Chap 5 - Time Frequency Wavelet

8/12/2019 Chap 5 - Time Frequency Wavelet

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Advanced signal processing

Dr. Mohamad KAHLIL

Islamic University of Lebanon


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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 transformAnalysis

Short time Fourier transformSynthesis

Discrete time STFT


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Signal processing

Signal Processing

Time-domain techniques Freq.-domain techniques

Filters Fourier T.

Non/Stationary Signals Stationary Signals

TF domain techniques

WAVELET

TRANSFORMS

STFT

CWT DWT MRA 2-D DWT SWT ApplicationsDenoising

Compression

Signal Analysis

Disc. Detection

BME / NDEOther


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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 itdoes not provide any information regarding the temporal (time)localization of these components. Why?

Stationary signals consist of spectral components that do not change intime

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 spectralcomponents

How do we find out which spectral component appears when?

FT only provides what spectral components exist , not where intime they are located.

Need some other ways to determine time locali zation of spectralcomponents


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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

tt

ttx

][)( 3215 xxxtx Concatenation


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Non-stationary Signals

5 Hz 20 Hz 50 Hz

Perfect knowledge of what

frequencies exist, but no

information 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 theentire signal, but cannot tell, at what time instances

these 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 window

4. 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

tjx dtettWtxtSTFT )()(),(

STFT of signal x(t):

Computed for each

window centered at t=t

Time

parameter

Frequency

parameterSignal 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 be

computed only

through the

support of thewindowing

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 thesignal using a sliding windowing function

How to choose the windowing function?

What shape? Rectangular, Gaussian,

Elliptic?How wide?

Wider window require less time stepslow time resolution

Also, window should be narrow enough to make sure that the

portion 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 resolution

Once 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 welltwo spikes in time can be

separated from each other in

the transform domain

Frequency resolution:Howwell 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.html


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STFT

.. ..

time

Amplitud

e

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

ttjx dtetWtxSTFT )()(),(

STFT of signal x(t):

Computed for each

window centered at t=localized s ectrum

Time

parameter Frequency

parameter

Signal to

be analyzed

Windowing

function

(Analysis window)

Windowing function

centered at t=

FT Kernel

(basis function)


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Alt t R t ti


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Alternate Representation

of STFT

deXetSTFT

dfefffXeftSTFT

tjtjx

ftj

f

tfjx

)~()()~,(

)~

()()~

,(

*~)(

2*~

2)(

STFT : The inverse FT of the windowed spectrum,with a phase factor

)

~

()( *

fffX


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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

Amplitude

Fre

quency

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

Time

Frequency


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300 Hz 200 Hz 100Hz 50Hz

STFT Example


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2/2

)( at

et

STFT Example


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a=0.001

STFT Example


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STFT Example


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Discrete Time Stft

n k

kFtjx enTtgkFnTSTFTtx

2)( )(),()(

dtenTttxkFnTSTFT kFtj

t

x

2*)(

)()(),(



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Timefrequency 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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TimeFrequency Resolution

Timefrequency resolution problem with STFT

Analysis window dictates both time and frequency

resolutions, once and for all

Narrow windowGood time resolution

Narrow band (wide window)Good frequency

resolution

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

good frequency resolution?


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Scale & Translation

Translationtime shift

f(t)f(a.t) a>0

If 00

If 0


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The Mother of All Oscillatory

Little Basis Functions

The kernel functions used in Wavelet transform are all obtained fromone prototype function, by scaling and translating the prototype

function.

This prototype is called the mother wavelet

a

1

)(1)(,a

bt

atba

Scale parameter

Translationparameter

)()(0,1 tt

Normalization factor to ensure that allwavelets have the same energy

dttdttdttba22

)0,1(

2

),( )()()(


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Continuous Wavelet Transform

dt

abttx

abaWbaCWTx )(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

Amplitu

de

b0

),10( 0bW

bN

),10( NbW

time

Amplitude

b0

),25( 0bW

bN

),25( NbW

dta

bt

txabaWbaCWTx

)(

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

dtt 2)(

Wave

let


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The CWT as a Correlation

Recall that in the L2space an inner product is defined as

then

Cross correlation:

then

dttgtftgtf )()()(),(

)(),(),( , ttxbaW ba

)(),(

)()()(

tytx

dttytxRxy

)(

)(),(),(

,,

0,

bR

bttxbaW

oax

a


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The CWT as a Correlation

Meaning of life:

W(a,b) is the cross correlation of the signal x(t) with themother wavelet at scale a, at the lag of b. If x(t) is similar

to the mother wavelet at this scale and lag, then W(a,b)

will be large.

Filtering Interpretation of


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Filtering Interpretation of

Wavelet Transform

Recall that for a given system h[n], y[n]=x[n]*h[n]

Observe that

Interpretation:For any given scale a (frequency ~ 1/a), theCWT W(a,b) is the output of the filter with the impulse

response to the input x(b), i.e., we have a

continuum of filters, parameterized by the scale factor a.

dthx

thtxty

)()(

)(*)()(

)(*)(),( 0, bbxbaW a

)(0, ba


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What do Wavelets Look Like???

