The document discusses various signal transformation techniques, primarily focusing on the Fourier Transform (FT), Short Time Fourier Transform (STFT), and Wavelet Transform (WT) for analyzing signals in time and frequency domains. FT provides frequency information but lacks time resolution, while STFT offers a time-frequency analysis with a trade-off in resolution depending on window size. WT enhances this by using different window sizes for different frequency components, thus allowing better time localization for high frequencies and is deemed more suitable for real-world signals.

1. Any signal in time domain is
concidered as raw signal. 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.
The commonly used transformation techniques are
• Discreet Fourier Transform
• Short time Fourier transform
• Wavelet transform
1.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. In this time amplitude graph time is
taken at the x Axis as an independent variable Whereas amplitude
of the signal is taken at y axis .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 frequency domain representation of the signal there is a graph
between frequency and amplitude .In this frequency Amplitude
graph The frequency of the signal is taken at the x axis as
independend variable whereas the Amplitude of the signal is taken
at y Axis.The frequency domain representation of the signal tells

2. us that in a given signal what different frequency components are
present and what are there respective amplitudes.But again
frequency domain representation gives no idea that at which time
these frequency componants are presents. In some applications
frequency domain representation is more important. Or we can say
that frequency domain representation gives more information
about any signal (for example in any audio music signal).
Suppose
x= 100*sin(2*pi*2*t)+50*cos(2*pi*3*t)
is a time Domain representation of a given signal and
y=fft(x)
is frequency doamin representaion of signal x
0 1 2 3 4 5 6 7 8 9 10
-200
-100
0
100
200
TIME
AMPLITUDE
0 1 2 3 4 5 6 7 8 9 10
-1000
0
1000
2000
3000
FREQUENCY
AMPLITUDE
TIME DOMAINREPRESENTATIONOF SIGNAL
FREQUENCY DOMAINREPRESENTATIONOF SIGNAL

3. As it is clear from fig (1) that time domain representation of signal
gives no idea about the frequency componants of signal it simply
shows that how the amplitude of the signal is varying with respect
to time whereas frequency domain representation of the signal
shows the different frequency componants presents in a signal and
their respective amplitudes but it gives no idea that at which time
instants these frequency componants presents.
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 componants 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.
Both domain of signal analysis has its own utilities and has its own
importance and having its own sets of advantages and
disadvantages.

4. 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) …..(3)
Equation (1) and equation (2) gives Fourier transform and inverse
Fourier transform of any signal the exponential term of equation
(1) can be expressed in terms of sine and cosine function shown by
equation(3)
with the help of equation (1) and equation (2) we can convert any
time domain signal into frequency domain signal and frequency
domain signal into time domain signal respectively.
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 f1 frequency components it simply adds the
magnitude of all f1 frequency components.
Again take an another frequency for example f2 and search it from
– infinity to + infinity over time if it find the f2 frequency

5. components it simply adds the magnitude of all f2 frequency
components
Again repeat the same process with f3 ,f4, f5……. 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 wheather this
perticular frequency componant present or not present in time
from minus infinte to plus infine.And if present then how many
time this perticular frequency componants presents and what is
the amplitude of this perticular frequency componant and then
simply add that perticular frequency componant and calculate the
amplitude of any perticular frequency componant.
Again take a second frequency componant and check that in time
from minus infinite to plus infinite how many times this perticular
frequency componant exists and what is amplitude of this
perticular frequency componant and then simply adds them.In this
way the fourier transform calculated the amplitude of every
frequency componants presents in a given signal and draw a graph
between frequency and amplitude
Again there are certain disadvantage of frequency domain
representation of the signal first disadvantage is that it gives no
idea about time .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

6. So again the D.F.T. prove its suitability for the signals which are
stationary in nature but this transform is not suitable for non
stationary signal
By stationary signal we simply means the signal in which the
frequency does not change with respect to time or we can say that
all frequency components exits for all the time
By non stationary signal we simply means the signal in which
frequency changes with respect to the time. or in which all the
frequency components does not exits for all the time interval .but
some frequencies are exits for some particular time interval
whereas some other frequency components exists for some other
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. But in spite of it the
Fourier transform of these two signals will be the same because
these two signals have the same frequency components of course
one signal contains all the frequency components all the time
whereas second signal contains these frequency components at
different time intervals but as we know that Fourier transform has
nothing to do with the time. No matter where these frequency
components exits over time the matter is only that whether they
occur or not and what are their amplitudes.

