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