3.4 Proofs for Section 2.4.1 “The Cyclic Cross-Correlogram”

In this section, proofs of lemmas and theorems presented in Section 2.4.1 on the bias and covariance of the cyclic cross-correlogram are reported.

In the following, all the functions are assumed to be Lebesgue measurable. Consequently, without recalling the measurability assumption, we use the fact that if the functions ϕ₁ and ϕ₂ are such that |ϕ₁| ≤ |ϕ₂|, ϕ₁ is measurable and ϕ₂ is integrable (i.e., ϕ₂ is measurable and |ϕ₂| is integrable), then ϕ₁ is integrable (Prohorov and Rozanov 1989, p. 82). Furthermore, if and , then |ϕ₁ϕ₂| ≤ |ϕ₁|||ϕ₂||_∞ almost everywhere and, hence, .

3.4.1 Proof of Theorem 2.4.6 Expected Value of the Cyclic Cross-Correlogram

By using (2.31c) and (2.118) one has

(3.38a)

(3.38b)

(3.38c)

(3.38d)

from which (2.128) immediately follows.

In (3.38b), the interchange of statistical expectation and integral operations is justified by the Fubini and Tonelli theorem (Champeney 1990, Chapter 3). In fact, the cross-correlation is uniformly (with respect to t and τ) bounded since

(3.39)

and, accounting for Assumption 2.4.3 a, for any z {x, y}

(3.40)

Therefore,

(3.41)

where Assumption 2.4.5 and the variable change s = (t − t₀)/T are used.

The interchange of sum and integral operations to obtain (3.38d) is justified even if is not finite by using the dominated convergence theorem (Champeney 1990, Chapter 4). Specifically, by denoting with an increasing sequence of finite subsets of such that for h < k and , and defining

(3.42)

it results that

(3.43)

In fact, the integrand function in the second term of equality (3.43) is bounded by a summable function of t not depending on k:

(3.44)

where, in the last inequality, Assumptions 2.4.3a and 2.4.5 are used.

3.4.2 Proof of Theorem 2.4.7 Covariance of the Cyclic Cross-Correlogram

For zero-mean stochastic processes x(t) and y(t) one obtains (Gardner 1985; Spooner and Gardner 1994)

(3.45)

where

(3.46)

is the covariance of the complex random variables z₁ and z₂ and the cumulant of complex processes is defined according to (Spooner and Gardner 1994, Appendix A) (see also Section 1.4.2). Thus, accounting for (2.118), one obtains

(3.47a)

(3.47b)

(3.47c)

By substituting (2.119) and (2.120) (Assumption 2.4.2) into (3.47c) and making the variable changes u₁ = u and u₂ = u − s, it results in

(3.48)

where

(3.49)

(3.50)

(3.51)

Finally, by making the variable change u/T = t′ + (t₂ + s)/T into (3.49)–(3.51) and using (2.125) and (2.133), (2.129)–(2.132) easily follow. In (3.49)–(3.51), for notation simplicity, , , , and .

The interchange of the orders of cov{ · } and integral operations to obtain (3.47b) is justified by the Fubini and Tonelli theorem (Champeney 1990, Chapter 3). In fact,

(3.52)

The interchange of statistical expectation and single integral can be justified as in the proof of Theorem 2.4.6 accounting for Assumptions 2.4.3a and 2.4.5. As regards the interchange of statistical expectation and double integral, we have

(3.53)

Thus, accounting for the uniform boundedness of the absolute fourth-order moments of x(t) and y(t) (Assumption 2.4.4) and Assumption 2.4.5, one has

(3.54)

Therefore, the Fubini and Tonelli theorem (Champeney 1990, Chapter 3) can be used to obtain (3.47b) and (3.47c).

The interchange of sum and integral operations to obtain (3.48) is justified even if the sets and are not finite. In fact, let us consider the term defined in (3.49). Denote with and two increasing sequences of finite subsets of and , respectively, such that and , and define

(3.55)

The result is that

(3.56)

where Assumptions 2.4.3a and 2.4.5 have been accounted for. Thus, the left-hand side of (3.56) is bounded by a summable function of (s, u) not depending on h and k. Therefore, the dominated convergence theorem (Champeney 1990, Chapter 4) can be applied as follows:

(3.57)

An analogous result can be found for the term defined in (3.50). As regards the term defined in (3.51), denote with an increasing sequence of finite subsets of such that , and define

(3.58)

The result is that

(3.59)

where Assumption 2.4.3b has been accounted for. Hence, the dominated convergence theorem can be applied similarly as in (3.57).