WHAT IS FAULT DETECTION?

Fault detection and isolation is a subfield of control engineering which concerns itself with
monitoring a system, identifying when a fault has occurred and pinpoint the type of fault
and its location.

Fault detection and isolation (FDI) techniques can be broadly classified into two categories.
These include 1) Model-based FDI and 2) Signal processing based FDI

MODEL-BASED FDI

In model-based FDI techniques some model of the system is used to decide about the occurrence
of fault. The model may be mathematical model, or a knowledge based model of the
system. Some of the model-based FDI techniques include

Observer-based approach

Parity-space approach

Parameter identification based methods

SIGNAL PROCESSING BASED FDI

In signal processing based FDI, some mathematical or statistical operations are performed on the
measurements, or some neural network is trained using measurements to extract the information
about the fault.

Mathematical transformations are applied to signals to obtain a further information from that
signal that is not readily available in the raw signal. A time-domain signal as a raw signal, and a signal
that has been "transformed" by any of the available mathematical transformations as a processed signal.

There are number of transformations that can be applied, among which the Fourier transforms
are probably by far the most popular.

Most of the signals in practice, are TIME-DOMAIN signals in their raw format. That is,
whatever that signal is measuring, is a function of time. In other words, when we plot the signal
one of the axes is time (independent variable), and the other (dependent variable) is usually the
amplitude. When we plot time-domain signals, we obtain a time-amplitude representation of
the signal. This representation is not always the best representation of the signal for most signal
processing related applications. In many cases, the most distinguished information is hidden in
the frequency content of the signal. The frequency SPECTRUM of a signal is basically the
frequency components (spectral components) of that signal. The frequency spectrum of a signal
shows what frequencies exist in the signal.

So how do we measure frequency, or how do we find the frequency content of a signal? The
answer is FOURIER TRANSFORM (FT). If the FT of a signal in time domain is taken, the

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frequency-amplitude representation of that signal is obtained. In other words, we now have a plot
with one axis being the frequency and the other being the amplitude. This plot tells us how much
of each frequency exists in our signal.

FOURIER TRANSFORM

• FT decomposes a signal to complex exponential functions of different frequencies.

Note that the exponential term in Eqn. (1) can also be written as:

Cos(2.pi.f.t)+j.Sin(2.pi.f.t).......(3)

The above expression has a real part of cosine of frequency f, and an imaginary part of sine of
frequency f. So what we are actually doing is, multiplying the original signal with a complex
expression which has sines and cosines of frequency f. Then we integrate this product. In other
words, we add all the points in this product. If the result of this integration (which is nothing but
some sort of infinite summation) is a large value, then we say that : the signal x(t), has a
dominant spectral component at frequency "f". This means that, a major portion of this signal
is composed of frequency f. If the integration result is a small value, than this means that the
signal does not have a major frequency component of f in it. If this integration result is zero, then
the signal does not contain the frequency "f" at all.

It is of particular interest here to see how this integration works: The signal is multiplied with the
sinusoidal term of frequency "f". If the signal has a high amplitude component of frequency "f",
then that component and the sinusoidal term will coincide, and the product of them will give a
(relatively) large value. This shows that, the signal "x", has a major frequency component of
"f".

However, if the signal does not have a frequency component of "f", the product will yield zero,
which shows that, the signal does not have a frequency component of "f". If the frequency "f", is
not a major component of the signal "x(t)", then the product will give a (relatively) small value.
This shows that, the frequency component "f" in the signal "x", has a small amplitude, in other
words, it is not a major component of "x".

Now, note that the integration in the transformation equation (Eqn. 1) is over time. The left hand
side of (1), however, is a function of frequency. Therefore, the integral in (1), is calculated for
every value of f.

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Fourier transform is not suitable if the signal has time varying frequency, i.e., the signal is
non-stationary. If only the signal has the frequency component "f" at all times (for all "t"
values), then the result obtained by the Fourier transform makes sense.

