Eigenvalue Probability Density Function Research Articles

Reduced Complexity Power Allocation Strategies for MIMO Systems With Singular Value Decomposition

We consider wireless multiple-input-multiple-output (MIMO) systems in fading environments, with frequency flat fading, channel state information at both transmitter and receiver sides, and linear precoding based on singular value decomposition (SVD). The optimal solution for these MIMO SVD systems, in terms of achievable rate, requires water filling to optimally allocate power to the different channel eigenmodes. Alternatively, reduced complexity power allocation methods can be employed. We propose two power allocation techniques that only require statistical knowledge of the channel matrix coefficients and do not need knowledge of the instantaneous values of the channel state. To study these power allocation methods, we introduce a new expression for the exact distribution of the eigenvalues of Wishart matrices, where the probability density function of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> th largest eigenvalue is given as a sum of terms of the form <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> δ. The expression is here used, in the context of MIMO SVD systems, to obtain the achievable rate for both zero-outage and nonzero-outage strategies. We show that low-complexity methods have performance very similar to water-filling methods.

Statistics of Reflection Eigenvalues in Chaotic Cavities with Nonideal Leads

The scattering matrix approach is employed to determine a joint probability density function of reflection eigenvalues for chaotic cavities coupled to the outside world through both ballistic and tunnel point contacts. Derived under assumption of broken time-reversal symmetry, this result is further utilized to (i)calculate the density and correlation functions of reflection eigenvalues, and (ii)analyze fluctuations properties of the Landauer conductance for the illustrative example of asymmetric chaotic cavity. Further extensions of the theory are pinpointed.

The Largest Eigenvalue of Real Symmetric, Hermitian and Hermitian Self-dual Random Matrix Models with Rank One External Source, Part I

We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric ($\beta$ = 1), Hermitian ($\beta$ = 2), and Hermitian self-dual ($\beta$ = 4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the one-band condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all $\beta$ > 0.

Ergodic Capacity Analysis of Multiple Input Multiple Output Systems with Dual Hop Amplify and Forward Relay Networks

Problem Statement: In MIMO system, capacity mainly depends on the channel and the antenna characteristics. The capacity can be improved by proper design of antenna elements and choosing appropriate array configuration and the directive elements give more capacity than omni directional elements. The dual polarized antennas are better than single polarized antennas as the former can provide two information channels. The mutual coupling also changes the capacity considerably. When the spacing between the elements is less than 0.5λ, the effect of mutual coupling is more. Approach: This study analyzes the capacity of Echelon, H-shaped; V-shaped free standing dipoles and printed dipole arrays of a MIMO system. The channel model for multi-polarized antennas is used in the simulations. The mutual coupling between the dipoles of different orientations is also included to make simulation more realistic. MIMO system with relay network is used to improve the coverage and reliability and to reduce the interference in wireless network. So we have analyzed the characterization of the ergodic capacity of amplify-and-forward (AF) MIMO dual-hop relay channels, assuming that the channel state information is available at the destination terminal only. Results: This study presents the results for channel capacity for different antenna configuration and the exact Ergodic capacity, closed form expression for the high SNR regime and tight closed form upper and lower bounds for amplify and forward relay system using the probability density function of an unordered eigenvalues. Conclusion/Recommendations: The Echelon, H-shaped; V-shaped free standing dipoles offer better capacity results in conventional MIMO system. The results produced are valid for all SNR values and for arbitrary numbers of antennas at the source, relay and destination. We also demonstrate a relationship between the conventional point-to-point MIMO Channels and dual-hop AF MIMO relay channel.

Pfaffian point process for the Gaussian real generalised eigenvalue problem

The generalised eigenvalues for a pair of N × N matrices (X 1, X 2) are defined as the solutions of the equation det (X 1 − λX 2) = 0, or equivalently, for X 2 invertible, as the eigenvalues of \({X_{2}^{-1}X_{1}}\). We consider Gaussian real matrices X 1, X 2, for which the generalised eigenvalues have the rotational invariance of the half-sphere, or after a fractional linear transformation, the rotational invariance of the unit disk. In these latter variables we calculate the joint eigenvalue probability density function, the probability p N,k of finding k real eigenvalues, the densities of real and complex eigenvalues (the latter being related to an average over characteristic polynomials), and give an explicit Pfaffian formula for the higher correlation functions \({\rho_{(k_1,k_2)}}\). A limit theorem for p N,k is proved, and the scaled form of \({\rho_{(k_1,k_2)}}\) is shown to be identical to the analogous limit for the correlations of the eigenvalues of real Gaussian matrices. We show that these correlations satisfy sum rules characteristic of the underlying two-component Coulomb gas.

