Rayleigh Multiple-input Multiple-output Channels Research Articles

Asymptotic Outage Analysis of Spatially Correlated Rayleigh MIMO Channels

The outage performance of multiple-input multiple-output (MIMO) technique has received intense attention in order to ensure the ultra-reliability requirement for broadcasting systems. In this article, the outage probability at high signal-to-noise ratio (SNR) is asymptotically studied for MIMO channels to thoroughly investigate the transmission reliability. To fully capture the spatial correlation effects, the MIMO fading channel matrix is modelled according to three types of Kronecker correlation structure, i.e., independent, semi-correlated and full-correlated Rayleigh MIMO channels. The outage probabilities under all three Kronecker models are expressed as weighted sum representations of generalized Fox's H functions. The tractable results empower the asymptotic outage analyses, which are conducted not only to reveal helpful insights into understanding the behavior of fading effects, but also to offer useful design guideline for MIMO configurations. Particularly, the asymptotic outage probability is proved to be a monotonically increasing and convex function of the transmission rate. In the absence of the channel state information (CSI), the transmitter tends to equally allocate the total transmit power among its antennas to enhance the reception reliability especially in high SNR regime. In the end, the analytical results are validated through extensive numerical experiments.

Blind Modulation Recognition of the Lower Order PSK Signals Under the MIMO Keyhole Channel

The presence of a keyhole can severely degrade the signal identification performance of the multiple input multiple output (MIMO) system. In this letter, a novel algorithm, namely, the direct modulation recognition (DMR) algorithm is presented to classify the lower order PSK constellations under the spatially correlated MIMO keyhole and Rayleigh channels. In the proposed algorithm, the ratios of higher order cumulants derived from the received signal are used directly as classifying features. The advantage of the DMR algorithm is that it works without the knowledge of the channel state information. Simulation results are presented to show the efficacy of the proposed DMR algorithm.

Finite-SNR Diversity–Multiplexing Tradeoff With Accurate Performance Analysis for Fully Correlated Rayleigh MIMO Channels

Zheng and Tse were the first to discover the fundamental diversity–multiplexing tradeoff (DMT) that occurs in multiple-input multiple-output (MIMO) systems. Their research was limited to infinite signal-to-noise ratio (SNR) situations. Later, research was also conducted on finite-SNR situations; however, this research had significant problems with coarse approximation. In this paper, we propose more accurate approximations of the finite-SNR DMT and the outage probability for Rayleigh fading MIMO channels to solve these problems. The proposed formulas provide more general solutions that are applicable not only to uncorrelated and semicorrelated channels but also to fully correlated channels, which have not been considered in the literature. Utilizing the proposed finite-SNR DMT, we further investigate the influence of the deviation of the finite-SNR DMT on system performance. We also derived an accurate SNR offset quantity, which provides a link for transformation of finite-SNR DMT to actual outage probability. Finally, we present computer simulation results for illustration and comparison to confirm the validity of our research work.

Analysis of MIMO Receiver Using Generalized Least Squares Method in Colored Environments

The classical detection techniques for multiple-input multiple-output (MIMO) systems are usually designed with the assumption that the additive complex Gaussian noise is uncorrelated. However, for closely spaced antennas, the additive noise is correlated due to the mutual antenna coupling. This letter analyzes an improved zero-forcing (ZF) technique for MIMO channels in colored environments. The additive noise is assumed to be correlated and the Rayleigh MIMO channel is considered doubly correlated. The improved ZF detector, based on the generalized least squares estimator (GLS), takes into account the noise covariance matrix and provides an unbiased estimator of the transmitted symbol vectors. We introduce some novel bounds on the achievable sum rate, on the normalized mean square error at the receiver output, and on the outage probability. The derived expressions are compared to Monte Carlo simulations.

Finite-SNR Diversity-Multiplexing Tradeoff for Rayleigh MIMO Channels

In this paper, exact diversity-multiplexing tradeoff (DMT) at finite signal to noise ratio (SNR) for multiple-input multiple-output (MIMO) systems with dual antennas at the transmitter (2 x N_r) and/or at the receiver (N_tx 2) is derived over Rayleigh fading channels. We derive the outage probability of the mutual information between transmitted and received signals versus SNR. It is shown that at realistic SNRs, achievable diversity gains are significantly lower than asymptotic values. While space-time codes (STCs) for MIMO systems are conventionally designed to achieve the asymptotic DMT frontier, this finite-SNR DMT could provide a new insight to design STCs for practical MIMO systems optimized at operational SNRs.

