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 of width larger than 1. However, there exists a (unique) generator with compact support contained in [− 1/2, 1/2), that is such that
(5.173) 
With this choice for the generator, the following useful expression holds for the periodic delta train in (4.188):
Due to the local invertibility of functions 
, each function 
has compact support contained in [− 1/2, 1/2) where is invertible, and has values in [− 1/2, 1/2). Let us denote by 
its inverse function. 
has in turn compact support contained in [− 1/2, 1/2) and values in [− 1/2, 1/2). Therefore, for every 
there exists only one pair (ν1, ν2) 
[p1 − 1/2, p1 + 1/2) × [p2 − 1/2, p2 + 1/2) such that 
or, equivalently 
. Consequently, a variable change in the argument of the Dirac delta leads to (Zemanian 1987, Section 1.7)
(5.175) 
Thus,
where the second and the third equalities are consequence of the finite support of 
(and 
). By substituting (5.176) into (5.174) we have
where
(5.178) 
By using (5.177) into (4.190a), we have that (4.190b) and (4.191b) immediately follow. The proof of (4.191a) is similar. 