ch9-time frequency wavelet

8/12/2019 Ch9-Time Frequency Wavelet

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Advanced signal processing

Dr. Mohamad KAHLIL


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Chapter 4: Time frequency and

wavelet analysis Definition

Time frequency

Shift time fourier transform

Winer-ville representations and others

Wavelet transform Scalogram

Continuous wavelet transform

Discrete wavelet transform: details and approximations applications


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The Story of Wavelets

Theory and Engineering Applications

Time frequency representation

Instantaneous frequency and group delay

Short time Fourier transform Analysis

Short time Fourier transform Synthesis

Discrete time STFT


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FT At Work

)52cos()(1 ttx =

)252cos()(2 ttx =

)502cos()(3 ttx =


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FT At Work

)(1X

)(2X

)(3X

)(1 tx

)(2 tx

)(3 tx


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FT At Work

)502cos(

)252cos(

)52cos()(4

t

t

ttx

+

+=

)(4X)(4 tx


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Stationary and Non-stationary

Signals FT identifies all spectral components present in the signal, however it does not provide

any information regarding the temporal (time) localization of these components. Why?

Stationary signals consist of spectral components that do not change in time all spectral components exist at all times

no need to know any time information

FT works well for stationary signals

However, non-stationary signals consists of time varying spectral components

How do we find out which spectral component appears when?

FT only provides what spectral components exist , not where in timethey are located.

Need some other ways to determine time localization of spectral

components


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Stationary and Non-stationary

Signals Stationary signals spectral characteristics do not change with time

Non-stationary signals have time varying spectra

)502cos(

)252cos(

)52cos()(4

t

t

ttx

+

+

=

][)( 3215 xxxtx = Concatenation


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Non-stationary Signals

5 Hz 20 Hz 50 Hz

Perfect knowledge of what

frequencies exist, but noinformation about where

these frequencies are

located in time


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FT Shortcomings

Complex exponentials stretch out to infinity in time

They analyze the signal globally, not locally

Hence, FT can only tell what frequencies exist in the

entire signal, but cannot tell, at what time instancesthese frequencies occur

In order to obtain time localization of the spectral

components, the signal need to be analyzed locally

HOW ?


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Short Time Fourier Transform

(STFT)1. Choose a window function of finite length

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

3. Truncate the signal using this window4. Compute the FT of the truncated signal, save.

5. Slide the window to the right by a small amount

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

For each time location where the window is centered, we obtain a different FT Hence, each FT provides the spectral information of a separate time-slice

of the signal, providing simultaneous time and frequency information


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STFT


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STFT

[ ]

=t

tj

x

dtettWtxtSTFT )()(),(

STFT of signal x(t):

Computed for each

window centered at t=t

Time

parameter

Frequency

parameter

Signal to

be analyzed

Windowing

function

Windowing function

centered at t=t

FT Kernel

(basis function)


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0 100 200 300-1.5

-1

-0.5

0

0.5

1

0 100 200 300-1.5

-1

-0.5

0

0.5

1

0 100 200 300-1.5

-1

-0.5

0

0.5

1

0 100 200 300-1.5

-1

-0.5

0

0.5

1

Windowed

sinusoid allows

FT to becomputed only

through the

support of the

windowing

function

STFT at Work


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STFT

STFT provides the time information by computing a different FTs forconsecutive time intervals, and then putting them together

Time-Frequency Representation (TFR)

Maps 1-D time domain signals to 2-D time-frequency signals

Consecutive time intervals of the signal are obtained by truncating the

signal using a sliding windowing function

How to choose the windowing function?

What shape? Rectangular, Gaussian,

Elliptic? How wide?

Wider window require less time steps low time resolution

Also, window should be narrow enough to make sure that theportion of the signal falling within the window is stationary

Can we choose an arbitrarily narrow window?


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Selection of STFT Window

Two extreme cases:

W(t)infinitely long: STFT turns into FT, providingexcellent frequency information (good frequency resolution), but no timeinformation

W(t)infinitely short:

STFT then gives the time signal back, with a phase factor. Excellenttime information (good time resolution), but no frequency information

Wide analysis windowpoor time resolution, good frequency resolution

Narrow analysis windowgood time resolution, poor frequency resolutionOnce the window is chosen, the resolution is set for both time and frequency.

[ ] =t

tjx dtettWtxtSTFT

)()(),(

1)( =tW

)()( ttW =[ ] tj

t

tjx etxdtetttxtSTFT

== )()()(),(


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Heisenberg Principle

4

1 ft

Time resolution: How well

two spikes in time can be

separated from each other in

the transform domain

Frequency resolution: How

well two spectral components

can be separated from each

other in the transform domain

Both time and frequency resolutions cannot be arbitrarily high!!!We cannot precisely know at what time instance a frequency component is

located. We can only know what interval of frequencies are present in which time

intervals

http://engineering.rowan.edu/~polikar/WAVELETS/WTpart2.html
http://engineering.rowan.edu/~polikar/WAVELETS/WTpart2.htmlhttp://engineering.rowan.edu/~polikar/WAVELETS/WTpart2.htmlhttp://engineering.rowan.edu/~polikar/WAVELETS/WTpart2.html


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STFT

.. ..

time

Am

plitude

Frequency ..

