MIMO Diversity in the Presence of Double Scattering

Abstract

Abstract-The potential benefits of multiple-antenna systems may be limited by two types of channel degradations-rank deficiency and spatial fading correlation of the channel. In this paper, we assess the effects of these degradations on the diversity performance of multiple-input multiple-output (MIMO) systems, with an emphasis on orthogonal space-time block codes (OSTBC), in terms of the symbol error probability (SEP), the effective fading figure (EFF), and the capacity at low signal-to-noise ratio (SNR). In particular, we consider a general family of MIMO channels known as double-scattering channels-i.e., Rayleigh product MIMO channels-which encompasses a variety of propagation environments from independent and identically distributed (i.i.d.) Rayleigh to degenerate keyhole or pinhole cases by embracing both rank-deficient and spatial correlation effects. It is shown that a MIMO system with n T transmit and n R receive antennas achieves the diversity of order n T n S n R = max(n T ; n S ; n R ) in a double-scattering channel with n S effective scatterers. We also quantify the combined effect of the spatial correlation and the lack of scattering richness on the EFF and the low-SNR capacity in terms of the correlation figures of transmit, receive, and scatterer correlation matrices. We further show the monotonicity properties of these performance measures with respect to the strength of spatial correlation, characterized by the eigenvalue majorization relations of the correlation matrices.Index Terms-Channel capacity, diversity, double scattering, fading figure, keyhole, multiple-input multiple-output (MIMO) system, orthogonal space-time block code (OSTBC), spatial fading correlation, symbol error probability (SEP).

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“…As we can see, (23) and (24) are GPPs. It can be easily proved that the average power constraint and the PDP constraint in (23) and (24) are active at their respective optimal solutions, from which we can deduce that (23) and (24) are also equivalent.…”

“…Similarly, when all the transmission powers P i in different ARQ rounds are distinct, we have M l = l, and τ i,l = 1, 1 ≤ i ≤ l. For this case, the characteristic coefficients can be written as [23]:…”

Cooperative Communications with Outage-Optimal Opportunistic Relaying

“…Then, using Theorem 2 in Appendix A, we obtain the conditional outage probability Pr { outage| D ℓ } as (9), shown at the bottom of the page, where [23]. Combining (6), (7), and (9) gives (10), shown at the bottom of the page.…”

“…are the distinct diagonal elements in decreasing order, τ i (A A A) is the multiplicity of µ i , and X i,j (A A A) is the (i, j)th characteristic coefficient of A A A [23]. 13 The cumulative distribution function (c.d.f.)…”

“…where the last equality follows from the fact that the sum of all the characteristic coefficients is equal to one [23], and Γ (n, z) is the incomplete gamma function defined by…”