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The behavior in terms of information theoretic metrics of the discrete-input, continuous-output noncoherent MIMO Rayleigh fading channel is studied as a function of spatial correlations. In the low SNR regime, the mutual information metric is considered, while at higher SNR regimes the cutoff rate expression is employed. For any fixed input constellation and at sufficiently low SNR, a fully correlated channel matrix is shown to maximize the mutual information. In contrast, at high SNR, a fully uncorrelated channel matrix (with independent identically distributed elements) is shown to be optimal, under a condition on the constellation which ensures full diversity. In the special case of the separable correlation model, it is shown that as a function of the receive correlation eigenvalues, the cutoff rate expression is a Schur-convex function at low SNR and a Schur-concave function at high SNR, and as a function of transmit correlation eigenvalues, the cutoff rate expression is Schur-concave at high SNR for full diversity constellations. Moreover, at sufficiently low SNR, the fully correlated transmit correlation matrix is optimal. Finally, for the general model, it is shown that the optimal correlation matrices at a general SNR can be obtained using a difference of convex programming formulation.