Optimal Spatial Correlations for the Noncoherent MIMO Rayleigh Fading Channel

Transcription

1 1 Optmal Spatal Correlatons for the Noncoherent MIMO Raylegh Fadng Channel Shvratna Gr Srnvasan and Mahesh K. Varanas e-mal: {srnvsg, Electrcal & Computer Engneerng Department Unversty of Colorado, Boulder, CO Ths work was supported n part by NSF grants CCF and CCF

2 1 Abstract The behavor n terms of nformaton theoretc metrcs of the dscrete-nput, contnuous-output noncoherent MIMO Raylegh fadng channel s studed as a functon of spatal correlatons. In the low SNR regme, the mutual nformaton metrc s consdered, whle at hgher SNR regmes the cutoff rate expresson s employed. For any fxed nput constellaton and at suffcently low SNR, a fully correlated channel matrx s shown to maxmze the mutual nformaton. In contrast, at hgh SNR, a fully uncorrelated channel matrx (wth ndependent dentcally dstrbuted elements) s shown to be optmal, under a condton on the constellaton whch ensures full dversty. In the specal case of the separable correlaton model, t s shown that as a functon of the receve correlaton egenvalues, the cutoff rate expresson s a Schur-convex functon at low SNR and a Schur-concave functon at hgh SNR, and as a functon of transmt correlaton egenvalues, the cutoff rate expresson s Schur-concave at hgh SNR for full dversty constellatons. Moreover, at suffcently low SNR, the fully correlated transmt correlaton matrx s optmal. Fnally, for the general model, t s shown that the optmal correlaton matrces at a general SNR can be obtaned usng a dfference of convex programmng formulaton. Index Terms Block fadng channels, noncoherent, MIMO, spatal correlaton, mutual nformaton, cutoff rate expresson, general SNR, schur-convexty, schur-concavty, global optmzaton, d.c. programmng, concave mnmzaton. I. INTRODUCTION Practcal MIMO channels exhbt correlatons between path gans of the antenna elements. It s therefore mportant to understand the effect of spatal correlatons on the channel capacty snce ths helps n optmally desgnng the transmt and receve antenna arrays. For ths reason, the effect of spatal correlatons on the capacty of the coherent MIMO Raylegh fadng channel where the channel realzatons are assumed to be known at the recever but only the long term statstcs are known at the transmtter s studed n several papers ncludng [1 5]. In ths paper, we consder the more challengng noncoherent MIMO Raylegh fadng channel where the transmtter and the recever have only knowledge of the long term statstcs and nether has knowledge of the channel realzatons. The ratonale for studyng the noncoherent model s ths. Snce n practce the channel s not known to the recever at the start of communcaton, an nformaton theoretc formulaton of the noncoherent problem whch mplctly accounts for the resources needed for (mplct) channel estmaton wthout constranng the transmsson scheme n any way s more fundamental than the coherent formulaton. Systems that assume coherent transmsson by argung that the

3 2 channel can be acqured at the recever by the use of plot-symbol asssted transmsson to perform explct channel estmaton ether (a) do not take nto account the resources (power and degrees of freedom) needed for plot transmssons or (b) when they do (as they should), they ncur a sgnfcant loss of optmalty n regmes nvolvng short coherence tmes and/or low SNRs, mplyng that explct plot-asssted channel estmaton s hghly sub-optmal n these regmes. We study the problem of noncoherent mult-antenna communcaton n the context of the general spatal correlaton model of [6], referred to n [2] as the Untary-Independent-Untary (UIU) model, whch subsumes the well known separable transmt and receve correlaton model of [1, 7] and the vrtual channel representaton model of [8]. Whle even the UIU Raylegh fadng model does not capture the most general form of correlatons, t s vewed as a reasonable compromse between valdty and analytcal tractablty. Justfcaton for the UIU Raylegh fadng model s gven n [9] based on physcal measurements. The partcular case of the separable model s ustfed n [1] as an approxmaton, whle [7] ncorporates physcal parameters lke the angle spread and antenna spacng n ths model. Consequently, we often specalze our results to the separable model and obtan sharper results for t. A summary of the man results n [2,3] whch compare the coherent capacty for the separable model wth transmt and receve correlatons wth the..d. fadng model, s as follows : 1. Receve correlaton reduces the capacty at every SNR. 2. For N t N r, transmt correlaton reduces the capacty at hgh SNR. 3. For N t > N r, transmt correlaton ncreases the low SNR capacty. The applcaton of the theory of maorzaton to such problems (c.f. [1,4,5]) helps n provdng a more complete understandng of how the correlatons affect the performance measure of nterest, and t ads comparson of correlated channels. The analyss n [4] s specfc to the coherent capacty of the MISO channel (N t > 1 and N r = 1) and t s shown that at a general SNR, the capacty s a Schur-convex functon wth respect to the egenvalues of the transmt correlaton matrx. Ths means that hgher transmt correlatons result n hgher capacty at all SNRs for the MISO channel. In [5], t s proved that the parwse error probablty (PEP) between every par of symbol matrces s a Schur-convex functon of the receve correlaton egenvalues. Ths means that hgher correlatons at the recever result n hgher PEPs at every SNR, whch ndcates that hgher correlatons are detrmental to error probablty.