Note that the Fourier transform tells whether a certain frequency component exists or not.
This information is independent of where in time this component appears. It is therefore very
important to know whether a signal is stationary or not, prior to processing it with the FT.

Consider an example:

x(t)=cos(2*pi*5*t)+cos(2*pi*10*t)+cos(2*pi*20*t)+cos(2*pi*50*t)

that is , it has four frequency components of 5, 10, 20, and 50 Hz., all occurring at all times.

And here is the FT of it. The frequency axis has been cut here, but theoretically it extends to
infinity (for continuous Fourier transform (CFT). Actually, here we calculate the discrete Fourier
transform (DFT), in which case the frequency axis goes up to (at least) twice the sampling
frequency of the signal, and the transformed signal is symmetrical. However, this is not that
important at this time.)

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Now, look at the following figure: Here the signal is again the cosine signal, and it has the same
four frequencies. However, these components occur at different times.

And here is the Fourier transform of this signal:

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The reason of the noise like thing in between peaks show that, those frequencies also exist in the
signal. But the reason they have a small amplitude , is because, they are not major spectral
components of the given signal, and the reason we see those, is because of the sudden change
between the frequencies. Especially note how time domain signal changes at around time 250
(ms).

From the above example, FT cannot distinguish the two signals very well. To FT, both signals
are the same, because they constitute of the same frequency components. Therefore, FT is not a
suitable tool for analyzing non-stationary signals, i.e., signals with time varying spectra.

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 LINEAR TIME FREQUENCY REPRESENTATIONS

THE SHORT TERM FOURIER TRANSFORM

There is only a minor difference between STFT and FT. In STFT, the signal is divided into small
enough segments, where these segments (portions) of the signal can be assumed to be stationary.
For this purpose, a window function "w" is chosen. The width of this window must be equal to
the segment of the signal where its stationarity is valid.

This window function is first located to the very beginning of the signal. That is, the window
function is located at t=0. Let's suppose that the width of the window is "T" s. At this time
instant (t=0), the window function will overlap with the first T/2 seconds (I will assume that all
time units are in seconds). The window function and the signal are then multiplied. By doing
this, only the first T/2 seconds of the signal is being chosen, with the appropriate weighting of
the window (if the window is a rectangle, with amplitude "1", then the product will be equal to
the signal). Then this product is assumed to be just another signal, whose FT is to be taken. In
other words, FT of this product is taken, just as taking the FT of any signal.

The result of this transformation is the FT of the first T/2 seconds of the signal. If this portion of
the signal is stationary, as it is assumed, then there will be no problem and the obtained result
will be a true frequency representation of the first T/2 seconds of the signal.

The next step, would be shifting this window (for some t1 seconds) to a new location,
multiplying with the signal, and taking the FT of the product. This procedure is followed, until
the end of the signal is reached by shifting the window with "t1" seconds intervals.

The following definition of the STFT summarizes all the above explanations in one line:

Please look at the above equation carefully. x(t) is the signal itself, w(t) is the window function,
and * is the complex conjugate. As you can see from the equation, the STFT of the signal is
nothing but the FT of the signal multiplied by a window function.

For every t' and f a new STFT coefficient is computed (Correction: The "t" in the parenthesis of
STFT should be "t'". I will correct this soon. I have just noticed that I have mistyped it).

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The following figure may help to understand this a little better:

The Gaussian-like functions in color are the windowing functions. The red one shows the
window located at t=t1', the blue shows t=t2', and the green one shows the window located at
t=t3'. These will correspond to three different FTs at three different times. Therefore, we will
obtain a true time-frequency representation (TFR) of the signal.

Probably the best way of understanding this would be looking at an example. First of all, since
our transform is a function of both time and frequency (unlike FT, which is a function of
frequency only), the transform would be two dimensional (three, if you count the amplitude too).
Let's take a non-stationary signal, such as the following one:

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n this signal, there are four frequency components at different times. The interval 0 to 250 ms is
a simple sinusoid of 300 Hz, and the other 250 ms intervals are sinusoids of 200 Hz, 100 Hz, and
50 Hz, respectively. Apparently, this is a non-stationary signal. Now, let's look at its STFT:

First of all, note that the graph is symmetric with respect to midline of the frequency axis.
Remember that, although it was not shown, FT of a real signal is always symmetric, since STFT
is nothing but a windowed version of the FT, it should come as no surprise that STFT is also
symmetric in frequency. The symmetric part is said to be associated with negative frequencies,
an odd concept which is difficult to comprehend, fortunately, it is not important; it suffices to
know that STFT and FT are symmetric.

