WLC31-2: Capacity of MIMO Rician Fading Channels ... - IEEE Xplore
Capacity of MIMO Rician Fading Channels with Transmitter and Receiver Channel State Information Amine Maaref and Sonia A¨ıssa INRS-EMT University of Quebec Montreal, QC, Canada Email: {maaref, aissa}@emt.inrs.ca
Abstract— This paper investigates the capacity of multipleinput multiple-output (MIMO) wireless systems when instantaneous channel state information (CSI) is available at both the transmitter and the receiver in a line-of-sight Rician fading environment. Specifically, an infinite series representation for the ergodic capacity of uncorrelated Rician fading MIMO channels is derived, assuming both transmitter and receiver CSI and a specular component of arbitrary rank. The ergodic capacity and its associated outage probability are expressed in terms of a cutoff level capturing the optimal eigen-mode power and rate adaptation. Moreover, an equation from which the cutoff value can be solved for numerically is derived for arbitrary numbers of transmit and receive antennas.
I. I NTRODUCTION While several authors have studied the capacity of MIMO Rician fading channels with instantaneous channel state information (CSI) at the receiver only (see e.g. [1]–[3] and references therein), and a fair amount of work has dealt with the MIMO channel capacity with statistical transmitter CSI (and receiver CSI) [4], [5], studies of the fundamental limits of MIMO fading channels with knowledge of the instantaneous CSI at both transmitter and receiver sides are few and far between. Moreover, exact results in that respect have so far been limited to single-input multiple-output (SIMO) systems [6]–[8] and MIMO diversity systems [9], [10], both of which being essentially “single-input” systems since they only convey a single data stream per channel use as opposed to spatial multiplexing systems where independent data streams can be simultaneously conveyed over the wireless channel. As regards the latter transmission scheme, exact results for the ergodic capacity have been derived for the MIMO independent and identically distributed (i.i.d.) Rayleigh fading channel in [11], then extended in [12] to encompass the delay-limited or zero-outage capacity which is achieved through the suboptimal eigen-mode channel inversion policy. The ergodic capacity with transmitter and receiver CSI for correlated MIMO Rayleigh fading channels was analyzed asymptotically in the number of antennas in [13], and for arbitrary numbers This work was partially supported by a Canada Graduate Scholarship (CGS) and a Discovery Grant (DG) from the Natural Sciences and Engineering Research Council (NSERC) of Canada.
of antennas in [14], [15], and more recently in [16], though no explicit expressions for either the ergodic capacity or for the equation from which to determine the optimal cutoff signalto-noise ratio (SNR) have been proposed. In this work, we investigate the impact of Rician fading on the ergodic capacity of spatial multiplexing MIMO systems with transmitter and receiver CSI and subject to a long-term average power constraint. In so doing, we derive explicit expressions for the capacity of MIMO flat-fading channels subject to uncorrelated but nonnecessarily identically distributed Rician fading. The remaining of this paper is organized as follows: Section II presents the system and channel models. In Section III, general results involving the capacity of MIMO fading channels with transmitter and receiver CSI are derived. Application of the foregoing results to the uncorrelated Rician fading (URiF) environment is performed in Section IV. Numerical results illustrating our theoretical analysis are provided in Section V, followed by concluding remarks in Section VI. II. S YSTEM AND C HANNEL M ODELS We consider a memoryless (frequency-flat) slowly varying MIMO system with nT transmit and nR receive antennas. The channel realizations are assumed to be perfectly tracked at the receiver and fed back to the transmitter through a zero-delay error-free feedback channel, hence perfect CSI is assumed at both the transmitter and the receiver. For each channel use, an independent realization of the channel matrix H[k] is drawn from a noncentral complex Gaussian distribution, whereby the discrete-time input-output relationship at time epoch k is captured by y[k] = H[k] x[k] + n[k]
(1)
where y[k] denotes the nR × 1 vector of received signals, x[k] is the nT ×1 vector of transmitted signals, n[k] is a nR ×1 white noise vector that follows a zero-mean circularly symmetric Gaussian distribution CN (0nR ,1 , N0 I nR ) with N0 > 0 being the noise power. At each time epoch, the nT transmitted signals are beamformed through spatial processing of p independent input data streams [5], [14] so that the transmitted vector x[k]
1-4244-0357-X/06/$20.00 ©2006 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
may be expanded as x[k] = T[k] P[k] s[k] =
p
Pi [k] si [k] ti [k]
(2)
i=1
T where s[k] = s1 [k], . . . , sp [k] is a zero-mean vector with identity covariance, denoting a sample of p independent data streams to be sent through the nT transmit antennas, P[k] = diag{P1 [k], . . . , Pp [k]} where Pi ≥ 0 for i = 1, . . . , p, is a di- agonal power assignment matrix and T[k] = t1 [k], . . . , tp [k] is a nT × p beamforming matrix with unit-energy columnvectors; i.e. ti [k]H ti [k] = 1 for i = 1, . . . , p, which maps the input data into the transmit antennas. Hereinafter, the time index k will be dropped for the sake of notational convenience. In order to capture the URiF environment, the columns of the channel matrix H, denoted as hj for j = 1, . . . , nT , are modelled as independent nonzero mean complex circularly symmetric vectors each distributed according Gaussian column K σ2 with K ≥ 0 denoting the to CN K+1 mj , K+1 I nR Ricean factor and mj for j = 1, . . . , nT is a series of deterministic nR × 1 column-vectors. Without loss of generality, power normalization letting σ 2 = 1 and adopting the standard H = n for MIMO channels, namely E tr HH T nR , implies
nT H = nT nR . Accordingly, the channel that tr j=1 mj mj matrix H can be written as a weighed sum of a deterministic line-of-sight matrix M = [m1 , . . . , mnT ] characterizing the specular component of H and a nR × nT zero-mean random matrix HSC capturing the scattered component of H, namely K 1 M+ HSC (3) H= K +1 K +1 √ SC SC = K + 1 hj − where HSC = [hSC 1 , . . . , hnT ] with hj √ K mj for j = 1, . . . , nT .
