The big picture

15 The big picture Contents 15.1 Introduction....................................................................................................... 463 15.2 Oscillatory spectrally correlated processes ............................................................. 463 15.3 Relationships among classes of nonstationary processes ......................................... 464 15.3.1 ACS, CS, and WSS processes ........................................................................ 464 15.3.2 GACS, SC, and ACS processes ...................................................................... 465 15.3.3 OSC, OACS, SC, and ACS processes ............................................................... 466 15.3.4 Oscillatory processes.................................................................................. 468

Abstract The new class of the oscillatory spectrally correlated processes is introduced. It includes, as special cases, all the previously considered classes of nonstationary signals, namely the almostcyclostationary, generalized almost-cyclostationary, spectrally correlated, and oscillatory almostcyclostationary signals. Inclusion relationships among these classes and those of the wide-sense stationary and of the Priestley oscillatory signals are enlightened. Keywords Oscillatory spectrally correlated, Relationships among process classes, Nonstationary processes, Autocorrelation, Loève bifrequency spectrum, Time-variant spectrum

15.1 Introduction In this chapter, relationships among the classes of nonstationary signals considered in the previous chapters are enlightened. Inclusion relationships among these classes are presented and the structure of their statistical functions in time and frequency domains are compared. The new class of the oscillatory spectrally correlated processes is introduced and its second-order characterization in the time and frequency domains is provided. This class of processes is shown to contain, as special cases, all the other classes of processes considered in the previous chapters.

15.2 Oscillatory spectrally correlated processes Let us consider a second-order harmonizable stochastic process y(t) that admits the representation (14.1) Cyclostationary Processes and Time Series. https://doi.org/10.1016/B978-0-08-102708-0.00027-3 Copyright © 2020 Elsevier Ltd. All rights reserved.

463

464

Cyclostationary Processes and Time Series  y(t) =

R

At (f ) ej 2πf t dZ(f ) .

(15.1)

The process y(t) is said to be oscillatory spectrally correlated (OSC) with respect to the family of oscillatory functions {At (f ) ej 2πf t } if Z(f ) is the integrated complex spectrum of an SC process x(t), that is (see (13.1a))      δ f2 − x(k) (f1 ) dμ(k) (15.2) E dZ(f1 ) dZ (∗) (f2 ) = x (f1 ) df2 , k∈I

where I is a countable set. The modulating functions At (f ), as functions of t, are in general (k) low-pass functions. In (15.2), μx (f ), k ∈ I, is a family of complex measures. The process x(t) is referred to as the underlying SC process. (k) If μx (f ) does not contain the singular component (Hurd and Miamee, 2007, p. 197), (k) (k) (k) (Cramér, 1940), then μx (f ) = Sx (f ) df , where Sx (f ) is the density of spectral correla(k) tion of the SC process x(t) (see (13.1a)). Sx (f ) contains Dirac deltas in correspondence of (k) the jumps of μx (f ). The (conjugate) autocorrelation function of y(t) is given by   E y(t + τ ) y (∗) (t)    (k) (∗) = At+τ (f1 ) At x(k) (f1 ) ej 2π[f1 +(−)x (f1 )] t ej 2πf1 τ dμ(k) (15.3) x (f1 ) k∈I

R

The Loève bifrequency spectrum of y(t) is given by   Sy (f1 , f2 )  E Y (f1 ) Y (∗) (f2 )    = A(f1 − ν; ν) A(∗) f2 − x(k) (ν); x(k) (ν) dμ(k) x (ν) k∈I

R

where

 A(f ; ν) 

R

At (ν) e−j 2πf t dt .

(15.4)

(15.5)

From (15.4) and (15.2) it follows that the Loève bifrequency spectrum of an OSC process y(t) can be seen as obtained by the Loève bifrequency spectrum of the underlying SC process x(t) by spreading around the support curves f2 = x(k) (f1 ), k ∈ I, the spectral (k) correlation densities Sx (f ).

15.3 Relationships among classes of nonstationary processes 15.3.1 ACS, CS, and WSS processes ACS processes in the wide sense have the autocorrelation function which is an almostperiodic function of time with coefficients depending on the lag parameter τ and frequencies, referred to as cycle frequencies, not depending on τ . Cycle frequencies are possibly incommensurate, that is, they are not necessarily harmonically related. Poly-cyclostationary

Chapter 15 • The big picture 465

FIGURE 15.1 Inclusion relationships for ACS, poly-CS, CS, and WSS process classes.