4 3 Wth regard to the noncoherent MIMO Raylegh fadng channel, a recent paper by Wu and Srkant [10] shows that at asymptotcally low SNR, a fully correlated channel maxmzes the relablty functon. In other SNR regmes however, lttle s known about the effect of spatal correlatons on the noncoherent MIMO channel performance. The man stumblng block n the analyss of the noncoherent channel s the absence of a closed form expresson for the capacty. Indeed, the problem of fndng the noncoherent MIMO capacty s one of the longstandng open problems, partal characterzatons of whch may be found n for nstance [11 14]. We therefore adopt the cutoff rate for our analyss and obtan useful nsghts n ths regard. The cutoff rate s a lower bound on capacty and was prevously used by [15] (and the references contaned theren n dfferent contexts), to analyze and characterze optmal constellatons for the peak-power constraned noncoherent MIMO..d. Raylegh fadng channel. In [16], the cutoff rate expresson s used as a crteron to desgn constellatons for the average-power constraned noncoherent MIMO Raylegh fadng channel at general SNRs. In ths paper, we maxmze the cutoff rate expresson wth respect to the channel correlaton matrces for arbtrary but fxed sgnal constellatons. We observe that the cutoff rate at suffcently low SNR behaves exactly the same way as the mutual nformaton upto second order, and hence the results hold for the mutual nformaton as well n ths regme. Our man results are as follows : 1. At suffcently low SNR, we prove that the mutual nformaton and cutoff rate expresson are maxmzed by a fully correlated channel matrx. For the separable model, the cutoff rate expresson s thus maxmzed by fully correlated transmt and receve correlaton matrces. In the separable case, we show the sharper result that the mutual nformaton s n fact a Schur-convex functon of the receve correlaton egenvalues. Ths ndcates that at low SNRs, t helps to have more correlatons at the receve antennas, whch s n contrast to results n the coherent case. 2. At asymptotcally hgh SNR, and under a condton that ensures that the constellaton acheves full dversty, we show that the cutoff rate expresson s maxmzed by a fully uncorrelated channel matrx. In the case of the separable model, we prove the sharper result that the cutoff rate expresson s a Schur-concave functon of the transmt and receve correlaton matrces. 3. We show how the optmal correlaton matrx may be obtaned at a general SNR, usng standard global optmzaton formulatons. In partcular, we transform such problems nto standard

5 4 global optmzaton problems lke dfference of convex (d.c.) programmng and concave mnmzaton, and ndcate algorthms that obtan the globally optmal soluton. Notaton : For an nteger N, I N s an N N dentty matrx. Matrces are denoted by the boldfaced captal letters, and vectors by bold faced small letters. The symbol denotes the Kronecker product. The matrces X T, X and X denote the transpose, complex-conugate, and conugate transpose respectvely. We denote the nner product between two vectors x and y by < x,y > = x y and the norm x = < x,x >. E[.] denotes the expectaton operator. dag(a 1, a 2,...,a N ) s an N N dagonal matrx wth dagonal elements a 1, a 2,...,a N. We use the notaton o(ρ) to mean that lm ρ 0 o(ρ) ρ = 0. II. SYSTEM MODEL We consder a communcaton system wth N t transmt antennas and N r receve antennas. We assume a block fadng channel where the channel matrx H IC Nt Nr s assumed to be constant for a duraton of T symbols, after whch t changes to an ndependent value. The channel matrx s assumed to be unknown to the transmtter and the recever, whle the channel statstcs are assumed to be known at the transmtter. The receved sgnal s R = γsh + W, (1) where S IC T Nt s the transmtted symbol matrx and W IC T Nr s the nose matrx. Here, the symbols {S} are normalzed such that E[tr(SS )] = 1, so that the average transmt power equals γ. It s assumed that W has..d. crcularly symmetrc CN(0, 1) entres. We next descrbe the form of the channel matrx H, whch has correlated, crcularly symmetrc, complex gaussan entres. A. Untary-Independent-Untary (UIU) Raylegh fadng model In the UIU Raylegh fadng model, the channel matrx s assumed to be of the form H = U t HU r, (2) where U t and U r are the transmt and receve untary matrces. The elements of H are uncorrelated and zero-mean, crcularly symmetrc, complex gaussan, but not necessarly wth the same varance. Defne h = vec(h). Assumng that Σ = E[vec( H)vec( H) ] = Λ, whch s a

6 5 non-negatve dagonal matrx, we have Σ = E[hh ] = (Ūr U t )Λ(Ūr U t ). (3) Let {λ } NtNr =1 be the egenvalues of Σ. The normalzatons n (1) are assumed to be such that NtN r =1 λ = N t N r. B. Separable Transmt and Receve Correlaton model For the separable transmt and receve correlaton model, H s represented by H = Σ 1/2 t H w Σ 1/2 r, (4) where H w has..d. crcularly symmetrc CN(0, 1) entres. The matrces Σ t and Σ r are the transmt and receve array correlaton matrces, wth egenvalues {λ t }Nt =1 and {λr }Nr =1, respectvely. Substtutng the egenvalue decompostons for Σ t = U t Λ t U t and Σ r = U r Λ r U r, we get that H = U t HU r, (5) where H = Λ 1/2 t U th w U r Λr 1/2. The normalzatons n (1) are assumed to be such that N t =1 λt = N t and N r =1 λr = N r. Snce U t H wu r has the same dstrbuton as H w, t can be seen that wth h = vec( H), Σ = E[ h h ] = Λ r Λ t. We may therefore obtan the correlaton matrx of h = vec(h) as E[hh ] = (Ūr U t )(Λ r Λ t )(Ūr U t ). (6) Comparng (3) and (6), we see that the man dfference between the separable and UIU Raylegh fadng model s that the egenvalue matrx of the channel correlaton matrx n the separable model s a Kronecker product, whle there s no such restrcton n the UIU Raylegh fadng model. C. Effectve channel model and output probablty densty functon (p.d.f.) The output of the channel n (1) usng ether assumpton on the channel matrx H can be wrtten as R = γsu t HU r + W. (7)