..

t0 t1 tk tk+1 tn

Th Sh Ti F i


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The Short Time Fourier

Transform Take FT of segmented consecutive pieces of a signal. Each FT then provides the spectral content of that time

segment only Spectral content for different time intervals

Time-frequency representation

[ ] =t

tjx dtetWtxSTFT )()(),(

STFT of signal x(t):

Computed for each

window centered at t=localized s ectrum

Timeparameter Frequency

parameter

Signal tobe analyzed

Windowing

function

(Analysis window)

Windowing function

centered at t=

FT Kernel

(basis function)


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Properties of STFT

Linear

Complex valued

Time invariant Time shift

Frequency shift

Many other properties of the FT also apply.


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Filter Interpretation of STFT

ftj

ftj

f

ettxfffX

dfefffXfffXF

2*

2**1

)()()~

()(

)~()()~()(

=

X(t) is passed through a bandpass filter with a center frequency of

Note that (f) itself is a lowpass filter.f~


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Filter Interpretation of STFT

x(t) ftjet 2)( X

ftje 2

),()( ftSTFTx

x(t) )( t ),()( ftSTFTx

X

ftje 2


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Resolution Issues

time

Am

plitude

Frequency

k

All signal attributes

located within the local

window interval

around t will appear

at t in the STFT

)( kt

n

)( kt


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Time-Frequency Resolution

Closely related to the choice of analysis window

Narrow window good time resolution

Wide window (narrow band) good frequencyresolution

Two extreme cases:

(T)=(t) excellent time resolution, no frequencyresolution

(T)=1 excellent freq. resolution (FT), no timeinfo!!!

How to choose the window length? Window length defines the time and frequency resolutions

Heisenbergs inequality

Cannot have arbitrarily good time and frequencyresolutions. One must trade one for the other. Theirroduct is bounded from below.


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Time-Frequency Resolution

Time

Frequency

Ti F Si l


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Time Frequency Signal

Expansion and STFT Synthesis

=f

tfjx fddetgfSTFTtx~

~

2)( ~)()~,()(

Synthesis window

Basis functions

Coefficients(weights)

Synthesized signal

Each (2D) point on the STFT plane shows how strongly a time

frequency point (t,f) contributes to the signal.

Typically, analysis and synthesis windows are chosen to beidentical.

=1)(*)( dtttg


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300 Hz 200 Hz 100Hz 50Hz

STFT Example


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302/2)( atet =

STFT Example


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31a=0.01

STFT Example


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32a=0.001

STFT Example


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33a=0.0001

STFT Example


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34a=0.00001

STFT Example


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Discrete Time Stft

=n k

kFtjx enTtgkFnTSTFTtx

2)( )(),()(

dtenTttxkFnTSTFT kFtjt

x 2*)( )()(),( =



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Time frequency resolution problem

Concepts of scale and translation

The mother of all oscillatory little basis functions

The continuous wavelet transform

Filter interpretation of wavelet transform

Constant Q filters


Theory and Engineering Applications


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Time Frequency Resolution

Time frequency resolution problem with STFT

Analysis window dictates both time and frequency

resolutions, once and for all

Narrow window Good time resolution

Narrow band (wide window)

Good frequencyresolution

When do we need good time resolution, when do we need

good frequency resolution?


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Continuous Wavelet Transform

== dta

bttx

abaWbaCWT

x

)(1

),(),()(

Normalization factor

Mother wavelettranslation

Scaling:

Changes the support of

the wavelet based on

the scale (frequency)

CWT of x(t) at scalea and translation b

Note: low scale high frequency


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Computation of CWT

),1( NbW

),5( NbW

time

Amplitude

b0

),1( 0bW

bN

time

Amplitude

b0

),5( 0bW

bN

time

Amplitude

b0

),10( 0bW

bN

),10( NbW

time

Amplitude

b0

),25( 0bW

bN

),25( NbW

== dta

bttx

abaWbaCWTx

)(1

),(),()(


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WT at WorkHigh frequency (small scale)

Low frequency (large scale)


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Why Wavelet?

We require that the wavelet functions, at a minimum,

satisfy the following:

= 0)( dtt


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The CWT as a Correlation

Recall that in the L2 space an inner product is defined as

then

Cross correlation:

then

>=< dttgtftgtf )()()(),(

>=< )(),(),( , ttxbaW ba

>==

ch9-time frequency wavelet

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