IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 9, SEPTEMBER 2005 3121 General Capacity Bounds for Spatially Correlated Rician MIMO Channels Matthew R. McKay, Student Member, IEEE, and Iain B. Collings, Senior Member, IEEE Abstract—This paper considers the capacity of spatially corre- lated Rician multiple-input multiple-output (MIMO) channels. We consider the general case with double-sided correlation and arbitrary rank channel means. We derive tight upper and lower bounds on the ergodic capacity. In the particular cases when the numbers of transmit and receive antennas are equal, or when the correlation is single sided, we derive more specific bounds which are computationally efficient. The bounds are shown to reduce to known results in cases of independent and identically distributed (i.i.d.) and correlated Rayleigh MIMO channels. We also analyze the outage characteristics of the correlated Rician MIMO channels at high signal-to-noise ratio (SNR). We derive the mean and variance of the mutual information and show that it is well approximated by a Gaussian distribution. Finally, we present numerical results which show the effect of the antenna configuration, correlation level (angle spreads), Rician -factor, and the geometry of the dominant Rician paths. Index Terms—Capacity, multiple-input multiple-output (MIMO), Rician, spatial fading correlation. I. INTRODUCTION M ULTIPLE-input multiple-output (MIMO) antenna sys- tems have recently attracted considerable research at- tention as they offer substantial capacity improvements over single-antenna systems without requiring additional power or bandwidth. For independent and identically distributed (i.i.d.) Rayleigh-fading channels, Telatar [1] demonstrated that as the number of transmit antennas and receive antennas grew large, the ergodic capacity increased linearly with . Many papers have examined special cases for the i.i.d. Rayleigh channel, including uniform input capacities [2] and achievable linear receiver capacities [3], [4]. Experimental evidence has demonstrated however, that the i.i.d. assumption is unrealistic in practice, for example, due to insufficient angular spread in- duced by the scattering environment or closely spaced antenna elements, or when there are dominant (possibly line-of-sight (LoS)) nonfading paths between the transmitter and receiver [5], [6]. In such situations, a spatially correlated Rician channel model is more appropriate. In this paper, we consider the capacity of Rician MIMO chan- nels with single-sided and double-sided spatial fading correla- tion. Single-sided correlation typically occurs when a small mo- Manuscript received October 21, 2004; revised March 15, 2005. This paper will be presented in part at the IEEE International Symposium on Information Theory, Adelaide, Australia, September 2005. The authors are with the Telecommunications Laboratory, School of Elec- trical and Information Engineering, University of Sydney, NSW 2006, Aus- tralia, and the ICT Centre, CSIRO, Australia (e-mail: mckay@ee.usyd.edu.au; i.collings@ee.usyd.edu.au). Communicated by M. Médard, Associate Editor for Communications. Digital Object Identifier 10.1109/TIT.2005.853325 bile unit with closely spaced antennas communicates with an el- evated base station with widely spaced antennas. Double-sided correlation occurs when the antennas are insufficiently spaced at both ends, or when there are a limited number of scatterers sur- rounding both the transmit and receive terminals. This can arise, for example, when base-station antennas have a spacing of less than 10 wavelengths [7]. This will often be the case in practice (especially for systems operating at frequencies 2 GHz, see [8]) due to specific mounting requirements and strict environ- mental regulations. The impact of spatial fading correlation on the MIMO ca- pacity has been well examined for the Rayleigh channel [6], [9]. Tight analytical bounds and exact results are now available for Rayleigh channels with both single-sided [8], [10], [11] and double-sided correlation [12]–[14]. It is well known that spatial fading correlation reduces the Rayleigh MIMO capacity [9]. There are relatively few analytical results on Rician capacity with multiple antennas, especially for channels with spatial cor- relation. For ergodic capacity, exact expressions were presented in [15] and [16], however in both cases the channel gains were assumed to be uncorrelated and the results were not given in closed form (requiring numerical integration). In [17], [18] and [19], [20], ergodic capacity bounds were presented for uncor- related and single-sided correlated Rician MIMO channels, re- spectively, assuming rank-1 channel mean matrices. For uncor- related channels with arbitrary-rank means, high signal-to-noise ratio (SNR) ergodic capacity results were recently reported in [21]. In this paper, we derive tight upper and lower bounds on the ergodic capacity with both single-sided and double-sided spatial fading correlation, and with channel mean matrices of arbitrary rank. The expressions are derived under the common assump- tion that the receiver has perfect channel knowledge, and the transmitter does not (either instantaneous or statistical). For both the upper and lower bounds, we first present expres- sions for double-sided correlation using properties of complex matrix-variate noncentral quadratic forms (most of which we derive here). The resulting bounds are given in terms of Hayakawa polynomials of a matrix argument. We then present more computationally efficient bounds in cases when the quadratic forms can be reduced to complex noncentral Wishart matrices. In general, results for noncentral Wishart matrices contain hypergeometric functions of a matrix argument which are typically expressed in terms of zonal polynomials, and are therefore difficult to evaluate. In this paper, we derive an alternate scalar representation for a particular hypergeometric function of a matrix argument, and therefore arrive at efficient expressions for the bounds. 0018-9448/$20.00 © 2005 IEEE