Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

3.8 Proofs for Section 2.5 “Sampling of GACS Processes”

3.8.1 Proof of Theorem 2.5.3 Aliasing Formula for the Cyclic Cross-Correlation Function

Equation (2.179b) is coincident with (2.177). Equation (2.179a) is obtained by specializing to second-order (Izzo and Napolitano 2003, eq. (49)) (or (Izzo and Napolitano 2005, eq. (4.30))). The proof is analogous to the proof of (1.174) for ACS processes.

Once (2.179a) is proved, an alternative proof of (2.179b) is obtained as follows. By substituting (2.39) into (2.179a) one has

(3.141)

from which (2.179b) easily follows, provided that condition

(3.142)

is satisfied in order to allow the interchange of the order of the sum operations (Johnsonbaugh and Pfaffenberger 2002, Theorem 29.4).