The problem with STFT is the fact whose roots go back to what is known as the Heisenberg
Uncertainty Principle . This principle originally applied to the momentum and location of
moving particles, can be applied to time-frequency information of a signal. Simply, this principle
states that one cannot know the exact time-frequency representation of a signal, i.e., one cannot
know what spectral components exist at what instances of times. What one can know are the
time intervals in which certain band of frequencies exist, which is a resolution problem.

The problem with the STFT has something to do with the width of the window function that is
used. To be technically correct, this width of the window function is known as the support of the
window. If the window function is narrow, than it is known as compactly supported . This
terminology is more often used in the wavelet world.

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Recall that in the FT there is no resolution problem in the frequency domain, i.e., we know
exactly what frequencies exist; similarly we there is no time resolution problem in the time
domain, since we know the value of the signal at every instant of time. Conversely, the time
resolution in the FT, and the frequency resolution in the time domain are zero, since we have no
information about them. What gives the perfect frequency resolution in the FT is the fact that the
window used in the FT is its kernel, the exp{jwt} function, which lasts at all times from minus
infinity to plus infinity. Now, in STFT, our window is of finite length, thus it covers only a
portion of the signal, which causes the frequency resolution to get poorer. What I mean by
getting poorer is that, we no longer know the exact frequency components that exist in the signal,
but we only know a band of frequencies that exist:

In FT, the kernel function, allows us to obtain perfect frequency resolution, because the kernel
itself is a window of infinite length. In STFT is window is of finite length, and we no longer
have perfect frequency resolution. You may ask, why don't we make the length of the window in
the STFT infinite, just like as it is in the FT, to get perfect frequency resolution? Well, than you
loose all the time information, you basically end up with the FT instead of STFT.

If we use a window of infinite length, we get the FT, which gives perfect frequency resolution,
but no time information. Furthermore, in order to obtain the stationarity, we have to have a short
enough window, in which the signal is stationary. The narrower we make the window, the better
the time resolution, and better the assumption of stationarity, but poorer the frequency resolution:

Narrow window ===>good time resolution, poor frequency resolution.

The window function we use is simply a Gaussian function in the form:

w(t)=exp(-a*(t^2)/2);

Where a determines the length of the window, and t is the time. The following figure shows four
window functions of varying regions of support, determined by the value of a . Please disregard
the numeric values of a since the time interval where this function is computed also determines
the function. Just note the length of each window. The above example given was computed with
the second value, a=0.001. I will now show the STFT of the same signal given above computed
with the other windows.

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First let's look at the first narrowest window. We expect the STFT to have a very good time
resolution, but relatively poor frequency resolution:

The above figure shows this STFT. The figure is shown from a top bird-eye view with an angle
for better interpretation. Note that the four peaks are well separated from each other in time. Also
note that, in frequency domain, every peak covers a range of frequencies, instead of a single

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frequency value. Now let's make the window wider, and look at the third window (the second
one was already shown in the first example).

Note that the peaks are not well separated from each other in time, unlike the previous case,
however, in frequency domain the resolution is much better. Now let's further increase the width
of the window, and see what happens:

Narrow windows give good time resolution, but poor frequency resolution. Wide windows give
good frequency resolution, but poor time resolution; furthermore, wide windows may violate the
condition of stationarity. The problems, of course, are a result of choosing a window function,

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once and for all, and use that window in the entire analysis. If the frequency components are well
separated from each other in the original signal, than we may sacrifice some frequency resolution
and go for good time resolution, since the spectral components are already well separated from
each other.