(poly-CS) processes are obtained as special case of ACS processes when the cycle frequencies are integer multiples of the reciprocals of two or more incommensurate periods (Remark 1.10). Cyclostationary (CS) processes in the wide sense are obtained as special case of poly-CS processes (and hence of ACS processes) when the cycle frequencies are all integer multiples of a fundamental one. Finally, WSS processes are obtained as further specialization when the only cycle frequency is zero. Therefore, the class of the ACS processes includes that of the poly-CS processes that includes that of the CS processes which in turn includes that of the WSS processes (Fig. 15.1). The same inclusion relationships are obtained if almost-cyclostationarity, cyclostationarity, and stationarity are considered in the strict sense for a fixed order or if the almostperiodicity or periodicity of the statistical functions are defined in one of the generalized senses considered in Section B.4. Note that a different inclusion relationship among the ACS, poly-CS, CS, and WSS process classes is obtained if the classification is made in the FOT approach on the basis of the existence of FOT probabilistic models. In fact, the existence of an almost-periodic distribution requires more stringent conditions on the time series with respect to those for the existence of the stationary distribution (Section 2.3.3). Therefore, time series for which a periodic, poly-periodic, or almost-periodic FOT probabilistic model exists are a subclass of the time series for which a stationary model exists.

15.3.2 GACS, SC, and ACS processes SC processes have the Loève bifrequency spectrum with spectral masses concentrated on a countable set of curves in the bifrequency plane. ACS processes are the special case when the support curves are lines parallel to the main diagonal (compare Fig. 13.1 (top) and Fig. 1.2 (bottom)).

466

Cyclostationary Processes and Time Series

FIGURE 15.2 GACS, SC, and ACS process classes.

Table 15.1 (Conjugate) autocorrelation Rx (t, τ ), Loève bifrequency spectrum Sx (f1 , f2 ), and time-variant spectrum Sx (t, f ) of WSS, ACS, GACS, and SC processes. Void boxes correspond to not easily interpretable expressions. Rx (t, τ ) = E{x(t + τ ) x (∗) (t)} (Conjugate) Autocorrelation

Sx (f1 , f2 ) = E{X(f1 ) X (∗) (f2 )} Loève Bifrequency Spectrum

Sx (t, f ) = Fτ →f [Rx (t, τ )] Time-Variant Spectrum

ACS

Rx (τ )  Rxα (τ ) ej 2π αt

Sx (f1 ) δ(f2 + (−)f1 )  Sxα (f1 )δ(f2 − (−)(α − f1 ))

Sx (f )  Sxα (f ) ej 2π αt

α∈A

α∈A

α∈A

GACS

 (k) (k) Rx (τ ) ej 2π αx (τ )t k∈I

SC

†

   (k) (k) Sx (f1 )δ f2 − x (f1 )

 (k) (k) Sx (f ) ej 2π [f +(−)x (f )]t

k∈I

k∈I

WSS

†

 k∈I R

(k) (k) Sx (f ) ej 2πf (t+τ ) e(−)j 2π x (f )t df

GACS processes have an almost-periodically time-variant autocorrelation function whose (generalized) Fourier series expansion has coefficients and frequencies (cycle frequencies) depending on the lag parameter (Chapter 12). ACS processes are obtained as special case when the frequencies are constant with respect to the lag parameter. This corresponds to a Loève bifrequency spectrum with spectral masses concentrated along lines with unit slope. Thus, the class of the ACS processes is the intersection of the classes of the GACS and SC processes (Fig. 15.2), (Table 15.1), (Napolitano, 2012, Sec. 4.2.2).

15.3.3 OSC, OACS, SC, and ACS processes OSC and OACS processes admit the representation (15.1). For OSC processes, Z(f ) is the integrated complex spectrum of an SC process while for ACS processes Z(f ) is the integrated complex spectrum of an ACS process. Since ACS processes are a subclass of the SC process class, OACS processes are obtained as special case of OSC processes (Fig. 15.3). If At (f ) = 1 in (15.1), OSC processes reduce to SC processes (Fig. 15.3). Specifically, the Loève bifrequency spectrum of an OSC process has spectral density significantly different from zero in strips around the support curves of the underlying SC process (Table 15.2). If At (f ) = 1, then A(f ; ν) = δ(f ) independent of ν and the Loève bifrequency spectrum (15.4) reduces to that of an SC process. If At (f ) = 1 in (15.1), OACS processes reduce to ACS processes (Fig. 15.3). Specifically, the Loève bifrequency spectrum of an OACS process has spectral density significantly

Chapter 15 • The big picture 467

FIGURE 15.3 Inclusion relationships for GACS, SC, ACS, MC, OACS, and OSC process classes. Gray area represents AM-TW ACS processes that do not reduce to ACS processes.

