4.8 Discrete-Time SC Processes

In this section, discrete-time spectrally correlated processes are defined and characterized (Napolitano 2011). For the sake of generality, a joint characterization of two processes x₁(n) and x₂(n) in terms of cross-statistics is provided. The characterization of a single process can be obtained as a special case by taking x₁ ≡ x₂.

The discrete-time processes x₁(n) and x₂(n) are said to be second-order jointly harmonizable if (Loève 1963)

(4.182)

with spectral covariance of bounded variation:

(4.183)

Under the harmonizability assumption, we have

(4.184)

where χ_i(ν) is the integrated spectrum of x_i(n). We can formally write dχ_i(ν) = X_i(ν) dν (Gardner 1985, Chapter 10.1.2), (Papoulis 1991, Chapter 12-4), where

(4.185)

is the Fourier transform of x_i(n) to be considered in the sense of distributions (Gelfand and Vilenkin 1964, Chapter 3), (Henniger 1970), provided that χ_i(ν) does not contain singular component (Hurd and Miamee 2007).

Definition 4.8.1 Let x₁(n) and x₂(n) be discrete-time complex-valued second-order jointly harmonizable stochastic processes. Their Loève bifrequency cross-spectrum is defined as

(4.186)

where subscript x denotes and, in the sense of distributions (Gelfand and Vilenkin 1964, Chapter 3), (Henniger 1970),

(4.187)

provided that χ_i(ν) and do not contain singular components.

Definition 4.8.2 Let x₁(n) and x₂(n) be discrete-time complex-valued second-order jointly harmonizable stochastic processes not containing any additive finite-strength sinewave component. The processes are said to be jointly spectrally correlated if their Loève bifrequency cross-spectrum can be expressed as

(4.188)

where is a countable set, , and and are complex-and real-valued, respectively, periodic functions of ν with period 1.

From (4.188) it follows that discrete-time jointly SC processes have spectral masses concentrated on the countable set of support curves

(4.189)

where mod 1 is the modulo 1 operation with values in [− 1/2, 1/2). Moreover, the spectral mass distribution is periodic with period 1 in both frequency variables ν₁ and ν₂.

Two equivalent representations for the Loève bifrequency cross-spectrum hold, as stated by the following result.

Theorem 4.8.3 Characterization of Discrete-Time Jointly Spectrally Correlated Processes (Napolitano 2011, Theorem 3.1). Let x₁(n) and x₂(n) be discrete-time jointly SC processes. Their Loève bifrequency cross-spectrum (4.188) can be expressed in the two equivalent forms

(4.190a)

(4.190b)

with

(4.191a)

(4.191b)

provided that is locally invertible in every interval of width 1, is the periodic replication with period 1 of one of the local inverses, and both and are differentiable.

Proof: See Section 5.9.

For discrete-time jointly SC processes with Loève bifrequency cross-spectrum (4.190a), (4.190b), the bounded variation condition (4.183) reduces to

(4.192)

From (4.190a) and (4.190b), according to notation in (Lii and Rosenblatt 2002), the following alternative representation for the Loève bifrequency cross-spectrum of discrete-time jointly SC processes can be easily proved.

(4.193a)

(4.193b)

where the functions , , , and are such that

(4.194)

(4.195)

(4.196)

(4.197)

Without lack of generality, it can be assumed that two support curves and , with k ≠ k′, intersect at most in a finite or countable set of points (ν₁, ν₂).

In the general case of support functions with nonunit slope for every k, analogously to the continuous-time case (see (4.20a) and 4.20b), two spectral cross-correlation density functions (4.191a) and (4.191b) should be considered, depending on which one of ν₁ and ν₂ is taken as independent variable in the argument of the Dirac deltas in (4.190a) and (4.190b)

From (4.182), (4.190a), and (4.190b), it follows that the second-order cross-moment of jointly SC processes can be expressed as

(4.198a)

(4.198b)

Second-order jointly ACS signals in the wide-sense are characterized by an almost-periodic cross-correlation function. That is (Section 1.3.8)

(4.199)

where the Fourier coefficients

(4.200)

are referred to as cyclic cross-correlation functions and

(4.201)

is the countable set of cycle frequencies in the principal domain [− 1/2, 1/2). By double Fourier transforming both sides of (4.199) with n₁ = n + m and n₂ = n, the following expression for the Loève bifrequency cross-spectrum is obtained

(4.202)

where

(4.203)

are the cyclic spectra. From (4.202) it follows that discrete-time jointly ACS processes are obtained as a special case of jointly SC processes when the spectral support curves are lines with slope ±1 in the principal frequency domain (ν₁, ν₂) [− 1/2, 1/2]². If the set contains the only element , then the cross-correlation function (4.199) does not depend on n (for the considered choice of (*)) and the processes x₁(n) and x₂(n) are jointly WSS. In such a case, the Loève bifrequency cross-spectrum has support contained in a diagonal of the principal frequency domain.

More generally, the processes x₁(n) and are said to exhibit joint almost-cyclostationarity at cycle frequency if the (conjugate) cross-correlation function is not necessarily an almost-periodic function of n but contains a finite-strength additive sinewave component at frequency . In such a case, the cyclic cross-correlation function (4.200) is nonzero for .

In (Akkarakaran and Vaidyanathan 2000), (Izzo and Napolitano 1998b), it is shown that multirate transformations, such as expansion and decimation, of ACS processes lead to ACS processes with different cyclostationarity properties. In contrast, in Section 4.10 it is shown that a discrete-time ACS process and its expanded or decimated version are jointly SC.