7 6 After post-multplyng (7) by U r, denotng SU t by X, and denotng RU r by Y, we get Y = γx H + N, (8) whch represents the suffcent statstcs of the receved sgnal. Clearly, E[tr(SS )] = E[tr(XX )], and hence the precoded constellaton {X} satsfes the same average-power constrant as the orgnal constellaton {S}. N has..d. crcularly symmetrc CN(0, 1) entres snce t has the same dstrbuton as W. We may hence consder (8) to be our effectve channel model, and use the notaton X to denote a constellaton matrx precoded by the transmt untary matrx. Note that ths amounts to a form of statstcal beamformng whch explots knowledge of the channel statstcs at the transmtter and s not to be confused wth channel realzaton dependent transmt beamformng whch s of course not feasble n the noncoherent channel. Applyng a vec operaton to (8), we get y = γ(i Nr X) h + n = γx h + n, where y = vec(y) and n = vec(n). The pdf of y condtoned on X beng sent s gven by p(y X) = 1 π TNr I + γx ΣX e y (I+γX ΣX ) 1y. III. OPTIMAL CORRELATIONS AT LOW SNR Throughout ths paper, we assume our model to be a dscrete nput (of cardnalty L) and contnuous output channel over whch a constellaton {X } L =1 wth correspondng pror probabltes {P } L =1 s used. In ths secton we obtan the optmal correlaton matrces that maxmze the mutual nformaton at suffcently low SNR. Rao and Hassb [17] derve the low SNR mutual nformaton for the contnuous nput and contnuous output channel, when the sgnals are subect to average and peak power constrants. Such regularty condtons are requred snce otherwse the optmal sgnals at low SNR have very large peak-powers. Wth a smlar analyss talored to the dscrete nput and contnuous output channel wth spatally correlated fadng, a smlar expresson for the mutual nformaton can be obtaned whch s I low = γ2 2 { E[tr{(XΛX ) 2 }] tr{(e[xλx ]) 2} + o(γ 2 ). (9) Under dfferent regularty condtons and for more general channels, the authors n [18] also obtan the mutual nformaton upto the second order. When the expresson for mutual nformaton

8 7 at low SNR n [18] s specalzed to the channel model assumed n ths paper, t can be seen to be dentcal to (9). The expresson for I low n (9) may be rewrtten as follows I low { = γ2 P tr(x ΛX 4 X ΛX ) + P tr(x ΛX X ΛX ) 2tr( } P X ΛX P X ΛX ) + o(γ 2 ) = γ2 P P tr{(x ΛX 4 X ΛX )2 } + o(γ 2 ). (10) Let λ denote the vector of dagonal elements of Λ, whch are the egenvalues of Σ. Let λ denote the th element of λ. We next maxmze (10) wth respect to λ subect to the constrant λ = N t N r. Theorem 1: lm γ 0 I low γ 2 s maxmzed by choosng all egenvalues of Σ to be zero except for one, e. λ = [0, 0,..., N t N r,...,0] T, where the poston of the non-zero element N t N r depends on the specfc constellaton used. Proof: Denote the k th column of X by x k, k = 1,..., N t N r. lm γ 0 I low γ 2 = 1 4 = 1 4 = 1 4 P P tr{(x ΛX X ΛX )2 } (11) { NtN } 2 r P P tr λ k (x k x k x k x k) k=1 { } P P tr λ k λ l (x k x k x k x k)(x l x l x l x l) k l (12) (13) Let A k = x k x k x kx k, k = 1,..., N tn r. Also, defne a kl = λ k λ l. We need to solve the { optmzaton problem max P k λ k =N t Nr k l λ } kλ l P P tr(a k A l ) λ k 0 k { } = max λ P P k λ l P P tr(a k A l ) (14) k l λ k λ l =N2 t N2 r λ k 0 k k l { } = Nt 2 a kl N2 r max P Nt 2 Nr 2 P tr(a k A l ), (15) P P a kl k l N t 2N2 =1 r λ k 0 k k l

9 8 whch can be vewed as a convex combnaton of terms of the form { wth weghts a kl N 2 t N2 r P P tr(a k A l ), k, l, }. Therefore, the maxmum should occur when all except one of the {a kl } are zero. The only non-zero weght (say) a kl = Nt 2N2 r corresponds to the ndces that acheve max k,l P P tr(a k A l ). We cannot however have a kl = λ k λ l = Nt 2 Nr 2 when k l snce k λ k = N t N r. Therefore, we frst show that the maxmum occurs only when k = l and then convenently obtan the maxmum of the convex combnaton. P P tr(a k A l ) = P P m P P m < A (m) k,a(m) l > (16) A (m) k A(m) l (17) P P A (m) k 2 A (m) l 2 (18) m m = P P tr(a 2 k )tr(a2 l ) (19) { } { } P P tr(a 2 k ) P P tr(a 2 l ) (20) { max P P tr(a 2 k), } P P tr(a 2 l). (21) The nequaltes n (17) and (18) are obtaned by applyng the Cauchy-Schwarz nequalty successvely. The nequalty n (20) s obtaned by recognzng that the geometrc mean f(x) = x1 x 2 s a concave functon on x R 2 ++ and by applyng the Jensen s nequalty on (19). The square root s well defned snce tr(a 2 ) 0 whenever A s Hermtan, whch s the case here. Fnally, (21) s obtaned by usng the fact that ab max {a, b} for postve a,b. The chan of nequaltes leads to the concluson that the max k,l P P tr(a k A l ) occurs only when k = l. For ths maxmzng ndex k (say), the convex combnaton n (15) s maxmzed by choosng λ k = N t N r and all other egenvalues to be zero, e., λ = [ N t N r... 0] T. The maxmum value of the mutual nformaton would be N 2 t N2 r γ 2 4 max k P P tr(a 2 k ). (22)

