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 trans-mission in a wireless channel. We assume that the underly-ing 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 probabil-ity P (e) of uncoded binary modulation, as well as the maxi-mum and minimum outage probability Pout. 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 sit-uation for pout is more complicated: the worst fading yields pout = 1, while the best fading has a Rayleigh or Rice density only for high SNR values. 1