By making use of the well-known determinant identity det [I + AB] = det [I + BA], it is easily seen that (5) can further be written as
1 1/2 H H 1/2 P T H HTP . I(x; y) = log2 det I p + N0 p (6) According to [18], the capacity of this channel is achieved by a complex circularly symmetric Gaussian distribution for the channel input x, which is fulfilled whenever s, in turn, is complex circularly symmetric Gaussian. Besides, the average capacity corresponds to the maximum, over the choice of the beamforming matrix T and the transmit power matrix P, of the instantaneous mutual information (6) averaged over all channel realizations, subject to the average power constraint (4). Considering the Hermitian matrix HH H involved in (6), we note that it can be spectrally decomposed as HH H = U diag(λ)UH where λnT = [λ1 , λ2 , . . . , λnT ]T is an nT -dimensional vector containing the set of unordered nonnegative eigenvalues λ1 , λ2 , . . . , λnT of HH H, and U is a nT × nT unitary matrix. Bearing in mind that any choice of T does not affect the average power constraint (4), then setting T = U on a per-channel use basis, has the merit of maximizing the instantaneous mutual information (6) owing to Hadamard’s inequality for positive-definite matrices [19, p. 1067], and yields the average channel capacity given by C
III. E IGEN -M ODE P OWER A DAPTATION S CHEMES : G ENERAL R ESULTS
=
In this section, we derive some general results pertaining to the ergodic capacity of MIMO systems with transmitter and receiver CSI subject to the long-term average power constraint E [tr(Q)] = E [tr(P)] ≤ P
Conditioned on the channel realization H, the instantaneous mutual information between the channel input x and output y when the receiver has perfect CSI is given in bits/s/Hz by [18]
1 H H HTPT H . (5) I(x; y) = log2 det I nR + N0 nR
(4)
where Q denotes the instantaneous input covariance matrix given by Q = E xxH = TPTH . It is suggested in [11] that the ergodic capacity of MIMO systems with transmitter and receiver CSI and subject to an average power constraint is achieved via space-time water filling over the unordered squared singular values of the random channel matrix H. Such an intuitively appealing solution is also reported in [15], [17], yet no adequate proof is available in the literature except for the recently derived [16, Theorem 1]. In this section, an alternative derivation upholding the latter result is provided. It has the advantage over the proof [16, Appendix] of clearly establishing the eigen-mode optimal power and rate adaptation policy that achieves the MIMO channel capacity with transmitter and receiver CSI.
max
P(λnT ):E[tr(P(λnT ))]≤P
E log2
P(λnT ) diag (λnT ) det I nT + γ ,(7) P nT
where γ = NP0 denotes the average SNR per receive antenna. It is obvious that the matrix HH H has exactly the same nonzero real positive eigenvalues as HHH , possibly along with some other eigenvalues all equally zero with probability one (w.p.1). Owing to the assumptions on H stated in Section II, HHH is a noncentral Wishart-type random 1 I nR matrix with nT degrees of freedom, scale matrix K+1 H and noncentrality parameter KMM [3]. In particular, when nT ≥ nR , HHH is noncentral Wishart distributed and has full rank nR w.p.1. We shall henceforth use the shorthand HHH ∼ 1 I nR , KMMH ) to denote such a distribution. On CW(nT , K+1 the other hand, when nT < nR , HHH is noncentral pseudoWishart distributed. Moreover HHH is not full rank and has rank nT w.p.1. Thus, letting m := min{nT , nR } and n := max{nT , nR }, it is seen that in both cases, HHH has exactly m nonzero eigenvalues, say λ1 , λ2 , . . . , λm , which are
1-4244-0357-X/06/$20.00 ©2006 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
real and positive w.p.1. Hence, (7) is equivalent to
m Pi (γi ) γi C= E log2 1 +
max P P(γ m ): m i=1 E[Pi (γi )]≤P i=1 (8) where γ m := [γ1 , γ2 , . . . , γm ]T with γi := γλi for i = 1, . . . , m. Owing to the symmetry between the set of unordered nonzero eigenvalues of a random matrix, the vector maximization problem (8), can be reexpressed in terms of a single unordered random variable, say γ1 , thus reducing to the standard scalar variational optimization problem C
=
P1 (γ):
+∞ 0
m
+∞
0
max
P1 (γ)pγ1 (γ) dγ≤P/m
P1 (γ) γ pγ1 (γ) dγ. (9) log2 1 + P
Obviously, the optimal power allocation scheme which maximizes (9) corresponds to time water-filling over the variations of each unordered eigen-mode i for i = 1, . . . , m, namely 1 1 Pi (γ) if γ ≥ γ0 − , = (10) γ0 γ 0, P if γ < γ 0
where γ0 is a cutoff SNR level below which no eigen-mode transmission is performed. γ0 is chosen so as to comply with the average power constraint (4), hence, it has to satisfy the following equation +∞ 1 1 1 (11) − pγ1 (γ) dγ = . γ γ m 0 γ0 The ergodic capacity of MIMO systems with transmitter and receiver CSI is achieved via simultaneous adaptation of the transmission power and rate to the variations over time of the nonzero unordered eigenvalues of the Hermitian random matrix HHH . Such an adaptation policy, referred to as eigenmode optimal power and rate adaptation (em-opra) is an extension of the single-mode opra policy first introduced by Goldsmith and Varaya in [20], and its ensuing capacity is given by +∞ γ m,n log2 (12) pγ1 (γ) dγ. Cem-opra = m γ 0 γ0 It should be noted that the optimal cutoff SNR for the em-opra policy is always less than m which can be seen by applying Jensen’s inequality to the concave function x → γ10 − x1 defined over the interval [γ0 , +∞) and using the identity (11), thus leading to +∞ pγ1 (γ)dγ 1 1 γ0 0 ≤ +∞ (13) ≤ − . γ0 m γ pγ1 (γ)dγ γ0
The right-hand side of (13) is shown in [16, Theorem 2] to be 1 upper bounded by E γ1 , thereby leading to
1 1 Prob(γ1 ≥ m) 1 1 + ≤ +E ≤ , (14) m E [γ1 ] γ0 m γ1