10 9 Theorem 1 mples that for any set of sgnals, at suffcently low SNR, a channel havng ust a sngle effectve egenchannel s optmal. Havng a sngle effectve channel would mply that the effectve dmensons avalable s one, but ths enables focusng power whch s more essental at low SNR, when there s no channel state nformaton at the recever. Placng the transmt and receve antennas densely reduces the resolvablty of the dfferent paths n the angular doman, thus resultng n H havng more correlated entres [19]. In terms of system desgn therefore, Theorem 1 suggests that havng closely spaced transmt and receve antenna arrays results n a hgher capacty for noncoherent MIMO communcatons n the low SNR regme. Ths s n contrast to the coherent MIMO scenaro at hgh SNR, where maxmzng the number of degrees of freedom n the channel s crucal. Smlar nsghts are also obtaned n [10] whle consderng the relablty functon at low SNR. We wll need the defntons of maorzaton, Schur-convex and Schur-concave functons from [20] for some of the ensung propostons. These defntons and propertes are provded n Appendx-A for the sake of completeness. If x and y are vectors of egenvalues of two correlaton matrces Σ 1 and Σ 2, x y would mean that Σ 2 s more correlated than Σ 1. Ths noton of maorzaton has been used n many papers studyng the effect of fadng correlatons on the MIMO channel. It provdes a more detaled characterzaton of the performance, and thereby helps to compare two correlated channels whenever possble usng ther respectve vectors of egenvalues. It should be noted that the noton of maorzaton need not relate any two vectors whose entres sum up to the same value. The results obtaned, hence pertan to those vectors of egenvalues that can be compared va maorzaton. The optmal correlatons for the separable model may be obtaned by solvng Λ r Λ t = Λ = dag(0,..., 0, N t N r, 0,..., 0). Therefore, the ontly optmal transmt and receve correlaton egenvalues are gven byλ t = dag(0,...,0, N t, 0,...,0) and Λ r = dag(0,...,0, N r, 0,..., 0) respectvely. The optmal matrces Λ t and Λ r have exactly one non-zero value each and ther postons depend on the specfc constellaton used. We ntroduce the followng notaton whch wll be used n the rest of ths paper. Let λ t and λ r be the vectors of egenvalues of the transmt and receve correlaton matrces, wth elements {λ t n }Nt n=1 and {λ r n n=1, respectvely. Usng the }Nr fact that for the separable model Λ = Λ r Λ t, the low SNR mutual nformaton expresson can

11 10 be smplfed to the followng form I low = γ2 4 { } (λ r n) 2 n P P tr{(x Λ t X X Λ t X ) 2 } + o(γ 2 ). (23) The followng propostons descrbe more conclusons that can be made wth respect to the low SNR mutual nformaton as a functon of λ r and λ t, for the separable model. Proposton 1: In the separable model, the mutual nformaton at low SNR s a Schur-convex functon of λ r. Proof: Snce { n (λr n )2 } s a Schur-convex functon of λ r [20], so s I low. Proposton 1 ndcates that hgher correlatons at the recever are benefcal n the low SNR regme. Ths result contrasts wth results n the coherent scenaro, where receve correlatons are detrmental to the performance at any SNR. An ntutve explanaton for ths dfference s that n the low SNR noncoherent channel, t helps the mplct channel estmaton when the fadng coeffcents across the receve antennas are hghly correlated. The mutual nformaton upto second order s not a Schur-convex functon of the transmt egenvalues at low SNR n general. Ths s because t depends on the specfc sgnals used and the expresson s not even a symmetrc functon of λ t. The followng proposton can however be proved by analytcally maxmzng (23) wth respect to λ t, for any fxed λ r. Proposton 2: The mutual nformaton at suffcently low SNR for the separable model s maxmzed by λ t = [0 0.. N t 0 0] T for any fxed λ r, where the poston of the non-zero element N t depends on the specfc constellaton used. Proof: The proof s along smlar lnes to that of Theorem 1 and the detals are left to the reader. IV. OPTIMAL CORRELATIONS AT HIGHER SNR REGIMES At a general SNR, the mutual nformaton s not known n closed form. As a result we use the cutoff rate expresson as our desgn crteron.

