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

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5.6 Proofs for Section 4.7.2 “Asymptotic Normality of the Frequency-Smoothed Cross-Periodogram”

In this section, proofs of lemmas and theorems presented in Section 4.7.2 on the asymptotic complex Normality of the frequency-smoothed cross-periodogram are reported.

Fact 5.6.1

Let

(5.122) equation

where [*]i represents the ith optional complex conjugation, and let us consider the k × 2 table

(5.123) equation

and a partition of its elements into disjoint sets {ν1, …, νp}. The cumulant cum{Z1(f11, f21), …, Zk(f1k, f2k)} can be expressed as (Leonov and Shiryaev 1959), (Brillinger 1965), (Brillinger and Rosenblatt 1967)

(5.124) equation

where νm (m = 1, …, p) are subsets of elements of the k × 2 table (5.123), img is the cumulant of the elements in νm, and the (finite) sum in (5.124) is extended over all indecomposable partitions of table (5.123), including the partition with only one element (see the discussion following (2.159) for details).

Let

(5.125) equation

The elements of the table (5.123) can be identified by the pair ...

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