Table 15.2 (Conjugate) autocorrelation Ry (t, τ ), Loève bifrequency spectrum Sy (f1 , f2 ), and time-variant spectrum Sy (t, f ) of MC, OACS and OSC processes y(t). Void boxes correspond to not easily interpretable expressions. For an OACS process y(t), x(t) denotes the underlying ACS process. For an OSC process y(t), x(t) denotes the underlying SC process.

MC

Ry (t, τ ) = E{y(t + τ ) y (∗) (t)} (Conjugate) Autocorrelation  ryα (τ ) mαy (t) ej 2π αt

Sy (f1 , f2 ) = E{Y (f1 ) Y (∗) (f2 )} Loève Bifrequency Spectrum  syα (f1 )Myα (f1 + (−)f2 − α)

Sy (t, f ) = Fτ →f [Ry (t, τ )] Time-Variant Spectrum  syα (f ) mαy (t) ej 2π αt

α∈A

α∈A

α∈A



OACS

ρyα (t, τ ) ej 2π αt

†

α∈A



Syα (t, f ) ej 2π αt

α∈A

OSC †† †††     (∗) f2 − (−)(α − ν); (−)(α − ν) dμαx (ν) † A(f1 − ν; ν) A α∈ A R

††



k∈I R

†††

(∗)

At+τ (f1 ) At

 k∈I R

  (k) (k) (k) x (f1 ) ej 2π [f1 +(−)x (f1 )] t ej 2πf1 τ dμx (f1 )

  (k) (k) (k) A(f1 − ν; ν) A(∗) f2 − x (ν); x (ν) dμx (ν)

different from zero in strips around the support lines of the underlying ACS process. It reduces to the Loève bifrequency spectrum of an ACS process when At (f ) = 1. Equivalently, the autocorrelation function of an OACS process is constituted by the superposition of amplitude- and angle-modulated sine waves. When At (f ) = 1, the modulation is not present and one obtains the autocorrelation of an ACS process which is given by the superposition of (non-modulated) sine waves (Tables 15.1 and 15.2). MC processes are obtained as subclass of the OACS processes when each evolutionary cyclic autocorrelation function factorizes into the product of a function of t and a function of τ (see (14.20)). If the function of t is constant, then we obtain the ACS processes. Thus

468

Cyclostationary Processes and Time Series

FIGURE 15.4 Inclusion relationships for WSS, Priestley’s, OACS, and OSC process classes.

the class of the OACS processes contains that of the MC processes that in turn contains that of the ACS processes (Fig. 15.3). (k) A GACS process with generalized (conjugate) cyclic autocorrelation functions Ryy (∗) (τ ) (k)

and lag-dependent (conjugate) cycle frequencies αyy (∗) (τ ) is a special case of OACS process (Fig. 15.3) with evolutionary (conjugate) cyclic autocorrelation functions ρyy (∗) (t, τ ) = (k)

j 2π

(k)

(τ ) t

(k)

(k)

yy (∗) Ryy (∗) (τ ) e , where αyy (∗) (τ )  αk + yy (∗) (τ ) and αk are the (conjugate) cycle frequencies of the underlying ACS process for the OACS process. As already observed in the general case of OACS processes, the case of slowly varying evolutionary (conjugate) cyclic (k) autocorrelation functions is of interest. This corresponds to the condition supτ | yy (∗) (τ )|  |αk | for αk = 0. Note that in (Napolitano, 2016a, Fig. 13), only OACS processes with slowly varying evolutionary cyclic autocorrelation functions are considered. AM-TW ACS processes are a special case of OACS processes (Section 14.3). In the absence of amplitude modulation and time warping AM-TW ACS processes reduce to ACS processes. If the evolutionary (conjugate) cyclic autocorrelation factorizes, AM-TW ACS processes are also MC processes. In some cases they are GACS (e.g., if the time-warping function is quadratic). In Fig. 15.3, the gray area represents only those AM-TW ACS processes that do not reduce to ACS processes.

15.3.4 Oscillatory processes OACS processes admit the representation (15.1) with Z(f ) integrated complex spectrum of an ACS process. Thus, OACS processes are obtained as special case of OSC processes when the support curves in (15.2) reduces to the support lines with unit slope in (14.5). If Z(f ) is an orthogonal-increment process, the only support line in the bifrequency plane is the main diagonal and one obtains the oscillatory processes in the sense of Priestley. The Priestley’s oscillatory processes reduce to the WSS processes when At (f ) = 1 (Fig. 15.4).