Array Gain in the DMT Framework for MIMO Channels

Following the seminal work by Zheng and Tse on the diversity and multiplexing tradeoff (DMT) of multiple-input multiple-output (MIMO) channels, in this paper, we introduce the array gain to investigate the fundamental relation between transmission rate and reliability in MIMO systems. The array gain gives information on the power offset that results from exploiting channel state information at the transmitter or as a consequence of the channel model. Hence, the diversity, multiplexing, and array gain (DMA) analysis is able to cope with the limitations of the original DMT and provide an operational meaning in the sense that the DMA gains of a particular system can be directly translated into a parameterized characterization of its associated outage probability performance. In this paper, we derive the best DMA gains achievable by any scheme employing isotropic signaling in uncorrelated Rayleigh, semicorrelated Rayleigh, and uncorrelated Rician block-fading MIMO channels. We use these results to analyze the effect of important channel parameters on the outage performance at different points of the DMT curve.

On the Application of Character Expansions for MIMO Capacity Analysis

To evaluate the unitary integrals, such as the well-known Harish–Chandra–Itzykson–Zuber integral, character expansions were developed by Balantekin, where the matrix integrand is a group member; i.e., a square matrix with a nonzero determinant. Recently, this method has been exploited to derive the joint eigenvalue distributions of the Wishart matrices; i.e., <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\bf H}{\bf H}^{\ast}$</tex></formula> where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\bf H}$</tex></formula> is the complex Gaussian random channel matrix of a multiple-input multiple-output (MIMO) system. The joint eigenvalue distributions are used to calculate the moment generating function of the mutual information (ergodic capacity) of a MIMO channel. In this paper, we show that the previous integration framework presented in the literature is not correct, and results in incorrect joint eigenvalue distributions for the Ricean and full-correlated Rayleigh MIMO channels. We develop a new framework to apply the character expansions for integrations over the unitary group, involving general rectangular complex matrices in the integrand. We derive the correct distribution functions and use them to obtain the capacity of the Ricean and correlated Rayleigh MIMO systems in a unified and straightforward approach. The integration technique proposed in this paper is general enough to be used for other unitary integrals in engineering, mathematics, and physics.

Regional Diversity-Multiplexing Tradeoff

The concept of regional diversity-multiplexing tradeoff (DMT) is introduced by extending the asymptotic outage probability expression for multiple-input multiple-output (MIMO) channels. It is shown that for both Rayleigh and Rician MIMO channels, the regional diversity gain is a linear function of the regional multiplexing gain and that the original DMT curve can be obtained from the set of regional DMT lines. As a result, vital information for capturing both finite and infinite signal-to-noise ratio characteristics in terms of DMT is provided.

Optimization of transmit-beam number and power allocation for generic correlated MIMO rayleigh channels

The optimal transmit-beam number and power allocation over generic correlated multiple-input multiple-output (MIMO) Rayleigh fading channels are investigated. By fully exploiting the algebraic structures and properties of the optimization problem, we obtain a procedure for determining the thresholds of optimal beam number and derive a set of elementary equations for optimal power allocation, which can be efficiently solved by using a Broyden-like method. A closed-form criterion for the optimality of beamforming is also proposed. Numerical results are presented for illustration.

Shannon Capacity and Symbol Error Rate of Space-Time Block Codes in MIMO Rayleigh Channels With Channel estimation Error

Space-time block coding (STBC) is an attractive solution for improving quality in wireless links. In this paper, we analyze the impact of channel estimation error on the ergodic capacity and symbol error rate (SER) for space-time block coded multiple-input multiple-output (MIMO) systems. We derive a closed-form capacity expression over MIMO Rayleigh channels with channel estimation error. Moreover, we derive an exact closed-form SER for general PAM/PSK/QAM with channel estimation error. We show that, as expected, channel estimation error introduces the capacity loss and diversity gain loss. Furthermore, as the number of transmit and receive antennas increases, the sensitivity of the STBC system to channel estimation error increases. Simulation results demonstrate the accuracy of our analysis.

Ergodic channel capacity of the spatial correlated rayleigh MIMO channel

The theoretical capacity of the spatial correlated Rayleigh multiple input multiple output (MIMO) channel is an important issue in MIMO technology. In this article, an ergodic channel capacity formula of the spatial correlated rayleigh MIMO channel is provided, which is deduced when two antennas exist at either the transmitter or the receiver. The multi-dimensional least-squares fit algorithm is employed to narrow the difference between the theoretical formula capacity and the practical capacity. Simulation results show that the theoretical capacity approaches the practical one closely.

