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Regularity Conditions, Compact Support, and Daubechies' Wavelets

Module by: Phil Schniter

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.
up_samplers.png
Figure 1
The properties of the ii-stage cascaded lowpass filter
,i1: H ( i ) z=k=0i-1Hz2k 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 limi H ( i ) ω i H ( i ) ω converges to a response in 2 2 ? In fact, there are some rather strict conditions on Hω 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 Hω 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=Hz2i-1 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=khk h ( i 1 ) n-2i-1k h ( i ) n k k h k h ( i 1 ) n 2 i 1 k Now define the function φ ( i ) t=2i2n 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 ab a b : [ a , b ) t=1iftab0iftab [ a , b ) t 1 t a b 0 t a b The definition of φ ( i ) t φ ( i ) t implies
t,tn2in+12i: h ( i ) n=2-i2 φ ( i ) t t t n 2 i n 1 2 i h ( i ) n 2 i 2 φ ( i ) t (2)
t,tn2in+12i: h ( i 1 ) n-2i-1k=2-i-12 φ ( i 1 ) 2t-k 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=2khk φ ( i 1 ) 2t-k . 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 limi 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 Hω H ω can be factored as
P,P1:Hω=21+ω2PRω P P 1 H ω 2 1 ω 2 P R ω (5)
for Rω R ω such that
sup ω |Rω|<2P-1 sup ω R ω 2 P 1 (6)
Here PP denotes the number of zeros that Hω 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:
,1: sup ω |Rω|<2P-1- 1 sup ω R ω 2 P 1 (7)
then limi 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-1N-1+1 N i 2 i 1 N 1 1 . By construction, then, φ ( i ) t φ ( i ) t will be zero outside the interval 02i-1N-1+12i 0 2 i 1 N 1 1 2 i . Assuming that the regularity conditions are satisfied so that limi φ ( i ) t=φt i φ ( i ) t φ t , it follows that φt φ t must be zero outside the interval 0N-1 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 0N-1 0 N 1 and gn g n is length NN, ψt ψ t will also be compactly supported on the interval 0N-1 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.
daubechies_a.png
Subfigure 2.1
daubechies_b.png
Subfigure 2.2
Figure 2
daubechies_c.png
Subfigure 3.1
Daubechies_d.png
Subfigure 3.2
Figure 3
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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