Capacity-achieving Input Covariance for Correlated Multi-Antenna Channels

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1 Capaity-ahieving Input Covariane for Correlated Multi-Antenna Channels Antonia M. Tulino Universita Degli Studi di Napoli Federio II 85 Napoli, Italy Angel Lozano Bell Labs (Luent Tehnologies) Holmdel, NJ 7733 Sergio Verdú Prineton University Prineton, NJ 854 Abstrat This paper haraterizes the apaity-ahieving input ovariane for multiantenna hannels with (zero-mean) arbitrary distribution. The solution aommodates a wide range of orrelation strutures, not neessarily separate transmitreeive. Our haraterization of the ovariane enompasses both its eigenvetors, whih are found expliitly, and its eigenvalues, for whih we present neessary and suffiient onditions as well as an iterative algorithm. In addition, we identify the orrelation strutures for whih an isotropi input ahieves apaity. Introdution While, in most instanes of wireless ommuniation, the reeiver an aurately trak the state of the fading hannel, the transmitter is often unable to perform suh traking. Statistial information about the hannel, on the other hand, is virtually always aessible sine the periods over whih the fading proess is basially stationary are several orders of magnitude larger than the duration of the fades. As a result, the most typial operating regime in mobile systems is that in whih (i) the reeiver has aess to the instantaneous state of the hannel, and (ii) the transmitter has only aess to its distribution. The input annot therefore depend on the state of the hannel, but only on its distribution. In multi-antenna hannels impaired by additive Gaussian noise and with perfet knowledge of the hannel at the reeiver, the apaity-ahieving input is zero-mean Gaussian and thus haraterizing its distribution boils down to determining its spatial ovariane. The earliest statement about this ovariane was made in ], where it was shown that, for hannels with zero-mean IID (independent identially distributed) Gaussian entries, the apaity-ahieving input is isotropi. Posterior findings have expanded this initial result in several diretions: For hannels with zero-mean Gaussian entries, orrelated only at the transmitter side, the eigenvetors of the apaity-ahieving input ovariane were established in, 3]. (The result was first proved for the multi-transmit single-reeive

2 ase in ], then extended to multiple unorrelated reeive antennas in 3].) In both ases, the eigenvalues were left to numerial optimization. For multi-transmit single-reeive hannels with non-zero-mean IID Gaussian entries, the eigenvetors of the input ovariane were haraterized in ] (see also 5]). Some relationships between the eigenvalues were also put forth, but the need for a numerial optimization was not fully eliminated. For the region of low signal-to-noise ratio (SNR), a omplete haraterization of the limiting ovariane, valid for arbitrary hannels, was given in 6]. This paper generalizes the results of ] 4] removing the various onstrains imposed therein. Speifially, the ontributions are as follows: We obtain the apaity-ahieving input ovariane for hannels with (i) arbitrary numbers of transmit and reeive antennas, (ii) zero-mean but otherwise arbitrary fading distribution, and (iii) very general orrelations not neessarily separable between the entries of the hannel matrix. Our haraterization of the input ovariane enompasses both the eigenvetors, whih are found expliitly, and the eigenvalues, for whih we present neessary and suffiient onditions as well as an expliit iterative algorithm. We identify the orrelation strutures for whih the result in ] holds, i.e., for whih an isotropi input ahieves apaity. Definitions Throughout the paper, ( ) i, indiates the (i,)-th entry of a matrix, ( ) indiates its -th olumn, and ( ) indiates the submatrix obtained by removing the -th olumn. Given transmit and n R reeive antennas and frequeny-flat fading, the baseband model we onsider is y = g H x + n where x and y are the input and output vetors while n is white Gaussian noise. The hannel is represented by the (n R ) zero-mean random matrix g H where the salar g is suh that Tr{HH }] = n R. The ovariane of the input, onveniently normalized, is denoted by Φ x ] xx ] where the normalization ensures that Tr{Φ}]=. The input is isotropi when Φ=I. The ergodi apaity is C = max log det I + SNR Φ:Tr{Φ}=n HΦH () T For the multi-transmit single-reeive ase, an iterative sheme to ompute these eigenvalues was proposed in 4]. This sheme, however, requires that an impliit equation be solved at eah iteration.

