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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*)(
)()(),(
The Story of Wavelets
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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
The Story of Wavelets
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.