Asymptotic Analysis of Rayleigh Product Channels: A Free Probability Approach

Abstract

The Rayleigh product channel model is useful in capturing the performance degradation due to rank deficiency of MIMO channels. In this paper, such a performance degradation is investigated via the channel outage probability assuming slowly varying channel with delay-constrained decoding. Using techniques of free probability theory, the asymptotic variance of channel capacity is derived when the dimensions of the channel matrices approach infinity. In this asymptotic regime, the channel capacity is rigorously proven to be Gaussian distributed. Using the obtained results, a fundamental tradeoff between multiplexing gain and diversity gain of Rayleigh product channels can be characterized by closed-form expression at any finite signal-to-noise ratio. Numerical results are provided to compare the relative outage performance between Rayleigh product channels and conventional Rayleigh MIMO channels. Index TermsCentral limit theorem; finite-SNR diversity-multiplexing tradeoff; free probability theory; MIMO; outage capacity; Rayleigh product channels.

Information-Theoretic Characterization of MIMO Systems With Multiple Rayleigh Scattering

“…all eigenvalues but one. This can be done by exploiting [33,Theorem I], which provides the result of the integration in terms of matrices G 1 ( ) and A 0 , with A 0 given by The CDF in (12) can be obtained by integrating (11). The matrix G 2 ( ), given by…”

“…Even after such a strong simplification, finite-size analysis of multiple scattering MIMO channels was considered to be impractical till very recently, due to the lack of closed-form characterization of the spectrum of the product of an arbitrary number of independent random matrices of finite size. A noticeable exception is the case of N = 2, i.e., the so-called double scattering channel [6], [10], [11], the simplified version of which is referred to as Rayleigh-product channel [12, and references therein]. A very large number of studies (not cited here for the sake of brevity) have appeared on this case since very early MIMO analysis.…”

“…At last, we mention that we do not address spatially correlated systems. The impact of spatial correlation on the average asymptotic capacity of double-scattering channels has been quantified as early as in [3], while the corresponding effect on the variance of the mutual information has been handled in [11]. Very recently, a relevant study [9] has appeared on multiple-cluster scattering with spatial correlation at the end of the link with the least number of antennas.…”

Distribution of linear statistics of singular values of the product of random matrices

“…One motivation for investigating the asymptotic distribution of products of random matrices follows from recent applications in wireless telecommunication, see [21]. Compare as well very recent results in [29]. Other immediate applications are in finance [10] and quantum entanglement [13], [30].…”

Impact of Residual Additive Transceiver Hardware Impairments on Rayleigh-Product MIMO Channels With Linear Receivers: Exact and Asymptotic Analyses

“…Morever, the authors in [28], considering the additive transceiver impairments, extended the analysis of [22] to massive MIMO for arbitrary SNR values. It is worthwhile to mention that the double scattering channel has been already investigated for massive MIMO systems, one of the prominent technologies for 5G of [31]- [33]. Moreover, note that the keyhole channel is a first step towards the double scattering channels which is a suitable model for characterizing the scattering limitations of higher frequencies envisaged in 5G.…”