Capacity Bounds for Gaussian MIMO relay channel withChannel State Information

Sebastien Simoens, Olga Munozl, Josep Vidal1, Aitor Del Coso2

Motorola Labs, Paris, France. E-mail: simoens@motorola.comlUniversitat Politecnica de Catalunya (UPC), Signal Processing and Communications group, Barcelona, Spain

2Centre Tecnologic de Telecomunicacions de Catalunya (CITC), Casteldefells, Barcelona, Spain

Abstract-In this paper, source and relay precoders are derivedwhich optimize upper and lower bounds on the Gaussian MIMOrelay channel capacity. First, the prior art on the cut-set upper­bound on capacity is extended by showing that the optimizationof the source and relay codebooks can be formulated as a convexproblem without having to introduce a scalar parameter thatcaptures their cross-correlation. Both the Full-Duplex and TimeDivision Duplex (TDD) relay channels are addressed, assumingperfect knowledge of all channels, and two procedures areproposed which solve the problem efficiently by relying onanalytical expressions of gradients, subgradients and projectionoperators: the first one solves the dual problem while the secondone applies the barrier method. Similar techniques are then usedto maximize the achievable rate of Decode-and-Forward (DF)TDD MIMO relaying strategies with either partial or fulldecoding at the relay. Sub-optimum precoders are also proposedwhich have a closed-form expression that can be obtained fromthe KKT conditions, thus reducing the computational complexityat the expense of a lower rate. Simulations in a cellular downlinkscenario show that the partial DF strategy can achieve a rate veryclose to capacity for realistic values of the Source to Relay signal­to-noise ratio. Finally, the availability of Channel StateInformation (CSI) in a real system is discussed.

Index Terms-eooperative, relaying, decode, forward, CSI,MIMO

I. INTRODUCTION

Cooperative Relaying is regarded as a promising technique fornext generation cellular wireless systems (e.g. [1]) and isbeing considered by several standardization groups (e.g.IEEE802.16j). This paper focuses on the one-way single relaychannel, where the Relay (R) cooperates with the Source (S)and the Destination (D) to maximize the information rate fromS to O. Although full-duplex relaying is briefly addressed,most of the paper considers a two-phase TOO relaying l

protocol with a relay-receive slot of relative duration denotedby IE [0;1] and a relay-transmit slot of duration 1-1. Thenumber of antennas at S, Rand D are respectively denoted byNs , NR and ND and each one may be greater than 1. Thechannels from S to D, S to Rand R to 0 are all assumed staticand are denoted respectively H o, HI and H 2 • Fullknowledge of all channels is assumed to be available at all

1 Half-Duplex relaying lends itself more easily to practical implementation,by removing the need to physically isolate the relay transmit and receivechains

nodes, except in section IV where this assumption, called "fullCSI" in the rest of the paper, is challenged.

The maximization of the Gaussian MIMO channel capacitywith perfect CSI at the source is solved by Telatar in [2]. Inthe landmark paper [3], the capacity of the full-duplex single­relay channel is studied: the Decode-and-Forward (OF)(Theorem 1) and Compress-and-Forward (CF) (Theorem 6)coding strategies are introduced and an upper-bound on therelay channel capacity (a.k.a. cut-set bound) is provided. InOF the relay decodes at least a part of the message transmittedby the source, whereas in CF the relay observation iscompressed and forwarded to the destination as a newcodeword. In this paper, we focus on OF. Neither OF nor CFis capacity-achieving in the general relay channel, but OFmeets the cut-set bound when R gets infinitely close to S,while CF outperforms OF when R gets close to 0, ultimatelyachieving the capacity when their distance goes to zero (seee.g.[4]). Other strategies exist like Amplify-and-Forward (AF)which can be viewed as a sub-case of Linear Relaying (LR).One advantage of LR is processing simplicity and we referthe reader to e.g. [5] for the optimization of linear relayingwith full MIMO CSI. A drawback of LR in the single TOOrelaying case is that the time-sharing is constrained to 1=1/2,which severely limits the achievable rate. This effect can bealleviated by two-path relaying (when multiple relays areavailable) or two-way relaying (which applies to bidirectionaltraffic) for which precoder optimization can also be performed[8].

