In this section, the problem of LTV filtering of SC processes is addressed. The class of LTV systems considered here is that of the FOT-deterministic linear systems characterized in Section 6.3.8. FOT-deterministic linear systems are defined as those linear systems that transform input almost-periodic functions into output almost-periodic functions.
In Section 6.3.8 it is shown that the system transmission function of a FOT-deterministic linear system can be written as
where Ω is a countable set. The functions ϕσ(·) are assumed to be invertible and differentiable, with inverse functions ψσ(·) also differentiable and referred to as frequency mapping functions. The functions Gσ(·) and Hσ(·) are linked by the relationships
(4.79)
with and denoting the derivative of ψσ(·) and ϕσ(·), respectively.
In Section 6.3.8 it is shown that the impulse-response function of FOT deterministic linear systems can be expressed as
By substituting (4.77a) and (4.77b) into the input/output relationship in the frequency domain (1.43) one has (Section 6.3.8)
(4.81a)
Analogously, by substituting (4.80a) and (4.80b) into the input/output relationship in the time domain (1.41) one has (Section 6.3.8)
(4.82b)
where ⊗ denotes convolution and
(4.83)
In other words, the output of FOT-deterministic LTV systems is constituted by frequency warped and then LTI filtered versions of the input (Figure 4.4).
It can be shown that the parallel and cascade concatenation of FOT-deterministic LTV systems is still a FOT-deterministic LTV system.
The subclass of FOT-deterministic LTV systems obtained by considering Ω containing only one element was studied, in the stochastic process framework, in (Franaszek 1967) and (Franaszek and Liu 1967) with reference to the continuous-time case and in (Liu and Franaszek 1969) with reference to the discrete-time case. The most important property of these systems, as evidenced in (Franaszek 1967), (Franaszek and Liu 1967), and (Liu and Franaszek 1969), is that they preserve in the output the wide-sense stationarity of the input random process (Section 4.13).
The class of FOT deterministic LTV systems includes that of the linear almost-periodically time-variant (LAPTV) systems (Section 1.3.3) which, in turn, includes, as special cases, linear periodically time-variant (LPTV) and linear time-invariant (LTI) systems. For LAPTV systems, the frequency mapping functions ψσ(f) are linear with unit slope, that is,
(4.84a)
(4.84b)
and then the impulse-response function can be expressed as (see (1.107))
(4.58)
Systems performing time-scale changing are FOT deterministic. The impulse-response function is given by
(4.86)
where s ≠ 0 is the time-scale factor, the set Ω contains just one element,
(4.87)
and
(4.88)
Decimators and interpolators are FOT deterministic discrete-time linear systems (Izzo and Napolitano 1998b) (Section 4.10).
Let x1(t) and x2(t) be jointly SC signals, that is
where , and let h1(t, u) and h2(t, u) be deterministic FOT systems, that is, accordingly with (4.77a) and (4.77b), with transmission functions
(4.90a)
(4.90b)
where is invertible, is its inverse, and both functions are assumed to be differentiable. According to (4.81b), we have
In Section 5.1 it is proved that
From (4.92) it follows that y1(t) and y2(t) are jointly SC. In particular, if x1 ≡ x2 and h1 ≡ h2 (and, hence, y1 ≡ y2), we obtain the notable result that FOT-deterministic linear systems transform SC signals into SC signals. That is, the class of the SC signals is closed under FOT-deterministic linear transformations.
By specializing (4.92) to jointly ACS signals and LAPTV systems one obtains the results of Section 1.3.3.
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