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The Wavelet Transform
An Introduction
By
Prof Vishnu Narayan Saxena
&
Prof Pooja Saxena

Dedication
Dedicated to my father Mr
...
N
...

And my cute daughter pari who developed
into me a sense of creativity, love and
responsibility

Acknowledgement
I want to express my extreme
thanks to technology which gave me a
beautiful and wonderful platform to share
my thoughts to knowledge seekers
...

I am grateful to the co-author of this
book and my wife Prof
...

I am also thankful to all the authors and
researchers who share their thoughts
publically on internet about this beautiful
and recent topic wavelet transform
...
If
we can arrange any topic in proper
sequence and in systematic way then
suddenly we found that it is very easy
...


Why wavelet Transform
Wavelet transform is a very beautiful tool for signal
processing which gives us high degree of freedom
and flexibility
...

But it gives no idea about the different frequency
components presents in any given signal
The time resolution of time domain representation is
very high but the frequency resolution of time
domain representation is zero
...


The frequency domain representation of the signal
gives us an information that how the amplitude of
any signal is varying with respect to frequency
Or
What are the amplitudes of different frequency
components presents in any signal
So frequency resolution of time domain signal is very
high but its time resolution is zero
So Fourier transform is again not able
to provide complete information about any signal
particularly for non stationary signals (in which
frequency changes with respect to time) Fourier
transform is not a suitable tool
...
T
...
T
...
In
sort time Fourier transform we simply multiply the
signal with a window function of small length
...
T
...
T
...
So S
...
F
...
solve the problem of
F
...
but up to certain extent and it gives a three

dimensional information about the signal (about time,
frequency and amplitude of any signal)
But the main drawback with S
...
F
...
is that in
S
...
F
...
the size of window remains constant and any
single window size cannot be suitable for all
frequency components present in any signal
...
T
...
T
...
e
...

Hence the wavelet transform has multi
resolution properties i
...
it uses different time
scales (scale is inversely proportional to the
frequency) for the analysis of different frequency
components
...


Again Wavelet Transform is most suitable tool for
studying the local behavior of any signal such as
discontinuities or spikes
In Fourier transform (F
...
) or Short time Fourier
transform(S
...
F
...
) any time domain signal is
converted into sinusoid of different amplitudes and
frequencies whereas in Wavelet transform signal is
converted into shifted and scaled version mother
wavelets
...

There are well defined families of standard wavelet
and we have also freedom of defining our own
wavelet according to our requirement
...
Vishnu N
...
The Propose of all Transformation
techniques are to convert time domain signal in a form so that
desired information can be extracted from these signal’s and
after the application of certain transform the resultant signal
is known as processed signal
...
DISCREAT FOURIER TRANSFORMS: The Discrete
Fourier transform is used for converting time domain signal
into frequency domain
In time domain representation of the signal there is a graph
between time and amplitude
...
the time
domain representation of the signal gives an information
about the signal that at which time instants what is the
amplitude of the signal or how amplitude of the signal is
varying with respect to time but it gives no information about

the Different frequency contents that are presents in the
signal
...
In this frequency
Amplitude graph The frequency of the signal is taken at the x
axis as independent variable whereas the Amplitude of the
signal is taken at y Axis
...
But again frequency domain
representation gives no idea that at which time these
frequency components are present
...
Or we
can say that frequency domain representation gives more
information about any signal (for example in any audio music
signal)
...


In the other word we can say in time domain representation
of signal the time resolution of signal is very high but its
frequency resolution is zero because it gives no idea about
different frequency components presents in a signal where as
in frequency domain representation of the signal the
frequency resolution of the signal is very high but its time
resolution is zero because it gives no idea about time
...


HOW FOURIER TRANSFORM CONVERT TIME
DOMAIN SIGNAL INTO FREQUENCY DOMAIN:
Suppose x(t) shows the time domain representation
of signal and X(f) shows the frequency domain signal
Then

Where
e2jpift = Cos(2*pi*f*t)+J*Sin(2*pi*f*t) …
...