12 11 A. The cutoff rate Consder a constellaton {X } L =1 wth correspondng pror probabltes {P } L =1. The cutoff rate for the dscrete nput (of cardnalty L) and contnuous output channel s gven by { } p(y )p(y )dy R 0 = max log P P. (24) {P } L =1,{X } L =1 For the system model gven n Secton II, the argument of max(.) n (24) s easly found n Appendx-B to be { I + γx ΛX 1/2 I + γx ΛX CR = log P P I + γ (X 2 ΛX + X ΛX ) 1/2 }. (25) We refer to CR n (25) as the cutoff rate expresson. It should be noted that the cutoff rate expresson s a lower bound on the mutual nformaton at any SNR. The next proposton s an extenson of a result for the..d. channel by Hero and Marzetta n [15] to the correlated channel. Proposton 3: In the low SNR regme, the cutoff rate expresson upto second order n γ may be expressed as CR low = γ2 8 Proof: Refer to Appendx-C. P P tr{(x ΛX X ΛX ) 2 } + o(γ 2 ). (26) The followng smple proposton shows that the low SNR cutoff rate expresson behaves dentcally to the low SNR mutual nformaton. Proposton 4: At suffcently low SNR, CR low = 1 2 I low + o(γ 2 ). Proof: The proposton follows by nspecton of (10) and (26). Snce CR low has the same behavor as the mutual nformaton at suffcently low SNR, the results n Secton III are vald for the cutoff rate expresson as well. B. Hgh SNR In ths secton, we optmze the cutoff rate expressson at hgh SNR over the egenvalues of the correlaton matrx. In the next theorem, we assume that [X X ] has full column rank e. a rank of 2N t, for whch n turn T 2N t s a necessary condton. These condtons are known to be suffcent to ensure that the constellaton acheves the maxmum possble dversty order n the channel [21]. If Λ s of full rank, the maxmum possble dversty order would be N t N r.

13 12 Theorem 2: Let [X X ] be of full column rank 2N t for every par, of constellaton matrces. Then, at asymptotcally hgh SNR the cutoff rate expresson s a Schur-concave functon of λ for λ k (0, N t N r ], k. Proof: Snce [X X ] s of full column rank, [X X ] s also of full column rank. We may wrte the cutoff rate expresson as CR = log P 2 +, I + γx ΛX P P 1/2 I + γx ΛX I + γ (X 2 ΛX + X ΛX ) 1/2. (27) Usng the dentty I + AB = I + BA, we smplfy CR as follows CR = log P 2 + I + γx P P X Λ 1/2 I + γx X Λ 1/2, I + γ [X 2 X ] [X X ] (I 2 Λ) (28) log P 2 + γx X Λ 1/2 γx X Λ 1/2 P P γ [X, 2 X ] [X X ] (I 2 Λ) (29) = log P 2 + γx P P X 1/2 Λ 1/2 γx X 1/2 Λ 1/2 γ [X, 2 X ] [X X ] Λ 2 (30) = log P P P c Λ,. (31) Now, 1 Λ s a Schur-convex functon of λ for λ k > 0. The condton λ k > 0 s needed so that Λ, whch occurs n the denomnator s non-zero. Therefore, snce c 0, and {P } L =1 are all non-negatve, and snce h(x) = log(x) s a decreasng functon n R ++, CR at hgh SNR s a Schur-concave functon of λ for λ k (0, N t N r ], k. By the theory of maorzaton and Theorem 2, we conclude that λ = [ ] T s the optmal choce of the egenvalue vector. At hgh SNR, Theorem 2 ndcates that the channel matrx should be made as close to..d. as possble. Ths has the benefcal effect of creatng as many ndependent paths as possble. The optmal correlatons for the separable model may be obtaned by solvng Λ r Λ t = Λ = dag(1, 1,..., 1) under the assumpton that n λt n = N t and n λr n = N r. Ths mples that the ontly optmal transmt and receve correlaton egenvalues are gven by Λ t = dag(1, 1,..., 1)

14 13 and Λ r = dag(1, 1,..., 1), respectvely. More nsghtful conclusons pertanng to the separable model can be made usng the theory of maorzaton, whch are stated n the followng proposton. Proposton 5: () At asymptotcally hgh SNR and for any λ t, the cutoff rate expresson s a Schur-concave functon of λ r. () Let [X X ] be of full column rank 2N t for every par, of constellaton matrces. Then, at asymptotcally hgh SNR and for any λ r, the cutoff rate expresson s a Schur-concave functon of λ t for λ t k (0, N t] k. Proof: Let N = rank(σ r ). () We may wrte the cutoff rate expresson as CR = log P 2 +, N I + γx Λ t X λ r n 1/2 I + γx Λ t X λ r n P P I + γ (X 2 Λ t X + X Λ t X )λr n n=1 1/2 (32). Let the non-zero egenvalues of A = 1 2 X Λ t X be {µ q} Q q=1 and those of A +A = 1 2 (X Λ t X + X Λ t X ) be {θ s} S s=1. Then the cutoff rate expresson may be wrtten as CR = log P 2 +, P P N n=1 [ Q ] 1/2 [ q=1 (1 + γλr n µ Q q) r=1 (1 + γλr n µ r) S s=1 (1 + γλr n θ s) whch, for asymptotcally hgh SNR (γ ), may be wrtten as CR = log P 2 +, P P ( 1 γ N N n=1 λr n ) S Q /2 Q /2 Q ] 1/2 q=1 µ N 2 Q q r=1 µ N 2 r S s=1 θ N s, (33). (34) ( N ) 1 Now, n=1 λr n s a Schur-convex functon of λ r by (3.E.1) [20], snce 1/λ r n s a log-convex functon of λ r n. Snce Q = rank(a 1/2 ), Q = rank(a 1/2 ), and S = rank([a 1/2 A 1/2 ]), clearly S Q and S Q. Ths makes S Q /2 Q /2 0. Hence ( N n=1 λr n ) (S Q /2 Q /2) s also a Schur-convex functon of λ r by (3.B.1) [20]. Groupng all terms wthn the summaton whch multply ths term and denotng t by c 0, we get that, c ( N n=1 λr n ) (S Q /2 Q /2) s also a Schur-convex functon, snce a non-negatve weghted combnaton of Schur-convex functons s also Schur-convex. Fnally, snce h(x) = log(x) s a decreasng functon n R ++,

