The noncoherent demodulator for binary FSK uses as the decision statistic the square root of the sum of the squares of the noisy outputs of the in-phase and quadrature matched filters.
a. A co-worker proposes that instead of using a noncoherent demodulator, one could instead estimate phase from the matched-filter outputs and correct for it. Specifically, if uR(t) and uI(t) are the matched-filter outputs, let
Prove that this gives a demodulator that is mathematically equivalent to the noncoherent demodulator.
b. Now the co-worker proposes that θ be estimated on two successive bit times, as above, and averaged. Will this give a better or worse demodulator if phase error is independent from bit to bit? Will this give a better or worse demodulator on a partially-coherent channel? A partially-coherent channel is one on which phase error changes very slowly compared to one bit duration.