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

Orthogonal Wavelets and Homework February 23

Properties of multiresolution subspaces V j ● ● ● ● ● ● ● ● ●

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 },... ● W j does not contain W k, j>k ( but V j does contain V k ) ● It is desirable to have V j to be orthogonal to W j and

Multiresolution Subspace Construction

Wavelet Equation (Mallat) ● W o Ϲ V 1 => 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 V 1 => φ(t)=2 ∑ k h [ k ] φ(2t-k) w here 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( e iw ), G( e iw ) 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(z i )=0 then P(1/z i )=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