Regularity Conditions, Compact Support, and Daubechies' Wavelets 2.8 2002/02/05 2005/10/05 15:12:07.632 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu Daubechies Wavelets (Blank Abstract) Here we give a quick description of what is probably the most popular family of filter coefficients h n and 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 .

The properties of the i-stage cascaded lowpass filter i 1 H ( i ) z k 0 i 1 H z 2 k in the limit i give an important characterization of the wavelet system. But how do we know that i H ( i ) ω converges to a response in ℒ 2 ? In fact, there are some rather strict conditions on 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 ℒ 2 . Below we present some "regularity conditions" on H ω that ensure convergence of the iterated synthesis lowpass filter. 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 by h ( i ) n . Writing H ( i ) z H z 2 i 1 H ( i − 1 ) z in the time domain, we have h ( i ) n k k h k h ( i − 1 ) n 2 i 1 k Now define the function φ ( i ) t 2 i 2 n n h ( i ) n ℐ [ n / 2 i , n + 1 / 2 i ) t where ℐ [ a , b ) t denotes the indicator function over the interval a b : ℐ [ a , b ) t 1 t a b 0 t a b The definition of φ ( i ) t implies t t n 2 i n 1 2 i h ( i ) n 2 i 2 φ ( i ) t 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 and plugging the two previous expressions into the equation for h ( i ) n yields φ ( i ) t 2 k k h k φ ( i − 1 ) 2 t k . Thus, if φ ( i ) t converges pointwise to a continuous function, then it must satisfy the scaling equation, so that i φ ( i ) t φ t . Daubechies showed that, for pointwise convergence of φ ( i ) t to a continuous function in ℒ 2 , it is sufficient that H ω can be factored as P P 1 H ω 2 1 ω 2 P R ω for R ω such that sup ω R ω 2 P 1 Here P denotes the number of zeros that H ω has at ω . Such conditions are called regularity conditions because they ensure the regularity, or smoothness of φ t . In fact, if we make the previous condition stronger: ℓ ℓ 1 sup ω R ω 2 P 1 ℓ then i φ ( i ) t φ t for φ t that is ℓ-times continuously differentiable. There is an interesting and important by-product of the preceding analysis. If h n is a causal length-N filter, it can be shown that h ( i ) n is causal with length N i 2 i 1 N 1 1 . By construction, then, φ ( i ) t will be zero outside the interval 0 2 i 1 N 1 1 2 i . Assuming that the regularity conditions are satisfied so that i φ ( i ) t φ t , it follows that φ t must be zero outside the interval 0 N 1 . In this case we say that φ t has compact support. Finally, the wavelet scaling equation implies that, when φ t is compactly supported on 0 N 1 and g n is length N, ψ t will also be compactly supported on the interval 0 N 1 . Daubechies constructed a family of H z with impulse response lengths N 4 6 8 10 … which satisfy the regularity conditions. Moreover, her filters have the maximum possible number of zeros at ω , and thus are maximally regular (i.e., they yield the smoothest possible 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 and we show φ t , Φ Ω , ψ t , and Ψ Ω for various members of the Daubechies' wavelet system. See Vetterli and Kovacivić for a more complete discussion of these matters.

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