Capacity of MIMO channels in the presence of co‐channel interference

This paper presents analytical results on the capacity of multiple‐input‐multiple‐output (MIMO) fading channels in the presence of co‐channel interference (CCI). We consider the scenario in which the desired and CCI users are all subject to Rayleigh fading. We assume that channel realizations of both the desired and CCI users are known at the receiver. Moreover, we consider the case where the transmitter does not have any CSI and as such equal‐power allocation among transmit antennas is used. Given this setup, we derive the moment generating function (MGF) and the mean of the mutual information (MI). We then study the complementary cumulative distribution function of the MI using a Gaussian approximation. Finally, we present and discuss numerical examples to illustrate the mathematical formalism and to show the effect of various parameters on the capacity of MIMO channels in the presence of CCI. Copyright © 2006 John Wiley & Sons, Ltd.     

同信道干扰下相关多输入多输出信道容量分析

在Rayleigh衰落环境下,研究了具有同信道干扰的多输入多输出(MIMO)信道容量问题,分析了通信用户发送端带有空间相关性的情况。假设接收端完美地知道信道状态信息而发送端不知道,基于矩阵变量分布理论,推导出MIMO信道互信息之矩生成函数的精确闭式表达式。利用该表达式进一步推导出MIMO遍历信道容量的精确表达式。用数值结果验证了分析结果的正确性,并给出各种参数对遍历信道容量的影响。     

Capacity of correlated MIMO Rayleigh channels

Abstract-This paper presents some exact results on the capacity of multiple-input-multiple-output (MIMO) channels subject to correlated Rayleigh fading when perfect channel state information (CSI) is known at the receiver. The authors focus on the semicorrelated scenario in which correlation exists either at the transmitter or at the receiver., They consider two cases: 1) the transmitter does not have any CSI and as such allocates power equally among transmitter antennas and 2) the transmitter only knows the statistical distribution of the channel. The first case derives the moment generating function (MGF) of the mutual information (MI) and then deduces from this MGF the mean MI. The authors also study the cumulative distribution function (CDF) of the MI, which can serve as an upper bound to the outage probability under the capacity versus outage formulation when the channel is nonergodic. The second case studies the capacity achieved by optimum power-loading and beamforming schemes based on covariance feedback. Numerical results illustrate that the full capacity of MIMO systems can be preserved even for relatively high values of correlation coefficients.     

Outage mutual information of space-time MIMO channels

We derive analytical expressions for the probability density function (pdf) of the random mutual information between transmitted and received vector signals of a random space-time independent and identically distributed (i.i.d.) multiple-input multiple-output (MIMO) channel, assuming that the transmitted signals from the multiple antennas are Gaussian i.i.d.. We show that this pdf can be well approximated by a Gaussian distribution, and such a Gaussian approximation is based on expressions for the given pdfs mean and variance that we derive. We prove that at high signal-to-noise ratio (SNR), every factor of 2 increase in SNR leads to an increase in outage rate in the amount of min(M,N) bits, where M and N denote the number of transmit and receive antennas, respectively. A simple expression for the moment generating function (MGF) of the mutual information pdf is also provided, based on which we establish normality of the pdf, when both M and N are large, and the SNR is large.     

Power Minimization of Central Wishart MIMO Block-Fading Channels

In this correspondence, a central Wishart multiple input multiple-output (MIMO) K-block fading channel with an information outage probability constraint is studied. Under this block-limited channel condition, we aim to minimize the transmit power required for attaining a given outage probability based on the statistical channel information at the transmitter (SCIT). Using Gaussian approximation to express the probability density function (pdf) of the instantaneous mutual information and by deriving analytically the mean and variance of the mutual information of the MIMO channel, the optimal power allocation can be obtained numerically by a simple one-dimensional sampling method such as dividing rectange (DIRECT).     