3 with SNR g x ] n R n ] whih orresponds to the average signal-to-noise ratio per reeive antenna when either the input is isotropi or the hannel entries are IID. The orrelation between the (i,)-th and (i, )-th entries of H is denoted by R H (i, ; i, ) H) i, (H) i, ]. Most of the multi-antenna literature, however, deals only with separable (also termed kroneker or produt) orrelations 7], onstrained as follows: Definition The orrelation of H is said to be separable if R H (i, ; i, ) = (Θ R ) i,i (Θ T ), where Θ R and Θ T are (n R n R ) and ( ) orrelation matries whose entries indiate the orrelation between reeive antennas and between transmit antennas, respetively. While simple and analytially friendly, the separable orrelation model has lear limitations. It usually suffies to represent the orrelation that arises with spatial diversity, due to antenna proximity, but it annot aommodate other diversity mehanisms suh as those that rely on polarization or pattern differenes 8]. The n R eigenvalues of a separable orrelation funtion are determined by those of Θ R and Θ T. This restrition on the number of degrees of freedom in the eigenvalues of R H, whih ultimately govern the apaity impat of orrelation, preludes the representation of many hannels of interest. In this paper, we onsider more general orrelations whose eigenvalues have full n R degrees of freedom. Speifially, our analysis enompasses any hannel that an be expressed as H = U HV () where H has unorrelated entries while U and V are (n R n R ) and ( ) deterministi unitary matries. The expansion in () orresponds to the Karhunen-Loève transform of H and thus 9]: The olumns of U and V must ontain the eigenfuntions of R H. From (), suh olumns are respetively the eigenvetors of HH ] and H H]. Denoted by λ k,l ( ) the (k,l)-th eigenvalue, the varianes of the entries of H are ( H) ] k,l = λ k,l (R H ). If the orrelation is separable, then U and V orrespond, respetively, with the eigenvetors of Θ R and Θ T. In addition, λ k,l (R H )=λ k (Θ R )λ l (Θ T ). Note that H and H have the same singular values and hene the same apaity. Moreover, if H is Gaussian (i.e. its entries are ointly Gaussian), then H is also Gaussian. In general, the average reeive signal-to-noise is ghx ] n ] = Tr{H H]Φ} n R SNR, whih depends on Φ.

4 Definition A (n R ) matrix B taking values in B R + is olumn-regular if the entries of every olumn exhibit the same empirial distribution, i.e. n R n R i= {(B) i, < ξ} does not depend on, with { } the indiator funtion. 3 Capaity-ahieving Input Covariane 3. igenvetors and Conditions on the igenvalues Let Φ be the input ovariane that ahieves apaity. Its haraterization boils down to determining (i) its eigenvetors, i.e. the diretions on whih signalling should take plae, and (ii) its eigenvalues, i.e. the power that should be alloated onto eah suh diretion. The following entral result proved in the Appendix identifies the former and lays down neessary and suffiient onditions to be satisfied by the latter. Theorem Consider a hannel with zero-mean but otherwise arbitrarily distributed entries that an be expressed as H=U HV with H having unorrelated entries while U and V ontain, respetively, the eigenvetors of HH ] and H H]. The apaity-ahieving input ovariane is Φ =VP V with P a diagonal matrix whose entries, onstrained suh that Tr{P }=, satisfy n R Tr { ( I + SNR ( H) ( H) ) ( I + SNR Note that U is immaterial in (3) and thus it does not affet P. ) }] = if (P ), > HP H if (P ), = In general, the power alloation P does not admit a waterfill interpretation. An iterative algorithm to find this power alloation, whih depends on the SNR, is provided in Setion 3.. At low and high SNR, however, the onditions in (3) simplify drastially: For SNR, the entire power budget should be alloated to the eigenspae within V orresponding to the maximal eigenvalue of H H] to ahieve seond-order optimality 6, Theorem ]. If the multipliity of suh eigenvalue is plural, the power should be evenly divided between the orresponding eigenvetors. For SNR, the power should be evenly divided among the eigevetors within V whose eigenvalues in H H] are nonzero. In the speial ase of separable orrelations: V is defined by the eigenspaes of the transmit orrelation, Θ T, as laimed (for the speial ase of Gaussian hannels with Θ R =I) in 3]. The reeive orrelation, Θ R, enters only the omputation of the powers in P. Moreover, in terms of the low- and high-snr asymptotes, it plays no role at all. (3)

