Coherency for the Binary Symmetric Channel

For the purpose of communication system planning, predictions of system performance based upon models of an ideal coherent channel are extensively used. Previous work has demonstrated the deleterious effects of radio frequency (RF) phase error on channel performance by machine computation of an integral derived by statistically averaging over the phase error. The required receiver performance in estimating the RF phase for detecting binary coherent signals is further examined. Upper and lower bounds are generated which measure the inaccuracy of the familiar performance formula derived from the idealized coherent channel model. The bounds are obtained analytically without resorting to machine computation. The Tikhonov probability distribution for the RF reference phase error is used. The principal result is the relation
\sigma_{\varpi}^{2} \leq (kE/N_{0})^{-1}
where
\sigma_{\varpi}^{2}
is the variance in the error of the derived carrier phase,
E/N_{0}
is the energy (per uncoded symbol) to noise density ratio required by the ideal coherent channel, and
k
is a factor relating to the tolerable degradation in error rate as given by the bounds. If
k
is on the order of 3 the symbol error rate no more than doubles; in general a
k
of 10 will suffice for no degradation in error rate.