We consider a multiple-input multiple-output (MIMO) detection problem with widely-spaced antennas at both the transmitter and the receiver, and we assume that target scattering is modeled as an exchangeable and unitarily-invariant process. We illustrate optimal signal design (i.e., space-time coding) at the transmitter for two criteria, i.e, the lower Chernoff bound (LCB) to the detection probability for fixed probability of false alarm and the mutual information (MI) between the observations and the target scattering matrix, under a semidefinite rank constraint and a transmit power constraint, showing that the Gaussian scattering assumption is robust. A by-product, of not secondary importance, of our derivation is the proof of a number of new properties concerning concavity and Schur-concavity of MI and LCB.