Modeling wireless channels under spherical invariance assumptions

We look for the probability distribution of the fading envelope yielding the worst and the best performance of digital transmission in a wireless channel. We assume that the underlying fading process is spherically invariant. Using a general representation theorem for such a process in conjunction with semidefinite programming techniques, we derive the envelope densities yielding the maximum and minimum error probability P(e) of uncoded binary modulation, as well as the maximum and minimum outage probability P_out. In particular, for P(e) we show that the worst fading yields P(e) = 0.5, while if the fading process has zero mean the most benign fading has a Rayleigh density, while if its mean is nonzero it has a Rice density with the appropriate Rice coefficient K. The situation for P_out is more complicated: the worst fading yields P_out = 1, while the best fading has a Rayleigh or Rice density only for high SNR values.