when combined with (13). For i ∈ {1, . . . , m}, whenever the instantaneous SNR γi , pertaining to eigen-mode i, falls below the cutoff SNR level γ0 , no data are transmitted through that eigen-mode. Accordingly, if all the γi simultaneously fall below γ0 , then an outage takes place and no data are transmitted across the MIMO channel. Care should be taken here, as the notion of outage when transmitter CSI is available is significantly different from the standard outage characterizing the performance of systems in fading environments with no transmitter CSI. Indeed, whereas the latter is indicative of lost data due to decoding errors, the former involves no data loss as the channel is not used all together [21]. The outage probability in the case of the em-opra can be quantified as follows: m,n = Prob(γmax ≤ γ0 ) (15) Pout where γmax = max(γ0 , γ1 , . . . , γm ) = γλmax , with λmax being the largest eigenvalue of the Wishart-type random matrix HHH . IV. A PPLICATION TO U NCORRELATED R ICIAN FADING If the MIMO channel is subject to URiF, then, as stated in Section III, the nR × nR random matrix HHH follows a Wishart-type distribution. By defining the m × m random matrices S and Ω such that KMMH , if nT ≥ nR HHH , if nT ≥ nR , Ω := S := H H H, if nT < nR KMH M, if nT < nR (16) it is easily seen that the distribution of the nonzero eigenvalues of HHH is determined by the eigenvalue distribution of the complex noncentral Wishart matrix S ∼ 1 I m , Ω). Assuming that Ω has t > 0 distinct CW(n, K+1 eigenvalues ωm−t+1 , . . . , ωm , denoted as the column-vector ω t = [ωm−t+1 , . . . , ωm ]T , then, following the approach in [3], [22], it can be shown that the PDF of a single unordered eigenmode SNR is given by
m ωt 1 (t) Km,n µK c (µK γ)d+i+j−2 e−µK γ pγ1 (γ) = j ij m i=1 2 (t) + cij (µK γ)d+i−1 e−µK γ 0 F 1 (d + 1; µK γωj ) (17) j
(K+1) ω t )−1 where µK := , (Km,n := γ m m−t m ωj m−t j=m−t+1 V (ω t )e j=1 Γ(d + j)Γ(j) j=m−t+1 ωj with V (ω t ) denoting the Van Der Monde determinant of order t associated with ω t and Γ(.) the standard Gamma
+∞ xk function [19, eq. (8.310.1)], 0 F 1 (b; x) := k=0 Γ(b+k)k! is (t)
a regularized confluent hypergeometric limit function, cij := th (−1)i+j det [Cij (ω t )]m−1 , with Cij (ω t ) denoting the (i, j)
minor of C(ω t ) :=
Γ(d+i+j−1) Γ(d+i)1 F 1 (d+i;d+1;ωj ) j≤m−t j>m−t th th
m
obtained by removing the i row and j column thereof
+∞ Γ(a+k)xk and 1 F 1 (a; b; x) := k=0 Γ(a)Γ(b+k)k! , d := n − m and we
1
m−t have introduced the shorthand notations j := j=1 and
2
m j := j=m−t+1 for convenience.
1-4244-0357-X/06/$20.00 ©2006 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
m 1 (t) γ0 = c [Γ(d + i + j − 1, µK γ0 ) − µK γ0 Γ(d + i + j − 2, µK γ0 )] ω t j ij Km,n i=1
+
m,n Cem-opra
(t) ω m 2 cij e j Qd+2i−1,d 2ωj , 2µK γ0 − 2µK γ0 Qd+2i−3,d 2ωj , 2µK γ0 i−1 j 2 i=1
m ωt E1 (µK γ0 ) Km,n + =m ln(2) ln(2) i=1
1
(t) c Γ(d j ij
+
2 j
Substituting (17) into (11), it is seen that the optimal cutoff value has to satisfy (21) at the top of this page, where +∞ Γ(a, x) := x ta−1 e−t dt denotes the upper incomplete Gamma function and Qp,q (a, b) is the normalized Nuttall Q function, defined as Qp,q (a, b) := Qp,q (a, b) /aq , where Qp,q (a, b) stands for the standard Nuttall Q-function [23, eq. (86)] 2 ∞ x + a2 xp exp − Qp,q (a, b) = Iq (ax) dx, 2 b 0 < a, b < ∞, p, q = 0, 1, 2, . . . . (23) with Iq (.) being a modified Bessel function of the first kind. Existence and uniqueness of a γ0 satisfying (21) can be easily proved following a similar approach to the one used by the authors in [24]. In order to evaluate the capacity of the em-opra transmission policy, we substitute (17) into (12), then replace 0 F 1 (.; .) with its infinite series representation, and exchange the integral and summation orders which yields (22) at the top of next page after some further manipulation, where E1 (µ) := +∞ e−µt dt denotes the exponential integral function of the t 1 first order. The outage probability hinges on the cumulative distribution function (CDF) of the largest eigenvalue of HHH [11]. Accordingly, it can be shown that the outage probability induced by the em-opra policy can be expressed as functions of γ0 as m,n = Prob(γmax ≤ γ0 ) = P (µK γ0 ) Pout
(27)
ω t det [D(x)] , and the elements of the where P (x) := Km,n m m × m matrix D(x) for i, j = 1, . . . , m are given by γ(d + i + j − 1, x), if j = 1, . . . , m − t Γ(d + i)1 F 1 (d + i; d + 1;√ ωj ) {D(x)}ij = eω j 2ωj , 2x , otherwise. − 2i−1 Qd+2i−1,d where γ(a, x) := Γ(a) − Γ(a, x). V. N UMERICAL R ESULTS In order to numerically illustrate the theoretical analysis developed in the previous sections, we hereafter consider an URiF channel where the specular component M in (3) is captured by a deterministic matrix of the type 1nR ,nT . Hence, the
+ i + j − 1)
+∞ (t) cij k=0
d+i+j−2 l=1
Γ(l, µK γ0 ) Γ(l + 1)
d+i+k−1 Γ(l, µK γ0 ) Γ(d + i + k)ωjk Γ(k + 1)Γ(d + 1 + k) Γ(l + 1)
(21)
(22)
l=1
noncentrality parameter Ω defined in (16) has a rank of unity and its unique nonzero eigenvalue is given ωm = KnT nR . Now, as established by (13), the cutoff level for the em-opra policy is always less than the minimum number of transmit and receive antennas, a fact which is obviously corroborated by Fig. 1 which depicts the cutoff value γ0 as a function of the average SNR γ for different antenna configurations and two values of the Ricean K-factor, namely K = 1 (solid lines) and K = 3 (dashed lines). The different values of γ0 have been found based on (21) using numerical search techniques implemented in the Matlab software starting from the initial search interval implied by (14). As can be observed from Fig. 1, γ0 is an increasing function of γ and converges towards m as γ increases. Fig. 2 shows the capacity of the em-opra policy for multiple-input single-output (MISO) or equivalently SIMO systems for two distinct values of the Rician K-factor, namely K = 1, and K = 10. Not surprisingly, it is seen that the capacity increases with increasing the K factor, an increase which seems to vanish as the diversity order of the MISO system increases. It is worthwhile to mention however that the capacity increase owing to higher values of the parameter K does not expand to MIMO configurations as witnessed by Fig. 3. All the contrary, we notice a steady decrease in capacity as the Rician K-factor gets larger which is due to the choice of Ω being ill-conditioned of unity rank. As a result, increasing K produces a gain that affects a single eigen-mode whereas the gains pertaining to all eigen-modes are increased whenever K decreases (owing to the scattered component of the channel). VI. C ONCLUSION In this paper, we have derived the capacity of uncorrelated Rician fading MIMO channels when channel state information is available at both the transmitter and the receiver. Numerical results have been provided which illustrate our theoretical analysis and quantify the capacity improvement owing to the knowledge of transmitter CSI in MIMO Rician fading channels. R EFERENCES [1] S. K. Jayaweera and H. V. Poor, “On the capacity of multiple antenna systems in Ricean fading,” IEEE Trans. Wireless Commun., vol. 4, no. 3, pp. 1102–1111, May 2005.
