/α, for all i [16, p. 43], and W , of thisp.d.f.doespossess a moreelegantstructure [16]. If we
i Q1
M λ h˘ + βi 2|. S|imilarly, µ Q µ=φˆ µ 2, and can avoid the p.d.f. and use its Laplace transform instead, we
i=1 i| wi √α| † 2 1k k
willbeabletoovercomethedifficulty.ItturnsoutthatLemma
P h†BQ2hB = [µ+√αhw]†VΦˆV†[µ+√αhw] 1 in Appendix I is the precise tool we need here.
=[( µ 0...0)+√αhˆ ]Φˆ [( µ 0...0)T +√αhˆ ] Based on Lemma 1, to show (17), it suffices to show that
k k †w k k w
=αWQ2, W Wˆ , subject to(16).
Q1 LT Q2
≤
w(hˆhwe1re..W.hˆQw2M,)Tφˆ1,|hˆVw1†h+wk√hµαaks|2th+esPamMi=e2dφiˆsit|rhˆibwuit|i2o.nHaesrhewhˆ,wan,d pL.edt.fM.ofWWQ1(s)(Wˆ[MW)ˆ.QA2c(sco)]rdbinegthtoeDLeafipnliatcioent1ra(nssefeorAmppoefndthixe
thus hˆ 2 is distributed as central chi-square of two degrees Q1 Q2
of fre|edwoim| (or simply, exponential), i=2,...,M. I), it is equivalent to show that MWˆQ2(s)≤MWQ1(s),∀s>
0,subject to(16), or,
Now, within the subset representedby Q , it can be shown
2
that, given any 0≤φˆ1 ≤1, among all Φˆ matrices [see (9)], log[MWˆQ2(s)/MWQ1(s)]≤0, ∀s>0, under(16). (18)
1 φˆ 1 φˆ
Φˆ∗ =diag φˆ1, − 1,..., − 1 (10) It can be shown that [16, p. 43]
{ M 1 M 1}
− − s
maximizesC(Q2);i.e.,tomaximizeC(Q2),equalpower 1M−φˆ11 log MWˆQ2(s)/MWQ1(s) =J(s)− αR(s),
must be allocated to v ...v [14]. Thus, we denote − h i
2 M
(1+λ s)...(1+λ s)
WˆQ2 ,φˆ1 hˆw1+ k√µαk 2+ M1−φˆ11 M |hˆwi|2. (11) J(s)=log(1+φˆ1s1) 1+ 1M−φˆ11sMM−1 (19)
(cid:12) (cid:12) − i=2  − 
Up to now, we(cid:12)(cid:12)(cid:12) have (cid:12)(cid:12)(cid:12) X (s)= φˆ1kµk2 M (cid:16)λi|βi|2 . (cid:17)  (20)
C(Q1)= 21EhB,hF{log[1+k1WQ1]}, (12) R 1+φˆ1s −Xi=1 1+λis
1 To show that (s) 0 for all s>0, note that M log(1+
C(Q2)= 2EhB,hF{log[1+k2WˆQ2]}, (13) tis) is a SchuJr-conc≤ave function in t = (t1...tMi=)T1, for all
P
s>0, and subject to (16),
where optimum equal power allocation among v ...v is
2 M
used in Q ,
2 1 φˆ 1 φˆ
Gh 2α φˆ1 − 1 ... − 1 (λ1...λM),
k = | F| , (14) M 1 M 1!≺
1 Mi=1λi|βi|2+α+ 1+Gγ|hF|2 i.e., the left-hand−side is −majorized by the right-hand
Gh 2α
k =P | F| . (15) side [14][18]. We can also show that (s) 0, s > 0,
2 φˆ1kµk2+α+ 1+Gγ|hF|2 byrepeatedlyusing(16).Thus,(18)holdRs,whi≥chimp∀liesthat
(17) holds. Since the construction of Q involves only (6),
Note that (12) and (13) differ not only in W and Wˆ , but 2
Q1 Q2 (10) and (16), none of which depends on h , and (17) holds
F
also in k and k , making further comparison prohibitively
1 2 for any h , we obtain
F
difficult. To proceed, we choose the only free (unspecified)
parameter in Q2, i.e., φˆ1, as follows: C(Q )= 1E E log(1+kW )
1 2 hF WQ1{ Q1 }
µ Q µ M λ β 2 M λ β 2
φˆ1 = k†µk12 = Pi=k1µki2| i| = Pi=k1βki2| i| , (16) ≤ 21EhF (cid:8)EWˆQ2{log(1+kWˆQ2)}(cid:9) =C(Q2). (21)
n o
such that k = k = k > 0 in (14) and (15). Since 0
1 2 BasedonthearbitrarinessofQ and(v ...v ),weconclude
miniλi ≤ µ†µQ12µ ≤ maxiλi ≤ 1, (16) is always valid. Th≤e that the optimum solution, as1it exists2, musMt have the same
choice of φˆkink (16) is crucial and will be shown to enable eigen-structureasQ .Also, asseenin theproof,equalpower
1 2
the final comparison. Naturally, our next step is to show that allocation among v ...v is necessary for optimality. At