5 3. Power Alloation: An Iterative Algorithm For arbitrary SNR and number of antennas, the set of powers that satisfy the onditions in (3) an be found via the algorithm below, whih is derived in the Appendix diretly from those onditions. It is useful to introdue, as an auxiliary quantity, the mean-square error exhibited at the output of a linear MMS reeiver by the signal transmitted along the -th eigenvetor in V. Defining suh error is given by ] B ( I + SNR ( H) (P) ( H) ) (4) MS = SNR + (P), ( H) B ( H). (5) In order to aommodate the iterative nature of the algorithm, we use ( ) (k) to index the suession of values taken by the power alloation P and by other quantities that depend on P. If no prior information about P is available, the most reasonable initial alloation is I, after whih eah reursion onsists of two steps:. For =,...,, let p (k+) = max { Tr SNR B (k) }] + MS (k) ( H) ( MS (k) (P) (k), B (k) ] n R ) ], (P) (k), ( H). (6). Obtain the new power alloation as P (k+) = nt = p (k+) diag{p (k+), p (k+),..., p (k+) }. (7) The saling performed by (7) simply ensures that the total transmitted power is held at the orret value throughout the reursions. It an be verified that the apaityahieving alloation, P, is the only fixed point of the algorithm. 3.3 Isotropi Input: When does it Ahieve Capaity? If P =I, then Φ =I and thus an isotropi input suffies. There are relevant hannel strutures beyond the IID Gaussian ase reported in ] for whih this is the ase. Proposition Let the entries of H in () be Gaussian. Denote by G a matrix suh that ( G) i, = ( H) i, ]. If G is olumn-regular as per Definition, then Φ =I ahieves apaity. The proof, as well as a more general but less intuitive ondition for non-gaussian hannels, an be found in the Appendix.

6 4 xamples The algorithm presented in Setion 3. exhibits remarkable onvergene speed. Moreover, onvergene appears to take plae irrespetive of the (nonzero) alloation used to initialize the reursions. In this final setion, we illustrate this proess. xample Consider a 3-antenna uniform linear transmit array with -wavelength antenna spaing and a broadside (trunated) Gaussian power azimuth spetrum with a root-meansquare spread. The orresponding transmit orrelation is (Θ T ) i, e ]. Further.5 (i ) onsider 4 unorrelated reeive antennas. Signalling over the eigenvetors of Θ T, the performane of the power alloation algorithm at SNR= 3 db and 6 db is depited in Fig.. xample Consider a (4 ) hannel H whose entries are Gaussian and independent with varianes ( ) 5/3 /3 5/3 /3 G = /3 5/3 /3 5/3 The performane of the power alloation algorithm at SNR=3 db is depited in Fig.. For both examples, the expetations in (6) are omputed as averages of independent realizations of H. Appendix Proof of Theorem Let Φ be the ovariane that maximizes (). Define P V Φ V with V the eigenvetor matrix of H H ]. Sine Tr{P }=Tr{Φ }, the apaity in terms of P is C = log det I + SNR HP H (8) where H HV. Let P =P d +P off diagonal. Further defining Q HP off with P d diag{p } and P off H (I + SNR the apaity in (8) an be expanded as C = log det I + SNR d HP H + ) d HP H log det having zeros along its I + SNR Q. (9) Next, we show that the seond term in the right-hand side of (9) is negative and, hene, that P must diagonal: ] I off HP log det I + SNR H Q = log det ( I + SNR d HP H ) () I ( ]) I off HP H log det ( I + SNR d HP H ) () I = log det I ) ] I + SNR d () HP H I = (3)

7 (k) (P), 3 SNR= -3 db 3 = =,3. ~ rate / bandwidth (bits/s/hz) (k) (P), SNR= -3 db SNR= 6 db =. ~ ~ P apaity.85 =.5 rate / bandwidth (bits/s/hz) 6 4 = P8 SNR= 6 db ~ apaity Figure : With =3 and n R =4 as per xample, values taken by P (k) for k=,..., 7 at SNR= 3 db and 6 db given the initialization P () =I. Also shown are the orresponding rates per unit bandwidth. The rightmost values within eah hart speify the atual P (obtained numerially) and the orresponding apaity.

8 . rate / bandwidth (bits/s/hz) (k) (P), rate / bandwidth (bits/s/hz) P8 3 = =,3,4 SNR= 3 db ~ ~ ~ apaity Figure : With =4 and n R = as per xample, values taken by P (k) for k=,..., 7 at SNR=3 db with initialization P () =diag{4,,, }. Also shown are the orresponding rates per unit bandwidth. The rightmost values speify the atual P =I (given by Proposition ) and the orresponding apaity. where () follows from ( A A det A A ]) = det ( A A A A ) det (A ), () applies Jensen s inequality, () follows from the diagonal struture of H H] and the off-diagonal struture of P off, and (3) is given by the unit determinant of a triangular matrix whose diagonal elements are. Altogether then, the apaity-ahieving ovariane is Φ =VP V where the eigenvetors in V equal those of H H] while P is the diagonal eigenvalue matrix that maximizes (8). The determination of P is thus a maximization problem for the onave funtion I(P) = log det I + SNR HP H (4) over the onvex set of diagonal real nonnegative matries P whose trae equals.