1-4244-0357-X/06/$20.00 ©2006 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
[2] M. Kang and M.-S. Alouini, “Capacity of MIMO Ricean channels,” IEEE Trans. Wireless Commun., vol. 5, no. 1, pp. 112–122, Jan. 2006. [3] A. Maaref and S. A¨ıssa, “On the capacity statistics of MIMO Ricean and Rayleigh fading channels,” in Proc. IEEE Int. Conf. on Commun. (ICC’06), Istanbul, Turkey, June 2006, in press. [4] E. Visotsky and U. Madhow, “Space-time transmit precoding with imperfect feedback,” IEEE Trans. Inform. Theory, vol. 47, no. 6, pp. 2632–2639, Sept. 2001. [5] S. A. Jafar and A. Goldsmith, “Transmitter optimization and optimality of beamforming for multiple antenna systems,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1165–1175, July 2004. [6] M.-S. Alouini and A. J. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1165–1181, July 1999. [7] J. Cheng and T. Berger, “Capacity of a class of fading channels with channel state information (CSI) feedback,” in Proc. 39th Ann. Allerton Conf. on Commun., Control, and Comp. (Allerton’01), Monticello, IL, Oct. 2001, pp. 1152–1160. [8] R. K. Mallik, M. Z. Win, J. W. Shao, M.-S. Alouini, and A. J. Goldsmith, “Channel capacity of adaptive transmission with maximal ratio combining in correlated Rayleigh fading,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1124–1133, July 2004. [9] A. Maaref and S. A¨ıssa, “Capacity of space-time block codes in MIMO Rayleigh fading channels with adaptive transmission and estimation errors,” IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2568–2578, Sept. 2005. [10] ——, “Closed-form expressions for the outage and ergodic shannon capacity of MIMO MRC systems,” IEEE Trans. Commun., vol. 53, no. 7, pp. 1092–1095, July 2005. [11] S. K. Jayaweera and H. V. Poor, “Capacity of multiple-antenna systems with both receiver and transmitter channel state information,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2697–2709, Oct. 2003. [12] L. S. Pillutla and S. K. Jayaweera, “Capacity of MIMO systems in Rayleigh fading with sub-optimal adaptive transmission schemes,” in Proc. IEEE Int. Symp. on Inform. Theory & Applications (ISITA’04), Parma, Italy, Oct. 2004. [13] A. M. Tulino, A. Lozano, and S. Verd´u, “MIMO capacity with channel state information at the transmitter,” in Proc. IEEE Int. Symp. on Spread Spectrum Techn. and Applications (ISSSTA’04), Sydney, Australia, Sept. 2004, pp. 22–26. [14] M. T. Ivrla˘c, W. Utschick, and J. A. Nossek, “Fading correlations in wireless MIMO communication systems,” IEEE J. Select. Areas Commun., vol. 21, no. 5, pp. 819–828, June 2003. [15] J. Liu, J. Chen, A. Høst-Madsen, and M. P. C. Fossorier, “Correlated MIMO Rayleigh fading systems with transmit channel state information,” in Proc. IEEE Vehic. Techn. Conf. (VTC-F’04), Los Angeles, CA, Sept. 2004, pp. 1513–1517. [16] L. W. Hanlen and A. Grant, “Optimal transmit covariance for ergodic MIMO channels,” submitted to IEEE Trans. Inform. Theory., available online at http://arxiv.org/abs/cs.IT/0510060. [17] D. Tse and P. Viswanath, Fundamentals of Wireless Communication, 1st ed. New York, NY: Cambridge University Press, 2005. [18] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. Telecommun. (ETT), vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999. [19] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, California: Academic Press, 2000. [20] A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 1986–1992, Nov. 1997. [21] A. J. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath, “Capacity limits of MIMO channels,” IEEE J. Select. Areas Commun., vol. 21, no. 5, pp. 684–702, June 2003. [22] G. Alfano, A. Lozano, A. M. Tulino, and S. Verd´u, “Mutual information and eigenvalue distribution of MIMO Ricean channels,” in Proc. IEEE Int. Symp. on Inform. Theory & Applications (ISITA’04), Parma, Italy, Oct. 2004, pp. 1040–1045. [23] A. H. Nuttall, “Some integrals involving the Q function,” Naval Underwater Systems Center, New London Lab., New London, CT, 4297, 1972. [24] A. Maaref and S. A¨ıssa, “On the achievable spectral efficiency of adaptive transmission with transmit-beamforming,” in Proc. IEEE Int. Conf. on Commun. (ICC’05), Seoul, Korea, May 2005, pp. 2287–2291.
Fig. 1. Cutoff value γ0 for the em-opra policy with different antenna configurations and two values of the Rician K-factor, namely, K = 1 (solid lines) and K = 3 (dashed lines).
Fig. 2. Capacity of the em-opra policy for MISO or SIMO systems (m = 1) and different values of n with K = 1 (solid lines) and K = 10 (dashed lines).
n=m=7
n=m=5
n=m=3
Fig. 3. Capacity of the em-opra policy for MIMO systems with n = m and different values of K.