In [6], the cut-set bound and the achievable rate of a partialOF strategy are expressed in the case of Gaussian TOOrelaying, assuming single antenna devices. In [7], an upper­bound and several lower-bounds are derived for the full­duplex MIMO relay channel with full CSI. In [9], twostrategies are introduced which improve the achievable rateon the full-duplex MIMO relay channel. The present paperaims at contributing to the study of MIMO relaying asfollows:

• Two alternative convex optimization procedures arepresented in section II, which compute the cut-setupper-bound on both full-duplex and TOO MIMOrelay channels without requiring the introduction of ascalar parameter to capture the cross-correlation as in[7].

• The achievable rates of OF strategies with either partialor full decoding at the relay are computed for the TOOMIMO relay channel in section III. Moreover, sub-

978-1-4244-2046-9/08/$25.00 ©2008 IEEE 441 SPAWC 2008

2

(3)

(2)

(1)

(4)

(8)

which is convex in (R~) ,R~)) for a given t .

B. Solution ofthe problems

The procedures presented in this section aim at solving (8) and(10) efficiently by relying on the convex optimizationliterature [10][11]. Let first focus on (8). It can be turned intoa differentiable problem by writing the equivalent epigraph[10] form of the problem:

CFD ::;-min(-£)RSR~O

s.t. £-CA ::;0; £-CB ::;0 (11)L...-v---' L...-v---'

~fi(E,RsR) ~f2(£,RSR)

~r(DsRsRD~)-PsJ::;O and ~r(DRRsRD~)-PRJ::;O

~f3(;,RSR) ~f4(;,RsR)

This problem is convex in (£,RSR ) on }R+xS+ where S+ is thecone of PSO matrices. For notational convenience weintroduce the following column vector:

v~[£ vec{RSR)TJT (12)

where vec{.) is the operator that stacks the colums of a matrixto generate a column vector.

Starting from (11), a first method to compute the cut-setbound consists in solving the dual problem:

~g(Jl) where g(Jl)~£.k~(-:£+J1T~{£'RSR~J (13)~L(v,Jl)

where (a) comes from equation (5) and (b) from the positive

semi-definiteness of R SR •

The cut-set bound (2) can be expressed as:CFD ::;maxmin(CA,CB )

RSR~O

CA =log2lINs+DsRsRD~ (H~Ho+HfHl)1

CB =log21IND+[Ho H 2]RsR [Ho H2]HIs.t. tr(DsRsRD~)::;~ and tr(DRRsRD~)::;~

where Ps and PR are the maximum transmit powers at Sand

R. The objective is concave on the PSO cone, because it is thepointwise minimum of two concave functions [10]. Thereforethe problem is convex with respect to a unique variable R SR •

In the case of TOO relaying, the cut-set bound can beexpressed as follows (see equation (77) in [6]):

{tI(X(l).y(l) y(l) Ix(l) =O)+(I-t)I(x(2)'y(2)lx(2») }

C ::; max min s' R' DRS 'D R ,roD ~t~l tI(x(l)'y(l)lx(l) =O)+(I-t)I(x(2) X(2)'y(2»)

p(xg),x~2),x~2») S' DRS , R , D

(9)

where the superscript (i) indicates the slot during which thesignal was transmitted or received. Using similar arguments asin the full-duplex case, the cut-set bound for the TOO MIMOrelay channel can be expressed as:

CroD ::; max min{CA,CB}~t$l

R~)~O,R~h:o

CA =tlog2lINs+R~)(H~Ho+H~Hl)I+(I-t)log2IIND +HoDsR~~D~H~1

CB =tlog2 II ND+HoR~)H~I+(I-t)log2IIND +[Ho H2]R~)[Ho H 2]HI

S.t. tr(R~»)::;~ ; tr(DsR~D~)::;~ and tr(DRR~~D~)::;~

(10)

(7)

optimum source and relay precoders are proposed withanalytical expressions derived from KKT conditions.

• The upper and lower bounds are benchmarked bysimulations in a cellular downlink scenario (section IV)before discussing on CSI availability in real systems.

We start by introducing the notations and formulating thecut-set bound as the solution of a convex optimizationproblem in the full-duplex and TOO relaying cases, beforeproposing methods to solve it.