The Wavelet transform (WT) solves the dilemma of resolution to a certain extent.

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 MULTIRESOLUTION ANALYSIS
Although the time and frequency resolution problems are results of a physical phenomenon (the
Heisenberg uncertainty principle) and exist regardless of the transform used, it is possible to
analyze any signal by using an alternative approach called the multiresolution analysis (MRA) .
MRA, as implied by its name, analyzes the signal at different frequencies with different
resolutions. Every spectral component is not resolved equally as was the case in the STFT.

MRA is designed to give good time resolution and poor frequency resolution at high frequencies
and good frequency resolution and poor time resolution at low frequencies. This approach makes
sense especially when the signal at hand has high frequency components for short durations and
low frequency components for long durations. Fortunately, the signals that are encountered in
practical applications are often of this type. For example, the following shows a signal of this
type. It has a relatively low frequency component throughout the entire signal and relatively high
frequency components for a short duration somewhere around the middle.

THE CONTINUOUS WAVELET TRANSFORM

The continuous wavelet transform was developed as alternative approaches to the short time
Fourier transform to overcome the resolution problem. The wavelet analysis is done in a similar
way to the STFT analysis, in the sense that the signal is multiplied with a function, {\it the
wavelet}, similar to the window function in the STFT, and the transform is computed separately
for different segments of the time-domain signal. However, there are two main differences
between the STFT and the CWT:

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1. The Fourier transforms of the windowed signals are not taken, and therefore single peak will
be seen corresponding to a sinusoid, i.e., negative frequencies are not computed.

2. The width of the window is changed as the transform is computed for every single spectral
component, which is probably the most significant characteristic of the wavelet transform.

The continuous wavelet transform is defined as follows

As seen in the above equation , the transformed signal is a function of two variables, tau and s ,
the translation and scale parameters, respectively. psi(t) is the transforming function, and it is
called the mother wavelet . The term mother wavelet gets its name due to two important
properties of the wavelet analysis as explained below:

The term wavelet means a small wave . The smallness refers to the condition that this (window)
function is of finite length ( compactly supported). The wave refers to the condition that this
function is oscillatory . The term mother implies that the functions with different region of
support that are used in the transformation process are derived from one main function, or the
mother wavelet. In other words, the mother wavelet is a prototype for generating the other
window functions.

The term translation is used in the same sense as it was used in the STFT; it is related to the
location of the window, as the window is shifted through the signal. This term, obviously,
corresponds to time information in the transform domain. we do not have a frequency parameter,
as we had before for the STFT. Instead, we have scale parameter which is defined as
$1/frequency$.

The Scale

The parameter scale in the wavelet analysis is similar to the scale used in maps. As in the case of
maps, high scales correspond to a non-detailed global view (of the signal), and low scales
correspond to a detailed view. Similarly, in terms of frequency, low frequencies (high scales)
correspond to a global information of a signal (that usually spans the entire signal), whereas high
frequencies (low scales) correspond to a detailed information of a hidden pattern in the signal
(that usually lasts a relatively short time). Cosine signals corresponding to various scales are
given as examples in the following figure.

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Fortunately in practical applications, low scales (high frequencies) do not last for the entire
duration of the signal, unlike those shown in the figure, but they usually appear from time to time
as short bursts, or spikes. High scales (low frequencies) usually last for the entire duration of the
signal.

Scaling, as a mathematical operation, either dilates or compresses a signal. Larger scales

In terms of mathematical functions, if f(t) is a given function f(st) corresponds to a contracted

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However, in the definition of the wavelet transform, the scaling term is used in the denominator,
and therefore, the opposite of the above statements holds, i.e., scales s > 1 dilates the signals
whereas scales s < 1 , compresses the signal.

COMPUTATION OF THE CWT

Interpretation of the above equation will be explained in this section. Let x(t) is the signal to be
analyzed. The mother wavelet is chosen to serve as a prototype for all windows in the process.
All the windows that are used are the dilated (or compressed) and shifted versions of the mother
wavelet. There are a number of functions that are used for this purpose. The Morlet wavelet and
the Mexican hat function are two candidates, and they are used for the wavelet analysis of the
examples.

