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The Story of Wavelets Theory and Engineering Applications. 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. - PowerPoint PPT Presentation
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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
The Story of WaveletsTheory and Engineering Applications
Time – Frequency Resolution
Time – frequency resolution problem with STFTAnalysis window dictates both time and
frequency resolutions, once and for allNarrow window Good time resolutionNarrow band (wide window) Good frequency
resolution When do we need good time resolution, when do we need
good frequency resolution?
Scale & Translation
Translation time shift f(t) f(a.t) a>0
If 0<a<1 dilation, expansion lower frequency If a>1 contraction higher frequency
f(t)f(t/a) a>0 If 0<a<1 contraction low scale (high
frequency) If a>1 dilation, expansion large scale (lower
frequency) Scaling Similar meaning of scale in maps
Large scale: Overall view, long term behaviorSmall scale: Detail view, local behavior
1:44,500,000 1:2,500,000
1:62,5001:375,500
frequencyscale
1
The Mother of All Oscillatory Little Basis Functions
The kernel functions used in Wavelet transform are all obtained from one prototype function, by scaling and translating the prototype function.
This prototype is called the mother wavelet
a
1
)(1
)(, a
bt
atba
Scale parameter
Translation parameter
)()(0,1 tt
Normalization factor to ensure that allwavelets have the same energy
dttdttdttba
22
)0,1(
2
),( )()()(
Continuous Wavelet Transform
dta
bttx
abaWbaCWTx )(
1),(),()(
Normalization factor
Mother wavelet translation
Scaling:Changes the support of the wavelet based on the scale (frequency)
CWT of x(t) at scalea and translation bNote: low scale high frequency
Computation of CWT
)1( NbW
)5( NbW
time
Am
plit
ude
b0
)1( 0bW
bN
time
Am
plit
ude
b0
)5( 0bW
bN
time
Am
plit
ude
b0
)10( 0bW
bN
)10( NbW
time
Am
plit
ude
b0
)25( 0bW
bN
)25( NbW
dta
bttx
abaWbaCWTx )(
1),(),()(
Why Wavelet?
We require that the wavelet functions, at a minimum, satisfy the following:
0)( dtt
dtt 2)(
Wave…
…let
The CWT as a Correlation
Recall that in the L2 space an inner product is defined as
then
Cross correlation:
then
dttgtftgtf )()()(),(
)(),(),( , ttxbaW ba
)(),(
)()()(
tytx
dttytxRxy
)(
)(),(),(
,,
0,
bR
bttxbaW
oax
a
The CWT as a Correlation
Meaning of life:
W(a,b) is the cross correlation of the signal x(t) with the mother 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.
wavelets
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), the
CWT 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
What do Wavelets Look Like???
Mexican Hat Wavelet Haar Wavelet Morlet Wavelet
Constant Q Filtering
A special property of the filters defined by the mother wavelet is that they are –so called – constant 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)
Constant Q
f0 2f0 4f0 8f0
B 2B 4B 8B
B B B B BB
f0 2f0 3f0 4f0 5f0 6f0
STF
TC
WT
B
fQ
Inverse CWT
a b
ba dadbtbaWaC
tx )(),(11
)( ,2
dC
)(
provided that
0)( dtt
Properties of Continuous Wavelet Transform
Linearity Translation Scaling Wavelet shifting Wavelet scaling Linear combination of wavelets
Example
Example
Example