As clear from equation (1) that in Fourier transform the
signal is integrated from – infinity to + infinity over time for

each frequency In the other word we can say that equation 1
take a frequency for example f1 and search it from –infinity
to + infinity over time if it find the f 1 frequency components
it simply adds the magnitude of all f1 frequency components
...
and so on
No matter in time axis where these frequency components
exits from – infinity to + infinity it will effect the result of
integration in the same way
For every frequency Fourier transform check that whether
this particular frequency component present or not present in
time from minus infinite to plus infinite
...

Again take a second frequency component and check that in
time from minus infinite to plus infinite how many times this

particular frequency component exists and what is amplitude
of this particular frequency component and then simply adds
them
...
The frequency transform of any signal
simply tells us that in any given signal what spectral
components are present and what are their respective
amplitudes but it gives no idea that in time axis where these
frequency components exists
So again the D
...
T
...
or in which all
the frequency components does not exist for all the time
interval
...

To understand the suitability of the DFT only for stationary
signal and not for non stationary signal take the following
example suppose there are two signals S1 and S2 the signal S1
has three frequency components f1,f2 and f3 all the times and
suppose the signal S2 contains the same frequency
components f1,f2 and f3 but for the different -different time
interval’s so we can say that these two signals are completely
different in nature
...
No matter where
these frequency components exits over time the matter is
only that whether they occur or not and what are their
amplitudes
...

And signal S4 contain frequency f3 for time interval t0 to t1,
frequency f2 for t1 to t2 and frequency f1 for time interval t2 to
t3
So we can say that these two signals are quite different
though both signals are having the same frequency
components in same amount but the time instances where
these frequency components exists are different so the overall
characteristics of above these two signals S3 and S4 will be
different but in spite of this the Fourier transform of these
two signals will be the same because Fourier transform has
nothing to do with time
...
S
...
F
...
is nothing it is simply the Fourier
transform of any signal multiplied by a window function
...
w*( t – t')]
...
6
...

So we can say that the STFT gives an idea about time
frequency and amplitude
But again the problem with STFT is that how to choose the
size of window (time interval of window) because if we
choose
A small size window than it will give good time resolution
but poor frequency resolution i
...
it gives good information
about the time but its frequency information is poor
...

As shown in fig that for all frequencies the size of window is
same
Once window size is chosen then we cannot change the size
of window
...
T
...
T
...
T
...
T
...
WAVELET TRANSFORM
By using wavelet Transform we can overcome the problem
with S
...
F
...
in wavelet transform we used different window
size for different frequency components
...
Wavelet transforms has multi resolution property
2
...
Means
wavelet transform provide higher time resolution for high
frequency means if any signal contain a very high frequency
component then with the help of wavelet transform we can
know that at which exact time interval (very small time
interval) these frequency components exists

Whereas the large scale (large window size or large time
scale) is used for the analysis of small frequency
components
...
So wavelet transform prove its suitability
for real time signals
...
Scale is inversely proportional to the frequency of the
signal it means the high frequency signals are resolved at low
scale whereas low frequency signals are resolved at high
scale because by choosing a single scale we cannot resolve
the all frequency components present in any signal or in other
word if we want to capture every detail of the signal then we
need to resolve the different frequency components present in
signal at different scale
...
If we can
resolve more this map then we will be able to see the
boundary of each nation
...
If we can further increased resolution then we will be
able to see the map of major cities of countries and If we
further increased resolution then we will be able to see the
different sector, blocks or different area of the city
...
Or in the other word we
can say that by choosing a single resolution level we cannot
capture the every detail in the map for capturing the different

detail of map we need to resolve the map at different levels
this is called multi resolution
...
Again if we can further zoom our camera then we
can focus at any particular home and can watch that
particular home how many windows are there in any
particular home or how many doors that particular home has
again we can focus at the roof of any particular home and
even focus at the face of any person who is standing at the
roof
...
So by zooming our
camera at different- different levels we can able to capturing
the different level of details by choosing a single zooming
level we can’t capture every details of any picture

The same thing with wavelet transform in wavelet transform
we resolve each frequency components at different -different
scales
...
So by choosing
any constant magnification factor we cannot watch all the

frequency components present in any signal
...


So we can draw a three dimensional plot between time
frequency and amplitude of the signal
Problem with Fourier transform is that
its time resolution is zero means it gives no idea about the
time
...
T
...
T
...
T
...
T
...

