ABSTRACT
Wavelets-based representations offer an important option for capturing localized effects in many signals.
This is achieved by employing representations via double integrals (continuous transforms), or via
double series (discrete transforms). Seminal to these representations are the processes of scaling and
shifting of a generating (mother) function. Over a period of several decades, wavelet analysis has been set
on a rigorous mathematical framework and has been applied to quite diverse fields. Wavelet families
associated with specific mother functions have proven quite appropriate for a variety of problems. In this
context, fast decomposition and reconstruction algorithms ensure computational efficiency, and rival
classical spectral analysis algorithms such as the fast Fourier transform (FFT). The field of vibration
analysis has benefited from this remarkable mathematical development in conjunction with vibration
monitoring, system identification, damage detection, and several other tasks. There is a voluminous body
of literature focusing on wavelet analysis. However, this chapter has the restricted objective of, on one
hand, discussing concepts closely related to vibration analysis, and on the other hand, citing sources that
can be readily available to a potential reader. In view of this latter objective, almost exclusively books and
archival articles are included in the list of references. First, theoretical concepts are briefly presented; for
more mathematical details, the reader may consult references [1-23]. Next, the theoretical concepts are
supplemented by vibration-analysis-related sections on time-varying spectra estimation, random field
synthesis, structural identification, damage detection, and material characterization. It is noted that most
of the mathematical developments pertain to the interval [0,1] relating to dimensionless independent
variables derived by normalization with respect to the spatial or temporal “lengths” of the entire signals.
A convenient way to introduce the wavelet transform is through the concept of time-frequency
representation of signals. In the classical Fourier theory, a signal can be represented either in the time or
in the frequency domain, and the Fourier coefficients define the average spectral content over the entire
duration of the signal. The Fourier representation is appropriate for signals that are stationary, in terms
of parameters which are deemed important for the problem in hand, but becomes inadequate for
nonstationary signals, in which important parameters may evolve rapidly in time.