Once the mother wavelet is chosen the computation starts with s=1 and the continuous wavelet
transform is computed for all values of s , smaller and larger than ``1''. However, depending on
the signal, a complete transform is usually not necessary. For all practical purposes, the signals
are bandlimited, and therefore, computation of the transform for a limited interval of scales is
usually adequate.

For convenience, the procedure will be started from scale s=1 and will continue for the
increasing values of s , i.e., the analysis will start from high frequencies and proceed towards low
frequencies. This first value of s will correspond to the most compressed wavelet. As the value of
s is increased, the wavelet will dilate.

The wavelet is placed at the beginning of the signal at the point which corresponds to time=0.
The wavelet function at scale ``1'' is multiplied by the signal and then integrated over all times.
The result of the integration is then multiplied by the constant number 1/sqrt{s} . This
multiplication is for energy normalization purposes so that the transformed signal will have the
same energy at every scale. The final result is the value of the transformation, i.e., the value of
the continuous wavelet transform at time zero and scale s=1 . In other words, it is the value that
corresponds to the point tau =0 , s=1 in the time-scale plane.

The wavelet at scale s=1 is then shifted towards the right by tau amount to the location t=tau ,
and the above equation is computed to get the transform value at t=tau , s=1 in the time-
frequency plane.

This procedure is repeated until the wavelet reaches the end of the signal. One row of points on
the time-scale plane for the scale s=1 is now completed.

Then, s is increased by a small value. Note that, this is a continuous transform, and therefore,
both tau and s must be incremented continuously . However, if this transform needs to be
computed by a computer, then both parameters are increased by a sufficiently small step size.
This corresponds to sampling the time-scale plane.

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The above procedure is repeated for every value of s. Every computation for a given value of s
fills the corresponding single row of the time-scale plane. When the process is completed for all
desired values of s, the CWT of the signal has been calculated.

The figures below illustrate the entire process step by step.

the signal and the wavelet function are shown for four different values of tau . The signal is a
truncated version of the signal shown in Figure 3.1. The scale value is 1 , corresponding to the
lowest scale, or highest frequency. Note how compact it is (the blue window). It should be as
narrow as the highest frequency component that exists in the signal. Four distinct locations of the
wavelet function are shown in the figure at to=2 , to=40, to=90, and to=140 . At every location,
it is multiplied by the signal. Obviously, the product is nonzero only where the signal falls in the
region of support of the wavelet, and it is zero elsewhere. By shifting the wavelet in time, the
signal is localized in time, and by changing the value of s , the signal is localized in scale
(frequency).

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If the signal has a spectral component that corresponds to the current value of s (which is 1 in
this case), the product of the wavelet with the signal at the location where this spectral
component exists gives a relatively large value. If the spectral component that corresponds to
the current value of s is not present in the signal, the product value will be relatively small, or
zero. The signal in Figure 3.3 has spectral components comparable to the window's width at s=1
around t=100 ms.

The continuous wavelet transform of the signal in above figure will yield large values for low
scales around time 100 ms, and small values elsewhere. For high scales, on the other hand, the
continuous wavelet transform will give large values for almost the entire duration of the signal,
since low frequencies exist at all times.

Now, let's take a look at an example, and see how the wavelet transform really looks like.
Consider the non-stationary signal .This is similar to the example given for the STFT, except at
different frequencies. As stated on the figure, the signal is composed of four frequency
components at 30 Hz, 20 Hz, 10 Hz and 5 Hz.

Figure below is the continuous wavelet transform (CWT) of this signal. Note that the axes are
translation and scale, not time and frequency. However, translation is strictly related to time,
since it indicates where the mother wavelet is located. The translation of the mother wavelet can
be thought of as the time elapsed since t=0 . The scale, however, has a whole different story.
Remember that the scale parameter s in equation is actually inverse of frequency. In other words,

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whatever we said about the properties of the wavelet transform regarding the frequency
resolution, inverse of it will appear on the figures showing the WT of the time-domain signal.

