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Conditions on h[n] and g[n]

Module by: Phil Schniter. E-mail the author

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

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

δm= φ 1 , 0 t, φ 1 , m t=nhnφtn,hφt2m=nhnh(φtn,φt2m) δ m φ 1 , 0 t φ 1 , m t n h n φ t n h φ t 2 m n h n h φ t n φ t 2 m
(1)
where δn+2m=φtn,φt2m δ n 2 m φ t n φ t 2 m
δm=n=hnhn2m δ m n h n h n 2 m
(2)

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

pm=hm*hm=nhnhnm p m h m h m n h n h n m
(3)
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

Power-Symmetry Property

Pz=HzHz-1 P z H z H z -1
(4)
1=1/2p=01Pz1/2ej2π2p=1/2Pz1/2+1/2Pz1/2 1 12 p 0 1 P z 12 2 2 p 12 P z 12 12 P z 12 Putting these together,
2=Hz1/2Hz1/2+Hz1/2Hz1/2 2 H z 12 H z 12 H z 12 H z 12
(5)
2=HzHz-1+HzHz-1 2 H z H z -1 H z H z -1 2=|Hejω|2+|Hej(πω)|2 2 H ω 2 H ω 2 where the last property invokes the fact that hnR 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 :

δm= φ 2 , 0 t, φ 2 , m t=nhn φ 1 , n t,h φ 1 , + 2 m t=nhnh( φ 1 , n t, φ 1 , + 2 m t)=n=hnhn2m δ m φ 2 , 0 t φ 2 , m t n h n φ 1 , n t h φ 1 , + 2 m t n h n h φ 1 , n t φ 1 , + 2 m t n h n h n 2 m
(6)
where δn+2m= φ 1 , n t, φ 1 , + 2 m t δ n 2 m φ 1 , n t φ 1 , + 2 m t . Using induction, we conclude that the previous condition will be necessary and sufficient for orthonormality of φ k , n t nZ φ k , n t n for all kZ k .

To find conditions on gn g n ensuring that the set ψ k , n t nZ 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

δm=n=gngn2m δ m n g n g n 2 m
(7)
2=GzGz-1+GzGz-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 kZ k . Note that, for any kZ k , W k V k W k V k is guaranteed by ψ k , n t nZ φ k , n t nZ 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=ngn φ k , n t,h φ k , + 2 m t=ngnh( φ k , n t, φ k , + 2 m t)=ngnhn2m 0 ψ k + 1 , 0 t φ k + 1 , m t n g n φ k , n t h φ k , + 2 m t n g n h φ k , n t φ k , + 2 m t n g n h n 2 m
(8)
for all mm where δn+2m= φ k , n t, φ k , + 2 m t δ n 2 m φ k , n t φ k , + 2 m t . In other words, a 2-downsampled version of gn*hn g n h n must consist only of zeros. This necessary and sufficient condition can be restated in the frequency domain as
0=1/2p=01Gz1/2e(j2π2p)Hz1/2ej2π2p 0 12 p 0 1 G z 12 2 2 p H z 12 2 2 p
(9)
0=Gz1/2Hz1/2+Gz1/2Hz1/2 0 G z 12 H z 12 G z 12 H z 12 0=GzHz-1+GzHz-1 0 G z H z -1 G z H z -1 The choice
Gz=±zPHz-1   odd P G z ± z P H z -1
(10)
satisfies our condition, since GzHz-1+GzHz-1=±zPHz-1Hz-1zPHz-1Hz-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
gn=±-1nhPn   . odd P g n ± -1 n h P n .
(11)
Recall that this property was satisfied by the analysis filters in an orthogonal perfect reconstruction FIR filterbank.

Note that the two conditions Gz=±zPHz-1   odd P G z ± z P H z -1 2=HzHz-1+HzHz-1 2 H z H z -1 H z H z -1 are sufficient to ensure that both φ k , n t nZ φ k , n t n and ψ k , n t nZ 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+GzGz-1 2 G z G z -1 G z G z -1 automatically.

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