1-4244-0357-X/06/$20.00 ©2006 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
Abstract— This paper investigates the capacity of multipleinput multiple-output (MIMO) wireless systems when instantaneous channel state information (CSI) is available at both the transmitter and the receiver in a line-of-sight Rician fading environment. Specifically, an infinite series representation for the ergodic capacity of uncorrelated Rician fading MIMO channels is derived, assuming both transmitter and receiver CSI and a specular component of arbitrary rank. The ergodic capacity and its associated outage probability are expressed in terms of a cutoff level capturing the optimal eigen-mode power and rate adaptation. Moreover, an equation from which the cutoff value can be solved for numerically is derived for arbitrary numbers of transmit and receive antennas.
I. I NTRODUCTION While several authors have studied the capacity of MIMO Rician fading channels with instantaneous channel state information (CSI) at the receiver only (see e.g. [1]–[3] and references therein), and a fair amount of work has dealt with the MIMO channel capacity with statistical transmitter CSI (and receiver CSI) [4], [5], studies of the fundamental limits of MIMO fading channels with knowledge of the instantaneous CSI at both transmitter and receiver sides are few and far between. Moreover, exact results in that respect have so far been limited to single-input multiple-output (SIMO) systems [6]–[8] and MIMO diversity systems [9], [10], both of which being essentially “single-input” systems since they only convey a single data stream per channel use as opposed to spatial multiplexing systems where independent data streams can be simultaneously conveyed over the wireless channel. As regards the latter transmission scheme, exact results for the ergodic capacity have been derived for the MIMO independent and identically distributed (i.i.d.) Rayleigh fading channel in [11], then extended in [12] to encompass the delay-limited or zero-outage capacity which is achieved through the suboptimal eigen-mode channel inversion policy. The ergodic capacity with transmitter and receiver CSI for correlated MIMO Rayleigh fading channels was analyzed asymptotically in the number of antennas in [13], and for arbitrary numbers This work was partially supported by a Canada Graduate Scholarship (CGS) and a Discovery Grant (DG) from the Natural Sciences and Engineering Research Council (NSERC) of Canada.
of antennas in [14], [15], and more recently in [16], though no explicit expressions for either the ergodic capacity or for the equation from which to determine the optimal cutoff signalto-noise ratio (SNR) have been proposed. In this work, we investigate the impact of Rician fading on the ergodic capacity of spatial multiplexing MIMO systems with transmitter and receiver CSI and subject to a long-term average power constraint. In so doing, we derive explicit expressions for the capacity of MIMO flat-fading channels subject to uncorrelated but nonnecessarily identically distributed Rician fading. The remaining of this paper is organized as follows: Section II presents the system and channel models. In Section III, general results involving the capacity of MIMO fading channels with transmitter and receiver CSI are derived. Application of the foregoing results to the uncorrelated Rician fading (URiF) environment is performed in Section IV. Numerical results illustrating our theoretical analysis are provided in Section V, followed by concluding remarks in Section VI. II. S YSTEM AND C HANNEL M ODELS We consider a memoryless (frequency-flat) slowly varying MIMO system with nT transmit and nR receive antennas. The channel realizations are assumed to be perfectly tracked at the receiver and fed back to the transmitter through a zero-delay error-free feedback channel, hence perfect CSI is assumed at both the transmitter and the receiver. For each channel use, an independent realization of the channel matrix H[k] is drawn from a noncentral complex Gaussian distribution, whereby the discrete-time input-output relationship at time epoch k is captured by y[k] = H[k] x[k] + n[k]
(1)
where y[k] denotes the nR × 1 vector of received signals, x[k] is the nT ×1 vector of transmitted signals, n[k] is a nR ×1 white noise vector that follows a zero-mean circularly symmetric Gaussian distribution CN (0nR ,1 , N0 I nR ) with N0 > 0 being the noise power. At each time epoch, the nT transmitted signals are beamformed through spatial processing of p independent input data streams [5], [14] so that the transmitted vector x[k]
1-4244-0357-X/06/$20.00 ©2006 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
may be expanded as x[k] = T[k] P[k] s[k] =
p
Pi [k] si [k] ti [k]
(2)
i=1
T where s[k] = s1 [k], . . . , sp [k] is a zero-mean vector with identity covariance, denoting a sample of p independent data streams to be sent through the nT transmit antennas, P[k] = diag{P1 [k], . . . , Pp [k]} where Pi ≥ 0 for i = 1, . . . , p, is a di- agonal power assignment matrix and T[k] = t1 [k], . . . , tp [k] is a nT × p beamforming matrix with unit-energy columnvectors; i.e. ti [k]H ti [k] = 1 for i = 1, . . . , p, which maps the input data into the transmit antennas. Hereinafter, the time index k will be dropped for the sake of notational convenience. In order to capture the URiF environment, the columns of the channel matrix H, denoted as hj for j = 1, . . . , nT , are modelled as independent nonzero mean complex circularly symmetric vectors each distributed according Gaussian column K σ2 with K ≥ 0 denoting the to CN K+1 mj , K+1 I nR Ricean factor and mj for j = 1, . . . , nT is a series of deterministic nR × 1 column-vectors. Without loss of generality, power normalization letting σ 2 = 1 and adopting the standard H = n for MIMO channels, namely E tr HH T nR , implies
nT H = nT nR . Accordingly, the channel that tr j=1 mj mj matrix H can be written as a weighed sum of a deterministic line-of-sight matrix M = [m1 , . . . , mnT ] characterizing the specular component of H and a nR × nT zero-mean random matrix HSC capturing the scattered component of H, namely K 1 M+ HSC (3) H= K +1 K +1 √ SC SC = K + 1 hj − where HSC = [hSC 1 , . . . , hnT ] with hj √ K mj for j = 1, . . . , nT .