15 14 and g(x) =, c ( N n=1 λr n) (S Q /2 Q /2) s Schur-convex, the composton h g s Schur-concave. Hence, CR s a Schur-concave functon of λ r at hgh SNR. () The proof of ths part s smlar to the proof of Theorem 2. The proof s omtted and detals are left to the reader. Under nput peak constrants, snce the noncoherent capacty scales as O(SNR 2 ) [17], the energy per bt ncreases wthout bound as the SNR tends to zero. Ths observaton, whch was frst made n [22], ndcates that t s very energy neffcent to operate at vanshngly small SNRs. Nevertheless, numercal results n [22] ndcate that the mnmum energy per bt typcally occurs n the non-asymptotc low SNR regme. Snce the noncoherent capacty s not known at general SNR, the nsghts that we obtan at asymptotcally low SNR offer engneerng gudelnes whch may stll hold at the SNR where the energy-effcency s maxmum. In any case, we next propose a technque to fnd the optmal correlatons at a general SNR. C. General SNR It can be seen that the cutoff rate expresson s non-convex n general wth respect to the transmt and receve egenvalues, and hence ths problem comes under the realm of determnstc global optmzaton [23]. In order to maxmze the cutoff rate at a general SNR, a globally optmal soluton can be obtaned by formulatng t as a dfference of convex programmng (d.c. programmng) problem [23]. We gve some defntons from [23] that we wll need n the theorem that follows. Defnton 1: A polyhedron s defned to be the set of ponts C = {x R n : Ax b, where A R m n and b R m. A bounded polyhedron s called a polytope. Defnton 2: A real valued functon f defned on a convex set A R n s called d.c. (dfference of convex) on A f, for all x A, f can be expressed n the form f(x) = p(x) q(x), (35) where p, q are convex functons on A. The representaton (35) s sad to be a d.c. decomposton of f. Defnton 3: A global optmzaton problem s called a d.c. programmng problem or d.c. program f t has the form

16 15 mn f 0 (x) s.t x A, f (x) 0 ( = 1,..., m), (36) where A s a closed convex subset of R n and all functons f, ( = 0, 1,..., m) are d.c on A. If the set of all constrants form a polytope, then the problem s called a d.c. program over a polytope. There are a number of algorthms gven n Chapter 4 of [23] to fnd the global mnmum of a d.c. program f the d.c. decomposton s known. Defnton 4: A concave mnmzaton problem s an optmzaton problem n the followng form : mn x X f(x), (37) where f(x) s a concave functon and X R n s a convex set. Defnton 5: A reverse convex set (or concave set) s a set whose complement s an open convex set. We state the next lemma whch s needed to prove the ensung theorem. Lemma 1: The functon f(µ,d 1,D 2 ) = det { (I + µ(ad 1 A + BD 2 B )) 1} defned over postve semdefnte dagonal matrces D 1, D 2 and non-negatve µ s a ontly log-convex functon of D 1 and D 2 for fxed µ. Proof: The functons ndcated are all compostons of the functon h(c) = log det(i + C) and lnear functons of the form g(d 1,D 2 ) = AD 1 A + BD 2 B. Snce h(c) s convex over postve semdefnte C, the composton f = h g s also convex [24]. Defnton 6: A functon f(x) s log-convex f log(f(x)) s convex. A lemma that wll be found useful n obtanng d.c. decompostons of complcated functons s next proved. It wll be nvoked n the ensung theorem to get d.c. decompostons. Lemma 2: Let h (x) and g (x) be log-convex functons = 1...., L over R n and c be ( ) non-negatve constants. Then f(x) = log c g (x) h s d.c. and has a d.c. decomposton (x) ( log c g (x) ) h (x) log h (x) (38) Proof:

17 16 ( ) g (x) f(x) = log c h (x) ( = log c g (x) h ) (x) h (x) (39) (40) The product of log-convex functons s log-convex, the sum of log-convex s log-convex and a postve constant tmes a log-convex functon s log-convex. Hence the argument of log(.) n (40) s the rato of log-convex functons. Therefore, a d.c. decomposton for f(x) s ( log c g (x) ) h (x) log h (x). Theorem 3: For a general SNR, the problem of maxmzng the cutoff rate wth respect to λ can be obtaned through ether () a d.c. program over a polytope, or () a concave mnmzaton program, or () a convex mnmzaton program wth an addtonal reverse convex constrant. Proof: The constrant set s n λ n = N t N r, whch s a closed and convex set. We can nstead use the nequalty constrant n λ n N t N r, snce the cutoff rate expresson s an ncreasng functon n γ and hence a soluton has to le on the boundary. The cutoff rate expresson may be wrtten as CR = log P 2 +, I + γ P P (X 2 ΛX + X ΛX ) 1 I + γx ΛX 1/2 I + γx ΛX 1/2 Maxmzng the expresson n (41) s equvalent to maxmzng the followng expresson due to the monotoncty of log(c + x) and log(x). log 2 > I + γ P P (X 2 ΛX + X ΛX ) 1 I + γx ΛX 1/2 I + γx ΛX 1/2 (41). (42) By Lemma 1, we have that I+ γ 2 (X ΛX +X ΛX ) 1, I+γX ΛX 1 and I+γX ΛX 1 are log-convex functons of Λ. A log-convex functon rased to a postve ndex s stll logconvex. Also, a postve constant tmes a log-convex functon s log-convex. Therefore, the expresson (42) can be seen to be n the form needed n Lemma 2, and hence a d.c. decomposton can be obtaned. We can further smplfy ths d.c. decomposton and transform t nto