A. Formulation ofthe problem

In full-duplex MIMO relaying, the signals received at therelay and destination can be written as in [7]:

YR=H1xS+DRYD =Hoxs+H2XR+DD

where circularly symmetric complex white Gaussian noise ofunit variance is assumed at the relay and destination, Le.DR -CN(O,INR ) and DD -CN(O,I ND ). The expression of the

cut-set bound is given by equation (3) in [7]:

CFD::; max min(I(xs;YD,YRIxR),I(xs,XR;YD))P(XS,XR)

where the maximization is performed over the distribution ofthe source and relay codebooks p(XS'xR). The authors in [7]

show that Gaussian codebooks are optimum and conclude thatthe optimization of (2) must be carried on w.r.t. three matricesRs~E[xsx~] , RR ~E[XRX~] and the cross-correlation

E[xsx~] . This optimization seems highly non-trivial and the

authors resort to introducing a scalar parameter p that

captures the cross-correlation in order to finally obtain(Theorem 3.1) an upper-bound which involves a maximizationonly over Rs , RR and p. This bound is however not

evaluated in the fixed channel case in [7] and its numericalcomputation seems complex because evaluating the objectiverequires to solve an optimization problem (equation (9) in [7]).

However, let define the joint covariance matrix:

R ~[Rs E[XSX~]]SR - E[XRX~] RR

Let also define the following matrices:

Ds ~[INs ONsxNR] and DR ~[ONRxNs INR ]

From (3) and (4), the following relationships hold:

Rs=DsRsRD~ and RR=DRRsRD~ (5)

Note that if RSR is Positive Semidefinite (PSO), then Rs and

RR are PSO too:

RSRtO=>RstOandRR~O (6)

Indeed, for any two vectors xe eNs and ye e NR , defining

i ~ D~x and Y~D:Y, the following holds:(a) (b)

xHRsX= xHDsR SR DsHX= i H RSRx ~ 0(a) (b)

yHRRY =yHDRRsRDRHy=yHRsRY ~ 0

II. THE CUT-SET BOUND WITH FULL MIMO CSI

442

and

(23)

L(V,J.1) denotes the Lagrangian and Cis the column vectorformed by stacking the 4 inequality constraints j;. Note thatfrom the weak Slater's conditions (5.27) in [10], strong duality(Le. zero duality gap) holds in (11) and therefore solving (13)or (11) yields the same value for the cut-set bound. Thecomputation ofthe dual function g(J.1) at a given point J.10 canbe performed by the Gradient Projection Method (GPM, seesection 2.3 in [11]). The gradient of the Lagrangian at apoint v0reads:

VL1YO ~(dLI vec(2~ JTJT (14)dE Yo ~YO

An analytical expression ofthe gradient can be obtained fromthe following formulas (see [12] and [13]):

dloglA+BXq [C(A+BXC)-lB]H. dtr(AXAH)_AHA (15)d~ , ~.

The GPM is an iterative algorithm which computes at stepk the following points:

V(k+1) =V(k) +a(k) (V(k) _V(k») and V(k) =[V(k) -S(k)V LIv<k)]+ (16)

where S(k) is a positive scalar, a(k)E (0,1] is the step sizeand [.]+ denotes the projection on the feasible set. Here, whenminimizing the Lagrangian, a projection on the PSO cone isrequired. This projection is applied by forming a Hermitian(but not necessarily PSO) matrix denoted by M from the(Ns +NR)2 last coordinates of the candidate pointV(k) -S(k)V Lly(k) then, considering its eigenvalue decompositionM=Udiag(~)UH ,the projected matrix is ([13]):

M+ =Udiag(max(A;,O»UH (17)

There exist various methods to select parameters S(k) and a(k)

in order to ensure convergence. In our simulations we used theArmijo Rule along the feasible direction (sec. 2.3 in [11]).

In order to solve the dual problem, it remains to maximizethe dual function with respect to J.1. From (13), the dualfunction may not be differentiable. However, as shown below,an analytical expression of a subgradient can be found (seealso sec. 6.3 of [11]). Since the dual function is concave in J.1,a vector h is a subgradient of g at J.10 if for all J.11 :

g(J.1I)~g(J.1o)+hT (fJ1-fJO) (18)

Let Vo and VI be minimizers of the Lagrangian at respectivelyJ.10 and J.11 . Then

g("l)~L(Vl'''I)~L(vo'''I) (19)

=> g(fJ1)-g(fJo)~L(VO,J.11)-L(vo,J.1o) =f (VO)T (PI-PO)

where the constraints f were rewritten as a function of v.From (19) and (18) it can be concluded that C(vo) is asubgradient of g at J.10, and the dual can be solved efficientlyby subgradient methods. We refer the reader to sec. 6.3 of [11]for step size selection strategies.

We have described a procedure that computes the dualusing the GPM and maximizes it via a subgradient method.Another alternative is to solve the primal problem using thebarrier method [10], which we only briefly describe here forlack of space. In this case, a (logarithmic) barrier of the form-m loge- J;) is added to the objective for each inequalityconstraint before minimization, with m a positive parameterthat is decreased at each outer iteration of the method. The

3

constraint R SR ~°can be taken into account by adding thebarrier -mloglRsRI (Theorem 5.1 in [14]). As before, ananalytical expression ofthe gradient can be found.