A Generalized Plasma and Interpolation Between Classical Random Matrix Ensembles

The eigenvalue probability density functions of the classical random matrix ensembles have a well known analogy with the one component log-gas at the special couplings \beta = 1,2 and 4. It has been known for some time that there is an exactly solvable two-component log-potential plasma which interpolates between the \beta =1 and 4 circular ensemble, and an exactly solvable two-component generalized plasma which interpolates between \beta = 2 and 4 circular ensemble. We extend known exact results relating to the latter --- for the free energy and one and two-point correlations --- by giving the general (k_1+k_2)-point correlation function in a Pfaffian form. Crucial to our working is an identity which expresses the Vandermonde determinant in terms of a Pfaffian. The exact evaluation of the general correlation is used to exhibit a perfect screening sum rule.

A finitization of the bead process

The bead process is the particle system defined on parallel lines, with underlying measure giving constant weight to all configurations in which particles on neighbouring lines interlace, and zero weight otherwise. Motivated by the statistical mechanical model of the tiling of an abc-hexagon by three species of rhombi, a finitized version of the bead process is defined. The corresponding joint distribution can be realized as an eigenvalue probability density function for a sequence of random matrices. The finitized bead process is determinantal, and we give the correlation kernel in terms of Jacobi polynomials. Two scaling limits are considered: a global limit in which the spacing between lines goes to zero, and a certain bulk scaling limit. In the global limit the shape of the support of the particles is determined, while in the bulk scaling limit the bead process kernel of Boutillier is reclaimed, after appropriate identification of the anisotropy parameter therein.

Tridiagonal realization of the antisymmetric Gaussian β-ensemble

The Householder reduction of a member of the antisymmetric Gaussian unitary ensemble gives an antisymmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter β, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of {qi}, the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the antisymmetric tridiagonal matrices. This proof uses the Dixon–Anderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real antisymmetric tridiagonal matrices, its eigenvalues, and {qi}. The third proof maps matrices from the antisymmetric Gaussian β-ensemble to those realizing particular examples of the Laguerre β-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Prüfer phases of the random matrices.

Spatial-Correlation-Based Antenna Grouping for MIMO Systems

We investigate antenna-grouping algorithms, which are hybrids of beamforming and spatial multiplexing. We partition transmit antennas into several groups and use beamforming in a group and spatial multiplexing between groups. With antenna grouping, we can achieve diversity gain through beamforming and spectral efficiency through spatial multiplexing. In this paper, we review existing criteria and present several new criteria in multiple-input-multiple-output (MIMO) antenna-grouping systems where the number of transmit antennas is larger than that of receive antennas. We then propose a novel low-complexity antenna-grouping algorithm, which reduces the computational complexity with little degradation of the bit error rate (BER). Comparing the BER and complexity with existing criteria, we demonstrate the benefits of the proposed algorithm. We also derive the BER lower bound of the proposed algorithm in independent and identically distributed (i.i.d.) channels by using the probability density function of the largest eigenvalue of a Wishart matrix.

A method to assess spatially variant noise in dynamic MR image series

Accurate measurement of spatially variant noise in MR images acquired using parallel imaging techniques is challenging. Image-based noise measurement methods such as the subtraction method proposed by the National Electrical Manufacturers Association or the multiple acquisition method often cannot be applied in vivo due to motion and/or dynamic contrast changes. Based on the Karhunen-Loeve transform and random matrix theory, we propose a novel method to accurately assess the noise variance in image series bearing temporal redundancy. The method fits the probability density function of eigenvalues from the temporal covariance matrix of the image series to the Marcenko-Pastur distribution. The accuracy of our method was validated using numerical simulation and an MR noise measurement experiment. The ability of this method to derive the g-factor map of a static phantom was validated against the multiple acquisition method. The method was applied to in vivo cardiac and brain image series and the results agreed with subtraction and multiple acquisition methods, respectively. This new image-based noise measurement method provides a practical means of retrospectively evaluating the noise level and/or g-factor map from multiframe image series.

Ergodic Capacity Analysis of MIMO Multi-Keyhole Channel in Rayleigh Fading

This paper analyzes the ergodic capacity of the MIMO multi-keyhole channel, assuming that the channel state information (CSI) is available only at the receiver. We first derive new closed-form expressions for marginal probability density function (pdf) of the single unordered eigenvalue as well as joint pdf of ordered eigenvalues of the channel matrix in a simple and general framework. With these statistical results, we then present an exact closed-form expression for the ergodic capacity. We analyze tight bounds on the exact capacity and propose a new tight lower bound. We also investigate the asymptotic capacity performances in low-signal-to-noise-ratio (SNR) and high-SNR regimes to gain further insights. All our results apply for arbitrary number of keyholes and antennas. Numerical simulations are presented to validate our theoretical analysis.

Derivation of an eigenvalue probability density function relating to the Poincaré disk

A result of Zyczkowski and Sommers (2000 J. Phys. A: Math. Gen. 33 2045–57) gives the eigenvalue probability density function for the top N × N sub-block of a Haar distributed matrix from U(N + n). In the case n ⩾ N, we rederive this result, starting from knowledge of the distribution of the sub-blocks, introducing the Schur decomposition and integrating over all variables except the eigenvalues. The integration is done by identifying a recursive structure which reduces the dimension. This approach is inspired by an analogous approach which has been recently applied to determine the eigenvalue probability density function for random matrices A−1B, where A and B are random matrices with entries standard complex normals. We relate the eigenvalue distribution of the sub-blocks to a many-body quantum state, and to the one-component plasma, on the pseudosphere.

