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Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications by Antonio Napolitano

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3.11 Proofs for Section 2.6.3 “Asymptotic Results as N→ ∞ and Ts → 0′′

3.11.1 Proof of Lemma 2.6.12

Condition (2.209) holding uniformly w.r.t. τ assures that Assumption 2.6.11 is verified. The numbers Mp, possibly depending on img, are independent of Ts. Under Assumption 2.6.11, the Weierstrass M-test (Johnsonbaugh and Pfaffenberger 2002, Theorem 62.6) assures the uniform convergence of the series of functions of Ts

equation

Therefore, the limit operation can be interchanged with the infinite sum

(3.207) equation

where, in the second equality the sufficient condition (2.209) for Assumption 2.6.11 is used.

3.11.2 Proof of Theorem 2.6.13 Mean-Square Consistency of the Discrete-Time Cyclic Cross-Correlogram

From Lemma 2.6.12 we have

(3.208) equation

(not necessarily uniformly with respect to img and m).

From Theorems 2.6.4 and 2.6.5 it follows that, for every fixed Ts, img, and m,

(3.209)

that is,

(3.210)

Therefore, for ...

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