By making use of the well-known determinant identity det [I + AB] = det [I + BA], it is easily seen that (5) can further be written as
1 1/2 H H 1/2 P T H HTP . I(x; y) = log2 det I p + N0 p (6) According to [18], the capacity of this channel is achieved by a complex circularly symmetric Gaussian distribution for the channel input x, which is fulfilled whenever s, in turn, is complex circularly symmetric Gaussian. Besides, the average capacity corresponds to the maximum, over the choice of the beamforming matrix T and the transmit power matrix P, of the instantaneous mutual information (6) averaged over all channel realizations, subject to the average power constraint (4). Considering the Hermitian matrix HH H involved in (6), we note that it can be spectrally decomposed as HH H = U diag(λ)UH where λnT = [λ1 , λ2 , . . . , λnT ]T is an nT -dimensional vector containing the set of unordered nonnegative eigenvalues λ1 , λ2 , . . . , λnT of HH H, and U is a nT × nT unitary matrix. Bearing in mind that any choice of T does not affect the average power constraint (4), then setting T = U on a per-channel use basis, has the merit of maximizing the instantaneous mutual information (6) owing to Hadamard’s inequality for positive-definite matrices [19, p. 1067], and yields the average channel capacity given by C
III. E IGEN -M ODE P OWER A DAPTATION S CHEMES : G ENERAL R ESULTS
=
In this section, we derive some general results pertaining to the ergodic capacity of MIMO systems with transmitter and receiver CSI subject to the long-term average power constraint E [tr(Q)] = E [tr(P)] ≤ P
Conditioned on the channel realization H, the instantaneous mutual information between the channel input x and output y when the receiver has perfect CSI is given in bits/s/Hz by [18]
1 H H HTPT H . (5) I(x; y) = log2 det I nR + N0 nR
(4)
where Q denotes the instantaneous input covariance matrix given by Q = E xxH = TPTH . It is suggested in [11] that the ergodic capacity of MIMO systems with transmitter and receiver CSI and subject to an average power constraint is achieved via space-time water filling over the unordered squared singular values of the random channel matrix H. Such an intuitively appealing solution is also reported in [15], [17], yet no adequate proof is available in the literature except for the recently derived [16, Theorem 1]. In this section, an alternative derivation upholding the latter result is provided. It has the advantage over the proof [16, Appendix] of clearly establishing the eigen-mode optimal power and rate adaptation policy that achieves the MIMO channel capacity with transmitter and receiver CSI.
max
P(λnT ):E[tr(P(λnT ))]≤P
E log2
P(λnT ) diag (λnT ) det I nT + γ ,(7) P nT
where γ = NP0 denotes the average SNR per receive antenna. It is obvious that the matrix HH H has exactly the same nonzero real positive eigenvalues as HHH , possibly along with some other eigenvalues all equally zero with probability one (w.p.1). Owing to the assumptions on H stated in Section II, HHH is a noncentral Wishart-type random 1 I nR matrix with nT degrees of freedom, scale matrix K+1 H and noncentrality parameter KMM [3]. In particular, when nT ≥ nR , HHH is noncentral Wishart distributed and has full rank nR w.p.1. We shall henceforth use the shorthand HHH ∼ 1 I nR , KMMH ) to denote such a distribution. On CW(nT , K+1 the other hand, when nT < nR , HHH is noncentral pseudoWishart distributed. Moreover HHH is not full rank and has rank nT w.p.1. Thus, letting m := min{nT , nR } and n := max{nT , nR }, it is seen that in both cases, HHH has exactly m nonzero eigenvalues, say λ1 , λ2 , . . . , λm , which are
1-4244-0357-X/06/$20.00 ©2006 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
real and positive w.p.1. Hence, (7) is equivalent to
m Pi (γi ) γi C= E log2 1 +
max P P(γ m ): m i=1 E[Pi (γi )]≤P i=1 (8) where γ m := [γ1 , γ2 , . . . , γm ]T with γi := γλi for i = 1, . . . , m. Owing to the symmetry between the set of unordered nonzero eigenvalues of a random matrix, the vector maximization problem (8), can be reexpressed in terms of a single unordered random variable, say γ1 , thus reducing to the standard scalar variational optimization problem C
=
P1 (γ):
+∞ 0
m
+∞
0
max
P1 (γ)pγ1 (γ) dγ≤P/m
P1 (γ) γ pγ1 (γ) dγ. (9) log2 1 + P
Obviously, the optimal power allocation scheme which maximizes (9) corresponds to time water-filling over the variations of each unordered eigen-mode i for i = 1, . . . , m, namely 1 1 Pi (γ) if γ ≥ γ0 − , = (10) γ0 γ 0, P if γ < γ 0
where γ0 is a cutoff SNR level below which no eigen-mode transmission is performed. γ0 is chosen so as to comply with the average power constraint (4), hence, it has to satisfy the following equation +∞ 1 1 1 (11) − pγ1 (γ) dγ = . γ γ m 0 γ0 The ergodic capacity of MIMO systems with transmitter and receiver CSI is achieved via simultaneous adaptation of the transmission power and rate to the variations over time of the nonzero unordered eigenvalues of the Hermitian random matrix HHH . Such an adaptation policy, referred to as eigenmode optimal power and rate adaptation (em-opra) is an extension of the single-mode opra policy first introduced by Goldsmith and Varaya in [20], and its ensuing capacity is given by +∞ γ m,n log2 (12) pγ1 (γ) dγ. Cem-opra = m γ 0 γ0 It should be noted that the optimal cutoff SNR for the em-opra policy is always less than m which can be seen by applying Jensen’s inequality to the concave function x → γ10 − x1 defined over the interval [γ0 , +∞) and using the identity (11), thus leading to +∞ pγ1 (γ)dγ 1 1 γ0 0 ≤ +∞ (13) ≤ − . γ0 m γ pγ1 (γ)dγ γ0
The right-hand side of (13) is shown in [16, Theorem 2] to be 1 upper bounded by E γ1 , thereby leading to
1 1 Prob(γ1 ≥ m) 1 1 + ≤ +E ≤ , (14) m E [γ1 ] γ0 m γ1
