osama fina2

8/8/2019 osama fina2

1/35

Introduction In previous sessions we studied

Wavelet Transform

Meaning of Transform

Types of transforms

Algorithms

Applications

Among which Fourier transform is probably by far the mostpopular

BUT , is it necessary to have both the time and the frequencyinformation at the same time?!

TIME FREQUENCY


2/35

Introduction For the given signals

Wavelet Transform

When in time these frequencies happened ??!!

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0

- 6

- 4

- 2

0

2

4

6

8

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0

- 8

- 6

- 4

- 2

0

2

4

6

8

FFT

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

FFT


3/35

Introduction

Wavelet Transform


4/35

Introduction

Time information is not required when the signal is so-calledstationary

Most of practical signals are non-stationary as in speech,radar, sonar, seismic and even two dimensional images arenon-stationary.

FT can be used for non-stationary signals, if we are onlyinterested in what spectral components exist, but not interestedwhere these occur.

ONE EARLIER SOLUTION: SHORT-TIME FOURIERTRANSFORM (STFT)

Wavelet Transform


5/35

Short Time Fourier Transform

Windowed F.T. or Short Time F.T. (STFT) : Segmenting the signal

into narrow time intervals, narrow enough to be considered

stationary, and then take the Fourier transform of each segment,Gabor 1946.

Followed by other Time Frequency Representations (TFRs), which

differed from each other by the selection of the windowing

function

Wavelet Transform


6/35


Wavelet Transform


7/35


1. Choose a window function of finite length

2. Place the window on top of the signal at t=0

3. Truncate the signal using this window

4. Compute the FT of the truncated signal, save.5. Incrementally slide the window to the right

6. Go to step 3, until window reaches the end of the signal

For each time location where the window is centered, weobtain a different FT

Hence, each FT provides the spectral information of a separatetime-slice of the signal, providing simultaneous time andfrequency information

Wavelet Transform


8/35

FTX



9/35

FTX



10/35

FTX



11/35

FTX



12/35

FTX



13/35

FTX



14/35

FTX



15/35

FTX



16/35

FTX



17/35

FTX



18/35

FTX



19/35

? A d!dt

tjx dtettWtxtS

[[ [ )()(),(

STFT of signal x(t):

Computed for each

window centered at t=t

Time

parameter

Frequency

parameterSignal to

beanalyzed

Windowing

function

Windowing function

centered at t=t

FT Kernel

(basis function)



20/35

The window size determines what we callRESOLUTION

We are concerned in Time & Frequency resolution. Resolution may also vary from window type to another. Once you choose a particular size for the time window - it will

be the same for all frequencies. For a window of infinite length we get FT which gives perfect

frequency resolution but no time resolution. The narrower the window the better time resolution but poorer

frequency resolution.

Wavelet Transform



21/35

Wavelet Transform

The problem with STFT is the fact whose roots go back towhat is known as the HeisenbergUncertainty Principle .

Simply, this principle states that one cannot know the exacttime-frequency representation of a signal

Many signals require a more flexible approach - so we canvary the window size to determine more accurately eithertime or frequency.

2/1. u!(( constt [



22/35


Wavelet Transform


23/35


Wavelet Transform


24/35


Wavelet Transform


25/35


Wavelet Transform


26/35


Wavelet Transform


27/35


Time and frequency resolution problems are results of aphysical phenomenon and exist regardless of the transformused.

Multi-resolution analysis (MRA)is alternative approach toanalyze the signal at different frequencies with differentresolutions. Every spectral component is not resolved equallyas was the case in the STFT.

MRA is designed to give good time resolution and poorfrequency resolution at high frequencies and good frequencyresolution and poor time resolution at low frequencies.

This approach makes sense as signals in practical applicationshave high frequency components for short durations and lowfrequency components for long durations.

Wavelet Transform


28/35

Continuous Wavelet Transform (CWT)

Wavelet Transform

Time

Frequency Better timeresolution;

Poor frequency

resolution

Better frequency

resolution

Poor time

resolution

Each box represents a equal portionResolution in STFT is selected once for entire analysis


29/35


Wavelet analysis is done in a similar way to the STFTanalysis.

Wavelet Transform

FT of the windowed signals are not taken. Width of the window is changed as the transform iscomputed for every spectral component.

There are two main differences between the STFT and theCWT:


30/35


Wavelet Transform


31/35


dtsttxsss xx

X]y!X=!X ]]

*1,,WT

Wavelet Transform

Wavelet

Small wave

Means the window function is of finite length.

MotherWavelet

A prototype for generating the other window functions All the used windows are its dilated or compressed and

shifted versions

Translation

(The location of

the window)

Scale

Mother Wavelet


32/35

Wavelet Transform

Wavelet analysis produces a time-scale view of thesignal.

Scalingmeans stretching or compressing of the signal


f t a

f t a

f t a

t

t

t

( )

( )

( )

sin( )

sin( )

sin( )

! !

! !

! !

;

;

;

scale factor (a) for sine waves:

f t a

f t a

f t a

t

t

t

( )

( )

( )

( )

( )

( )

! !

! !

! !

=

=

=

;

;

;

1

2 1

24 1

4

Scale factor works exactly the same with wavelets:


33/35


Step 1: The wavelet is placed at the beginning of the signal,and set s=1 (the most compressed wavelet)

Wavelet Transform

Computation of the CWT

signal

Step 2: The wavelet function at scale 1 is multiplied by thesignal, and integrated over all times; then multiplied by 1/s;

Step 3: Shift the wavelet to t=, and get the transform value att= and s=1;

Step 4:Repeat the procedure until the wavelet reaches the endof the signal

Step 5: Scale s is increased by a sufficiently small value, theprevious steps are repeated for all s


34/35


Wavelet Transform

Time

Frequency Better timeresolution;

Poor frequency

resolution

Better frequency

resolution

Poor time

resolution

Each box represents a equal portionResolution in STFT is selected once for entire analysis


35/35


Wavelet Transform

Perfect timeresolution

No frequency

resolution

No time resolutionPerfect frequency

resolution

Constant time &

frequency resolution

Variable time &

frequency resolution

Documents

osama fina2