8. Again take an another example suppose there are two signals S3
and S4
Signal S3 contains frequencies f1 for time interval t0 to t1,frequency
f2 for time interval t1 to t2,and frequency f3 for time interval t2 to t3.
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 quie different though both
signals are having the same frequency componants in same amount
but the time instances where these frequency componants exists
are different so the overall characteristics of above these two
signals S3 and S4 will be different but inspite of this the Fourier
transform of these two signals will be the same because fourier
transform has nothing to do with time.
Fourier transform simply watch that what frequency componants
any signal has and what are their respective amplitudes

10. 2 SHOT TIME FOURIER TRANSFORM:
The Short time Fourier Transform is a modified version of Fourier
transform. S.T.F.T. is nothing it is simply the Fourier transform of
any signal multiplied by a window function.
STFTX
(w)
(t, f) = ∫t [x(t). w*( t – t')].e-j2Πf t
dt …….6.3
The basic idea behind the STFT is that any non stationary signal
can be considered stationary for a short time interval. 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.e. it gives good information about the
time but its frequency information is poor.
Again if we choose large size window than it will provide us a
very good frequency information but the time information is poor
again if we choose the large size of window then the signal can not
be considered stationary(because the signal is stationary only for
the short time interval)
Again at the same time we can’t get good time and frequency
resolution either we get good time resolution or good frequency
resolution
The small window size is suitable for high frequencies whereas
Large window size is suitable for low frequencies

11. But the problem with STFT is that the window size remain same
for the all analysis we can’t choose different -different window
Size for the analysis of different frequency components .
As shown in fig that for all frequencies the size of window is same
Once window size is choosen then we can not change the size of
window .And any single window size can not suitable for different
different frequency componants
Again in S.T.F.T. it is very toufh task to choose the size of window
If we choose a narrow window size then it will provide a good
time resolution and bad frequency resolution
If we choose a large window size then it will give good frequency
resolution but poor time resolution ( though we can’t choose very
large otherwise it will against the concept of Short time fourier
transform)

12. Hence we can say that there is some resolution problem with
S.T.F.T. at the same time we can’t obtain both time as well as
frequency resolution means we can not know Exactly at which
perticular time instant which perticular frequency componants
Exists We can know only that in which time intervals(not time
instants) what frequency specturm occurs(not exact frequency)
Again if we want to increase the time resolution (by decresing
window size) then frequency resolution get decreased
And if we want to increased frequency resolution(by increasing
window size)then time resolution get decreased
So there is a trade off between the time resolution and frequency
resolution

13. WAVELET TRANSFORM:
By using wavelet Transform we can overcome the problem
with S.T.F.T. in wavelet transform we used different window
size for different frequency componants.Low scale(small
window size or small time scale) is used for higher frequencies
and higher scale (Large Window size or large time scale) is
used for low frequencies
The main advantage of wavelet transforms are
1. Wavelet transforms has multiresolution properity
2. Better Spectral localization properity
What is Multiresolution properity:
Mutiresolution properity means different frequency componants
presents in any signal are resolved at different scale(different time
scale or different window size)
Scale is inversly proportional to the frequency means small scale is
used for higher frquencies whereas lage scale is used for small
frequencies
Small scale (Small window Size or Small time scale)is used for the
analysis of higher frequency components.Means wavelet transform
provide higher time resolution for high frequency means if any signal
contain a very high frequency componant then with the help of wavelet
transform we can know that at which exact time interval (very small time
interval) these frequency componants exists
whereas the large scale (large window size or large time scale)is used for
the analysis of small frequency components.
Wavelet transform provide good time resolution for higher frequencies
whereas for small frequencies time resolution is not so good
But if we study real word signals then we find that genrally higher
frequencies are occurs only for very short time interval whereas small
frequency componants are presents for long time interval.So wavelet
transform prove its suitability for real time signals.

