Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

5.9 Proofs for Section 4.8 “Discrete-Time SC Processes”

5.9.1 Proof of Theorem 4.8.3 Characterization of Discrete-Time Spectrally Correlated Processes

At first, let us observe that, analogously to the case of continuous-time signals, the functions in the right-hand side of (4.188) in general are not unambiguously determined. By opportunely selecting the support of the functions , the corresponding functions can be chosen to be locally invertible in intervals [p − 1/2, p + 1/2), with p integer, that is, their restrictions to these intervals are invertible. In addition, since are in the argument of a periodic delta train with period 1, they can always be chosen with values in [− 1/2, 1/2).

Every periodic function can be expressed as the periodic replication, with period 1, of a or generator function :

(5.172)

The generator, in general, is not unambiguously determined and can have support ...