Mexican Hat Wavelet Haar Wavelet

Morlet Wavelet


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Constant Q Filtering

A special property of the filters defined by the motherwavelet is that they areso calledconstant Q filters.

Q Factor:

We observe that the filters defined by the mother wavelet

increase their bandwidth, as the scale is reduced (center

frequency is increased)

bandwidth

frequencycenter

w (rad/s)


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Constant Q

f0 2f0 4f0 8f0

B 2B 4B 8B

B B B B BB

f0 2f0 3f0 4f0 5f0 6f0

STFT

CWT

B

fQ


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Inverse CWT

a b

ba dadbtbaWaC

tx )(),(11

)( ,2

dC

)(

provided that

0)( dtt

Properties of


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Properties of

Continuous Wavelet Transform

Linearity

Translation

Scaling

Wavelet shifting

Wavelet scaling

Linear combination of wavelets


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Example

l


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Example

l


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Example

Spectrogram &


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Spectrogram &

Scalogram

Spectrogramis the square magnitude of the STFT,

which provides the distribution of the energy of the signal in the time-frequency

plane.

Similarly, scalogramis the square magnitude of the CWT, and provides the

energy distribution of the signal in the time-scale plane:

2

20

2)()(

)()(

),(),(

t

ftj

xxS

dtetttx

ftSTFTftSPECP

2

2)()(

)()(

1

),(),(

t

xxW

dta

bttx

a

baCWTbaSCALP

E


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Energy

It can be shown that

which implies that the energy of the signal is the same whether you are

in the original time domain or the scale-translation space. Compare this

the Parsevals theorem regarding the Fourier transform.

xEdttxdbdaa

baCWT

2

2

2

)(),(

CWT in


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CWT in

Terms of Frequency

Time-frequency version of the CWT can also be defined, though note that this formis not standard,

where is the mother wavelet, which itself is a bandpass function centered at t=0 in

time andf=f0in frequency. That isf0is the center frequency of the mother wavelet.

The original CWT expression can be obtained simply by using the substitution a=f0/

f and b=

In Matlab, you can obtain the pseudo frequency corresponding to any given scale

by wherefsis the sampling rate and

Tsis the sampling period.

dttf

ftxfffCWT

t

x

0

0 )(,

a

Tf

fa

ff so

s

o

Discretization of


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Discretization of

Time & Scale Parameters

Recall that, if we use orthonormal basis functions as ourmother wavelets, then we can reconstruct the original

signal by

where W(a,b)is the CWT ofx(t)

Q: Can we discretize the mother wavelet a,b(t)in such a

way that a finite number of such discrete wavelets can still

form an orthonormal basis (which in turnallows us to

reconstruct the original signal)? If yes, how often do we

need to sample the translation and scale parameters to be

able to reconstruct the signal?

A: Yes,but it depends on the choice of the wavelet!

)(),,()( , tbaWtx ba

D di G id


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Dyadic Grid

Note that, we do not have to use a uniform sampling rate

for the translation parameters, since we do not need as high

time sampling rate when the scale is high (low frequency).

Lets consider the following sampling grid:

log a

b where ais sampled on a logscale, and b is sampled at a

higher rate when ais small,

that is,

where a0 and b0 are constants,

and j,kare integers.

00

0

bakbaa

j

j

D di G id


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Dyadic Grid

If we use this discretization, we obtain,

A common choice for a0 andb0 are a0 = 2and b0 = 1, which lend

themselves to dyadic sampling grid

Then, the discret(ized) wavelet transform (DWT) pair can be given as

kj

kj

d

dtttxbaW

,

, )()(),(

a

bt

a

tba 1

)(,

002/

0

0

00

0

,

1)(

kbtaa

a

bkat

at

jj

j

j

jkj

DWT

j k

kjkj tdc

tx )(1

)( ,,

Inverse DWT

Zkjktt jjkj ,22)( 2/,

N t th t


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Note that

We have only discretized translation and scale parameters, a and b; time hasnot been discretized yet.

Sampling steps of bdepend on a. This makes sense, since we do not need as

many samples at high scales (low frequencies)

For small a0, say close to 1, and for small b0, say close to zero, we obtain a

very fine sampling grid, in which case, the reconstruction formula is verysimilar to that of CWT

For dense sampling, we need not place heavy restriction on (t) to be ablereconstructx(t), whereas sparse sampling puts heavy restrictions on (t).

It turns out that a0 = 2and b0 = 1 (dyadic / octave sampling)provides a nice

trade-off. For this selection, many orthonormal basis functions (to be used as

mother wavelets) are available.

a b

ba dadbtbaWaC

tx )(),(11

)( ,2

j k

kjkj tdc

tx )(1

)( ,,

Di t W l t T f


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Discrete Wavelet Transform

We have computed a discretized version of the CWT,

however, we still cannot implement the given DWT as itincludes a continuous time signal integrated over all times.

We will later see that the dyadic grid selection will allow

us to compute a truly discrete wavelet transform of a given

discrete time signal.

Documents

Chap 5 - Time Frequency Wavelet