3.12 Proofs for Section 2.6.4 “Concluding Remarks”

3.12.1 Proof of Theorem 2.6.21 Asymptotic Discrete-Time Cyclic Cross-Correlogram

Let t₀ = n₀T_s, T = (2N + 1)T_s fixed.

(3.241)

where

(3.242)

The stochastic function ψ(t) is mean-square Riemann-integrable in (t₀ − T/2, t₀ + T/2). That is, in the limit as the sampling period T_s approaches zero (and, hence, N→ ∞ so that (2N + 1)T_s = T is constant), in (3.241) we have

(3.243)

In fact, a necessary and sufficient condition such that (3.243) holds is (Loève 1963, Chapter X) (see also Theorem 2.2.15)

(3.244)

and the summability of can be proved under Assumptions 2.4.2–2.4.5 by following the proof of Theorem 2.4.7 (see (3.47b), (3.47c), and (3.52)–(3.54)).

Remark 3.12.1 Let

(3.245)

(3.246)

(3.247)

Then,

(3.248)

is not a useful bound to prove that as T_s → 0 and T→ ∞ since the limits

(3.249)

(3.250)

are not uniform with respect to T and T_s, respectively. This is in agreement with the fact that in Theorem 2.6.13 we have first N→ ∞ and then T_s → 0.