3.11 Proofs for Section 2.6.3 “Asymptotic Results as N→ ∞ and T_s → 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 M_p, possibly depending on , are independent of T_s. 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 T_s

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

(3.207)

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)

(not necessarily uniformly with respect to and m).

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

(3.209)

that is,

(3.210)

Therefore, for and we have

(3.211)

with ₁ and ₂ arbitrarily small. Note that the limit is not necessarily uniform in .

Equation (3.209) holds for fixed (and finite) T_s > 0. Therefore, in (2.211) we have that first N→ ∞ and then T_s → 0. This order of the two limits is in agreement with the result of (Dehay 2007) where T_s = N^−δ with 0 < δ < 1.

3.11.3 Noninteger Time-Shift

Let us consider a discrete-time signal x(n) with Fourier transform X(ν).

(3.212)

Definition 3.11.1 The time-shifted version of x(n) with noninteger time-shift μ is defined as

(3.213)

Fact 3.11.2 If x(n) is a finite N-length sequence defined for n {0, 1, ..., N − 1}, the time-shifted version of x(n) with noninteger time-shift μ, can be expressed by the exact interpolation formula

(3.214)

In (3.214),

(3.215)

is the discrete Fourier transform (DFT) of x(n) and

(3.216a)

(3.216b)

with referred to as Dirichlet kernel. According to Definition 3.11.1, we have

(3.217)

Alternatively, the time-shifted version of x(n) with noninteger time-shift μ, can be expressed by the exact interpolation formula

(3.218)

3.11.4 Proof of Theorem 2.6.16 Asymptotic Expected Value of the Hybrid Cyclic Cross-Correlogram

Use a simple modification of Lemma 2.6.12 with mT_s replaced by τ and then use (2.217a)

3.11.5 Proof of Theorem 2.6.17 Rate of Convergence of the Bias of the Hybrid Cyclic Cross-Correlogram

Let us consider the set

(3.219)

defined in (2.222). Since is assumed to be finite and the functions α_k(τ) are bounded for finite τ, the sampling period T_s can be chosen sufficiently small such that

(3.220)

that is, mod f_s is not necessary in the definition (3.219). Moreover, if , then α_k(τ) ≠ α. Thus, for T_s sufficiently small (depending on α and τ) the result is that

(3.221)

In addition, since is finite, for T_s sufficiently small (α − α_k(τ))T_s = 0 only if α = α_k(τ). Therefore, for T_s sufficiently small (depending on α and τ) we have

(3.222)

finite set not depending on T_s. Consequently,

(3.223)

Since for T_s small we have that finite set independent of T_s, with regard to the quantity defined in (2.223), for T_s small we have

(3.224)

where in the second equality inf is substituted by min since (and also ) is finite, and the third equality holds for T_s sufficiently small (depending on α and τ) such that (3.221) holds. Therefore,

(3.225)

provided that the functions α_k(τ) are bounded for finite τ. In the second equality, lim and min operations can be interchanged since min is over a finite set not depending on T_s (in general, lim and inf operations cannot be inverted if inf is over an infinite set). In addition,

(3.226)

Let us define for notation simplicity

It results that

(3.227)

Under the assumption of finite , Assumption 2.6.11 is verified and the thesis of Lemma 2.6.12 is also verified since in the right-hand side of

(3.228)

the sum is identically zero for T_s sufficiently small (depending on τ), that is, for f_s sufficiently large so that, for fixed τ,

(3.229)

where the maximum exists since the number of lag-dependent cycle frequencies is finite and each function α_k(τ) is assumed to be bounded for finite τ.

From the counterpart for the H-CCC of Theorem 2.6.7, it follows that (see (3.199))

(3.230)

where and

(3.231)

Consequently, from (3.226), (3.227), and (3.230), we have

(3.232)

where and the order of the two limits cannot be inverted.

3.11.6 Proof of Theorem 2.6.18 Asymptotic Covariance of the Hybrid Cyclic Cross-Correlogram

Let us consider the limit

(3.233)

where , , and are defined in (2.196), (2.197), and (2.198), respectively, with the replacements and .

As regards the term , we have

(3.234)

with the right-hand side coincident with defined in (2.147). In (3.234), we used the fact that if α_k(τ) are bounded and and are finite sets, for T_s sufficiently small, one has

(3.235)

In addition, the integrand function has been assumed to be Riemann integrable.

Note that the right-hand side of (3.234) can be nonzero only if in a set of values of s with positive Lebesgue measure.

Analogous results hold for terms , and .

3.11.7 Proof of Lemma 2.6.19 Rate of Convergence to Zero of Cumulants of Hybrid Cyclic Cross-Correlograms

From (3.203) it follows that

(3.236)

where the sums over r_i, i = 1, ..., k − 1, and n in the second term range over sets {r_i,min, ..., r_i,max} and {n_min, ..., n_max} with extremes r_i,min, r_i,max, n_min, n_max depending on N and such that, as N→ ∞, r_i,min and n_min approach −∞ and r_i,max and n_max approach +∞.

The rhs of (3.236) does not depend on , m_i. Therefore, the same inequality holds by replacing in the lhs with and we also have

(3.237)

Thus, for k 2 and > 0, we obtain (2.230), where the order of the two limits cannot be interchanged (that is, NT_s→ ∞).

Note that the (k − 1)-dimensional Riemann sum in the second line converges to the (k − 1)-dimensional Riemann integral in the third line if the Riemann-integrable function ϕ is sufficiently regular. In fact, the limit as T_s → 0 of the infinite sum in the second line is not exactly the definition of a Riemann integral over an infinite (k − 1)-dimensional interval. However, the function ϕ in Assumption 2.6.8 can always be chosen such that this Riemann integral exists.

3.11.8 Proof of Theorem 2.6.20 Asymptotic Joint Normality of the Hybrid Cyclic Cross-Correlograms

By following the guidelines of the proof of Theorem 2.6.10, let

(3.238)

From Theorem 2.6.17 holding for γ > 1 (or γ = 1 if a(t) = rect(t)), we have

(3.239)

From Theorem 2.6.18, we have that

is finite. From Theorem 3.13.3, we have that

is finite. From Lemma 2.6.19 with and k 3, we have

(3.240)

Thus, according to the results of Section 1.4.2, for every fixed α_i, τ_i, n_0i the random variables , i = 1, ..., k, are asymptotically (N→ ∞ and T_s → 0 with NT_s→ ∞) zero-mean jointly complex Normal (Picinbono 1996; van den Bos 1995).