14. WHAT IS SCALE:
Scale is inversely propertional to the frequency of any signal
large scale is used for the analysis of small frequency componants
presents in any signal Whereas Small scale is used for the analysis of high
frequency componants of any signal
What is Spectral localisation
Spectral localization properity means that wavelet transform tells us that
what frequency componants are present in any given signal and at time
axis where these frequency componants are presents
The Wavelet transform has
multiresolution capability it means it resolve the different-different

15. frequency componants of the signal at different scales.Scale is
inversely propertional 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 can not resolve the all frequency componants
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
componants present in signal at different scale.
To understand the concept of scale take the following example we
can draw the map of glope which shows only where Sea(water)
exist And where earth exists.If we can resolve more this map then
we will be able to see the boundary of each nation.If We can more
resolve this map then we will be able to see the boundaries of
states of each nation.If we can further increased resolution then we
will be able to see the map of major cities of countaries and If we
further increased resolution then we will be able to see the
different sector ,blocks or different area of the city. So as we are
increasing resolution then we are able to know each detail of map
or each detail of locations .Or in the other word we can say that by
choosing a single resolution level we can not capture the every
detail in the map for capturing the different detail of map we need
to resolve the map at different different levels this is called
multiresolution.
Again take an another example
suppose we are taking the photograph of city from the peak of any
mountain then the whole city will look like a dense population area
or look like that the group of several houses again if we zoom our
camera then we can focus any sector of city we can also watch the
roads,garden,electric cables etc.Again if we can further zoom our
camera then we can focus at any perticular home and can watch
that perticular home how many windows are there in any perticular
home or how many doors that perticular home has again we can
focus at the roof of any perticular home and even focus at the face

16. of any persion who is standing at the roof.we can also zoom at the
name-plate infront of any house and can read the name of
people.So by zomming our cemera at different different levels we
can able to capturing the different level of details By choosing a
single zooming level we cant capture every details of any picture
The same thing with wavelet transform in wavelet transform we
resolve each frequency componants at different different scales.
With the help of above equation we can understand that In the
process of taking wavelet Transform what exctly going on first we
choose a perticular frequency suppose f1 and choose a suitalbe
scale for it and search this frequency in time from minus infinity
to plus infinty and keeping record of both amplitude of frequency
as well as time duration where we found that perticular frequency
contents
Then increases the frequency to f2 and accordingly choose a
suitable scale (window size) and search this frequency from minus
infinity to plus infinity in time and keep track of both amplitude of
frequency contents and the time duration where we found this
frequency contents
In this way we collect all the information about any signal what
frequency contents it has what are the amplitude of these frequency
contents and in time where these frequency contents exists
It is just like to watch any signal with a microscope or lens with
different different megnification factor. So by choosing any

17. constant megnification factor we can not wacth all the frequency
componants present in any signal.we need to vary the
megnification factor of the lens of microscope as the frequency of
signal varies.
So we can draw a three dimentional 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.fourier
transform simply tells us that what different frequency componants
are present in a given signal but it gives no idea that at time axis
where these different frequency componants exists
S.T.F.T. solves the above problem of fourier transform but up to
certain extend S.T.F.T. also provide the time resolution but in
STFT the window size remain constant so the time resolution of
STFT is also constant
In case of wavelet transform the window size is
adjusting means we can analysis the same sagnal at different time
scale and according to the frequencies which are present in any
signal we can choose different different time scale for the analysis
of the different –different frequency componants
Scale is inversely proportional to the fequency so for the analysis
of low frequencies we choose the large time scale whereas for the
analysis of high frequency componants we choose low time scale
The small window size gives High time resolution but poor
frequency resolution
The large window size gives high frequency resolution but bad
time reolution

