3.5 Proofs for Section 2.4.2 “Mean-Square Consistency of the Cyclic Cross-Correlogram”

In this section, proofs of results presented in Section 2.4.2 on the mean-square consistency of the cyclic cross-correlogram are reported.

Lemma 3.5.1 Let a(t) be such that Assumption 2.4.5 is satisfied. We have the following.

Proof: Since , item (a) is a consequence of the properties of the Fourier transforms (Champeney 1990). In Assumption 2.4.5 it is also assumed that there exists γ > 0 such that

(3.62)

In reference to item (b), observe that from (2.125) we have

(3.63)

where, in the third equality, the variable change s = t/T is made. Thus, for f = 0,

(3.64)

(see (2.126)). For f ≠ 0, since , we have

(3.65)

for the Riemann-Lebesgue theorem (Champeney 1990, Chapter 3).

Furthermore, in the sense of distributions (generalized functions) (Champeney 1990; Zemanian 1987) from (2.127), it follows that

(3.66)

where δ(·) denotes Dirac delta.

Lemma 3.5.2 Let us consider the function defined in (2.133). Under Assumption 2.4.5, we have

(3.67)

Proof: For β = 0, one obtains

(3.68)

where the interchange of limit and integral operations is allowed by the dominated convergence theorem. In fact, accounting for Assumption 2.4.5, the magnitude of the integrand function |a(t + s/T)a^*(t)| is bounded by the summable function |a(t)|||a||_∞ not depending on T. The third equality is a consequence of the a.e. continuity of a(t). The last integral in (3.68) exists since .

Let us consider, now, the case β ≠ 0. We have

(3.69)

Thus,

(3.70)

In reference to the first integral in right-hand side of (3.70) it results that

(3.71)

That is, the integrand function is bounded by a summable function independent of T. Thus, by the dominated convergence theorem (Champeney 1990, Chapter 4) we have

(3.72)

since a(t) is continuous a.e. (Assumption 2.4.5). In regard to the second integral in (3.70), since and β ≠ 0, the Riemann-Lebesgue theorem (Champeney 1990, Chapter 3) can be applied:

(3.73)

Therefore, for β ≠ 0 one obtains

(3.74)

Analogously, for the function r_aa(β, s) defined in (3.136) it results

(3.75)

Fact 3.5.3 Signal-Tapering Window Versus Lag-Product-Tapering Window. The data-tapering window in Assumption 2.4.5, strictly speaking, is a lag-product-tapering window depending on the lag parameter τ. Let h_T(t) be the signal-tapering window. We have

(3.76)

That is, the lag-product-tapering window can be expressed in terms of the signal-tapering window h_T(t) as follows:

(3.77)

Its Fourier transform is

(3.78)

where is the Fourier transform of h_T(t) and (−) is an optional minus sign linked to the optional complex conjugation (*).

An analogous relation can be found for the normalized window a(t) in (2.125). Let

(3.79)

By Fourier transforming both sides we have

(3.80)

where A_τ(f) and are the Fourier transforms of a_τ(t) and , respectively, and in the last equality the variable change ν = λT is made. Thus, the rate of convergence to zero of A_τ(f) as |f|→ ∞ can be expressed in terms of the rate of convergence of .

The functions and r_aa(β, s) defined in (2.133) and (3.136), respectively, can be expressed in terms of the normalized signal-tapering window.

(3.81)

where [*] represents an optional complex conjugation (different from (*)).

For a rectangular signal-tapering window , the continuous-time lag-product tapering window and its Fourier transform are given by

(3.82)

(3.83)

respectively. Therefore for |τ| < T

(3.84)

That is,

(3.85)

which should be accounted for in the computation of the bias in Theorem 2.4.12.

Analogous considerations can be made for the discrete-time lag-product-tapering window in Definition 2.6.1. Let

(3.86)

(3.87)

be the discrete-time casual rectangular window and the indicator function, respectively. The discrete-time casual counterparts of (3.82) and (3.83) are

(3.88)

(3.89)

It results that

(3.90)

Thus, for N→ ∞,

(3.91)

3.5.1 Proof of Theorem 2.4.11 Asymptotic Expected Value of the Cyclic Cross-Correlogram

From (2.128) it follows that

(3.92)

where, in the last equality, (3.61) is used. Thus, accounting for (2.39), (2.143) immediately follows.

In (3.92), the interchange of limit and sum operations is justified since the function series

is uniformly convergent. In fact, from Lemma 3.5.1 the Fourier transform A(f) of a(t) is continuous and bounded. Hence, accounting for Assumption 2.4.3a, one obtains

(3.93)

with the right-hand side bounded and not depending on T. Thus, the function series is uniformly convergent due to the Weierstrass criterium (Johnsonbaugh and Pfaffenberger 2002, Theorem 62.6; Smirnov 1964).

3.5.2 Proof of Theorem 2.4.12 Rate of Convergence of the Bias of the Cyclic Cross-Correlogram

Accounting for (2.141), the cyclic cross-correlation function (2.39) can be written as

(3.94)

Moreover, from expression (2.128) of the expected value of the cyclic cross-correlogram, we have

(3.95)

and, hence,

(3.96)

where the fact that (see (2.126)) is used. Therefore,

(3.97)

Due to Assumption 2.4.5, there exits γ > 0 such that (3.62) holds. Then, accounting for (3.62) and (3.63), it follows that for , that is α_n(τ) ≠ α, one obtains

(3.98)

From (3.97), (3.98), and Assumption 2.4.3a, (2.145) immediately follows.

