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.

The properties of the ii-stage cascaded lowpass filter in the limit i→∞ i give an important characterization of the wavelet system. But how do we know that limi→∞ H ( i ) ⅇⅈω i H ( i ) ω converges to a response in ℒ 2 ℒ 2 ? In fact, there are some rather strict conditions on Hⅇⅈω 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 ℒ 2 . Below we present some "regularity conditions" on Hⅇⅈω H ω that ensure convergence of the iterated synthesis lowpass filter.

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=2i2∑n h ( i ) n ℐ [ n / 2 i , n + 1 / 2 i ) t 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 ℐ [ a , b ) t denotes the indicator function over the interval ab a b : ℐ [ a , b ) t=1ift∈ab0ift∉ab ℐ [ a , b ) t 1 t a b 0 t a b The definition of φ ( i ) t φ ( i ) t implies and plugging the two previous expressions into the equation for h ( i ) n h ( i ) n yields 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 ℒ 2 ℒ 2 , it is sufficient that Hⅇⅈω H ω can be factored as for Rⅇⅈω R ω such that Here PP denotes the number of zeros that Hⅇⅈω 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: then limi→∞ φ ( i ) t=φt i φ ( i ) t φ t for φt φ t that is ℓℓ-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 N∈46810… 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.