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 δn−ℓ+2m=<φt−n,φt−ℓ−2m> 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 p2m=δm p 2 m δ m . In the zz-domain, this yields the pair of conditions 1=1/2∑p=01Pz1/2ⅇⅈ2π2p=1/2Pz1/2+1/2P-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=HzHz-1+H-zH-z-1 ⇔ 2 H z H z -1 H z H z -1 ⇔ 2=|Hⅇⅈω|2+|Hⅇⅈπ−ω|2 ⇔ 2 H ω 2 H ω 2 where the last property invokes the fact that hn∈ℝ h n and that real-valued impulse responses yield conjugate-symmetric DTFTs. Thus we find that hn 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 hn 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 δn−ℓ+2m=< 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 gn 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 gn g n replacing hn 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=GzGz-1+G-zG-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 δn−ℓ+2m=< 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 gn*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=Gz1/2Hz-1/2+G-z1/2H-z-1/2 ⇔ 0 G z 12 H z 12 G z 12 H z 12 ⇔ 0=GzHz-1+G-zH-z-1 ⇔ 0 G z H z -1 G z H z -1 The choice satisfies our condition, since GzHz-1+G-zH-z-1=±z-PH-z-1Hz-1∓z-PHz-1H-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 Gz G z and Hz 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 ∀odd P:Gz=±z-PH-z-1 odd P G z ± z P H z -1 2=HzHz-1+H-zH-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=GzGz-1+G-zG-z-1 2 G z G z -1 G z G z -1 automatically.

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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 GMT-5.