Asymptotic Performance and Exact Symbol Error Probability Analysis of Orthogonal STBC in Spatially Correlated Rayleigh MIMO Channel

Space-time block coding is an attractive solution for improving quality in wireless links. In general, the multiple-input multiple-output (MIMO) channel is correlated by an amount that depends on the propagation environment as well as the polarization of the antenna elements and the spacing between them. In this paper, asymptotic performance and exact symbol error probability (SEP) of orthogonal space-time block code (STBC) are considered in spatially correlated Rayleigh fading MIMO channel. We derive the moment generating function (MGF) of effective signal-to-noise ration (SNR) after combining scheme at the receiver. Using the MGF of effective SNR, we calculate the probability density function (pdf) of the effective SNR and derive exact closed-form SEP expressions of PAM/PSK/QAM with M-ary signaling. We prove that the diversity order is given by the product of the rank of the transmit and receive correlation matrix. Moreover, we quantify the loss in coding gain due to the spatial correlation. Simulation results demonstrate that our analysis provides accuracy.

On the capacity of orthogonalised correlated MIMO channels under different adaptive transmission techniques

Orthogonal space-time block codes (STBCs) are known to orthogonalise the multiple-input multiple-output (MIMO) wireless channel, thus reducing the space-time vector detection to a simpler scalar detection problem. The capacity of STBCs over correlated Rayleigh and Ricean flat-fading MIMO channels under different adaptive transmitting techniques is studied. Three adaptive schemes known as optimal power and rate allocation, total channel inversion with fixed rate policy and its truncated variant are studied. Taking into account the effect of channel correlation, closed-form expressions are obtained for the capacity of orthogonalised Rayleigh and Ricean MIMO channels under these adaptive transmission techniques in order to avoid Monte-Carlo simulations.

Exact BER analysis of OSTBCs in spatially correlated MIMO channels

The bit-error rate (BER) performance of orthogonal space-time block codes (STBCs) in correlated multiple-input multiple-output (MIMO) channels is studied. We first derive closed-form exact BER equations for pulse amplitude modulation, quadrature amplitude modulation, and phase-shift keying constellations in a correlated Rayleigh MIMO channel. The BER expressions can be easily evaluated without any numerical methods. Then we consider correlated MIMO channels where the line of sight components exist and they suffer from shadowing, namely, correlated shadowed Rician MIMO channels. For three practical channel scenarios, the BER is analyzed, and closed-form BER equations are presented.

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.

Lower Bound on Training-Based Channel Estimation Error for Frequency-Selective Block-Fading Rayleigh MIMO Channels

A lower bound on the error correlation matrix of training-based channel estimators is derived for multiple-input multiple-output (MIMO) systems over block-fading frequency-selective channels with symbol-spaced receivers. The bound is obtained in a constructive way by evaluating the asymptotic performance of an estimator that fully exploits the algebraic structure of the multipath channel. In particular, the estimator is assumed to be able to estimate the long-term features of the channel consistently (e.g., second order statistics of fading, delays, angles) while tracking the fast-varying fading fluctuations by Wiener filtering. Known estimators that are able to attain the bound under simplified settings are referred to, and general guidelines for designing novel estimators are discussed. Based on the simple analytical expression of the bound, the impact of channel estimation error on the link capacity is investigated for different system parameters and channel characteristics (e.g., Doppler shift, spatial correlation of fading). Numerical results are provided to corroborate the analysis.

Closed-Form Capacity Expressions of Orthogonalized Correlated MIMO Channels

Space-time block codes are known to orthogonalize the multiple-input multiple-output (MIMO) wireless channel, thus reducing the space-time vector detection to a simpler scalar detection problem. The capacity over orthogonalized ergodic correlated Rayleigh and Ricean flat-fading MIMO channels has so far only been given in integral form. This letter derives a closed form capacity expression over such channels, hence avoiding numerical integrations or Monte Carlo simulations.

Performance degradation of differential space-time modulation in fast frequency flat fading Rayleigh MIMO channels

The performance degradation of differential unitary space-time modulation of any data rate, RD, due to time selectivity is quantified for any number of transmit antennas, M, for fast frequency flat fading Rayleigh multiple-input multiple-output (MIMO) channels. The degradation bounds related to the Doppler fading parameter are tabulated and demonstrated for varying RD and M.