when combined with (13). For i ∈ {1, . . . , m}, whenever the instantaneous SNR γi , pertaining to eigen-mode i, falls below the cutoff SNR level γ0 , no data are transmitted through that eigen-mode. Accordingly, if all the γi simultaneously fall below γ0 , then an outage takes place and no data are transmitted across the MIMO channel. Care should be taken here, as the notion of outage when transmitter CSI is available is significantly different from the standard outage characterizing the performance of systems in fading environments with no transmitter CSI. Indeed, whereas the latter is indicative of lost data due to decoding errors, the former involves no data loss as the channel is not used all together [21]. The outage probability in the case of the em-opra can be quantified as follows: m,n = Prob(γmax ≤ γ0 ) (15) Pout where γmax = max(γ0 , γ1 , . . . , γm ) = γλmax , with λmax being the largest eigenvalue of the Wishart-type random matrix HHH . IV. A PPLICATION TO U NCORRELATED R ICIAN FADING If the MIMO channel is subject to URiF, then, as stated in Section III, the nR × nR random matrix HHH follows a Wishart-type distribution. By defining the m × m random matrices S and Ω such that KMMH , if nT ≥ nR HHH , if nT ≥ nR , Ω := S := H H H, if nT < nR KMH M, if nT < nR (16) it is easily seen that the distribution of the nonzero eigenvalues of HHH is determined by the eigenvalue distribution of the complex noncentral Wishart matrix S ∼ 1 I m , Ω). Assuming that Ω has t > 0 distinct CW(n, K+1 eigenvalues ωm−t+1 , . . . , ωm , denoted as the column-vector ω t = [ωm−t+1 , . . . , ωm ]T , then, following the approach in [3], [22], it can be shown that the PDF of a single unordered eigenmode SNR is given by
m ωt 1 (t) Km,n µK c (µK γ)d+i+j−2 e−µK γ pγ1 (γ) = j ij m i=1 2 (t) + cij (µK γ)d+i−1 e−µK γ 0 F 1 (d + 1; µK γωj ) (17) j
(K+1) ω t )−1 where µK := , (Km,n := γ m m−t m ωj m−t j=m−t+1 V (ω t )e j=1 Γ(d + j)Γ(j) j=m−t+1 ωj with V (ω t ) denoting the Van Der Monde determinant of order t associated with ω t and Γ(.) the standard Gamma
+∞ xk function [19, eq. (8.310.1)], 0 F 1 (b; x) := k=0 Γ(b+k)k! is (t)
a regularized confluent hypergeometric limit function, cij := th (−1)i+j det [Cij (ω t )]m−1 , with Cij (ω t ) denoting the (i, j)
minor of C(ω t ) :=
Γ(d+i+j−1) Γ(d+i)1 F 1 (d+i;d+1;ωj ) j≤m−t j>m−t th th
m
obtained by removing the i row and j column thereof
+∞ Γ(a+k)xk and 1 F 1 (a; b; x) := k=0 Γ(a)Γ(b+k)k! , d := n − m and we
1
m−t have introduced the shorthand notations j := j=1 and
2
m j := j=m−t+1 for convenience.
1-4244-0357-X/06/$20.00 ©2006 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
m 1 (t) γ0 = c [Γ(d + i + j − 1, µK γ0 ) − µK γ0 Γ(d + i + j − 2, µK γ0 )] ω t j ij Km,n i=1
+
m,n Cem-opra
(t) ω m 2 cij e j Qd+2i−1,d 2ωj , 2µK γ0 − 2µK γ0 Qd+2i−3,d 2ωj , 2µK γ0 i−1 j 2 i=1
m ωt E1 (µK γ0 ) Km,n + =m ln(2) ln(2) i=1
1
(t) c Γ(d j ij
+
2 j
Substituting (17) into (11), it is seen that the optimal cutoff value has to satisfy (21) at the top of this page, where +∞ Γ(a, x) := x ta−1 e−t dt denotes the upper incomplete Gamma function and Qp,q (a, b) is the normalized Nuttall Q function, defined as Qp,q (a, b) := Qp,q (a, b) /aq , where Qp,q (a, b) stands for the standard Nuttall Q-function [23, eq. (86)] 2 ∞ x + a2 xp exp − Qp,q (a, b) = Iq (ax) dx, 2 b 0 < a, b < ∞, p, q = 0, 1, 2, . . . . (23) with Iq (.) being a modified Bessel function of the first kind. Existence and uniqueness of a γ0 satisfying (21) can be easily proved following a similar approach to the one used by the authors in [24]. In order to evaluate the capacity of the em-opra transmission policy, we substitute (17) into (12), then replace 0 F 1 (.; .) with its infinite series representation, and exchange the integral and summation orders which yields (22) at the top of next page after some further manipulation, where E1 (µ) := +∞ e−µt dt denotes the exponential integral function of the t 1 first order. The outage probability hinges on the cumulative distribution function (CDF) of the largest eigenvalue of HHH [11]. Accordingly, it can be shown that the outage probability induced by the em-opra policy can be expressed as functions of γ0 as m,n = Prob(γmax ≤ γ0 ) = P (µK γ0 ) Pout
(27)
ω t det [D(x)] , and the elements of the where P (x) := Km,n m m × m matrix D(x) for i, j = 1, . . . , m are given by γ(d + i + j − 1, x), if j = 1, . . . , m − t Γ(d + i)1 F 1 (d + i; d + 1;√ ωj ) {D(x)}ij = eω j 2ωj , 2x , otherwise. − 2i−1 Qd+2i−1,d where γ(a, x) := Γ(a) − Γ(a, x). V. N UMERICAL R ESULTS In order to numerically illustrate the theoretical analysis developed in the previous sections, we hereafter consider an URiF channel where the specular component M in (3) is captured by a deterministic matrix of the type 1nR ,nT . Hence, the
+ i + j − 1)
+∞ (t) cij k=0
d+i+j−2 l=1
Γ(l, µK γ0 ) Γ(l + 1)
d+i+k−1 Γ(l, µK γ0 ) Γ(d + i + k)ωjk Γ(k + 1)Γ(d + 1 + k) Γ(l + 1)
(21)
(22)
l=1
noncentrality parameter Ω defined in (16) has a rank of unity and its unique nonzero eigenvalue is given ωm = KnT nR . Now, as established by (13), the cutoff level for the em-opra policy is always less than the minimum number of transmit and receive antennas, a fact which is obviously corroborated by Fig. 1 which depicts the cutoff value γ0 as a function of the average SNR γ for different antenna configurations and two values of the Ricean K-factor, namely K = 1 (solid lines) and K = 3 (dashed lines). The different values of γ0 have been found based on (21) using numerical search techniques implemented in the Matlab software starting from the initial search interval implied by (14). As can be observed from Fig. 1, γ0 is an increasing function of γ and converges towards m as γ increases. Fig. 2 shows the capacity of the em-opra policy for multiple-input single-output (MISO) or equivalently SIMO systems for two distinct values of the Rician K-factor, namely K = 1, and K = 10. Not surprisingly, it is seen that the capacity increases with increasing the K factor, an increase which seems to vanish as the diversity order of the MISO system increases. It is worthwhile to mention however that the capacity increase owing to higher values of the parameter K does not expand to MIMO configurations as witnessed by Fig. 3. All the contrary, we notice a steady decrease in capacity as the Rician K-factor gets larger which is due to the choice of Ω being ill-conditioned of unity rank. As a result, increasing K produces a gain that affects a single eigen-mode whereas the gains pertaining to all eigen-modes are increased whenever K decreases (owing to the scattered component of the channel). VI. C ONCLUSION In this paper, we have derived the capacity of uncorrelated Rician fading MIMO channels when channel state information is available at both the transmitter and the receiver. Numerical results have been provided which illustrate our theoretical analysis and quantify the capacity improvement owing to the knowledge of transmitter CSI in MIMO Rician fading channels. R EFERENCES [1] S. K. Jayaweera and H. V. Poor, “On the capacity of multiple antenna systems in Ricean fading,” IEEE Trans. Wireless Commun., vol. 4, no. 3, pp. 1102–1111, May 2005.