The two previously described optimization procedures canbe directly applied to compute CFD' but computing CTDDrequires to run a one-dimensional maximization over t on topof the convex optimization. Fortunately, the search over tcan be performed by efficient techniques such as the GoldenSection Search. Indeed, it can be checked numerically that thefollowing function of t is unimodal (i.e. monotonicallyincreasing until a maximum, then monotonically decreasing):

max min{CA,CB} (20)R~l)~O,Rg>~O

In this section, the cut-set upper-bound on capacity wasexpressed in the full-duplex and TOO MIMO relaying cases,and two numerical evaluation procedures were proposed. Thenext section will address the maximization of achievable rates.

III. OECODE-AND-FoRWARD WITH FULL CSI

A. The Partial DF strategy

In a partial OF scheme [4][6], S transmits a first message (Va atrate ~ using signal x~) (at) during the first slot. The relay(R) transmits X~2) (at) during the second slot while Stransmits X~2) (~,~ )= X~~6 (~ )+ x~~i (~). This strategy isoften called partial OF because R only decodes the part ~ ofthe message (~,~) to cooperate with S during the secondslot. Because we assume a synchronized scenario, the signalsX~2) (at) and X~~6 (at) are correlated, whereas ~ is mappedonto an independent signal x~~i (~ ) transmitted at rate R1

using superposition coding. The destination successivelydecodes ~ and ~ . The derivation of the achievable rate is astraightforward extension of the proof of Prop. 2 in [6] andleads to:RDF = max Ro+RI = max min(RA,RB ) (21)

t,p(x~),x~~b,x~~~,x~») t, RY)~O, R~3~0, R~~,o~O

RA ~tlog2lINR +H1R~)Hfl+(I-t)log2IIND +HoDsRi~~D~H~1 (22)

RB~tlog2IIND+HoR~)H~1

+(1-t)log2IIND+HoR~~lH~ +[Ho H2]R~~,0[Ho H 2 ]HI

where R(I) ~E[x(1) (X(l»)HJ R(2) ~E[X(2)(X(2»)HJS s S , S,l S,l S,l

R(2) ~E[[(X(2»)T (X(2»)TJT

[(X(2»)H (X(2»)HJ] and the transmitSR,O s,o R S,O R

power constraints can be written as:tr(R~»)~Ps ; tr(DRR~loD~)~PR (24)

tr(R~D+tr(DsR~,oD~)~~ (25)

The problem (21) is convex and the maximization of theachievable rate of the partial DF strategy can be carried onusing the methods described in the previous section.

B. Sub-optimum strategies

We now consider a sub-optimum strategy, that we call full OF(FDF) which is defined as a variant of partial DF in which therelay decodes the whole message and the source does not

443

(36)

(35)

superimpose a new message during the second slot (Le.R1=0 ). In this case, the achievable rate simplifies as:

RFDF = max min(tRA,tRBl+(l-ORB2) (26)t,R~)~,R~h:o ' ,

RA ~log21INR +HIR~)H~I (27)

RB,1~log2IIND +HoR~)H~1 (28)

RB,2 ~log21IND +[Ho H2]R~~[Ho H2]HI (29)

s.t. tr(R~»)~Ps ; tr(DRR~~DZ)~PR ; tr(DsR~~D~)~Ps (30)

For a given t, it can be noticed that RFDF is an increasingfunction of RB ,2' which only depends on R~~. Therefore theoptimization (26) can be carried on in two steps:

RB2~maxRB2 (31), R~~~ '

RFDF = max min(tRA,tRB1+(l-ORB2) (32)t,R~)to ' ,

Although RFDF can be solved using the procedures describedin section II.B, we decide to evaluate sub-optimum precodersat the Source and Relay with a structure that further reducesthe optimization complexity.

