Here we derive sufficient conditions on the coefficients used in the scaling equation and wavelet scaling equation that ensure, for every k∈Z k , that the sets φ k , n ⁢t n∈Z φ k , n t n and ψ k , n ⁢t n∈Z ψ k , n t n have the orthonormality properties described in The Scaling Equation and The Wavelet Scaling Equation.

For φ k , n ⁢t n∈Z φ k , n t n to be orthonormal at all kk, we certainly need orthonormality when k=1 k 1 . This is equivalent to

where δ⁢n−ℓ+2⁢m=φ⁢t−n · φ⁢t−ℓ−2⁢m δ n ℓ 2 m φ t n φ t ℓ 2 m

There is an interesting frequency-domain interpretation of the previous condition. If we define

then we see that our condition is equivalent to p⁢2⁢m=δ⁢m p 2 m δ m . In the zz-domain, this yields the pair of conditions 1=1/2⁢∑p=01P⁢z1/2⁢ej⁢2⁢π2⁢p=1/2⁢P⁢z1/2+1/2⁢P⁢−z1/2 1 12 p 0 1 P z 12 2 2 p 12 P z 12 12 P z 12 Putting these together, ⇔ 2=H⁢z⁢H⁢z-1+H⁢−z⁢H⁢−z-1 ⇔ 2 H z H z -1 H z H z -1 ⇔ 2=|H⁢ej⁢ω|2+|H⁢ej⁢(π−ω)|2 ⇔ 2 H ω 2 H ω 2 where the last property invokes the fact that h⁢n∈R h n and that real-valued impulse responses yield conjugate-symmetric DTFTs. Thus we find that h⁢n h n are the impulse response coefficients of a power-symmetric filter. Recall that this property was also shared by the analysis filters in an orthogonal perfect-reconstruction FIR filterbank.

Given orthonormality at level k=0 k 0 , we have now derived a condition on h⁢n h n which is necessary and sufficient for orthonormality at level k=1 k 1 . Yet the same condition is necessary and sufficient for orthonormality at level k=2 k 2 :

where δ⁢n−ℓ+2⁢m= φ 1 , n ⁢t ·  φ 1 , ℓ + 2 m ⁢t δ n ℓ 2 m φ 1 , n t φ 1 , ℓ + 2 m t . Using induction, we conclude that the previous condition will be necessary and sufficient for orthonormality of φ k , n ⁢t n∈Z φ k , n t n for all k∈Z k .

To find conditions on g⁢n g n ensuring that the set ψ k , n ⁢t n∈Z ψ k , n t n is orthonormal at every kk, we can repeat the steps above but with g⁢n g n replacing h⁢n h n , ψ k , n ⁢t ψ k , n t replacing φ k , n ⁢t φ k , n t , and the wavelet-scaling equation replacing the scaling equation. This yields

⇔ 2=G⁢z⁢G⁢z-1+G⁢−z⁢G⁢−z-1 ⇔ 2 G z G z -1 G z G z -1 Next derive a condition which guarantees that W k ⊥ V k ⊥ W k V k , as required by our definition W k = V k ⊥ W k V k ⊥ , for all k∈Z k . Note that, for any k∈Z k , W k ⊥ V k ⊥ W k V k is guaranteed by ψ k , n ⁢t n∈Z ⊥ φ k , n ⁢t n∈Z ⊥ ψ k , n t n φ k , n t n which is equivalent to for all mm where δ⁢n−ℓ+2⁢m= φ k , n ⁢t ·  φ k , ℓ + 2 m ⁢t δ n ℓ 2 m φ k , n t φ k , ℓ + 2 m t . In other words, a 2-downsampled version of g⁢n*h⁢−n g n h n must consist only of zeros. This necessary and sufficient condition can be restated in the frequency domain as ⇔ 0=G⁢z1/2⁢H⁢z−1/2+G⁢−z1/2⁢H⁢−z−1/2 ⇔ 0 G z 12 H z 12 G z 12 H z 12 ⇔ 0=G⁢z⁢H⁢z-1+G⁢−z⁢H⁢−z-1 ⇔ 0 G z H z -1 G z H z -1 The choice satisfies our condition, since G⁢z⁢H⁢z-1+G⁢−z⁢H⁢−z-1=±⁢z−P⁢H⁢−z-1⁢H⁢z-1∓z−P⁢H⁢z-1⁢H⁢−z-1=0 G z H z -1 G z H z -1 ∓ ± z P H z -1 H z -1 z P H z -1 H z -1 0 In the time domain, the condition on G⁢z G z and H⁢z H z can be expressed Recall that this property was satisfied by the analysis filters in an orthogonal perfect reconstruction FIR filterbank.

Note that the two conditions G⁢z=±⁢z−P⁢H⁢−z-1   odd P G z ± z P H z -1 2=H⁢z⁢H⁢z-1+H⁢−z⁢H⁢−z-1 2 H z H z -1 H z H z -1 are sufficient to ensure that both φ k , n ⁢t n∈Z φ k , n t n and ψ k , n ⁢t n∈Z ψ k , n t n are orthonormal for all kk and that W k ⊥ V k ⊥ W k V k for all kk, since they satisfy the condition 2=G⁢z⁢G⁢z-1+G⁢−z⁢G⁢−z-1 2 G z G z -1 G z G z -1 automatically.

Content actions

Footer

This work is licensed by Phil Schniter under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.

Last edited by Charlet Reedstrom on Sep 20, 2005 7:59 pm -0500.