18. In wavelet transform we choose a small window size for high
frequency so it provide a very high time reolution for higer
frequency
again a large window size is choosen for low frequencies so it
provide good frequency resolution for low frequency
In practice all the practical signals has low frequency componants
for long time duration whereas has high frequency componants for
a very small time duration so wavelet transform is very suitable for
all real word signals.
The main problem with sort time fourier transform is that the size
of window function in STFT is same for the analysis of all spectral
componants of any signal.
So if we choose a small size window then it gives good time
approximation but poor frequency approximation whereas if we
choose a large window size then it gives good frequency
approximation but very poor time approximation again problem
with choosing a large window size is that for long window size a
signal can not be concidred statonary.again we can say that no
single window size is suitable for all the frequency componants
presents in any signal
Above Equation shows the wavelet transfor of any signal It is
clear from the above equation that we can change the scale by
varying the value of s .Now it is also clear from the above equation
that the different different scales are used for the analysis of
different different frequency componants as their suitabily for the
different different frequency componants it is just like to use
different different filters (high pass and low pass filter with
different cut off frequencies) for different different frequency
componants what exact we are doing in the process of taking
wavelet transform we passes the different frequency componants
with a filter of different cut off frequency.
Equation (1) gives us a theoritical 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

19. different frequency componants.Now how we can converted this
theortical approach in to practice
Process of taking Wavelet transform of any signal
In the process of wavelet transform the original signal( S ) is first
decompose into Approximate Cofficients and Detailed Cofficients by
simply passing the signal through low pass filter and high pass filter
respectively.
The output of low pass filter is called Approximate[A1](Low freqency
componants) cofficient of the signal
The output of High pass filter is called Detailed [D1](High freqency
componants) Cofficients Of the signal.
This Approximate cofficient[A1] again passed through a low pass and
high pass filter
And again Decompose the signal into Approximate[A 2] and Detailed
Cofficients[D 2]
Further Approximate Componants[A2] can be decomposed into
Approximate cofficients[A3] And Detailed Cofficients[D3]
The number of Decomposition levels depends on the length of signal and
our requirements
S =A1+D1 [First level Wavelet Decomposition]…(a)
A1=A2+D2 [Second level Wavelet Decomposition]…(b)
A2=A3+D3 [Third level Wavelet Decomposition]…(c)
S=A3+D3+D2+D1 ………………………………….(1)

20. The Origional Signal S can be reconstruct with the help of
A3,D3,D2 and D1.
Fig(a) Wavelet Decomposition of signal
So it is clear that with the help of equation(1) and equation (a),(b),(c)
We can decompose any original signal sequences in to wavelet decomposition.
And with these wavelet decomposition again we can construst the origional
signal.
The number of sample in next decomposition level is half as
compaired to previous stage.Supose 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.
Signal(S)
A-1D-1
D-2 A-2
A-3D-3

21. So wavelet transform is highly suitable for the analysis of local
behaviour of the signal such as spikes or discontinuties. Because at
the point of discontinuty the frequencies changes very fast only for
a very littile time so by choosing suitable time scale we can also
study or analysis these sudden changes.
Again an another advantage of wavelet transform over fourier
transform is that the fourier transform convert the signal into the
different sinusoids of different different frequencies
The shape of sine and cosine waves are predefined and
predectable.wheareas in wavelet transform we convert the signal
into the mother wavelets of different amplitude and scale the local
behavour of any signal can be discribed in better way by using
wavelets
Again with W.T. we have a freedom to choose the shape of
wavelets (mother wavelet).there are lot of standard wavelets
families (wavelet families contain different wavelets of different
orders) suitable for different applications.
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 intrgation of wavelet
function should be zero or the mean value of wavelet function
should be zero
∫ Ψ(x).d(x)=0
[2] Scale function: Scale function capture the low frequencies
information(approximate ) presents in any signal.the intregation of
scale function should be one it means its average value is one.
∫Ǿ(x).d(x)=1

