On analytical derivations of the condition number distributions of dual non-central Wishart matrices

Abstract

Abstract-In this paper, we explore the statistical characterization of Multiple-Input Multiple-Output (MIMO) channel correlation matrices with the main focus being on their condition number statistics. More specifically, novel expressions are derived for the probability density function (PDF) and cumulative distribution function (CDF) of the MIMO condition number. Contrary to the majority of related studies, where only the common case of Rayleigh fading was considered, our investigation is extended to account for the generalized case of Ricean fading where a deterministic Line-of-Sight (LoS) component exists in the communication link. The overall analysis is based on the principles of random matrix theory and particularly of dual complex noncentral Wishart matrices; the latter represent a practical class of MIMO systems, namely dual-branch systems which are equipped with two transmit and receive antenna elements. All the derived formulae are validated through extensive simulations with the attained accuracy being remarkably good.

On the condition number distribution of complex wishart matrices

“…The SCN analysis of [41] was extended in [47] to allow for matrices of arbitrary size and arbitrary correlation structures; however, the final result for the SCN PDF was given as a complicated expression involving an infinite series of zonal polynomials, and is thus difficult to manage. Finally, the exact SCN distributions for dual complex non-central Wishart matrices with two DoF were recently presented in [48].…”

Reduced Complexity Detection for Ricean MIMO Channels Based on Condition Number Thresholding

“…In [17], we found that when κ ≤ 6.46 the instantaneous capacity of the HR channel is greater than the ergodic capacity of a common (2×2) i.i.d. Rayleigh channel and hence we have adopted a threshold of κ = 5 as a reasonable indicator of the channel rank and multipath richness.…”

“…In such a case though, the (λ 1 − λ 2 ) term in the denominator of (2) becomes zero making the analysis invalid (division by zero). In this light, we consider a suboptimum HR LoS channel model in order to guarantee that λ 1 = λ 2 ; the LoS matrix component then reads [17] …”

“…In [17], the authors showed that the probability density function (PDF) of z, f z (z), can be written as a weighted summation of polynomials according to (5), shown at the top of the next page. In (5), Γ(n) denotes the Gamma function which, for the case of an integer index, can be rewritten as Γ(n) = (n − 1)!.…”

Energy-Efficiency Optimization for MIMO-OFDM Mobile Multimedia Communication Systems With QoS Constraints

“…Based on the Wishart matrix theory [15]- [18], numerous channel models have been proposed in the literature for MIMO communication systems [19]- [26]. A closed-form joint probability density function (PDF) of eigenvalues of Wishart matrix was derived for evaluating the performance of MIMO communication systems [19].…”