4.3 Linear Time-Variant Filtering of SC Processes

4.3.1 FOT-Deterministic Linear Systems

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

(4.77a)

(4.77b)

where Ω is a countable set. The functions ϕ_σ(·) are assumed to be invertible and differentiable, with inverse functions ψ_σ(·) also differentiable and referred to as frequency mapping functions. The functions G_σ(·) and H_σ(·) are linked by the relationships

(4.78)

(4.79)

with and denoting the derivative of ψ_σ(·) and ϕ_σ(·), respectively.

In Section 6.3.8 it is shown that the impulse-response function of FOT deterministic linear systems can be expressed as

(4.80a)

(4.80b)

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)

(4.81b)

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)

(4.82a)

(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).

Figure 4.4 FOT-deterministic LTV systems: Realization and corresponding input/output relations in time and frequency domains. The first system is LTV operating frequency warping, the second system is a LTI filter.

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).

4.3.1.1 LAPTV Systems

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 ψ_σ(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)

4.3.1.2 Time-Scale Changing

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).

4.3.2 SC Signals and FOT-Deterministic Systems

Let x₁(t) and x₂(t) be jointly SC signals, that is

(4.89)

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

(4.91)

In Section 5.1 it is proved that

(4.92)

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.