18 17 other standard global optmzaton problems as follows. The argument of log(.) n (42) can be wrtten as 2 > P P I + γ (X 2 ΛX + X ΛX ) 1 k (I + γx k ΛX k) (I + γx l ΛX l ) 1/2 l,k>l I + γx ΛX. 1/2 I + γx ΛX 1/2 > By the propertes of log-convex functons, and snce the sum and product of log-convex functons s stll log-convex, the last expresson s a rato of two log-convex functons. Therefore, takng log(.) gves a d.c. decomposton for CR. We wll next express the maxmzaton of (42) as a concave mnmzaton problem. We need to maxmze the log(.) of the last expresson, whch s 1 = max tr(λ) N tn r 2 log ( I + γx ΛX I + γx ΛX ) 1 q(λ), (43),> where q(λ) s convex over Λ. An addtonal varable t s now ntroduced to get the equvalent optmzaton problem max tr(λ) N t Nr q(λ) t = mn tr(λ) N t Nr q(λ) t 1 2 log,> ( I + γx ΛX I + γx ΛX ) 1 t (44) t log ( I + γx ΛX I + γx ΛX ) (45),> Snce q(λ) t s a convex functon, q(λ) t 0 s a convex set. Snce the ntersecton of convex sets s convex, the constrant set s convex. Further, snce the obectve functon n (45) s concave, the optmzaton s a concave mnmzaton problem over Λ and t by defnton. The addtonal varable t may also be ntroduced n place of the other convex functon n (43), to get the equvalent problem max t q(λ) = tr(λ) N t Nr t r(λ) where r(λ) =,> 1 2 log ( I + γx ΛX I + γx ΛX mn tr(λ) N t Nr t r(λ) q(λ) t, (46) ). Snce r(λ) s a convex functon of Λ, t r(λ) s a reverse convex constrant of Λ. Snce q(λ) t s a convex functon of Λ, ths form of the optmzaton problem s a convex mnmzaton problem wth an addtonal reverse convex constrant (or concave constrant). All the three forms ndcated are global optmzaton problems and algorthms are avalable to solve them. An example of an algorthm that solves the d.c. program s the Smplcal Branch

19 18 and Bound algorthm(secton 4.6) [23]. Alternatvely, one could convert the problem nto a canoncal d.c. program and then use the Edge Followng Algorthm (Secton 4.5) [23]. More algorthms may be found n [23] and another reference [25]. The concave mnmzaton problem s known [23] to have a soluton at an extreme pont of the constrant regon and ths fact s exploted n algorthms to solve t. Several algorthms to solve ths problem are gven n Chapter 3 of [23] and n [25]. The formulaton nvolvng a convex mnmzaton problem wth an addtonal reverse convex constrant can be solved by the branch and bound algorthm gven n [26]. Snce the number of transmt and receve antennas s relatvely small n practce, the problem of fndng the optmalλ whch nvolves N t N r varables, can be solved numercally wth tractable complexty n many practcal cases of nterest. D. A Numercal Example In Fgure 1, we gve a numercal example usng the UIU Raylegh fadng model. In ths fgure, we compare the smulated (va Monte-Carlo smulatons) mutual nformatons of a systematc untary constellaton at dfferent values of γ on a fully correlated channel, an..d. channel and a channel usng the optmal correlatons. The constellaton used has 8 ponts and the parameters N t = 3, N r = 3 and T = 6. At relatvely low SNR, the performance wth optmal correlatons concdes wth that of the fully correlated channel. At hgh SNRs, the performance wth optmal correlatons concdes wth that of the..d. fadng channel. These smulatons are hence n concordance wth the analytcal results n Theorems 1 and 2. At moderate SNRs, gans of upto 2.5 db are observed when usng the optmal correlatons as compared to the better of the..d. or fully correlated case. Sgnfcant mprovements are observed for the optmal correlatons over the..d. fadng case. V. CONCLUSIONS We consdered the problem of fndng the optmal correlaton matrces of a noncoherent spatally correlated MIMO Raylegh fadng channel at dfferent SNR regmes. In the low SNR regme, we use the mutual nformaton as our desgn crteron, whle at hgher SNR regmes we use the cutoff rate expresson. At suffcently low SNR, we showed that a fully correlated channel matrx maxmzes the mutual nformaton. Ths ndcates that t s best to focus power along

20 19 one effectve channel n the low SNR regme. Therefore, systems wth more densely packed antenna arrays that result n hgh spatal correlatons have a hgher capacty at low SNR. At asymptotcally hgh SNR, we showed that a fully uncorrelated channel matrx s optmal under a condton on the constellaton whch ensures full dversty. Ths ndcates that n the hgh SNR regme, t helps to create as many ndependent parallel channels as possble. In the case of separable correlatons, we showed that the cutoff rate expresson s Schur-convex wth respect to the receve correlaton egenvalues at suffcently low SNR and Schur-concave at hgh SNR. Ths ndcates that t s benefcal to have hgh receve correlatons at suffcently low SNR, whle t helps to have the receve correlaton matrx as close to..d. as possble at hgh SNR. At suffcently low SNR, the fully correlated transmt correlaton matrx s optmal for any fxed receve correlaton matrx. We show that the cutoff rate expresson s Schur-concave wth respect to the transmt correlaton egenvalues at hgh SNR. Ths ndcates that t helps to have the transmt correlaton matrx as close to..d. as possble at hgh SNR. We also show how the problem of fndng the egenvalues of the optmal correlaton matrx at a general SNR can be formulated and solved by usng standard global optmzaton algorthms. APPENDIX A. Maorzaton, Schur-convex and Schur-concave functons The followng two defntons are from [20]. Defnton 7: For x,y R n, x s sad to be maorzed by y, denoted by x y, f and k =1 x [] n x [] = =1 k y [], k = 1,..., n 1, =1 n =1 y [] where x [] and y [] denote the th largest components of x and y respectvely. Defnton 8: A real valued functon f defned on a set A R n s sad to be Schur-convex on A f for any x,y A, x y f(x) f(y). Smlarly, f s defned to be Schur-concave on A f for any x,y A, x y f(x) f(y). Snce the vector [n ] T (wth the n occurng at any poston) maorzes every other nonnegatve vector whose elements add up to n, every Schur-convex functon of such vectors attans

