Fourier transform of a time-scaled, delayed, and frequency-shifted signal:
(7.383)
(7.384)
(7.385)
By using (7.261), the result is that
where is the continuous-time infinite average with respect to t, and, in the third equality, the variable change t′ = st − da is made, so that
This proof can be equivalently carried out in the FOT approach by simply removing the expectation operator E{ · } in (7.386) and (7.387).
(7.264) is obtained by Fourier transforming both sides of (7.263).
By reasoning as for (7.386), we have
(7.388)
(7.268) is obtained by Fourier transforming both sides of (7.267).
Accounting for (7.262) and (7.272), one obtains
from which (7.273) follows by a variable change in the argument of the Dirac delta (Zemanian 1987, Section 1.7).
Using (7.261), we have
(7.390)
from which (7.275) immediately follows. The same result is obtained by taking the double inverse Fourier transform (see (1.15)) of both sides of (7.273) and then making the variable changes t1 = t + τ, t2 = t.
Accounting for (7.262), one obtains
(7.391)
from which (7.276) immediately follows using (7.272).
Accounting for (7.284) and (1.88), we have
where the last equality is obtained by putting and into (1.88).
Under assumption (1.89), . Thus, if for k1 ≠ k2, then each term in the sum over α in (7.392) is a finite-strength additive sinewave component of the cyclic autocorrelation function only when k1 = k2 = k so that there is no dependence on t in the argument of . Thus we have the first and second equality in (7.286).
By taking the Fourier coefficient at frequency β in (7.286) we have
(7.393)
from which (7.287) easily follows observing that the last time average on t equals to the Kronecker delta .
Note that (7.287) can be obtained as special case of (Izzo and Napolitano 2002b, eq. (90)) for N = 2 with only the second optional complex conjugation present, by using (Izzo and Napolitano 1998a, eq. (36)) that expresses reduced-dimension (generalized) cyclic temporal moment functions in terms of (generalized) cyclic temporal moment functions.
Replicas in (7.325) are separated by 1 in both ν1 and ν2 variables. Thus, from (7.326a) it follows that fs ≥ 2B|s| is a sufficient condition such that replicas do not overlap. Note that, even if replicas do not overlap, the mapping νi = fi/fs, i = 1, 2, does not link (7.273) and (7.325) for νi [− 1/2, 1/2]. A sufficient condition to assure such a mapping or, equivalently, that only replica (n1, n2) in (7.325) lies in the square (ν1, ν2) (n1 − 1/2, n1 + 1/2) × (n2 − 1/2, n2 + 1/2), according to (7.326b) is , that is, , where (7.298) is accounted for.
According to (7.326a), if , the mapping ν1 = f1/fs for ν1 [− 1/2, 1/2] between the densities in (7.325) with n1 = n2 = 0 and those in (7.273) is assured provided that the support of replica with n1 = n2 = 0 does not intercept that of replicas with n1 = 0, n2 = ± 1. That is, on every support line
the following implication
must hold, where (7.394) is obtained by rearranging the argument of the Dirac delta in (7.325). From (7.394) with substituted into we have
(7.396)
where the bound for comes form (1.176b). Thus, inequality in (7.95) is satisfied provided that that is, fs ≥ 2(2B|s| + |fa|).
These results agree with those in (Napolitano 2010b, Theorem 3) since (−)κ = κ.
By setting and into the cyclic-spectrum support expression (1.176a), we have
(7.397)
where the last inclusion relationship holds provided that inequality fs ≥ȅ4(B|s| + |fa|) is satisfied. In fact, accounting for (7.298), such an inequality is equivalent to .
Replicas in (7.329) are separated by 1 in both and ν domains. Consequently, condition fs ≥ 4(B|s| + |fa|) assures that only replica with p = 0 and q = 0 gives nonzero contribution in the principal domain . Then, observing that only replicas with p = q = 0 are coincident in (7.329) and (7.330), equation (7.331) immediately follows.
Replicas in the aliasing formula (7.332) are separated by 1 in both ν1 and ν2 variables. Thus, from the support bound (7.333a) it follows that B|s|/fs ≤ 1/2 and B/fs ≤ 1/2 that is,
(7.398)
is a sufficient condition such that replicas do not overlap. Moreover, from (7.333b), it follows that a sufficient condition to obtain that only the replica (n1, n2) lies in the square (ν1, ν2) (n1 − 1/2, n1 + 1/2) × (n2 − 1/2, n2 + 1/2) is and B/fs ≤ 1/2. Thus, accounting for (7.298), for the replica with n1 = n2 = 0 we have (7.334).
Due to (7.333a), the mapping ν1 = f1/fs1 for ν1 [− 1/2, 1/2] between the densities with n1 = n2 = 0 in (7.332) and those in (7.276) is assured provided that the support replica with n1 = n2 = 0 does not intercept that of the replica with n1 = 0, n2 = ± 1. That is, on every support line
the following implication
must hold. From (7.399) with substituted into we have
(7.401)
Thus, inequality in (7.400) is satisfied provided that 3B/fs ≤ 1 − B/fs, that is, fs ≥ 4B. Then, accounting for (7.334), we obtain (7.336).
From (7.339) (with y(n) = xs(n), A = 1, ϕ = 0, , and d = 0) it follows that
Thus, if or if for |α| > fs/2, then for the sum in the rhs of (7.339) equals the rhs of (7.402) and, hence, . A sufficient condition to assure for |α| > fs/2 is that the bandwidth of xas(t) is less than fs/2 (Section 1.3.9), (Napolitano 1995; Gardner et al. 2006). That is, B|s| ≤ fs/2, i.e., (7.340).
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