MIMO rayleigh-product channels with co-channel interference

This paper presents an analytical performance investigation of an interference-limited double scattering multiple-input multiple-output (MIMO) channel employing optimum combining. Our main contribution is the derivation of the closed-form expressions for the cumulative distribution function (c.d.f.) and probability density function (p.d.f.) of the maximum eigenvalue of the resultant channel matrix after optimal combining when the transmit, receive and scattering correlation matrices are identities (hence, the channel is referred to as a Rayleigh-product channel). These results allow us to obtain the outage probability of the optimal combining system in a Rayleigh-product channel with co-channel interferences. Furthermore, we investigate, in depth, an important special case of double scattering channel, the keyhole channel, where based on various closed-form expressions for exact and asymptotic measures derived, we examine three key performance metrics, namely, the ergodic capacity, the outage performance and the symbol-error-rate (SER). The analytical results derived are validated by Monte-Carlo simulations.

Polynomial PDF of the Smallest Eigenvalue of Complex Central Wishart Matrix with Correlation at the Side with the Smallest Number of Antennas

In the literature, the marginal probability density function (pdf) of the smallest eigenvalue of Wishart matrix has been known in the form of determinants having elements with special functions. In this letter, a polynomial expression is derived on the pdf of the smallest eigenvalue of complex central Wishart matrix with correlation at the side with the smallest number of antennas.

A Random Matrix Decimation Procedure Relating β = 2/(r + 1) to β = 2(r + 1)

Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case $r=1$ of a family of inter-relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as $\beta$-ensembles. The inter-relations give that the joint distribution of every $(r+1)$-st eigenvalue in certain $\beta$-ensembles with $\beta = 2/(r+1)$ is equal to that of another $\beta$-ensemble with $\beta = 2(r+1)$. The proof requires generalizing a conditional probability density function due to Dixon and Anderson.

Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble

The partly symmetric real Ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti-symmetric Gaussian random matrices. Such matrices typically have both real and complex eigenvalues. For a fixed number of real eigenvalues, an earlier work has given the explicit form of the joint eigenvalue probability density function. We use this to derive a Pfaffian formula for the corresponding summed up generalized partition function. This Pfaffian formula allows the probability that there are exactly k eigenvalues to be written as a determinant with explicit entries. It can be used too to give the explicit form of the correlation functions, provided certain skew orthogonal polynomials are computed. This task is accomplished in terms of Hermite polynomials, and allows us to proceed to analyze various scaling limits of the correlations, including that in which the matrices are only weakly non-symmetric.

New Closed-Form of the Largest Eigenvalue PDF for Max-SNR MIMO System Performances

Multiple-input multiple-output (MIMO) maxi-mum-SNR (max-SNR) system employs the maximum ratio combiner (MRC) at the receiver side and the maximum ratio transmitter (MRT) at the transmitter side. Its performances highly depend on MIMO channel characteristics, which vary according to both the number of antennas and their distribution between the transmitter and receiver sides. By using the decomposition of the ordered Wishart distribution in the uncorrelated Rayleigh case, we derived a closed-form expression of the largest eigenvalue probability density function (PDF). The final result yields to an expression form of the PDF where polynomials are multiplied by exponentials; it is worth underlining that, though this form had been previously observed for given couples of antennas, to date no formally-written closed-form was available in the literature for an arbitrary couple. Then, this new expression permits one to quickly and easily get the well known largest eigenvalue PDF and use it to determine the binary error probability (BEP) of the max-SNR.

A note on biorthogonal ensembles

We study multiple orthogonal polynomials in the context of biorthogonal ensembles of random matrices. In these ensembles, the eigenvalue probability density function factorizes into a product of two determinants while the eigenvalue correlation functions can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term, whose multiple polynomials are related to the modified Bessel function of the first kind.

Integrable Structure of Ginibre’s Ensemble of Real Random Matrices and a Pfaffian Integration Theorem

In the recent publication [E. Kanzieper and G. Akemann, Phys. Rev. Lett. 95, 230201 (2005); arXiv: math-ph/0507058], an exact solution was reported for the probability "p_{n,k}" to find exactly "k" real eigenvalues in the spectrum of an "n" by "n" real asymmetric matrix drawn at random from Ginibre's Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly "k" real eigenvalues. In the particular case of "k=0", all correlation functions of complex eigenvalues are determined.

Distribution of ordered eigenvalues of Wishart matrices

Simple, exact and computationally efficient expressions for the cumulative distribution function and the probability density function of the lth largest eigenvalue of a Wishart matrix are presented. The results are important in the performance analysis of multiple-input, multiple-output systems operating over Rayleigh fading channels.