22. WAVELETS FAMILIES:
Some standard wavelet families are
Family name short name
Haar haar
Daubechies db
Symlets sym
Coiflets coif
BiorSplines bior
ReverseBior rbio
Meyer meyr
Dmeyer dmey
Gaussian gaus
Mexican_hat mexh
Morlet morl
Complex Gaussian cgau
Shannon shan
Frequency B-Spline fbsp
Complex Morlet cmor
Different wavelet of standard families
S.N. Standard Wavelet
family
Different wavelet
1 Haar Haar
2 Daubechies db1 db2 db3 db4 db5 db6 db7 db8
db9 db10
3 Symlets sym sym2 sym3 sym4 sym5 sym6 sym7

23. sym8
4 Coiflets coif1 coif2 coif3 coif4 coif5
5 BiorSplines bior1.1 bior1.3 bior1.5 bior2.2
bior2.4 bior2.6 bior2.8 bior3.1
bior3.3 bior3.5 bior3.7 bior3.9
bior4.4 bior5.5 bior6.8
6 ReverseBior rbio1.1 rbio1.3 rbio1.5 rbio2.2
rbio2.4 rbio2.6 rbio2.8 rbio3.1
rbio3.3 rbio3.5 rbio3.7 rbio3.9 rbio4.4
rbio5.5 rbio6.8
7 Meyer Meyr
8 DMeyer Dmey
9 Gaussian gaus1 gaus2 gaus3 gaus4
gaus5 gaus6 gaus7 gaus8
10 Mexican_hat Mexh
11 Morlet Morl
12 Complex Gaussian cgau1 cgau2 cgau3 cgau4
cgau5
13 Shannon shan1-1.5 shan1-1 shan1-0.5 shan1-0.1
shan2-3
14 Frequency B-
Spline
fbsp1-1-1.5 fbsp1-1-1 fbsp1-1-0.5 fbsp2-
1-1
fbsp2-1-0.5 fbsp2-1-0.1
15 Complex Morlet cmor1-1.5 cmor1-1 cmor1-0.5 cmor1-1
cmor1-0.5 cmor1-0.1
The choice of any perticular wavelet depend on that perticular
application and property of wavelet. The most common properties
of any wavelets are The support of scaling and wavelet
function(with compact support or without compact
support),Regularity,Orthoganality or Biorthogobnality,Number of
zero moment,Wavelet with FIR filter or without FIR filter etc

24. [1]the support of wavelet function Ψ(t) and scaling function Ǿ(x):-
This is the most important criteria that the wavelet function and
scaling function have compact support or not have compact
support.This properity of wavelet decides the localisation of
wavelet in time and frequency domain.the support of wavelet
function decides
the speed of convergence to 0 of wavelet function (Ψ(t) or Ψ(w))
when the time t or the frequency w goes to infinity.
[2] wavelet with F.I.R. filter or without F.I.R. filter:
[3] Symmetry of wavelets: wavelet is symmetric,near symmetric or
asymetric
[4] Orthogonality or biorthogonality
[5] Regularity of wavelet wich decide the smoothness of
reconstructing signal or image is a very important criteria.
[6] the number of vanishing moment (zero moment)for wavelet
function Ψ or scaling function Ǿ this is very useful for
compression purpose.
[7] the scaling function exist or does not exist
[8] an explict mathamatical expression available or not for scaling
function (if exist) and wavelet function
[9]Continue or discreate

25. Wavelet with filters Wavelet without filter
Classification of wavelet families
With compact
Support
Without compact
Support
Orthogonal Bi-Orthogonal Orthogonal
Real Complex
Db
Haar
Sym
Coif
Bior Meyr
Dmey
btlm
Gaus
Mexh
morl
Cgau
Shan
Fbsp
cmor