Orthogonal Waveletsand

Homework

February 23

Properties of multiresolution subspaces V

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Multiresolution Subspace Construction

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● An ordinary analog signal may have components in all of the above subspaces: ≠ 0 for all k

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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

OrthogonalityCondition:

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 orthogonality 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 orthogonal 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)

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● It is a necessary condition. It is not a sufficient condition for ω=0

Iterative scaling function computation in time-domain

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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

Homework1) 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