Lecture 49
Lecture 49 –– In search of scaling In search of scaling coefficients
coefficients coefficients coefficients
Dr Aditya Abhyankar Dr. Aditya Abhyankar
Framework Framework Framework Framework
y Gave us power to move up or down the p p ladder
y We can now indeed zoom-in or zoom-e ca ow ee oo o oo out of any part of the signal
y This makes the entire analysis ‘scalable’!!
y This makes the entire analysis scalable !!
y Scalability stems out of multi-resolution framework !
framework !
Framework Framework Framework Framework
y Leads us to two questionsq
1) How do we go about selecting the mother wavelet and scale of analysis?ot e wave et a sca e o a a ys s?
2) What is the procedure to calculate scaling and wavelet coefficients?
scaling and wavelet coefficients?
How Fourier Works!!
How Fourier Works!!
How Fourier Works!!
How Fourier Works!!
[ ] 0 6 i (2 3 ) 0 8 (2 8 ) [ ] 0.6 sin(2 3 ) 0.8 cos(2 8 )
y n = π n + π n
How Fourier Works!!
How Fourier Works!!
How Fourier Works!!
How Fourier Works!!
cos(2 8 )π n
[ ] 0.6 sin(2 3 ) 0.8 cos(2 8 )
y n = π n + π n
How Fourier Works!!
How Fourier Works!!
How Fourier Works!!
How Fourier Works!!
[ ], cos(2 8 )
y n π n
< >
cos(2 8 )π n
[ ] 0 6 i (2 3 ) 0 8 (2 8 )
2.0004e+003
[ ] 0.6 sin(2 3 ) 0.8 cos(2 8 )
y n = π n + π n
Application Application Application Application
y Detecting hidden jump discontinuityg j p y
y Consider function
, 0 1
t t 2
⎧ ≤ <
⎪⎪ 2
( ) 1
1, 1
2 g t
t t
= ⎨⎪
⎪ − ≤ <
⎪⎩
y Clear jump at t=0.5
, 2
⎪⎩
j p
Application Application Application Application
y Detecting hidden jump discontinuityg j p y
y Let’s integrate
⎧ 2 1
2 , 0 2
( ) ( )
t t
h t g t dt
⎧ ≤ <
=
∫
= ⎨⎪⎪ 2 ( ) ( )1 1, 1
2 2 2
h t g t dt
t t t
= = ⎨
⎪ − + ≤ <
⎪⎩
∫
y Cusp jump at t=0.5
⎩
p j p
Application Application Application Application
y Detecting hidden jump discontinuityg j p y
y Let’s integrate again
⎧ 3 1
6 , 0 2
( ) ( )
t t
f t h t dt
⎧ ≤ <
=
∫
= ⎨⎪⎪ 3 2 ( ) ( )1 1, 1
6 2 2 8 2
f t h t dt
t t t
t
= = ⎨
⎪ − + − ≤ <
⎪⎩
∫
y Appears smooth to eye
⎩
pp y
Coefficients Coefficients Coefficients Coefficients
y Who gives us coefficients of scaling g g equation?
y Haaraa
In search of coefficients In search of coefficients In search of coefficients In search of coefficients
y We can thinks of using three guiding g g g theorems !
In search of coefficients In search of coefficients In search of coefficients In search of coefficients
y We can thinks of using three guiding g g g theorems !
y Theorem 1:eo e :
∑
For the scaling equation ( ) 2 (2 ), with non-vanishing coefficients { } only for ,
k k M k k N
x h x k
h N k M
φ = φ −
≤ ≤
∑
g { } y ,
its ( ) is with a compact support contained in interval [ , ]
k k N
x N M
φ
=
In search of coefficients In search of coefficients In search of coefficients In search of coefficients
y We can thinks of using three guiding g g g theorems !
y Theorem 2:eo e :
If the scaling function ( ) has compact support on φ x 0 -1 and if, { ( - )} are linearly independent, then n ( ) 0, for n<0 and -1.
x N x k
h h n n N
φ
≤ ≤
= = >
Hence N is the length of the sequence.
In search of coefficients In search of coefficients In search of coefficients In search of coefficients
y We can thinks of using three guiding theorems !
theorems !
y Theorem 3:
If h li ffi i { }h If the scaling coefficieents { }
satisfy the condition for existence and hk
orthogonality of ( ), then
( ) k 2 (2 )
x
x g x k
φ
ϕ = ∑ φ −
where, ( 1)
k
k
k N k
g h −
∞
= ± −
, -
and, ( - ) ( - )∞ ϕ x l φ x k dx δl k 0,l k
∞
= = ≠
∫
Properties of scaling coefficients Properties of scaling coefficients Properties of scaling coefficients Properties of scaling coefficients
Properties of scaling coefficients Properties of scaling coefficients Properties of scaling coefficients Properties of scaling coefficients
Thank You!
Thank You!
Thank You!
Thank You!
Questions ??