In (3.97), the interchange of limit and sum operations can be justified by the Weierstrass criterium (Johnsonbaugh and Pfaffenberger 2002, Theorem 62.6; Smirnov 1964). In fact, accounting for (2.142), from (3.62) it follows that

(3.99)

uniformly with respect to n, . That is, there exists T_α,τ independent of n, such that for T > T_α,τ one obtains

(3.100)

for some K_α,τ independent of n. Hence,

(3.101)

where the first L^∞-norm is for functions of T and the second for functions of τ, and, in the last inequality, Assumption 2.4.3a has been accounted for. Therefore, for the Weierstrass criterium, the series of functions of T

(3.102)

is uniformly convergent and the order of limit and sum operations in (3.97) can be reversed.

Note that, in general, K_α,τ depends on both α and τ. That is, the convergence of the bias is not uniform with respect to α and τ.

As a final remark, results in Fact 3.5.3 should be used for a refinement of this proof in the case of rectangular signal-tapering window. In fact, from (3.84) it follows that also the term

(3.103)

should be accounted for in the computation of the bias in (3.96)–(3.98). That is, from (3.96)–(3.98) it follows

(3.104)

when γ = 1 as for the rectangular signal-tapering window.

3.5.3 Proof of Theorem 2.4.13 Asymptotic Covariance of the Cyclic Cross-Correlogram

Let us consider the covariance expression (2.129). As regards the term in (2.130), defined

(3.105)

one obtains

(3.106)

where is defined in (2.147) and Lemma 3.5.2 has been accounted for.

The interchange of the order of limit and sum operations in (3.106) is allowed since the double series over n′ and n^′′ of functions of T is uniformly convergent. In fact, we have

(3.107)

with the right-hand side not depending on T. In the last inequality in (3.107), the following inequality, which is a consequence of Assumption 2.4.5, is used

(3.108)

Consequently,

(3.109)

where, in the inequality, Assumptions 2.4.3a and 2.4.8a have been used. Then, from (3.109), it follows that the series of function of T

is uniformly convergent for the Weierstrass criterium (Johnsonbaugh and Pfaffenberger 2002, Theorem 62.6; Smirnov 1964).

The interchange of limit and integral operations in (3.106) is allowed by the dominated convergence theorem (Champeney 1990, Chapter 4). In fact, from (3.105) and (3.107) it follows that

(3.110)

That is, accounting for Assumption 2.4.8a, the integrand function in the second line of (3.106) is bounded by a summable function of s not depending on T.

Analogously, it can be shown that

(3.111)

where is defined in (2.148).

As regards the term defined in (2.132), defined

(3.112)

one obtains

(3.113)

where is defined in (2.149), and Lemma 3.5.2 has been accounted for.

The interchange of the order of limit and sum operations in (3.113) is allowed since the series over n of functions of T is uniformly convergent. In fact, accounting for (3.108), we have

(3.114)

with the right-hand side not depending on T. Consequently,

(3.115)

where, in the last inequality, Assumption 2.4.8b has been used. Then, from (3.115), it follows that the series of functions of T

is uniformly convergent for the Weierstrass criterium (Johnsonbaugh and Pfaffenberger 2002, Theorem 62.6; Smirnov 1964).

The interchange of limit and integral operations in (3.113) is allowed by the dominated convergence theorem (Champeney 1990, Chapter 4). In fact, from (3.112) and (3.114) it follows that

(3.116)

That is, accounting for Assumption 2.4.8b, the integrand function in the second line of (3.113) is bounded by a summable function of s not depending on T.

The proof of Theorem 2.4.13 can be carried out with minor changes by substituting Assumption 2.4.8a with the following:

Assumption 3.5.4 For any choice of z₁, ..., z₄ in {x, x^*, y, y^*} and it results

(3.117)

By reasoning as in the comments following Assumption 2.4.8a, we have that, accounting for (2.121), the function series , i, j {1, ..., 4}, are uniformly convergent due to the Weierstrass criterium (Johnsonbaugh and Pfaffenberger 2002, Theorem 62.6; Smirnov 1964). In addition, in order to satisfy (3.117), the function should be vanishing sufficiently fast as |s|→ ∞ so that the product is summable. A sufficient condition is that there exists > 0 such that and, hence, the result is that . In such a case, the process memory can vanish more slowly than as required to satisfy Assumption 2.4.8a (see the comments following Assumption 2.4.8).

If Assumption 3.5.4 is made instead of Assumption 2.4.8a, then (3.107), (3.109), and (3.110) should be replaced by

(3.118)

(3.119)

(3.120)

respectively.

3.5.4 Proof of Corollary 2.4.14

Let us consider the term defined in (2.130). One has

(3.121)

where, in the second inequality, accounting for (2.133), the inequality

(3.22)

is used. The right-hand side of (3.121) is bounded due to Assumptions 2.4.3a, 2.4.5, and 2.4.8a. Analogously, with reference to the term defined in (2.131), we get

(3.123)

with the right-hand side bounded and independent of τ₂ since the integral over does not depend on shifts of the integrand function. Finally, with reference to the term defined in (2.132), one has

(3.124)

with the right-hand side uniformly bounded with respect to τ₁ and τ₂ due to Assumption 2.4.9.