Joint and Marginal Eigenvalue Distributions of (Non)Central Complex Wishart Matrices and PDF-Based Approach for Characterizing the Capacity Statistics of MIMO Ricean and Rayleigh Fading Channels

This paper characterizes the eigenvalue distributions of full-rank Hermitian matrices generated from a set of independent (non)zero-mean proper complex Gaussian random vectors with a scaled-identity covariance matrix. More specifically, the joint and marginal cumulative distribution function (CDF) of any subset of unordered eigenvalues of the so-called complex (non)central Wishart matrices, as well as new simple and tractable expressions for their joint probability density function (PDF), are derived in terms of a finite sum of determinants. As corollaries to these new results, explicit expressions for the statistics of the smallest and largest eigenvalues, of (non)central Wishart matrices, can be easily obtained. Moreover, capitalizing on the foregoing distributions, it becomes possible to evaluate exactly the mean, variance, and other higher order statistics such as the skewness and kurtosis of the random channel capacity, in the case of uncorrelated multiple-input multiple-output (MIMO) Ricean and Rayleigh fading channels. Doing so bridges the gap between Telatar's initial approach for evaluating the average MIMO channel capacity (Telatar, 1999), and the subsequently widely adopted moment generating function (MGF) approach, thereby setting the basis for a PDF-based framework for characterizing the capacity statistics of MIMO Ricean and Rayleigh fading channels.     

Capacity of MIMO Rician channels

This paper presents exact results on the capacity of multiple-input-multiple-output (MIMO) Rician channels when perfect channel state information (CSI) is assumed at the receiver but the transmitter has neither instantaneous nor statistical CSI. It first derives the exact expression for the average mutual information (MI) rate of MIMO Rician fading channels when the fading coefficients are independent but not necessarily identically distributed. The results for the independent and identically distributed (i.i.d.) MIMO Rician and Rayleigh fading channels are also obtained as special cases. These results are derived using a different approach than the one used by Telatar for the i.i.d. Rayleigh case. The complementary cumulative distribution function (CCDF) of the MI is also obtained using a Gaussian approximation. The CDF of MI can serve as an upper bound to the outage probability of nonergodic MIMO Rician channels. Numerical results confirm that for a fixed channel gain, a strong tine-of-sight component decreases the channel capacity due to the lack of scattering.     

Effects of Spatial Correlation on MIMO Adaptive Antenna System With Optimum Combining

In this paper, we investigate the effects of the spatial fading correlation on the performance of a multiple-input multiple-output (MIMO) adaptive antenna system with optimum combining (OC) in the presence of multiple cochannel interferers over a correlated Rayleigh fading channel. Based on the Khatri's distribution functions of quadratic forms in complex Gaussian random matrices, we develop a unified determinant representation of those joint eigenvalue distributions. Taking into account the spatial correlation among the antenna elements at the transmitter or receiver, we derive the closed-form formulas for the probability density function and outage probability of the maximum output signal-to-interference ratio (SIR) in an interference-limited MIMO-OC system. Furthermore, the average output SIR and error probability are also investigated. From numerical examples, we show that a new theoretical approach gives a simple and accurate way to assess the performance of the MIMO-OC system over arbitrarily correlated fading channels     

Performance analysis of maximum ratio transmission with imperfect channel estimation

Maximal ratio transmission (MRT) is designed assuming the availability of perfect channel state information (CSI) at both the transmitter and the receiver. However, perfect CSI is not available in practice. This paper investigates the impact of Gaussian estimation errors on the MRT performance in independently and identically distributed (i.i.d.) Rayleigh fading channels. We derive the cumulative distribution function (cdf), the probability density function (pdf) and the moment generating function (mgf) of the MRT output signal-to-noise ratio (SNR) with imperfect CSI, enabling the evaluation of some useful performance metrics such as the average error rate and the outage performance. Numerical and simulation results are provided to show the impact of imperfect CSI on the MRT performance.     

General Capacity Bounds for Spatially Correlated Rician MIMO Channels

This paper considers the capacity of spatially correlated 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$K$-factor, and the geometry of the dominant Rician paths.     

Characterizing the statistical properties of mutual information in MIMO channels

We consider Gaussian multiple-input multiple-output (MIMO) frequency-selective spatially correlated fading channels, assuming that the channel is unknown at the transmitter and perfectly known at the receiver. For Gaussian codebooks, using results from multivariate statistics, we derive an analytical expression for a tight lower bound on the ergodic capacity of such channels at any signal-to-noise ratio (SNR). We show that our bound is tighter than previously reported analytical lower bounds, and we proceed to analytically quantify the impact of spatial fading correlation on ergodic capacity. Based on a closed-form approximation of the variance of mutual information in correlated flat-fading MIMO channels, we provide insights into the multiplexing-diversity tradeoff for Gaussian code books. Furthermore, for a given total number of antennas, we consider the problem of finding the optimal (ergodic capacity maximizing) number of transmit and receive antennas, and we reveal the SNR-dependent nature of the maximization strategy. Finally, we present numerical results and comparisons between our capacity bounds and previously reported bounds.     

