The Story of Wavelets Theory and Engineering Applications
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Transcript The Story of Wavelets Theory and Engineering Applications
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
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 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 behavior
Small scale: Detail view, local behavior
1:44,500,000
1:375,500
scale
1
frequency
1:2,500,000
1:62,500
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
1
t b
a,b (t )
(
)
a
a
Translation
parameter
Scale parameter
1
a
Normalization factor to ensure that all
wavelets have the same energy
(a,b) (t ) dt (1,0) (t ) dt (t ) dt
2
2
2
1,0 (t ) (t )
Continuous Wavelet Transform
Mother wavelet
1
( )
CWTx (a, b) W (a, b)
translation
t b
x(t )
dt
a
a
Normalization factor
CWT of x(t) at scale
a and translation b
Note: low scale high frequency
Scaling:
Changes the support of
the wavelet based on
the scale (frequency)
b0
Amplitude
Amplitude
W (1 b0 )
W (5 b0 )
b0
bN
W (1 bN )
time
W (5 bN )
bN
W (10 b0 )
W (10 bN )
b0
Amplitude
Amplitude
Computation of CWT
time
bN
W (25 b0 )
W (25 bN )
b0
bN
1
t b
CWTx( ) (a, b) W (a, b)
x
(
t
)
dt
a
a
time
time
Why Wavelet?
We require that the wavelet functions, at a minimum,
satisfy the following:
(t )dt 0
Wave…
(t ) dt
2
…let
The CWT as a Correlation
Recall that in the L2 space an inner product is defined as
f (t ), g (t ) f (t ) g (t )dt
then
W (a, b) x(t ), a,b (t )
Cross correlation:
Rxy ( ) x(t ) y (t )dt
x(t ), y (t )
then
W (a, b) x(t ), a,0 (t b)
Rx, a ,o (b)
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.
Filtering Interpretation of
Wavelet Transform
Recall that for a given system h[n], y[n]=x[n]*h[n]
y (t ) x(t ) * h(t )
x( )h(t )d
Observe that W (a, b) x(b) * a,0 (b)
Interpretation:For any given scale a (frequency ~ 1/a), the
CWT W(a,b) is the output of the filter with the impulse
response a,0 (b) to the input x(b), i.e., we have a
continuum of filters, parameterized by the scale factor a.
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:
center frequency
bandwidth
w (rad/s)
We observe that the filters defined by the mother wavelet
increase their bandwidth, as the scale is reduced (center
frequency is increased)
Constant Q
B
B
B
B
B
2f0
3f0
4f0
5f0
6f0
STFT
B
f0
2B
4B
8B
CWT
B
f0
2f0
4f0
8f0
f
Q
B
Inverse CWT
1
x(t )
C
a b a
C
provided that
1
2
W (a, b) a,b (t )dadb
( )
d
(t )dt 0
Properties of
Continuous Wavelet Transform
Linearity
Translation
Scaling
Wavelet shifting
Wavelet scaling
Linear combination of wavelets
Example
Example
Example