1-4244-0357-X/06/$20.00 ©2006 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
[2] M. Kang and M.-S. Alouini, “Capacity of MIMO Ricean channels,” IEEE Trans. Wireless Commun., vol. 5, no. 1, pp. 112–122, Jan. 2006. [3] A. Maaref and S. A¨ıssa, “On the capacity statistics of MIMO Ricean and Rayleigh fading channels,” in Proc. IEEE Int. Conf. on Commun. (ICC’06), Istanbul, Turkey, June 2006, in press. [4] E. Visotsky and U. Madhow, “Space-time transmit precoding with imperfect feedback,” IEEE Trans. Inform. Theory, vol. 47, no. 6, pp. 2632–2639, Sept. 2001. [5] S. A. Jafar and A. Goldsmith, “Transmitter optimization and optimality of beamforming for multiple antenna systems,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1165–1175, July 2004. [6] M.-S. Alouini and A. J. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1165–1181, July 1999. [7] J. Cheng and T. Berger, “Capacity of a class of fading channels with channel state information (CSI) feedback,” in Proc. 39th Ann. Allerton Conf. on Commun., Control, and Comp. (Allerton’01), Monticello, IL, Oct. 2001, pp. 1152–1160. [8] R. K. Mallik, M. Z. Win, J. W. Shao, M.-S. Alouini, and A. J. Goldsmith, “Channel capacity of adaptive transmission with maximal ratio combining in correlated Rayleigh fading,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1124–1133, July 2004. [9] A. Maaref and S. A¨ıssa, “Capacity of space-time block codes in MIMO Rayleigh fading channels with adaptive transmission and estimation errors,” IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2568–2578, Sept. 2005. [10] ——, “Closed-form expressions for the outage and ergodic shannon capacity of MIMO MRC systems,” IEEE Trans. Commun., vol. 53, no. 7, pp. 1092–1095, July 2005. [11] S. K. Jayaweera and H. V. Poor, “Capacity of multiple-antenna systems with both receiver and transmitter channel state information,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2697–2709, Oct. 2003. [12] L. S. Pillutla and S. K. Jayaweera, “Capacity of MIMO systems in Rayleigh fading with sub-optimal adaptive transmission schemes,” in Proc. IEEE Int. Symp. on Inform. Theory & Applications (ISITA’04), Parma, Italy, Oct. 2004. [13] A. M. Tulino, A. Lozano, and S. Verd´u, “MIMO capacity with channel state information at the transmitter,” in Proc. IEEE Int. Symp. on Spread Spectrum Techn. and Applications (ISSSTA’04), Sydney, Australia, Sept. 2004, pp. 22–26. [14] M. T. Ivrla˘c, W. Utschick, and J. A. Nossek, “Fading correlations in wireless MIMO communication systems,” IEEE J. Select. Areas Commun., vol. 21, no. 5, pp. 819–828, June 2003. [15] J. Liu, J. Chen, A. Høst-Madsen, and M. P. C. Fossorier, “Correlated MIMO Rayleigh fading systems with transmit channel state information,” in Proc. IEEE Vehic. Techn. Conf. (VTC-F’04), Los Angeles, CA, Sept. 2004, pp. 1513–1517. [16] L. W. Hanlen and A. Grant, “Optimal transmit covariance for ergodic MIMO channels,” submitted to IEEE Trans. Inform. Theory., available online at http://arxiv.org/abs/cs.IT/0510060. [17] D. Tse and P. Viswanath, Fundamentals of Wireless Communication, 1st ed. New York, NY: Cambridge University Press, 2005. [18] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. Telecommun. (ETT), vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999. [19] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, California: Academic Press, 2000. [20] A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 1986–1992, Nov. 1997. [21] A. J. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath, “Capacity limits of MIMO channels,” IEEE J. Select. Areas Commun., vol. 21, no. 5, pp. 684–702, June 2003. [22] G. Alfano, A. Lozano, A. M. Tulino, and S. Verd´u, “Mutual information and eigenvalue distribution of MIMO Ricean channels,” in Proc. IEEE Int. Symp. on Inform. Theory & Applications (ISITA’04), Parma, Italy, Oct. 2004, pp. 1040–1045. [23] A. H. Nuttall, “Some integrals involving the Q function,” Naval Underwater Systems Center, New London Lab., New London, CT, 4297, 1972. [24] A. Maaref and S. A¨ıssa, “On the achievable spectral efficiency of adaptive transmission with transmit-beamforming,” in Proc. IEEE Int. Conf. on Commun. (ICC’05), Seoul, Korea, May 2005, pp. 2287–2291.
Fig. 1. Cutoff value γ0 for the em-opra policy with different antenna configurations and two values of the Rician K-factor, namely, K = 1 (solid lines) and K = 3 (dashed lines).
Fig. 2. Capacity of the em-opra policy for MISO or SIMO systems (m = 1) and different values of n with K = 1 (solid lines) and K = 10 (dashed lines).
n=m=7
n=m=5
n=m=3
Fig. 3. Capacity of the em-opra policy for MIMO systems with n = m and different values of K.
1-4244-0357-X/06/$20.00 ©2006 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