1) Sub-optimum Source precoder during 1st slotLet first consider the optimization of R~) and introduce the

SVD of Hoand HI:

Ho=UoAoV: and H1=UIA1Vt (33)

withAo~diag(~,i) and AI~diag(~,;). We arbitrarily impose

the following structure to the source covariance matrix:R~)=Vodiag(po)V: +Vldiag(PI)VI

H (34)'--v----'" '---y---J

~Ro ~Rl

The structure (34) stems from the intuition that the sourceshall transmit part of its power on the eigenmodes of thechannel to the relay and the rest on the eigenmodes of thechannel to the destination (note the similarity with theprecoder optimization in [8]). Let Loand LI be the number of

non-zero singular values of Ho and HI. Inserting (34) into

(27) and (28) gives:

RA=log2II NR+HIRoH~ +HIRIH~I(a) I HI(b)~Ll 2 A

~log2 I NR +H1R1H1 = LJi=llog2(I+~,iPI,i)=JI

(a) I HI(b)~Lo 2 A

RB,l ~log2 I ND +HoRoHO = LJi=llog2(I+~,iPo,i)=Jo

Where (a) comes from the Minkowski determinant inequality

and (b) comes from (34). Inserting these lower bounds on

RAand RB,1 into (32) yields a lower-bound on RFDF :

RFDF ~maxmin(tJI,tJo+(I-t)RB,2)t,Po,Pl

s.t. Po~O; PI~O; lLPo+litPI =Ps

From the epigraph form of (36), writing the KKT conditionsleads to:

PO,i =(a-I/~i)+ ;PI,i =(fJ-I/ ~~i)+ (37)

The solution (37) can be obtained by a water-filling algorithmwith two water levels a and fJ which are not independentdue to the total source power constraint in (36). A fraction

4

~ ~ (1 T Pi )/Ps of it will be transmitted on the source-relaychannel eigenmodes, while the rest 1-~ will be transmittedon the eigenmodes of the source-destination channel. Findinga and fJ amounts to finding the optimum i1. E [0;1]. It can bechecked that J1is non-decreasing with ~ while Jois non­increasing. Therefore, the optimum 11 can be found byequating the left-hand and right-hand sides of the min(.) in(36), provided that (I-t)RB,2 ~tJI at ~ =1. Otherwise thesolution is trivial (Le. A=0 or A=1). In summary, wepropose in this section a sub-optimum approach which has lowcomplexity. It consists in transmitting a fraction of the sourcepower on the eigenmodes of the channel to the relay and therest on the eigenmodes of the channel to the destination, afterdetermining the optimum balance. Note that equality occurs in(35) when the two channels tum out to be orthogonal, and inthis case the solution (34) is optimum. Also note thatoptimality is approached anyway at large Ns because (notshown here for lack of space) assuming Ho and HI havei.i.d. Rayleigh components, the distribution of the ratioIIHf HoII: /(1IHoII~ IIH ill~ ) concentrates around 0 as Ns grows.

2) Sub-optimum Source and Relay precoder during 2nd slot

Let now consider the optimization of R~~ . Introducing the

SVD [H2Ho]=UAVH and the change of variable

R~VHR~~V , the problem (31) can be rewritten as:

RB2 =maxlog2lI N +ARAHI' R~O D (38)

s.t. tr(DsVRVHDf)~ps ; tr(DRVRVHD~)~PR

As in appendix B of [7], we arbitrarily enforce a diagonalstructure R =diag(P2). This turns the matrix optimization

problem (38) into the following vector optimization:L

maxLlog2(1+~2P2,i)P2 ;=1 (39)

s.t. aTP2 ~Ps ; bTP2 ~PR ; P2 ~o

where the components ofvectors a and b can be computed as

a i ~ L;~IIVNR+j,i12 and bi ~L;:IIVj ,i12. The problem (39) can

then be solved at a much lower computational cost than (31).Note that if the individual power constraints in (38) are

replaced by a sum-power constraint tr(R~i )=tr(R) ~Ps +PR ,

then Hadamard determinant inequality can be applied as in [2]to show that the optimum R is diagonal. In other words,under a total transmit power constraint at the source and relay,transmitting on the eigenmodes ofthe joint channel [H2H o] isoptimum.

IV. ApPLICATION To THE DOWNLINK OF CELLULAR SYSTEMS

In this section, we illustrate the application ofthe upper andlower bounds on the TDD MIMO relay channel in a cellulardownlink scenario. Let assume a Base Station (BS) equippedwith Ns = 2 antennas per sector. The BS is in Line-Of-Sight(LOS) with a dual antenna NR =2 fixed Relay Station (RS),

444

5

Figure 1: Upper and Lower bounds on TDD MIMO relaying channelcapacity with Source and Relay precoder optimization in a 2x2x2 antennaconfiguration.

