Old Section 36 (c)

Old Section 36 (c) PhL 3.19.13 I don't think this section has any use. I don't think the PSK remarks are correct since there you are really talking about a phase in the time domain, not the ω domain. I don't know that the significance of the phase of X(ω) is, so why try to compute its statistical average! (c) derivation of <phase of X(ω)> Finally, we might ask if there is something that can be said for the average phase of a statistical pulse train. We already have statements about the average full spectrum, and about the average magnitude squared. If we define the phase of X(ω) by X(ω)= |X(ω)|eiφ with -π < φ ≤ π, then it is easy to show from (36.2) that φi(ω) = tan-1 (36.8) where we extend the definition of tan-1 to the range (-π.π). So here is our not-very-satisfying result, <φ(ω)> = (1/I)[ tan-1 ] (36.9) For random coefficients, one might expect <φ(ω)> = 0 since it is hard to imagine what else it could be. On the other hand, one could certainly find seemingly random coefficients for which this is not true. For example, in N-ary phase shift keying, data words are encoded in bit patterns which cause the signal phase to jump between fixed points on a circular constellation in phase space. If a data word is repeated many times, one would expect <φ(ω)> those that would be used to generate an N-ary phase-shift-keyed (N-PSK) signal for a constant , in which case the average phase should peak at N distinct values.