2 M
this moment, the only parameter in Q available for further
E log(1+kW ) E log(1+kWˆ ) , (17) 2
WQ1{ Q1 }≤ WˆQ2{ Q2 } optimizationisφˆ1.Thoughφˆ1 chosenasin(16)issufficientto
when h is given and φˆ is chosen as per (16). guarantee (21) for a specific Q1, it can be potentially further
F 1
optimized to obtain φ as in Q . Therefore, for (4), Q
A straightforwardmethod to show the aboveis to calculate opt opt
[see (6)-(7)] is the optimum with φ numerically optimized
the expectationson bothsides of the inequality.The difficulty
according to the fading statistics of h and h . (cid:3)
here is that the calculation involves the probability density B F
function (p.d.f.) of a convex combination of M non-central Remark 1 The fading distribution of the relay-destination
chi-squarerandomvariables,whichistoocomplicatedtoserve channel h has no effect on the optimum transmit directions
F
our purpose [15]. On the other hand, the Laplace transform (eigenvectorsofthecovariancematrix)atthesource.However,
itdoesaffectthe optimumvalueofφ. Infact,φ isdetermined
by solving: max 1E log[1+Gh 2αc˜(φ)] , where 2
0≤φ≤1 2 { | F| } 1.8 OSupbti−moupmtimum
c˜(φ)= φ|hˆw1φ+kµkµαkk2|+2+α+M1−−1φ1+GPγ|hMiF=|22|hˆwi|2. nel use)11..46
n
a
This problem can be readily solved using one-dimensional ch1.2
search methods [19]. It is also interesting to see that the Nats / 1
capacitydependsonµonlythroughitsEuclideanlength µ . e (
Remark 2 When the relay power G , (4)kbek- Rat0.8
cthoemseasm:emmaxaQth(cid:23)e0m,atrt(iQca)=l1pr12oEblheBm{laosg[i1n+[9γ, Thh†BeQo→rheBm∞]}3,.1w].hTichhuiss, apacity / 0.6
C0.4
ourresultsubsumesasaspecialcasetheoptimum(traditional)
0.2
MISO precoding with channel mean feedback, and it is not
surprising to see the result in Theorem 1 and that in [9, 0
0 5 10 15 20 25
Theorem3.1]sharethesamestructure.3Inparticular,ourproof γ (dB)
here can also serve to prove [9, Theorem 3.1].
Fig. 1. Comparison of optimum and sub-optimum solutions for M = 2,
IV. OPTIMALITY OF BEAMFORMING ALONGµ α = 0.1, µ = (0.3518+j0.2496 −0.4039−j1.0437)T (kµk2/α =
14.3851),G=15dB.
Beamforming along the source-relay channel mean µ is
optimum if and only if all the source transmit power is
allocated to µ/ µ , and thus φ=1 and Q is rank-one.
opt
k k
Theorem 2 Assume that h is distributed as (0,1).4 signal-to-noise ratio (SNR). Here the “optimum” refers to
F c
N
Given γ, G, µ and α, beamformingin the direction of µ can Q which achieves capacity, and the “sub-optimum” refers
opt
achieve capacity if and only if to a sub-optimum Q (see the proof of Theorem 1) with
1
numerically optimized power allocation. The optimality of
1
E Z + E Zexp(Z)Γ(0,Z) Q is clearly shown in Fig. 1. At high γ, the difference
{ } G { } opt
E Z2exp(Z)Γ(0,Z) +D , (22) between the rates using the optimum and the sub-optimum
2
≤ { } diminishes. From Fig. 2, capacity increases with µ with
k k
where the expectation is taken with respect to the random otherparametersfixed.Similarobservationscanbemadewith
variable Z with the following probability density function differentsets of parameters,which are not presented here due
D µ 2 1 D to space constraint.
