Orthogonal Wavelets and Homework February 23 Properties of - - PowerPoint PPT Presentation

Orthogonal Wavelets and Homework

February 23

Properties of multiresolution subspaces Vj

Multiresolution Subspace Construction

• An ordinary analog signal may have components in all of the

above subspaces: ≠ 0 for all k

Wavelet subspaces

• Wo = span{ ψ(t-l), integer l },...
• Wj does not contain Wk, j>k (but Vj does contain

Vk)

• It is desirable to have Vj to be orthogonal to

Wj and

Multiresolution Subspace Construction

Wavelet Equation (Mallat)

• Wo Ϲ V1 =>

ψ(t)=√2 ∑k d[k]φ(2t-k)

• d[k]=√2 < ψ(t), φ(2t-k) >, ψ(t)=2∑k g[k]φ(2t-k)
• g[k]= √2 d[k] is a discrete-time half-band high-pass

filter

• Example: Haar wavelet

ψ(t) = φ(2t) – φ(2t-1) => d[0]=√2/2 , d[1]= -√2/2

• g and d are simple discrete-time high-pass filters

Scaling Equation

• Subspace Vo is a subset of V1 =>

φ(t)=2 ∑k h[k]φ(2t-k) where h[k]=√2 < φ(t), φ(2t-k) >

• h[k]= √2 c[k] is a half-band discrete-time

low-pass filter with passband: [0,π/2]

• In wavelet equation g[k] is a high-pass filter

with passband [π/2,π]

Fourier transforms of wavelet and scaling equations

Orthogonality Condition:

H(eiw), G(eiw) are the discrete-time Fourier transforms of h[k] & g[k], respectively.

Orthogonality of scaling and wavelet functions

In the j=0-th scale (Vo and Wo) we have the

• rthogonality conditions (1) and (2)

Conditions (2) and (3) is true for all multiresolution

• scales. Condition (1) is true only within a given

scale.

Orthogonality conditions lead to some interesting results

• Condition (1) together with the scaling (dilation)

equation

(4)

• Use the Fourier Transform of the scaling

equation in (4) and

• At this point we remember the symmetric half-

band filter p(n) with the property that P(z) + P (-z) =2 or P(ω) + P(ω+π)=2

From PR Filterbank to

• rthogonal wavelets
• Since p(n)=p(-n), if P(zi)=0 then P(1/zi)=0.

Therefore P(z) = C(z) C(z-1) where C(z) contains the zeros inside the unit circle and C(z-1) contains the zeros outside the unit circle. P(ω) = C(ω) C(-ω) = |C(ω)|2 (p(n) is real) P(ω+π) = |C(ω+π)|2

• Therefore we have |C(ω)|2 + |C(ω+π)|2 =2 and

|H(ω)|2 + |H(ω+π)|2 =1

Multiresolution Framework Construction

• Factorize P(z) into C(z) and C(z-1)
• The low-pass filter H(z)= √2 C(z)
• Obtain the high-pass filter G(z) using

alternating flip

• Construct the Fourier transform of scaling

function which converges if H(ω=π) =0 or Σ h(n)=1

• Wavelet: W(ω) = G( ω/2) Φ( ω/2)

Theorem (Daubechies)

• It is a necessary condition. It is not a

sufficient condition for ω=0

Iterative scaling function computation in time-domain

• , if H(π) =0 or Σ h(n)=1

Daubechies 4 (D4) wavelet and the corresponding scaling function

• D4 and D12 plots:
• Wavelets and scaling functions get smoother as the

number of filter coefficients increase

• D2 is Haar wavelet

Homework

1) a) Given x(t)=1 for 0< t <3.5. Project x(t) onto the subspace Vo which is constructed from φ(t)=1, for 0< t <1. b) Project x(t) onto the subspace V1 constructed from φ(2t)=1, for 0< t <1/2. Which projection produces a better approximation to the original signal? 2) a)Find the orthogonal filterbank coefficients (both lowpass and highpass) constructed from the half- band filter p[n]={-1/16,0,9/16,p(0)=1,9/16,0,-1/16}. Use alternating flip to obtain the high-pass filter.

Homework (cont'd)

3) Show that the filter coefficients satisfy the time-domain orthogonality conditions; where c is the low-pass, d is the high-pass filter and summations are with respect to n.

Some important relations