9 This maximum is haraterized by a set of Kuhn-Tuker onditions, whih we derive following the footsteps of, Setion 5.]. Hene, we impose that the derivative of (4) in the diretion from P to any alternative matrix P be nonpositive. Letting d dµ log det I + SNR HPµ H = P µ = ( µ)p + µp for µ, the one-side derivative of (4) with respet to µ at µ= + is Tr and, therefore, we impose that Tr { ( I + HP H ) ( ) }] I + SNR HP H I (5) { ( I + HP H ) ( ) }] I + SNR HP H I for every P in the set. Following onsiderations analogous to those in ], the Kuhn- Tuker onditions that haraterize P beome equivalent to the neessary and suffiient onditions in (3). Derivation of Iterative Algorithm Using (4) and (5) and with the aid of the matrix inversion lemma, { ( ) }] Tr I + SNR SNR HP H = Tr {B }] (P), MS( H) B( H) ] whih an be used to expand (3) into where we have further used Tr {B }] + (P), ( MS]) n R MS( H) B ( H) ] (P), (6) SNR SNR (P), ( H) ( I + SNR HP H ) ( H) = MS. The algorithm is based on seleting, at every reursion, the largest of the two quantities at either side of (6), with an additional saling step that ensures that Tr{P}=. Proof of Proposition Given any diagonal matrix P suh that Tr{P}=, denote by P {m} its yli shift by m positions, i.e. another diagonal matrix suh that (P) {m}, = (P), with =( m) mod nt. Clearly, Tr{P {m} }= for any shift m=,..., and m= P {m} = I

10 Denote by ve( H) the n R -dimensional vetor obtained by staking up the olumns of H. Invoking Jensen s inequality, log det I + SNR H H = m= log det log det I + SNR {m} HP H ( I + SNR {m} HP H (7) where (7) holds if the oint distribution of ve( H) equals that of ve( H {m} ), with the olumns of H {m} being yli shifted versions of the olumns of H. It follows that I is the apaity-ahieving input ovariane whenever f ve( H) ( ) = f ve( H{m} )( ). (8) Suffiient ondition for (8) is that the olumns of H be independent and marginally identially distributed. If H is Gaussian, then this ondition reverts to the olumnregularity of G. Referenes ] I.. Telatar, Capaity of multi-antenna Gaussian hannels, ur. Trans. Teleom, vol., pp , Nov ]. Visotsky and U. Madhow, Spae-time transmit preoding with imperfet feedbak, I Trans. Inform. Theory, vol. 47, pp , Sep.. 3] S. A. Jafar, S. Vishwanath and A. J. Goldsmith, Channel apaity and beamforming for multiple transmit and reeive antennas with ovariane feedbak, I Intern. Conf. Communiations (ICC ), vol. 7, pp. 66-7,. 4] H. Bohe and. Jorswiek, Optimal power alloation for MISO systems and omplete haraterization of the impat of orrelation on the apaity, I Intern. Conf. on Aoustis, Speeh and Signal Pro. (ICCASP 3), vol. 4, pp , Apr. 3. 5] A. Narula, M. J. Lopez, M. D. Trott and G. W. Wornell, ffiient Use of Side Information in Multiple-Antenna Data Transmission over Fading Channels, I J. Selet. Areas in Commun., vol. 6, No. 8, pp , Ot ] S. Verdú, Spetral ffiieny in the Wideband Regime, I Trans. Inform. Theory, vol. 48, No. 6, pp , June. 7] D.-S. Shiu, G. J. Foshini, M. J. Gans and J. M. Kahn, Fading Correlation and Its ffets on the Capaity of Multielement Antenna Systems, I Trans. Commun., vol. 48, No. 3, Mar.. 8] A. M. Tulino, S. Verdú and A. Lozano Capaity of antenna arrays with spae, polarization and pattern diversity, I Inform. Theory Workshop (ITW 3), Apr. 3. 9] W. K. Pratt, Digital Image Proessing, New York: Wiley, 978. ] S. Verdú, Multiuser Detetion, Cambridge University Press, 998. ] T.-S. Chu, L. Greenstein, A Semiempirial Representation of Antenna Diversity Gain at Cellular and PCS Base Stations, I Trans. Commun., vol. 45, June 997. ] T. M. Cover and J. A. Thomas, lements of Informatioheory, New York: Wiley, 99.