Analytical Derivation of the Inverse Moments of One-Sided Correlated Gram Matrices With Applications

Abstract

This paper addresses the development of analytical tools for the computation of the moments of random Gram matrices with one side correlation. Such a question is mainly driven by applications in signal processing and wireless communications wherein such matrices naturally arise. In particular, we derive closed-form expressions for the inverse moments and show that the obtained results can help approximate several performance metrics such as the average estimation error corresponding to the Best Linear Unbiased Estimator (BLUE) and the Linear Minimum Mean Square Error LMMSE or also other loss functions used to measure the accuracy of covariance matrix estimates.

Numerically Stable Evaluation of Moments of Random Gram Matrices With Applications

“…This kind of matrices arise in covariance matrix estimation and more precisely in exponentially weighted sample covariance matrix (more details can be found in [1], section III-B) . Note that for moderate to large values of q, the eigenvalues of Λ given by (1 − ξ) , (1 − ξ) ξ, · · · , (1 − ξ) ξ q−1 are very close to each other, which might cause singularity issues when using the formula in (1). We consider two different configurations config 1 and config 2 corresponding respectively to (n t = 3, q = 5) and (n t = 3, q = 20).…”

“…This clearly demonstrates the efficiency and the accuracy of our method in calculating the positive moments. 1 This can be seen by using the fact that P (14) with ξ = 0.85. Both settings are considered: config 1 : n t = 3, q = 5 and config 2 : n t = 3, q = 20 for different moment order values, p.…”

“…To validate our procedure, we compute the positive moments of the Gram matrix W in the case where the correlation matrix Λ follows the following model [1]…”

“…G RAM random matrices with one-sided correlation naturally arise in the context of signal processing [1] and wireless communications [2,3]. For instance, in signal processing, inverse moments of this kind of matrices are used to evaluate the performance of linear estimators such as the best linear unbiased estimator (BLUE) and optimize the design of some covariance matrix estimators [1,4]. In wireless communications, Gram random matrices arise as a key element in the computation of the ergodic capacity of amplify and forward (AF) multiple input multiple output (MIMO) dualhop systems [3].…”

“…G RAM random matrices with one-sided correlation naturally arise in the context of signal processing [1] and wireless communications [2,3]. For instance, in signal processing, inverse moments of this kind of matrices are used to evaluate the performance of linear estimators such as the best linear unbiased estimator (BLUE) and optimize the design of some covariance matrix estimators [1,4].…”