State-of-the-art TDD systems often rely on uplink!downlinkchannel reciprocity to enable CSIT exploitation withouthaving to explictly feed back the channel coefficients.However, in cooperative relaying it is hard ifnot impossible toavoid explicit signaling because for instance the BS cannotestimate the RS-MS channel by means of reciprocity.Quantized codebooks [15] allow to reduce the amount offeedback. For instance, the MS can estimate Ho and H 2 andfeedback the index of the best codeword to the BS and RS.Some preliminary simulation results (not detailed here for lackof space) obtained with Grassmannian beamformingcodebooks in the Ns =2;NR =2;ND =1 case indicate that only 4bits of feedback (Le. a codebook of size 16) are required to

and both cooperate to transmit to a dual antenna MobileStation (MS) ND = 2. The MS is in NLOS from both the BSand the RS. The MIMO channels are modeled by i.i.d.complex gaussian components in the NLOS case, and a 10dBRice factor is introduced in the LOS case. The SNR on theBS-MS link (SNRo) and on the RS-MS link SNR2 areassumed equal SNRo =SNR2 =OdB. On Fig. 1, the averageachievable rate of the partial DF and full DF (FDF) strategiesare plotted as a function of the SNR on the BS-RS link( SNR1 ). The 2x2 MIMO channel capacity is also plotted forreference with (solid line) and without (dashed line) CSI at thetransmitter side, in order to highlight the waterfilling gain. Itcan be checked that single-hop transmission outperforms non­cooperative DF relaying even at high SNR1 , but isoutperformed by all cooperative DF strategies (even the sub­optimum FDF with vector optimization) when SNR1 exceeds athreshold around 3 dB. As expected, the partial DFoutperforms FDF and both achieve rates quite close to the cut­set bound at high SNR1 (only 0.2 bit per channel use away atSNR1 =20dB ). Also note that the cooperative DF strategiesexploit a larger total transmit power during the 2nd slot.However, it can be verified (not plotted here) that cooperativeDF still improves upon both single-hop and non-cooperativeDF under the same sum-power constraint.

[1] Nosratinia, T. E. Hunter, A. Hedayat, "Cooperative Communication inWireless Networks", IEEE Communications Magazine, Oct 2004

[2] E. Telatar, "Capacity of multi-antenna gaussian channels" EuropeanTrans. Telecommun., vol. 10, pp. 585--595, 1999

[3] T. M. Cover, A.A. EI Gamal, "Capacity theorems for the relay channel",IEEE Trans. on Information Theory, vol. IT-25, No 5, Sep 1979

[4] G. Kramer, M. Gastpar, P. Gupta, "Cooperative Strategies and CapacityTheorems for Relay Networks", IEEE Trans. on Information Theory,Vol 51. No 9. Sep 2005

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[9] C.K. Lo, S. Vishwanath, R.W. Heath Jr. "Rate Bounds for MIMO RelayChannels Using Precoding" Proc. IEEE Globecom '05. Nov 2005.

[10] S. Boyd and L. Vandenberghe "Convex Optimization" CambridgeUniversity Press, 2004. ISBN 0521833787. Available at:http://www. stanford.edu'~boyd/cvxbook '

[11] D. P. Bertsekas "Nonlinear Programming". Athena Scientific. ISBN 1­886529-00-0. 2nd edition, 1999.

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[14] MJ. Todd "Semidefinite Optimization" in Acta Numerica 10, pp 515­560. Cambridge Univ. Press. 2001.

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ACKNOWLEDGMENT

REFERENCES

This work was performed in the framework of the FP7 ICT ROCKETproject, which is funded by the European Commission.

approach the average FDF achievable rate by less than 0.5 bitper channel use. The design of quantized codebooks forcooperative relaying thus seems a promising topic.

v. CONCLUSION

The cut-set bound on the MIMO relay channel capacity can beefficiently computed in the full-duplex and TDD cases byconvex optimization techniques, assuming full knowledge ofall channels. Moreover, the same techniques can be applied toderive the achievable rate of partial DF with source and relayprecoders optimization. Sub-optimum strategies and precoderswith lower complexity can also be considered at the expenseof lower rate. Finally, the exploitation of partial and statisticalchannel knowledge and the application of quantizedcodebooks seem promising topics in order to approach in areal system the achievable rate gains derived in this paper.

302510 15 20SNR on Source-Relay link (dB)

• __•• No Relay, no CSIT

- No Relay

3.5 -e- Non coop. OF-+- FOF, Vect. Optim.--A- FOF, Mat. Optim.--a- Partial OF___ Cut-Set Bound

Q) 3~

Q)ccjg 2.5o

445