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Understanding Wavelets & Filter Banks: Mallat Pyramid Algorithm & Multiresolution Analysis, Slides of Banking and Finance

The concepts of wavelets and filter banks through the mallat pyramid algorithm. It covers the goal of finding wavelet coefficients, the relationship between different functions, and the multiresolution decomposition and reconstruction equations. The document also discusses the importance of filter banks in implementing multiresolution analysis (mra) equations.

Typology: Slides

2012/2013

Uploaded on 07/29/2013

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Course 18.327 and 1.130
Wavelets and Filter Banks
Mallat pyramid algorithm
φ
φφ
φ
φφ
Pyramid Algorithm for Computing
Wavelet Coefficients
Goal: Given the series expansion for a function fj(t) in Vj
fj(t) = aj[k]φj,k(t)
k
how do we find the series
fj-1(t) = aj-1[k]φj-1,k(t)
k
in Vj-1 and the series
gj-1(t) = bj-1[k]wj-1,k(t)
k
in Wj-1 such that
fj(t) = fj-1(t) + gj-1(t) ?
2
1
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Download Understanding Wavelets & Filter Banks: Mallat Pyramid Algorithm & Multiresolution Analysis and more Slides Banking and Finance in PDF only on Docsity!

Course 18.327 and 1.

Wavelets and Filter Banks

Mallat pyramid algorithm

��� φφφ

φ φφ

Pyramid Algorithm for Computing

Wavelet Coefficients

Goal: Given the series expansion for a function f

j

(t) in V

j

f

j

(t) = � a

j

[k]φ

j,k

(t)

k

how do we find the series

f

j-

(t) = � a

j-

[k]φ

j-1,k

(t)

k

in V

j-

and the series

g

j-

(t) = � b

j-

[k]w

j-1,k

(t)

k

in W

j-

such that

f

j

(t) = f

j-

(t) + g

j-

(t)?

2

3

Example: suppose that φφφφ(t) = box on [0,1]. Then

functions in V

1

can be written either as a combination of

φφφφ(2t)

φφφφ(2t-1)

or as a combination of

φφφφ(t)

φ φφ

φ(t-1)

4

plus a combination of

w(t)

w(t-1)

Easy to see because φφφφ(2t) = ½[φφφφ(t) + w(t)]

φφ φφ(2t –1) = ½[φφφφ(t) - w(t)]

��� φφφ

√√√ ���  ��� φφφ

 

∞∞∞

∞∞∞

 

∞∞∞

∞∞∞

 



Multiresolution decomposition equations

a

j-

[n] = � f(t)φ

j-1,n

(t) dt

= √ 2 � h

0

[] � f(t)φ

j,2n + 

(t) dt

 - ∞

= √√√√ 2 ���� h

0

[] a

j

[2n + ]

So

a

j-

[n] = √√√√ 2 ���� h

0

[k-2n]a

j

[k]

k

→ Convolution with h

0

[-n] followed by downsampling

7

∞∞∞

∞ ∞∞

Similarly

b

j-

[n] = � f(t) w

j-1,n

(t) dt

which leads to

b

j-

[n] = √√√√ 2 ���� h

1

[k – 2n] a

j

[k]

k

8

φφφ

� �� φφφ ��� φφφ ��� φφφ

��� ��� φφφ φφφ

��� ��� φφφ

∞∞∞

∞ ∞∞

∞ ∞∞

∞∞∞

∞ ∞∞

∞∞∞

∞ ∞∞

∞∞∞

∞∞∞

∞∞∞

Multiresolution reconstruction equation

Start with

f

j

(t) = f

j-

(t) + g

j-

(t)

Multiply by φ

j,n

(t) and integrate

∞ ∞

� f

j

(t) φ

j,n

(t) dt = � f

j-

(t)φ

j,n

(t)dt + � g

j-

(t)φ

j,n

(t) dt

  • ∞ - ∞ - ∞

So

a

j

[n] = � a

j-

[k] � φ

j-1,k

(t) φ

j,n

(t) dt +

k

� b

j-

[k] � w

j-1,k

(t) φ

j,n

(t) dt

k

9

10

 φφφφ

j-1,k

(t) φφφφ

j,n

(t) dt = √√√√ 2 ���� h

0

[] ���� φφφφ

j,2k+

(t) φφφφ

j,n

(t) dt

= √√√√ 2 ���� h

0

[] δδδδ[2k +  - n]

= √√√√2 h

0

[n – 2k]

 



  • ∞∞∞∞

∞∞∞∞

  • ∞∞∞∞

∞∞∞∞

 



Similarly

���� w

j-1,k

(t)φφφφ

j,n

(t) dt = √√√√2 h

1

[n –2k]

Result:

a

j

[n] = √√√√ 2 ���� a

j-

[k]h

0

[n - 2k] +

√√√√ 2 ���� b

j-

[k]h

1

[n – 2k]

∞ ∞∞

  • ∞∞∞∞

k

k