Asymptotic analysis of MIMO wireless systems with spatial correlation at the receiver

With a unified approach, this paper investigates the asymptotic performance or, equivalently, the large-system properties, of various point-to-point systems with antenna-array-based multiple-input multiple-output (MIMO) channels having spatial correlations. Using the replica method originally developed for statistical physics, we provide analytical solutions to the input-output mutual information of MIMO systems: 1) at the transmit side, with arbitrary inputs, and 2) at the receive side, with either the optimum space-time joint decoding or various suboptimum spatial equalizers followed by a bank of temporal decoders. Important physical meanings revealed though the analytical solutions to those more practical combinations, such as how the input-output mutual information is affected by the channel spatial correlations, are highlighted along our derivation. Moreover, we provide a novel waterfilling algorithm to determine the data-rate-maximizing transmit signal covariance matrix when only the channel long-term spatial correlations are available at the transmitter.     

A general moment generating function for MRC diversity over fading channels with Gaussian channel gains

For maximal ratio combining (MRC) diversity over correlated fading channels with Gaussian channel gains, we utilize unitary diagonalization of the channel covariance matrix to decorrelate the physical channels into uncorrelated virtual channels to obtain the moment generating function (MGF) of the received signal‐to‐noise ratio (SNR). The MGF thus obtained has a compact form and can be universally applied to various popular fading models. In addition to the advantage of simple derivation procedure, this general MGF can be readily modified to express various scenarios of channel power distributions as well as joint fading models. To demonstrate these advantages, we use the generalized Ricean fading as a specific example to compare our derivation and our MGF expression with an existing work in the literature. Again, we present numerical simulations for MRC reception of binary phase shift keying (BPSK) signals over Nakagami fading to compare with existing results appearing in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

Performance analysis of space-time block codes over keyhole Nakagami-m fading channels

In multiple-input-multiple-output (MIMO) fading environments, degenerate channel phenomena, called keyholes or pinholes, may exist under the realistic assumption that the spatial fading is uncorrelated at the transmitter and the receiver, but the channel has a rank-deficient transfer matrix. In this paper, we analyze the exact average symbol error rate (SER) of orthogonal space-time block codes (STBCs) with M-PSK and M-QAM constellations over Nakagami-m fading channels in the presence of the keyhole. We derive the moment generating function (MGF) of instantaneous signal-to-noise ratio (SNR) after space-time block decoding (signal combining) in such channels. Using a well-known MGF-based analysis approach, we express the average SER of the STBC in the form of single finite-range integrals whose integrand contains only the derived MGF. Numerical results show that the keyhole significantly degrades the SER performance of the STBC from idealistic behaviors in independent identically distributed MIMO channels.     

Performance of joint transmit and receive antenna selection in dual hop amplify‐and‐forward relay network over Nakagami‐m fading channels

In this paper, performance of joint transmit and receive antenna selection in each hop of dual hop amplify‐and‐forward relay network is analyzed over flat and asymmetric Nakagami‐m fading channels. In the network, source, relay, and destination are equipped with multiple antennas. By considering relay location, we derive exact closed‐form cumulative distribution function, moment generating function, moments of end‐to‐end signal‐to‐noise ratio and closed form symbol error probability expressions for fixed and channel state information‐based relay gains. We also derive the asymptotical outage probability and symbol error probability expressions to obtain diversity order and array gain of the network. Analytical results are validated by the Monte Carlo simulations. Copyright © 2012 John Wiley & Sons, Ltd.     

