Conditions on h[n] and g[n] 2.2 2003/01/17 2005/09/20 19:59:05.231 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu scaling equation symmetry wavelet Here we derive sufficient conditions on the coefficients used in the scaling equation and wavelet scaling equation that ensure, for every k , that the sets φ k , n t n and 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 to be orthonormal at all k, we certainly need orthonormality when k 1 . This is equivalent to δ m φ 1 , 0 t φ 1 , m t n h n φ t n ℓ h ℓ φ t ℓ 2 m n h n ℓ h ℓ φ t n φ t ℓ 2 m where δ n ℓ 2 m φ t n φ t ℓ 2 m δ m n h n h n 2 m There is an interesting frequency-domain interpretation of the previous condition. If we define p m h m h m n h n h n m then we see that our condition is equivalent to p 2 m δ m . In the z-domain, this yields the pair of conditions Power-Symmetry Property P z H z H z -1 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 12 H z 12 H z 12 H z 12 ⇔ 2 H z H z -1 H z H z -1 ⇔ 2 H ω 2 H ω 2 where the last property invokes the fact that h n and that real-valued impulse responses yield conjugate-symmetric DTFTs. Thus we find that 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 , we have now derived a condition on h n which is necessary and sufficient for orthonormality at level k 1 . Yet the same condition is necessary and sufficient for orthonormality at level k 2 : δ m φ 2 , 0 t φ 2 , m t n h n φ 1 , n t ℓ h ℓ φ 1 , ℓ + 2 m t n h n ℓ h ℓ φ 1 , n t φ 1 , ℓ + 2 m t n h n h n 2 m where 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 for all k . To find conditions on g n ensuring that the set ψ k , n t n is orthonormal at every k, we can repeat the steps above but with g n replacing h n , ψ k , n t replacing φ k , n t , and the wavelet-scaling equation replacing the scaling equation. This yields δ m n g n g n 2 m ⇔ 2 G z G z -1 G z G z -1 Next derive a condition which guarantees that ⊥ W k V k , as required by our definition W k V k ⊥ , for all k . Note that, for any k , ⊥ W k V k is guaranteed by ⊥ ψ k , n t n φ k , n t n which is equivalent to 0 ψ k + 1 , 0 t φ k + 1 , m t n g n φ k , n t ℓ h ℓ φ k , ℓ + 2 m t n g n ℓ h ℓ φ k , n t φ k , ℓ + 2 m t n g n h n 2 m for all m where 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 must consist only of zeros. This necessary and sufficient condition can be restated in the frequency domain as 0 12 p 0 1 G z 12 2 2 p H z 12 2 2 p ⇔ 0 G z 12 H z 12 G z 12 H z 12 ⇔ 0 G z H z -1 G z H z -1 The choice odd P G z ± z P H z -1 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 In the time domain, the condition on G z and H z can be expressed odd P g n ± -1 n h P n . Recall that this property was satisfied by the analysis filters in an orthogonal perfect reconstruction FIR filterbank. Note that the two conditions odd P G z ± z P 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 and ψ k , n t n are orthonormal for all k and that ⊥ W k V k for all k, since they satisfy the condition 2 G z G z -1 G z G z -1 automatically.