11.3 Time-Dependent Spectra Estimation of Stochastic Processes

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Wavelets-based approaches are significant tools for joint time – frequency analysis of problems related to

vibrations of mechanical and structural systems. This applies both to the characterization of the system

excitation, the system identification, and the system response determination. Several examples exist in

nature of stochastic phenomena with a time-dependent frequency content. The frequency content of

earthquake records, for instance, evolves in time due to the dispersion of the propagating seismic waves

[51,52]. Further, sudden changes in the wave frequency at a given location of the sea surface are often

induced by fast-moving meteorological fronts [53]. Also, a rapid change in the frequency content is

generally associated with waves at the breaking stage. Similarly, turbulent gusts of time-varying frequency

content are often embedded in wind fields.

Appropriate description of such phenomena is obviously crucial for design and reliability assessments.

In an early attempt, concepts of traditional Fourier spectral theory were generalized to provide spectral

estimates, such as the Wigner– Ville method [25,54] or the CGT of Equation 11.1. However, it soon

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became clear that the extension of the traditional concept of a spectrum is not unique, and proposed

time-varying spectra could have contradictory properties [6,55].

Wavelet analysis is readily applicable for estimating time-varying spectral properties, and a significant

effort has been devoted to formulating “wavelet energy principles” that work as alternatives to classical

Fourier methods. Measures of a time-varying frequency content were first obtained by “sectioning,” at

different time instants, the wavelet coefficients mean square map [49,56 – 58]. Developing consistent

spectral estimates from such sections, however, is not straightforward. From a theoretical point of view, it

either requires an appropriate wavelet-based definition of time-varying spectra, or it must relate to wellestablished

notions of time-varying spectra. From a numerical point of view, it involves certain

difficulties in converting the scale axis to a frequency axis, especially when the wavelet functions are not

orthogonal in the frequency domain; that is, when the frequency content of wavelet functions at adjacent

scales do overlap.

Early investigations on wavelet-based spectral estimates may be found in references such as

[44,59 – 64], where wavelet analysis was applied in the context of earthquake engineering problems. In a

particular approach, a modified Littlewood Paley (MLP) wavelet basis can be introduced, whose mother

wavelet is defined in the frequency domain by the equation

C^ ðvÞ ¼

1 ffiffiffiffiffiffiffiffiffiffiffiffiffi

p2ðs 2 1Þp ; p #

􀀈 􀀈

v

􀀈 􀀈

# sp;

0; elsewhere

8><

>:

ð11:52Þ

In Equation 11.52, the symbol s denotes a scalar factor, to be adjusted depending on the desired

frequency resolution. The MLP wavelets are orthogonal in the frequency domain, that is, wavelets at

adjacent scales span nonoverlapping intervals. The MLP wavelets have been used in conjunction with a

discretized version of the CWT proposed by Alkemade [65] for a finite-energy process f ðtÞ

f ðtÞ ¼

X

i; j

KDb

aj

Wf ðaj; biÞcaj ;bi ðtÞ ð11:53Þ

where aj ¼ s j; Db is a time step, and K is a constant parameter depending on s:

In many instances, Equation 11.53 can be construed as representing realizations of a stochastic process,

and in this case, the following estimate of its instantaneous mean-square value of f ðtÞ has been

constructed

E½f 2ðtÞ􀀉

􀀈 􀀈

t¼bi ¼ K

X

j

E½Wf ðaj; biÞ􀀉2

aj ð11:54Þ

where E½·􀀉 is the mathematical expectation operator over the ensemble of realizations. From Equation

11.54, and based on the orthogonality properties of the MLP wavelets, the following quantity

Sf ðvÞ

􀀈 􀀈 t

¼

bi ¼

X

j

K

E½Wf ðaj; biÞ􀀉2

aj

􀀈 􀀈

C^ aj ;bi ðvÞ

􀀈 􀀈

2

ð11:55Þ

where the symbol C^ aj ; bi ðvÞ denotes the Fourier transform of the wavelet function caj ; bi ðtÞ, can be

taken as a measure of the time-varying power spectral density (PSD) of the process f ðtÞ: Based on

Equation 11.55, closed-form expressions can be derived between the input and the output PSDs [63].

In this context, linear-response statistics, such as the instantaneous rate of crossings of the zero level or

the instantaneous rate of occurrence of the peaks, have been estimated with considerable accuracy.

Analysis of nonlinear systems has also been attempted by an equivalent statistical linearization

procedure [61,66].

