Time Domain AnalysisPrior to the discovery of the FFT and the implementation
of the first real time spectral analysers, vibration analysis was predominantly
performed by looking at the time waveform of the signal. Although this
enabled rudimentary detection and diagnosis of faults by examining the
major repetitive components of a signal, complex signals with a multitude
of components could not be accurately assessed.

Several techniques can be used to enhance the characteristics that are
otherwise not readily observable from the time waveform. These include
time-synchronous averaging, and auto-correlation of the signal. Time synchronous
averaging uses the average of the signal over a large number of cycles,
synchronous to the running speed of the machine. This attenuates any contributions
due to noise or non-synchronous vibrations. The auto-correlation function
is the average of the product .
Application of the auto-correlation function on the time series allows
us to indirectly obtain information about the frequencies present in the
signal. However these techniques only provide a limited amount of additional
information. The need to distinguish between components of a similar nature
or hidden within a complex vibration signal led to the mathematical representation
of these signals in terms of their orthogonal basis functions, a field
of mathematics whose origins date back to Joseph Fourier's investigations
into the properties of heat transfer.

*Figure 8 Baron Jean Baptiste Joseph Fourier*

The advent of the Fourier Series in the early 1800's by Joseph Fourier
(1768-1830) provided the foundations for modern signal analysis, a well
as the basis for a significant proportion of the mathematical research
undertaken in the 19th and 20th centuries. Fourier introduced the concept
that an arbitrary function, even a function which exhibits discontinuity's,
could be expressed by a single analytical expression. At the time this
idea had its detractors within the mathematical fraternity, including some
of the more prominent mathematicians of the time, Biot, Laplace and Poisson.
However, his research has since been vindicated and has provided the kernel
to many advances in mathematics, science and engineering. Fourier was obsessed
with heat after his return to France from the warmer shores of Egypt where
he partook in Napoleons' Campaign. This probably explains his fervent fascination
in the phenonomem of heat transfer. In his principal composition, "The
Analytic Theory of Heat", Joseph Fourier established the partial differential
equations governing heat diffusion and solved it by using an infinite series
of trigonometric functions, the Fourier series. For a continuous function
of period 2P, the Fourier series is given by;

where the Fourier coefficients are calculated by,

For a more detailed and informative exposition on the life of Baron Jean
Baptiste Joseph Fourier I suggest visiting the WWW
site, maintained by David
A. Keston from the University of Glasgow, or another site that provides
an excellent index of bibliographies on famous mathematicians including
Joseph
Fourier.The characteristic of the Fourier transform that has made it
such a valuable tool is its ability to decompose any periodic function,
such as machine vibrations or a complex sound into a set of orthonormal
basis functions, of sines and cosines. The coefficients of these orthonormal
basis functions represent the contribution of the sine and cosine components
of the signal at all frequencies. This allows the signal to be analysed
in terms of its frequency componentsas shown below.

FIGURE 9 THE FOURIER TRANSFORM OF A SIGNAL

.A major development which revolutionised the computational implementation
of the Fourier transform was the introduction of the fast Fourier transform
(FFT) by Cooley and Tukey in 1965, which enabled the implementation of
the first real time spectral analysers [CIZE85]. The FFT improved the computational
efficiency of the Fourier transform of a signal represented by n discrete
data points, from an order of n x n to n x log(n) arithmetic operations.
Despite the functionality of the Fourier transform, especially in regard
to obtaining the spectral analysis of a signal, there are several shortcomings
of this technique. The first of these is the inability of the Fourier transform
to accurately represent functions that have non-periodic components that
are localised in time or space, such as transient impulses. This is due
to the Fourier transform being based on the assumption that the signal
to be transformed is periodic in nature and of infinite length. Another
deficiency is its inability to provide any information about the time dependence
of a signal, as results are averaged over the entire duration of the signal.
This is a problem when analysing signals of a non-stationary nature, where
it is often beneficial to be able to acquire a correlation between the
time and frequency domains of a signal. This is often the case when monitoring
machine vibrations. Some typical examples include;

- Engine start up or shutdown (Variable speed and resonant transients)
- Vibrations from cranes or excavators (Variable load and speed, transient vibrations, slow rotational speeds)
- Helicopter gear transmission systems (Variable transmission path, variable speed)
- Internal combustion engines (Variable transmission path, variable speed)
- To demonstrate the deficiencies of the Fourier transform, two examples are shown below. Figures 10 and 11 show how two signals, which bear no resemblance in the time domain can produce almost identical power spectrums. The blue line is a stationary signal created by excitation of a linear system by white noise, whereas the red line represents the same system being driven by an initial impulse. As it can be clearly seen it would be impossible to reconstruct or even determine the original time domain signals from their power spectrums.

FIGURE 10 TIME DOMAIN SIGNALS

FIGURE 11 POWER SPECTRUM

The second example displays the problem of spectral smearing encountered
during the start up of an engine. As depicted in Figure 12, spectral smearing
substantially effects the results obtained by conventional spectral analysis.
Two plots are shown, the red line represents the engine during start up,
and the blue line is the same engine after a steady condition has been
achieved. The vibrations evident in this figure are due to an unbalanced
shaft and a damaged stator. The vibrational frequency of the unbalanced
shaft is linked to the rotational frequency of the shaft which results
in the spectra being smeared during start up, whereas the fault frequency
of the damaged stator is twice the line frequency (ie. 2 x 50Hz = 100Hz).