Why is the Discrete Wavelet Transform Needed?

Although the discredited continuous wavelet transform enables the computation of the
continuous wavelet transform by computers, it is not a true discrete transform. As a matter of
fact, the wavelet series is simply a sampled version of the CWT, and the information it provides
is highly redundant as far as the reconstruction of the signal is concerned. This redundancy, on
the other hand, requires a significant amount of computation time and resources. The discrete
wavelet transform (DWT), on the other hand, provides sufficient information both for analysis
and synthesis of the original signal, with a significant reduction in the computation time.

The DWT is considerably easier to implement when compared to the CWT. The basic concepts
of the DWT will be introduced in this section along with its properties and the algorithms used to
compute it. As in the previous chapters, examples are provided to aid in the interpretation of the
DWT.

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THE DISCRETE WAVELET TRANSFORM (DWT)
The foundations of the DWT go back to 1976 when Croiser, Esteban, and Galand devised a
technique to decompose discrete time signals. Crochiere, Weber, and Flanagan did a similar
work on coding of speech signals in the same year. They named their analysis scheme as
subband coding. In 1983, Burt defined a technique very similar to subband coding and named it
pyramidal coding which is also known as multiresolution analysis. Later in 1989, Vetterli and
Le Gall made some improvements to the subband coding scheme, removing the existing
redundancy in the pyramidal coding scheme. Subband coding is explained below.

The Subband Coding and The Multiresolution Analysis

The main idea is the same as it is in the CWT. A time-scale representation of a digital signal is
obtained using digital filtering techniques. Recall that the CWT is a correlation between a
wavelet at different scales and the signal with the scale (or the frequency) being used as a
measure of similarity. The continuous wavelet transform was computed by changing the scale of
the analysis window, shifting the window in time, multiplying by the signal, and integrating over
all times. In the discrete case, filters of different cutoff frequencies are used to analyze the signal
at different scales. The signal is passed through a series of high pass filters to analyze the high
frequencies, and it is passed through a series of low pass filters to analyze the low frequencies.

The resolution of the signal, which is a measure of the amount of detail information in the signal,
is changed by the filtering operations, and the scale is changed by upsampling and
downsampling (subsampling) operations. Subsampling a signal corresponds to reducing the
sampling rate, or removing some of the samples of the signal. For example, subsampling by two
refers to dropping every other sample of the signal. Subsampling by a factor n reduces the
number of samples in the signal n times.

Upsampling a signal corresponds to increasing the sampling rate of a signal by adding new
samples to the signal. For example, upsampling by two refers to adding a new sample, usually a
zero or an interpolated value, between every two samples of the signal. Upsampling a signal by a
factor of n increases the number of samples in the signal by a factor of n.

Although it is not the only possible choice, DWT coefficients are usually sampled from the CWT
on a dyadic grid, i.e., s0 = 2 and  0 = 1, yielding s=2j and  =k*2j .

since the signal is a discrete time function, This sequence will be denoted by x[n], where n is an
integer. The procedure starts with passing this signal (sequence) through a half band digital
lowpass filter with impulse response h[n]. Filtering a signal corresponds to the mathematical
operation of convolution of the signal with the impulse response of the filter. The convolution
operation in discrete time is defined as follows:

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A half band lowpass filter removes all frequencies that are above half of the highest frequency in
the signal. For example, if a signal has a maximum of 1000 Hz component, then half band
lowpass filtering removes all the frequencies above 500 Hz.

The unit of frequency is of particular importance at this time. In discrete signals, frequency is
expressed in terms of radians. Accordingly, the sampling frequency of the signal is equal to 2 π
radians in terms of radial frequency. Therefore, the highest frequency component that exists in a
signal will be π radians, if the signal is sampled at Nyquist’s rate (which is twice the maximum
frequency that exists in the signal); that is, the Nyquist’s rate corresponds to π rad/s in the
discrete frequency domain. Therefore using Hz is not appropriate for discrete signals. However,
Hz is used whenever it is needed to clarify a discussion, since it is very common to think of
frequency in terms of Hz. It should always be remembered that the unit of frequency for discrete
time signals is radians.