Wavelet analysis for spectral estimation has also been pursued by Kareem et al., who have used the

squared wavelet coefficients of a DWT to estimate the PSD of stationary processes [56]. To improve the

frequency resolution of the DWT, where only adjacent octave bands can be accounted for, a CWT can be

implemented based on a complex Morlet wavelet basis. The latter is preferable due to the one-to-one

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correspondence between the scale a and the center frequency (Equation 11.51), which allows minimizing

the overlap between spectral estimates at adjacent scales. Further, the product of wavelet coefficients can

be used as a measure of the cross-correlation between two nonstationary signals xðtÞ and yðtÞ [56].

This concept can be refined by the introduction of a wavelet coherence measure [57,58] expressed by the

equation

ðcWða; bÞÞ2 ¼

􀀈 􀀈

SW

xy ða; bÞ

􀀈 􀀈

2

SW

xx ða; bÞSW

yy ða; bÞ ð11:56Þ

In this equation, the local spectrum SW

ij ða; bÞ is defined as

SW

ij ða; bÞ ¼

ð

T

W iða; bÞWjða; bÞdt ð11:57Þ

where the time integration window T depends on the desired time resolution. The local spectrum

(Equation 11.57), owing to the time average over T; allows smoothing of potential measurement noise

effects. Measures of higher-order correlation can also be introduced [56,58], such as the wavelet

bicoherence

ðbW

xxy ða1; a2; bÞÞ2 ¼

􀀈 􀀈

BW

xxy ða1; a2; bÞ

􀀈 􀀈

2

ð

T

􀀈 􀀈

Wx ða1; tÞWx ða2; tÞ

􀀈 􀀈

2

dt

ð

T

􀀈 􀀈

Wyða~; tÞ

􀀈 􀀈

2

dt ð11:58Þ

where 1=a~ ¼ 1=a1 þ 1=a2; and

BW

xxy ða1; a2; bÞ ¼

ð

T

Wxða1; tÞWxða2; tÞWyða~; tÞdt ð11:59Þ

Related remedies can be adopted to suppress spurious correlations induced by statistical noise, based on a

reference noise map created from artificially simulated signals [58].

Signal energy representation concepts have been examined in Ref. [67] by using quasi-orthogonal

Daubechies wavelets in the frequency domain to simulate earthquake ground motion accelerations.

Further, Massel has used wavelet analysis to capture time-varying frequency composition of sea-surface

records due to fast-moving atmospheric fronts in deep water, wave growth, and breaking or

disintegration of mechanically generated wave trains [68]. In this regard, absolute value wavelet maps

and a spectral measure called global wavelet energy spectrum, defined by the equation

E3ðaÞ ¼

ð1

0

E1ða; bÞdb ð11:60Þ

are used. The symbol E1ðt; bÞ denotes a time-scale energy density

E1ða; bÞ ¼

􀀈 􀀈

Wf ða; bÞ

􀀈 􀀈

2

a ð11:61Þ

The scale in Equation 11.61 is readily translated into frequency by selecting the Morlet wavelet basis.

Spanos and Failla [69] have applied wavelet analysis to estimate the evolutionary power spectral

density (EPSD) of nonstationary oscillatory processes defined as [70]

f ðtÞ ¼

ð1

21

Aðv; tÞeivt dZðvÞ ð11:62Þ

The symbol Aðv; tÞ denotes a slowly varying time- and frequency-dependent modulating function, and

ZðvÞ is a complex random process with orthogonal increments such that E½􀀈 􀀈

dZðvÞ

􀀈 􀀈

2􀀉 ¼ Sf0 f0 ðvÞdv;

where Sf0 f0 ðvÞ is the two-sided PSD of the zero-mean stationary process

f0ðtÞ ¼

ð1

21

eivt dZðvÞ ð11:63Þ

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The two-sided EPSD of f ðtÞ is then taken as

Sff ðv; tÞ ¼

􀀈 􀀈

Aðv; tÞ

􀀈 􀀈

2

Sf0 f0 ðvÞ ð11:64Þ

Due to its localization properties, the wavelet transform of f ðtÞ (Equation 11.62) may be approximated as

an oscillatory stochastic process. That is,

Wf ða; bÞ <

ð1

21

Aðv; bÞeivb dZ0ðvÞ ð11:65Þ

where dZ0ðvÞ ¼

ffiffiffiffiffi

p2paC^

–

ðvaÞdZðvÞ: Based on Equation 11.65, the following integral relation is found

between the mean-squared wavelet coefficients at each scale a and the EPSD of f ðtÞ: That is,

E½Wf ða; bÞ2􀀉 ¼ 4pa

ð1

0

􀀈 􀀈

C^ ðvaÞ

􀀈 􀀈

2

Sff ðv; bÞdv ð11:66Þ

A sufficient number of Equations 11.66, one for each scale a; can be solved by a standard solution

algorithm, applicable for both orthogonal and nonorthogonal bases in the frequency domain. This

procedure has proved quite accurate using both the Littlewood – Paley and the real Morlet wavelet bases.

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