FIGURE 12 SPECTRAL SMEARING OF SIGNAL DURING ENGINE START UP

A variety of alternative schemes have been developed to improve the description
of non-stationary vibration signals. These range from developing mathematical
models of the signal, to converting the signal into a pseudo-stationary
signal through angular sampling, and time-frequency analysis of the vibrations.
Before delving into the mathematics and properties of each of these methods
a brief exposition of the events leading to our current understanding of
time-frequency analysis shall be made.

Time-Frequency Signal AnalysisAs noted by Jean Ville in 1947 [VILL47] there
are two basic approaches to time-frequency analysis. The first approach
is to initially cut the signal into slices in time, and then to analyse
each of these slices separately to examine their frequency content. The
other approach is to first filter different frequency bands, and then cut
these bands into slices in time and anlyse their energy content. The first
of these approaches is used for the construction of the short time Fourier
transform and the Wigner-Ville transform, while the second leads to focus
of this thesis, the wavelet transform.The wavelet transform is a mechanism
used to dissect or breakdown a signal into its constituent parts, thus
enabling analysis of data in different frequency domains with each components
resolution matched to its scale. Alternatively this may be seen as a decomposition
of the signal into its set of basis functions (wavelets), analogous to
the use of sines and cosines in Fourier analysis to represent other functions.
These basis functions are obtained from dilations or contractions (scaling),
and translations of the mother wavelet. The important difference that distinguishes
the wavelet transform from Fourier analysis is its time and frequency localisation
properties. When analysing signals of a non-stationary nature, it is often
beneficial to be able to acquire a correlation between the time and frequency
domains of a signal. In contrast to the Fourier transform, the wavelet
transform allows exceptional localisation in both the time domain via translations
of the mother wavelet, and in the scale (frequency) domain via dilations.
Although the wavelet transform has come into prominence during the last
decade, the founding principles behind wavelets can be traced back as far
as 1909 when Alfred Haar [HAAR10] discovered another orthonormal system
of functions, such that for any continuous function f(x), the series

converges to f(x) uniformly over the interval .
Haar's research led to the simplest of the orthogonal wavelets, a set of
rectangular basis functions depicted in Figure 13. The Haar basis function
was of limited use due to it being discontinuous in nature. This resulted
in it being inefficient in modelling smooth signals, as many levels need
to be included to obtain an accurate representation.The theories expounded
by Haar were extended on in the 1930's by Levey in the study of Brownian
motion where he used the Schauder basis to examine local regularity properties
that were not accessible via the Fourier transform. A Similar problem was
being tackled by Littlewood and Paley, who were attempting to localise
the contributing energies of a function. They were interested in whether
the energy of a function was spread evenly over its entire interval or
concentrated about a few points. In order to reveal this information which
is hidden within the Fourier coefficients, Littlewood and Paley discovered
a series of manipulations that could be applied to the Fourier series to
retrieve this information. They introduced the dyadic block ,
a sequence of operators that act essentially as a bank of band pass filters
with an interval of separation of approximately an octave. The dyadic block
was defined by Littlewood and Paley as,

This allows us to rewrite the Fourier series in terms of its dyadic blocks,

Other contributions to the mathematics laying the groundwork for wavelets
during the 1930's include the works of Antoni Zygmund, Phillip Franklin
and Lusin.

FIGURE 13 HAAR WAVELETS: LEVELS 0, 1, 2

However it was not until 1946 that the first time-frequency wavelets (Gabor
wavelets) were introduced by Dennis Gabor [GABO46], an electrical engineer
researching into communication theory. Gabors idea was to break a wave
up into segments, and then analyse the individual segments of the wave
(wavelets), each of which had a well defined frequency band and position
in time. Although Gabor's wavelets worked for continuous decomposition's
of signals they were limited in their usefulness as corresponding wavelets
for discrete systems did not exist. Shortly after Gabor's work, Jean Ville
[VILL47] proposed another approach for obtaining a mixed signal representation.
Ville's work was tied into the research of Hermann Wigner (1932), a physicist
working in the field of quantum mechanics, and led to the development of
the Wigner-Ville transform, given by:

,where f(t) is the original
time signal. Unfortunately the Wigner-Ville transform renders imperfect
information about the energy distribution of the signal in the time-frequency
domain, and an atomic decomposition of a signal based on the Wigner-Ville
transform does not exist [MEYE93].The next major step forward did not come
until 1975 when Coifman and Weiss extended Lusin's work to invent what
we now consider the 'atoms' and 'molecules' which were to form the basic
building blocks of a function space, and the rules of assembly necessary
to reconstruct the function space in its entirety from these 'atoms'. The
first synthesis of these theories leading up to wavelet analysis, and the
impetus for the enormous interest in wavelet theory during the 1980's and
1990's came from the research of Grossmann *(theoretical physicist) *and
Morlet *(geophysicist)*. They employed wavelets to analyse earthquakes
and model the process of sound waves travelling through the Earth's crust,
introducing the concept of using wavelets to analyse signals of a non-stationary
nature. Grossmann and Morlet defined a wavelet as a "function in L()
whose Fourier transform satisfies
the criterion almost
everywhere" [MEYE93].Following on from this work Yves Meyer, a mathematician
researching into harmonic analysis, developed a family of wavelets that
he showed to be the most efficient for modelling complex phenomena. The
final transition from continuous signal processing to discrete signal processing
was achieved by Stephane Mallat [MALL89] and Ingrid Daubechies of Bell
Labs [DAUB88]. Since then there has been a proliferation of activity with
comprehensive studies expanding on the wavelet transform and its implementation
into many fields of endeavour. Applications that have been explored include
multi-resolution signal processing, image and data compression, telecommunications,
fingerprint analysis, numerical analysis and speech processing.