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
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# Book wavelets

Aug 18, 2015

## Engineering

#### domain of signal analysis

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Transcript

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

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

AM

PLI

TU

DE

0 1 2 3 4 5 6 7 8 9 10-1000

0

1000

2000

3000

FREQUENCY

AM

PLI

TU

DE

TIME DOMAIN REPRESENTATION OF SIGNAL

FREQUENCY DOMAIN REPRESENTATION OF SIGNAL

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.

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

components it simply adds the magnitude of all f2 frequency components

Again repeat the same process with f3 ,f 4, f 5……. 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

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.

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

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

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)

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

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 are1. Wavelet transforms has multiresolution properity2. 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 frequenciesSmall 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 goodBut 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.

WHAT IS SCALE:Scale is inversely propertional to the frequency of any signallarge 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 localisationSpectral 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

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

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 contentsThen 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 contentsIn 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 existsIt is just like to watch any signal with a microscope or lens with different different megnification factor. So by choosing any

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 existsS.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 componantsScale 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

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

different frequency componants.Now how we can converted this theortical approach in to practice

Process of taking Wavelet transform of any signalIn 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 signalThe 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)

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

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 frequenciesThe 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

WAVELETS FAMILIES:Some standard wavelet families are

Family name short nameHaar 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 Haar2 Daubechies db1 db2 db3 db4 db5 db6 db7 db8

db9 db10 3 Symlets sym sym2 sym3 sym4 sym5 sym6 sym7

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 Meyr8 DMeyer Dmey9 Gaussian gaus1 gaus2 gaus3 gaus4

gaus5 gaus6 gaus7 gaus8 10 Mexican_hat Mexh11 Morlet Morl12 Complex Gaussian cgau1 cgau2 cgau3 cgau4

cgau5 13 Shannon shan1-1.5 shan1-1 shan1-0.5 shan1-0.1

shan2-314 Frequency B-

Spline fbsp1-1-1.5 fbsp1-1-1 fbsp1-1-0.5 fbsp2-1-1fbsp2-1-0.5 fbsp2-1-0.1

15 Complex Morlet cmor1-1.5 cmor1-1 cmor1-0.5 cmor1-1cmor1-0.5 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

[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 decidesthe 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

Wavelet with filters Wavelet without filter

Classification of wavelet families

With compact Support

Without compact Support

Orthogonal Bi-Orthogonal Orthogonal

Real Complex

DbHaarSymCoif

Bior MeyrDmeybtlm

GausMexhmorl

CgauShanFbspcmor