After passing the signal through a half band lowpass filter, half of the samples can be eliminated
according to the Nyquist’s rule, since the signal now has a highest frequency of π /2 radians
instead of π radians. Simply discarding every other sample will subsample the signal by two,
and the signal will then have half the number of points. The scale of the signal is now doubled.
Note that the lowpass filtering removes the high frequency information, but leaves the scale
unchanged. Only the subsampling process changes the scale. Resolution, on the other hand, is
related to the amount of information in the signal, and therefore, it is affected by the filtering
operations. Half band lowpass filtering removes half of the frequencies, which can be interpreted
as losing half of the information. Therefore, the resolution is halved after the filtering operation.
Note, however, the subsampling operation after filtering does not affect the resolution, since
removing half of the spectral components from the signal makes half the number of samples
redundant anyway. Half the samples can be discarded without any loss of information. In
summary, the lowpass filtering halves the resolution, but leaves the scale unchanged. The signal
is then subsampled by 2 since half of the number of samples are redundant. This doubles the
scale.

This procedure can mathematically be expressed as:

Having said that, we now look how the DWT is actually computed: The DWT analyzes the
signal at different frequency bands with different resolutions by decomposing the signal into a
coarse approximation and detail information. DWT employs two sets of functions, called scaling
functions and wavelet functions, which are associated with low pass and highpass filters,
respectively. The decomposition of the signal into different frequency bands is simply obtained
by successive highpass and lowpass filtering of the time domain signal. The original signal x[n]
is first passed through a halfband highpass filter g[n] and a lowpass filter h[n]. After the filtering,
half of the samples can be eliminated according to the Nyquist’s rule, since the signal now has a
highest frequency of π /2 radians instead of π . The signal can therefore be subsampled by 2,

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simply by discarding every other sample. This constitutes one level of decomposition and can
mathematically be expressed as follows:

where yhigh[k] and ylow[k] are the outputs of the highpass and lowpass filters, respectively, after
subsampling by 2.

This decomposition halves the time resolution since only half the number of samples now
characterizes the entire signal. However, this operation doubles the frequency resolution, since
the frequency band of the signal now spans only half the previous frequency band, effectively
reducing the uncertainty in the frequency by half. The above procedure, which is also known as
the subband coding, can be repeated for further decomposition. At every level, the filtering and
subsampling will result in half the number of samples (and hence half the time resolution) and
half the frequency band spanned (and hence double the frequency resolution). where x[n] is the
original signal to be decomposed, and h[n] and g[n] are lowpass and highpass filters,
respectively. The bandwidth of the signal at every level is marked on the figure as "f".

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he Subband Coding Algorithm As an example, suppose that the original signal x[n] has 512
sample points, spanning a frequency band of zero to  rad/s. At the first decomposition level, the
signal is passed through the highpass and lowpass filters, followed by subsampling by 2. The
output of the highpass filter has 256 points (hence half the time resolution), but it only spans the
frequencies π/2 to π rad/s (hence double the frequency resolution). These 256 samples constitute
the first level of DWT coefficients. The output of the lowpass filter also has 256 samples, but it
spans the other half of the frequency band, frequencies from 0 to π/2 rad/s. This signal is then
passed through the same lowpass and highpass filters for further decomposition. The output of
the second lowpass filter followed by subsampling has 128 samples spanning a frequency band
of 0 to π/4 rad/s, and the output of the second highpass filter followed by subsampling has 128
samples spanning a frequency band of π/4 to π/2 rad/s. The second highpass filtered signal
constitutes the second level of DWT coefficients. This signal has half the time resolution, but
twice the frequency resolution of the first level signal. In other words, time resolution has
decreased by a factor of 4, and frequency resolution has increased by a factor of 4 compared to
the original signal. The lowpass filter output is then filtered once again for further
decomposition. This process continues until two samples are left. For this specific example there
would be 8 levels of decomposition, each having half the number of samples of the previous
level. The DWT of the original signal is then obtained by concatenating all coefficients starting
from the last level of decomposition (remaining two samples, in this case). The DWT will then
have the same number of coefficients as the original signal.

