Condition (2.209) holding uniformly w.r.t. τ assures that Assumption 2.6.11 is verified. The numbers Mp, possibly depending on , 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
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.
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 Ts, , and m,
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) Ts > 0. Therefore, in (2.211) we have that first N→ ∞ and then Ts → 0. This order of the two limits is in agreement with the result of (Dehay 2007) where Ts = N−δ with 0 < δ < 1.
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
In (3.214),
is the discrete Fourier transform (DFT) of x(n) and
(3.216a)
with referred to as Dirichlet kernel. According to Definition 3.11.1, we have
Alternatively, the time-shifted version of x(n) with noninteger time-shift μ, can be expressed by the exact interpolation formula
(3.218)
Use a simple modification of Lemma 2.6.12 with mTs replaced by τ and then use (2.217a)
Let us consider the set
defined in (2.222). Since is assumed to be finite and the functions αk(τ) are bounded for finite τ, the sampling period Ts can be chosen sufficiently small such that
(3.220)
that is, mod fs is not necessary in the definition (3.219). Moreover, if , then αk(τ) ≠ α. Thus, for Ts sufficiently small (depending on α and τ) the result is that
In addition, since is finite, for Ts sufficiently small (α − αk(τ))Ts = 0 only if α = αk(τ). Therefore, for Ts sufficiently small (depending on α and τ) we have
(3.222)
finite set not depending on Ts. Consequently,
(3.223)
Since for Ts small we have that finite set independent of Ts, with regard to the quantity defined in (2.223), for Ts 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 Ts sufficiently small (depending on α and τ) such that (3.221) holds. Therefore,
(3.225)
provided that the functions αk(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 Ts (in general, lim and inf operations cannot be inverted if inf is over an infinite set). In addition,
Let us define for notation simplicity
It results that
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 Ts sufficiently small (depending on τ), that is, for fs 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 αk(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))
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.
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
with the right-hand side coincident with defined in (2.147). In (3.234), we used the fact that if αk(τ) are bounded and and are finite sets, for Ts 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 .
From (3.203) it follows that
where the sums over ri, i = 1, ..., k − 1, and n in the second term range over sets {ri,min, ..., ri,max} and {nmin, ..., nmax} with extremes ri,min, ri,max, nmin, nmax depending on N and such that, as N→ ∞, ri,min and nmin approach −∞ and ri,max and nmax approach +∞.
The rhs of (3.236) does not depend on , mi. 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, NTs→ ∞).
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 Ts → 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.
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 αi, τi, n0i the random variables , i = 1, ..., k, are asymptotically (N→ ∞ and Ts → 0 with NTs→ ∞) zero-mean jointly complex Normal (Picinbono 1996; van den Bos 1995).
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