The main problem with sort time Fourier transforms is that
the size of window function in STFT is same for the analysis
of all spectral components of any signal
...
Again
we can say that no single window size is suitable for all the
frequency components presents in any signal

Above Equation shows the wavelet transform of any
signal It is clear from the above equation that we can change
the scale by varying the value of s
...

Equation (1) gives us a theoretical approach about the
wavelet transform that for different time interval how we can
change the time scale or how we can change the time scale
for the analysis of different frequency components
...

The output of low pass filter is called Approximate [A1](Low
frequency components) coefficient of the signal
The output of High pass filter is called Detailed [D1](High
frequency components) Coefficients Of the signal
...
(a)
A1=A2+D2 [Second level Wavelet Decomposition]…(b)
A2=A3+D3 [Third level Wavelet Decomposition]…
...
(1)
The Origional Signal S can be reconstruct with the help of
A3,D3,D2 and D1
...
And with these wavelet
decomposition again we can construct the original signal
...
Suppose The
original signal S has N samples then A1 and D1 will have
N/2 Samples and A2 and D2 will have N/4 Samples
...

Because at the point of discontinuity the frequencies changes
very fast only for a very little time so by choosing suitable
time scale we can also study or analysis these sudden
changes
...
Whereas in wavelet transform we convert the
signal into the mother wavelets of different amplitude and
scale the local behavior of any signal can be described in
better way by using wavelets
Again with W
...
we have a freedom to choose the shape of
wavelets (mother wavelet)
...


Again with wavelet transform we have freedom to design our
own wavelet hence we can define our own wavelet by
defining Two functions
[1]Wavelet function:
[2]Scale function:
[1] Wavelet function: wavelet function capture the details
(high frequencies) present in any signal and the integration of
wavelet function should be zero or the mean value of wavelet
function should be zero
∫ Ψ(x)
...
the
integration of scale function should be one it means its
average value is one
...
d(x) =1

WAVELETS FAMILIES:
Some standard wavelet families are
Family name
Haar
Daubechies
Symlets
Coiflets
BiorSplines
ReverseBior
Meyer
Dmeyer
Gaussian
Mexican_hat
Morlet
Complex Gaussian
Shannon
Frequency B-Spline
Complex Morlet

short name
haar
db
sym
coif
bior
rbio
meyr
dmey
gaus
mexh
morl
cgau
shan
fbsp
cmor

Different wavelet of standard families

The choice of any perticular wavelet depends on that
particular application and property of wavelet
...
This property of wavelet decides the
localization of wavelet in time and frequency domain
...

[2] Wavelet with F
...
R
...
I
...
filter:
[3] Symmetry of wavelets: wavelet is symmetric, near
symmetric or asymmetric
[4] Orthogonal or bi orthogonal
[5] Regularity of wavelet which decides the smoothness of
reconstructing signal or image is a very important criterion
...

[7] The scaling function exists or does not exist
[8] An explicit mathematical expression available or not for
scaling function (if exist) and wavelet function
[9]Continue or discreet

Classification of wavelet families

Wavelet with filters

With compact
Support

Orthogonal

Db
Haar
Sym
Coif

Wavelet without filter

Without
compact
Support

Bi-Orthogonal

Bior

Orthogonal

Meyr
Dmey
btlm

Real

Complex

Gaus
Mexh
morl

Cgau
Shan
Fbsp
cmor

Comparison between Fourier Transform and Wavelet
Transform
S
...
Fourier Transform
Wavelet Transform
1

The mathematical expression for The mathematical expression
Fourier transform is
for wavelet transform is
Where x(t) is time domain signal Where s is scale and Tao
and X( f ) is the frequency domain is translation parameter x(t) is
signal
original signal
...
2

Suitable for stationary signal

3
...


Fourier transform has zero time
resolution and very high frequency
resolution

5
...
T
...
e
...
T
...


Fourier transform is not suitable Wavelet transform is very
for studying the local behavior of suitable for studying the local
signal
behavior of the signal for
example discontinuity or spikes

7

In Fourier transform the input can In wavelet transform the input
be a real or complex function but can be a real or complex
its output is always complex
function but its output may be
real or complex
In Fourier analysis the signal is In wavelet analysis the signal is
converted into sine and cosine converted into scaled and
waves of various amplitude and translated version of mother
frequencies the shape of sine wavelet which is very irregular
and cosine wave are well and cannot be predicted

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