The frequencies that are most prominent in the original signal will appear as high amplitudes in
that region of the DWT signal that includes those particular frequencies. The difference of this
transform from the Fourier transform is that the time localization of these frequencies will not be
lost. However, the time localization will have a resolution that depends on which level they
appear. If the main information of the signal lies in the high frequencies, as happens most often,
the time localization of these frequencies will be more precise, since they are characterized by
more number of samples. If the main information lies only at very low frequencies, the time
localization will not be very precise, since few samples are used to express signal at these
frequencies. This procedure in effect offers a good time resolution at high frequencies, and good
frequency resolution at low frequencies. Most practical signals encountered are of this type.

One important property of the discrete wavelet transform is the relationship between the impulse
responses of the highpass and lowpass filters. The highpass and lowpass filters are not
independent of each other, and they are related by

where g[n] is the highpass, h[n] is the lowpass filter, and L is the filter length (in number of
points). Note that the two filters are odd index alternated reversed versions of each other.
Lowpass to highpass conversion is provided by the (-1)n term. Filters satisfying this condition are
commonly used in signal processing, and they are known as the Quadrature Mirror Filters
(QMF). The two filtering and subsampling operations can be expressed by

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The reconstruction in this case is very easy since halfband filters form orthonormal bases. The
above procedure is followed in reverse order for the reconstruction. The signals at every level are
upsampled by two, passed through the synthesis filters g’[n], and h’[n] (highpass and lowpass,
respectively), and then added. The interesting point here is that the analysis and synthesis filters
are identical to each other, except for a time reversal. Therefore, the reconstruction formula
becomes (for each layer)

However, if the filters are not ideal halfband, then perfect reconstruction cannot be achieved.
Although it is not possible to realize ideal filters, under certain conditions it is possible to find
filters that provide perfect reconstruction. The most famous ones are the ones developed by
Ingrid Daubechies, and they are known as Daubechies’ wavelets.

Note that due to successive subsampling by 2, the signal length must be a power of 2, or at least
a multiple of power of 2, in order this scheme to be efficient. The length of the signal determines
the number of levels that the signal can be decomposed to. For example, if the signal length is
1024, ten levels of decomposition are possible.

Interpreting the DWT coefficients can sometimes be rather difficult because the way DWT
coefficients are presented is rather peculiar. To make a real long story real short, DWT
coefficients of each level are concatenated, starting with the last level. An example is in order to
make this concept clear:

Suppose we have a 256-sample long signal sampled at 10 MHZ and we wish to obtain its DWT
coefficients. Since the signal is sampled at 10 MHz, the highest frequency component that exists
in the signal is 5 MHz. At the first level, the signal is passed through the lowpass filter h[n], and
the highpass filter g[n], the outputs of which are subsampled by two. The highpass filter output is
the first level DWT coefficients. There are 128 of them, and they represent the signal in the [2.5
5] MHz range. These 128 samples are the last 128 samples plotted. The lowpass filter output,
which also has 128 samples, but spanning the frequency band of [0 2.5] MHz, are further
decomposed by passing them through the same h[n] and g[n]. The output of the second highpass
filter is the level 2 DWT coefficients and these 64 samples precede the 128 level 1 coefficients in
the plot. The output of the second lowpass filter is further decomposed, once again by passing it
through the filters h[n] and g[n]. The output of the third highpass filter is the level 3 DWT
coefficiets. These 32 samples precede the level 2 DWT coefficients in the plot.

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CONCLUSION:
This is all about the literature survey up to which I have investigated regarding why wavelet
transform is used & what are its advantages in comparison to other mathematical transformation
so that further it can be used in my project i.e. analyzing the sensor output with & without fault
both in time-domain & frequency-domain simultaneously using wavelet transform.

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