Capacity of multiple-antenna fading channels: spatial fading correlation, double scattering, and keyhole

The capacity of multiple-input multiple-output (MIMO) wireless channels is limited by both the spatial fading correlation and rank deficiency of the channel. While spatial fading correlation reduces the diversity gains, rank deficiency due to double scattering or keyhole effects decreases the spatial multiplexing gains of multiple-antenna channels. In this paper, taking into account realistic propagation environments in the presence of spatial fading correlation, double scattering, and keyhole effects, we analyze the ergodic (or mean) MIMO capacity for an arbitrary finite number of transmit and receive antennas. We assume that the channel is unknown at the transmitter and perfectly known at the receiver so that equal power is allocated to each of the transmit antennas. Using some statistical properties of complex random matrices such as Gaussian matrices, Wishart (1928) matrices, and quadratic forms in the Gaussian matrix, we present a closed-form expression for the ergodic capacity of independent Rayleigh-fading MIMO channels and a tight upper bound for spatially correlated/double scattering MIMO channels. We also derive a closed-form capacity formula for keyhole MIMO channels. This analytic formula explicitly shows that the use of multiple antennas in keyhole channels only offers the diversity advantage, but provides no spatial multiplexing gains. Numerical results demonstrate the accuracy of our analytical expressions and the tightness of upper bounds.     

Monotonicity results for coherent MIMO Rician channels

The dependence of the Gaussian input information rate on the line-of-sight (LOS) matrix in multiple-input multiple-output (MIMO) coherent Rician fading channels is explored. It is proved that the outage probability and the mutual information induced by a multivariate circularly symmetric Gaussian input with any covariance matrix are monotonic in the LOS matrix D, or more precisely, monotonic in D/sup /spl dagger//D in the sense of the Loewner partial order. Conversely, it is also demonstrated that this ordering on the LOS matrices is a necessary condition for the uniform monotonicity over all input covariance matrices. This result is subsequently applied to prove the monotonicity of the isotropic Gaussian input information rate and channel capacity in the singular values of the LOS matrix. Extensions to multiple-access channels (MAC) are also provided.     

Capacity of Time-Varying Channels With Causal Channel Side Information

We derive the capacity of time-varying channels with memory that have causal channel side information (CSI) at the sender and receiver. We obtain capacity of block-memoryless and asymptotically block-memoryless channels with block-memoryless or weakly decorrelating side information. Our coding theorems rely on causal generation of the codewords relative to the causal transmitter CSI. The CSI need not be perfect, and we consider the case where the transmitter and receiver have the same causal CSI as well as the case where the transmitter CSI is a deterministic function of the receiver CSI. For block-memoryless and asymptotically block-memoryless channels, our coding strategy averages mutual information density over multiple transmission blocks to achieve the maximum average mutual information. We apply the coding theorem associated with the block-memoryless channel to determine the capacity and optimal input distribution of intersymbol interference (ISI) time-varying channels with causal perfect CSI about the time-varying channel. The capacity of this channel cannot be found through traditional decomposition methods     

On the relationship between mutual information and bit error probability for some linear dispersion codes

In this paper, we derive the relationship between the bit error probability (BEP) of maximum a posteriori (MAP) bit detection and the bit minimum mean square error (BMMSE). By using this result, the relationship between the mutual information and the BEP is derived for multiple-input multiple-output (MIMO) communication systems with the bit-linear linear-dispersion (BLLD) codes for the Gaussian channel. From the relationship, the lower and upper bounds on the mutual information can be derived.     

The Cost of Uncorrelation and Noncooperation in MIMO Channels

We investigate the capacity loss for using uncorrelated Gaussian input over a multiple-input multiple-output (MIMO) linear additive-noise channel. We upper-bound the capacity loss by a universal constant C* which is independent of the channel matrix and the noise distribution. For a single-user MIMO channel with n_t inputs and n_r outputs C* = min [ 1/2, n_r/n_t log₂ (1+n_t/n_r) ] bit per input dimension (or 2C* bit per transmit antenna per second per hertz), under both total and per-input power constraints. If we restrict attention to (colored) Gaussian noise, then the capacity loss is upper-bounded by a smaller constant C_G = n_r/2n_r log₂ (n_t/n_r) for n_r ges n_t/e, and C_G = 0.265 otherwise, and this bound is tight for certain cases of channel matrix and noise covariance. We also derive similar bounds for the sum-capacity loss in multiuser MIMO channels. This includes in particular uncorrelated Gaussian transmission in a MIMO multiple-access channel (MAC), and "flat" Gaussian dirty-paper coding (DPC) in a MIMO broadcast channel. In the context of wireless communication, our results imply that the benefit of beamforming and spatial water-filling over simple isotropic transmission is limited. Moreover, the excess capacity of a point-to-point MIMO channel over the same MIMO channel in a multiuser configuration is bounded by a universal constant.