1 1
pZ(z)=αγz2 exp − kαk + αγ z −1 Fig.3givessimulationresultstocorroborate(22).Consider
(cid:26) (cid:20) (cid:18) (cid:19)(cid:21)(cid:27) the case with two antennas at the source (M =2). Given the
I0 (2 µ (D1/z) 1)/(α√γ) ; 0<z D1. parametersα,µ,andG,theoptimumφcanbedeterminedfor
× k k − ≤
aInndthDe 2ab,ov(cid:16)e,αDγw+1e1hpa1v−e dγekGfiµnk2edexDp(1D,1)Γ(α(γ(cid:17)0,+D11)+,γΓk(µak,x2))/Gis a(cid:0)foEsrpm(cid:8)eZicni2gfiecixsγpo,(paZtni)mdΓlu(em0t,f,Zt(hγ)e(cid:9))n+=φDE={2Z(cid:1)1.}[Ao+rc,Gc11oE−rd{φiZn=ge0xto]p,((aZ2n2d))Γ,f(i(0fγ,b)Ze≤)a}m−0-.
the complementaryh incomplete Gamma functioin [20, Eqs. This consistency is clearly reflected in Fig. 3.
(6.5.3),(6.5.15)],andI (x)isthezeroth-ordermodifiedBessel
0
function of the first kind [20, Eqs. (9.6.10), (9.6.16)].
Proof: Due to space limitation, the proof is omit- VI. CONCLUDING REMARKS
ted. Numerical methods are required to evaluate E Z ,
E Zexp Z Γ(0,Z) and E Z2exp Z Γ(0,Z) . (cid:3) { } The optimum source covariance matrix of a noncoherent
{ { } } { { } } half-duplex AF MISO relay channel has been determined
Remark 3 When G , (22) coincides with [10,
→ ∞ with channel mean feedback at the source. We have used a
Theorem 4, n =1].
R new method based on the Laplace transform order of two
V. NUMERICAL EXAMPLES nonnegative random variables. Our results subsume as an
asymptoticcasetheoptimumprecodingforatraditionalMISO
We now provide simulation results to corroborate the
link. The superiority of the optimum transmit strategy over
analytical results. We choose the number of antennas at the
sub-optimumoneshasbeenshownbysimulationsandisseen
source M to be 2. Fig. 1 shows capacity versus γ. Since
more pronouncedat low to medium transmit SNR. Necessary
the noise power is normalized to one, γ denotes the transmit
and sufficient condition for optimality of beamforming has
3Similar observations have also been reported in[5]with channel covari- also been derived. It is expected that the powerful Laplace
ancefeedback. transformorderingapproachusedinthispaperwillfindmany
4Note that the result in Theorem 1 holds with any fading distribution of
applications pertaining to stochastic optimization problems in
hF.However,theconditionforbeamformingtobeoptimumdoesdependon
thedistribution ofhF. wireless communications and signal processing [22].
dx),d>0,isapositivefunctioninxwhenx>0withitsfirst-
2
orderderivativebeingcompletelymonotone(seeDefinition2).
1.8 Details can be found in [14]. (cid:3)
1.6
ACKNOWLEDGMENT
e)
el us1.4 The authorswould like to thank Dr. P. Dharmawansa,Prof.
n
an1.2 R. K. Mallik, Prof. M. R. McKay, and Prof. K. B. Letaief for
h
s / c 1 helpful discussions.
at
N
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1 2 [16] J.G.Proakis, DigitalCommunications, McGraw-Hill, 2000.
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{ } ≥ { } ∀
ThenT issaidtobesmallerthanT intheLaplacetransform tions,AcademicPress,1994.
1 2
[18] A. W. Marshall, I. Olkin, Inequalities: Theory of Majorization and Its
order, denoted by T T .
Definition 2 [17, 1p.≤9L6T] 2 A function q: R+ R is said [19A]pEp.liKca.tiPo.nCsh,oAncga,dSe.mHic.Z˙Parkes,sA,n19In7t9r.oductiontoOptimization,3rdEdition,
→
to be completely monotoneif all its derivativesq(n) exist and Wiley,2008.
( 1)nq(n)(x) 0, for all x > 0 and n = 0,1,2,... (all [20] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions,
− ≥ DoverPublications, 1965.
nonnegative integer values). [21] M. Ding, S. D. Blostein, et al., “A general framework for MIMO
Lemma 1 Let T and T be two nonnegative random transceiver design with imperfect CSI and transmit correlation,” in Proc.
1 2
variables, and let d be any positive constant (d > 0). If IEEEPIMRC,pp.182-186,Sep.2009.
[22] M.DingandQ.T.Zhang,“StochasticoptimizationbasedontheLaplace
T1 LT T2, then E log(1+dT1) E log(1+dT2) . transformorderwithapplications toprecoderdesigns,”toappearinProc.
≤ { }≤ { }
Proof: The proof involves Theorem 3.B.4 (a) (p. 97) and IEEEICASSP2011,May2011.
Eq.(3.B.2)(p.96)of[17],andisbasedonthefactthatlog(1+