21 20 ts maxmum at [n ] T. Smlarly, every Schur-concave functon attans ts maxmum at [ ] T among all non-negatve vectors whose elements add up to n. B. Dervaton of cutoff rate The ntegral p(y ) p(y )dy n (24) s known as the Bhattacharya coeffcent ρ between hypotheses and. For the noncoherent MIMO Raylegh fadng channel, ρ s ρ = Γ [ ] 1/2 p (y) p (y)dy p (y) = I + γx ΛX 1/2 I + γx ΛX 1/2E X [exp ( y F y)], (47) where F = 1 2 (I+γX ΛX ) (I+γX ΛX ) 1. The expectaton n (47) can be evaluated usng the man result n [27] to get ρ = I + γx ΛX 1/2 I + γx ΛX 1/2 1 I + 1(I + γx 2 2 ΛX )(I + γx ΛX ) 1 1/2 = I + γx ΛX 1/2 I + γx ΛX 1 (I + γx 2 ΛX ) + 1(I + γx 2 ΛX ) (48) Substtutng these expressons n (24) we get (25). In the specal case of separable correlatons, we may smplfy (48) further to obtan the followng expresson : ρ = N r n=1 I + γx ΛX λr n 1/2 I + γx ΛX λr n I + γ (X 2 ΛX + X ΛX )λr n 1/2 (49) Equaton (49) follows from (48) usng the relatons I + γλ r n X ΛX = I TNr + (I Nr γx )(Λ r Λ)(I Nr λ r n X ) = I Nr I T + (I Nr γx ΛX λ r n) = I Nr (I T + γx ΛX λr n ), and smplfyng.

22 21 C. Dervaton of low SNR cutoff rate In ths appendx, we derve the low SNR cutoff rate. The cutoff rate expresson may be wrtten as { { I + γ CR = log P P exp log (X 2 ΛX + X ΛX ) }} I + γx ΛX (50) 1/2 I + γx ΛX 1/2 { } = log P P e {1 2 log I+γX ΛX +1 2 log I+γX ΛX log I+ γ 2 (X ΛX +X ΛX ) }.(51), Now apply the formula log I + γa = γtr(a) γ2 2 tr(a2 ) + o(γ 2 ), whch s vald for any Hermtan matrx A and small γ. Wth ths approxmaton and some smplfcaton, we get that { } = log P P e γ2 8 tr{(x ΛX X ΛX )2 } + o(γ 2 )} (52) CR low { = log ( ) } P P 1 γ2 8 tr{(x ΛX X ΛX ) 2 } + o(γ 2 ) (53) { { }} γ 2 = log 1 P P 8 tr{(x ΛX X ΛX ) 2 } + o(γ 2 ) = γ2 8 (54) P P tr{(x ΛX X ΛX )2 } + o(γ 2 ). (55) In (53), we have used the approxmaton exp( x) = 1 x + o(x) whch holds for small x. In (55), we have used the approxmaton log(1 x) = x o(x) whch s true for small x. REFERENCES [1] C. Chuah, D. Tse, J. Kahn, and R. Valenzuela, Capacty Scalng n MIMO Wreless Systems under Correlated Fadng, IEEE Trans. Inform. Theory, vol. 48, no. 3, pp , Mar [2] A. M. Tulno, A. Lozano, and S. Verdu, The Impact of Antenna Correlaton on the Capacty of Multantenna Channels, IEEE Trans. Inform. Theory, vol. 51, no. 7, pp , July [3] A. Lozano, A. M. Tulno, and S. Verdu, Multple-Antenna Capacty n the Low-Power regme, IEEE Trans. Inform. Theory, vol. 49, no. 10, pp , Oct [4] E. A. Jorsweck and H. Boche, Optmal Transmsson Strateges and Impact of Correlaton n Multantenna Systems wth Dfferent Types of Channel State Informaton, IEEE Trans. Sgnal Processng, vol. 52, no. 12, pp , Dec [5] J. Wang, M. K. Smon, M. P. Ftz, and K. Yao, On the performance of space-tme codes over spatally correlated Raylegh fadng channels, IEEE Trans. Commun., vol. 52, no. 6, pp , June [6] J. H. Kotecha and A. M. Sayeed, Transmt Sgnal Desgn for Optmal Estmaton of Correlated MIMO Channels, IEEE Trans. Wreless Commun., vol. 52, no. 2, pp , Feb

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24 Mutual Informaton vs. gamma Mutual nformaton (bts/t) fully correlated..d. optmal gamma (db) Fg. 1. Mutual nformaton plot for systematc untary desgn wth L = 8, T = 6, N t = 3 and N r = 3.