In this paper, we assess the combined effects of rank deficiency and spatial fading correlation on the diversity performance of multiple-input multiple-output (MIMO) systems in terms of the symbol error probability, 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 Rayleigh to degenerate keyhole cases by embracing both rank-deficient and spatial correlation effects. We 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.
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