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Regularity Conditions, Compact Support, and Daubechies' Wavelets

Module by: Phil Schniter. E-mail the author

Summary: (Blank Abstract)

Here we give a quick description of what is probably the most popular family of filter coefficients hn h n and gn g n — those proposed by Daubechies.

Recall the iterated synthesis filterbank. Applying the Noble identities, we can move the up-samplers before the filters, as illustrated in Figure 1.

Figure 1
Figure 1 (up_samplers.png)

The properties of the ii-stage cascaded lowpass filter

H ( i ) z=k=0i1Hz2k  ,   i1    i 1 H ( i ) z k 0 i 1 H z 2 k
(1)
in the limit i i give an important characterization of the wavelet system. But how do we know that limit  i H ( i ) ejω i H ( i ) ω converges to a response in 2 2 ? In fact, there are some rather strict conditions on Hejω H ω that must be satisfied for this convergence to occur. Without such convergence, we might have a finite-stage perfect reconstruction filterbank, but we will not have a countable wavelet basis for 2 2 . Below we present some "regularity conditions" on Hejω H ω that ensure convergence of the iterated synthesis lowpass filter.

note:

The convergence of the lowpass filter implies convergence of all other filters in the bank.

Let us denote the impulse response of H ( i ) z H ( i ) z by h ( i ) n h ( i ) n . Writing H ( i ) z=Hz2i1 H ( i 1 ) z H ( i ) z H z 2 i 1 H ( i 1 ) z in the time domain, we have h ( i ) n=kkhk h ( i 1 ) n2i1k h ( i ) n k k h k h ( i 1 ) n 2 i 1 k Now define the function φ ( i ) t=2i2nn h ( i ) n [ n / 2 i , n + 1 / 2 i ) t φ ( i ) t 2 i 2 n n h ( i ) n [ n / 2 i , n + 1 / 2 i ) t where [ a , b ) t [ a , b ) t denotes the indicator function over the interval a b a b : [ a , b ) t={1  if  t a b 0  if  t a b [ a , b ) t 1 t a b 0 t a b The definition of φ ( i ) t φ ( i ) t implies

h ( i ) n=2i2 φ ( i ) t  ,   t n2i n+12i    t t n 2 i n 1 2 i h ( i ) n 2 i 2 φ ( i ) t
(2)
h ( i 1 ) n2i1k=2i12 φ ( i 1 ) 2tk  ,   t n2i n+12i    t t n 2 i n 1 2 i h ( i 1 ) n 2 i 1 k 2 i 1 2 φ ( i 1 ) 2 t k
(3)
and plugging the two previous expressions into the equation for h ( i ) n h ( i ) n yields
φ ( i ) t=2kkhk φ ( i 1 ) 2tk . φ ( i ) t 2 k k h k φ ( i 1 ) 2 t k .
(4)
Thus, if φ ( i ) t φ ( i ) t converges pointwise to a continuous function, then it must satisfy the scaling equation, so that limit  i φ ( i ) t=φt i φ ( i ) t φ t . Daubechies showed that, for pointwise convergence of φ ( i ) t φ ( i ) t to a continuous function in 2 2 , it is sufficient that Hejω H ω can be factored as
Hejω=21+ejω2PRejω  ,   P1    P P 1 H ω 2 1 ω 2 P R ω
(5)
for Rejω R ω such that
sup ω |Rejω|<2P1 sup ω R ω 2 P 1
(6)
Here PP denotes the number of zeros that Hejω H ω has at ω=π ω . Such conditions are called regularity conditions because they ensure the regularity, or smoothness of φt φ t . In fact, if we make the previous condition stronger:
sup ω |Rejω|<2P1  ,   1    1 sup ω R ω 2 P 1
(7)
then limit  i φ ( i ) t=φt i φ ( i ) t φ t for φt φ t that is -times continuously differentiable.

There is an interesting and important by-product of the preceding analysis. If hn h n is a causal length-NN filter, it can be shown that h ( i ) n h ( i ) n is causal with length N i =2i(N1)+1 N i 2 i 1 N 1 1 . By construction, then, φ ( i ) t φ ( i ) t will be zero outside the interval 0 2i(N1)+12i 0 2 i 1 N 1 1 2 i . Assuming that the regularity conditions are satisfied so that limit  i φ ( i ) t=φt i φ ( i ) t φ t , it follows that φt φ t must be zero outside the interval 0 N1 0 N 1 . In this case we say that φt φ t has compact support. Finally, the wavelet scaling equation implies that, when φt φ t is compactly supported on 0 N1 0 N 1 and gn g n is length NN, ψt ψ t will also be compactly supported on the interval 0 N1 0 N 1 .

Daubechies constructed a family of Hz H z with impulse response lengths N46810 N 4 6 8 10 which satisfy the regularity conditions. Moreover, her filters have the maximum possible number of zeros at ω=π ω , and thus are maximally regular (i.e., they yield the smoothest possible φt φ t for a given support interval). It turns out that these filters are the maximally flat filters derived by Herrmann long before filterbanks and wavelets were in vogue. In Figure 2 and Figure 3 we show φt φ t , ΦΩ Φ Ω , ψt ψ t , and ΨΩ Ψ Ω for various members of the Daubechies' wavelet system.

See Vetterli and Kovacivić for a more complete discussion of these matters.

Figure 2
(a)
Figure 2(a) (daubechies_a.png)
(b)
Figure 2(b) (daubechies_b.png)
Figure 3
(a)
Figure 3(a) (daubechies_c.png)
(b)
Figure 3(b) (Daubechies_d.png)

References

  1. I. Daubechies. (1988, Nov). Orthonormal bases of compactly supported wavelets. Commun. on Pure and Applied Math, 41, 909-996.
  2. O. Herrmann. (1971). On the approximation problem in nonrecursive digital filter design. IEEE Trans. on Circuit Theory, 18, 411-413.
  3. M. Vetterli and J. Kovacivić. (1995). Wavelets and Subband Coding. Englewood Cliffs, NJ: Prentice Hall.

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