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Course 18.327 and 1.130
Wavelets and Filter Banks
Multiresolution Analysis (MRA):
Requirements for MRA;
Nested Spaces and
Complementary Spaces;
Scaling Functions and Wavelets
2
Scaling Functions and Wavelets
Continuous time:
φ
(t) Box function
1
0
1
t
1
0
1
1
0
t
t
1/2
1/2
φ
(2t) Scaling
φ
(2t  1)
Scaling +
Shifting
1
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φ
φ
φ
∑
φ
φ
φ
φ
≤
<
φ
φ
∫
φ
∑
∫
φ
∑
∫
φ τ
τ
∑
∫
φ
≠
For this example:
φ
(t) =
(2t) +
(2t – 1)
More generally:
N
Refinement equation
(t) = 2
∑
h
0
[k]
φ
(2t – k)
or
k=0
Twoscale difference
equation
φ
(t) is called a scaling function
The refinement equation couples the representations
of a continuoustime function at two time scales. The
continuoustime function is determined by a discrete
time filter, h
0
[n]! For the above (Haar) example:
h
0
[0] = h
0
[1] = ½
(a lowpass filter)
3
Note: (i) Solution to refinement equation may not
always exist. If it does…
(ii)
(t) has compact support i.e.
(t) = 0 outside 0
t
N
(comes from the FIR filter, h
0
[n])
(iii)
(t) often has no closed form solution.
(iv)
(t) is unlikely to be smooth.
Constraint on h
0
[n]:
N
(t)dt = 2
∑
h
0
[k]
(2t – k)dt
k=0
N
= 2
h
0
[k] • ½
φ
(
τ
)d
k=0
So
N
∑
h
0
[k] = 1
Assumes
(t)dt
≠
0
k=0
4
2
∑
φ
5
Now consider:
1
0
1/2
1
t
1
0
1/2
t
w(t)
Square wave
of finite length 
Haar wavelet
1
0
1/2
t
φ
(2t)
Scaled

φ
(2t – 1)
Scaled + shifted
+ sign flipped
w(t)
φ
(2t)
φ
(2t – 1)
=

More generally:
N
w(t) = 2
∑
h
1
[k]
(2t – k)
Wavelet equation
k=0
For the Haar wavelet example:
h
1
[0] = ½
h
1
[1] = ½
(a highpass filter)
6
3
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Some observations for Haar scaling function and wavelet
1. Orthogonality of integer shifts (translates):
1
0
2
1
0
t
t
1
1
φ
(t)
φ
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This note was uploaded on 12/04/2011 for the course ESD 18.327 taught by Professor Gilbertstrang during the Spring '03 term at MIT.
 Spring '03
 GilbertStrang

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