12.11 Regularity conditions, compact support, and daubechies'

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, asillustrated in .

The properties of the $i$ -stage cascaded lowpass filter

Let us denote the impulse response of $H^{\left(i\right)}(z)$ by $h^{\left(i\right)}(n)$ . Writing $$H^{\left(i\right)}(z)=H(z^{2^{(i-1)}})H^{(i-1)}(z)$$ in the time domain, we have $$h^{\left(i\right)}(n)=\sum h(k)h^{(i-1)}(n-2^{(i-1)}k)$$ Now define the function $${\phi }^{\left(i\right)}(t)=2^{\left(\frac{i}{2}\right)}\sum h^{\left(i\right)}(n){ℐ}_{[n/2^i,n+1/2^i)}(t)$$ where ${ℐ}_{\left[a,b\right)}(t)$ denotes the indicator function over the interval $\left[a , b\right)$ : $${ℐ}_{\left[a,b\right)}(t)=\begin{cases}1 & \text{if $t\in \left[a , b\right)$}\\ 0 & \text{if $t\notin \left[a , b\right)$}\end{cases}$$ The definition of ${\phi }^{\left(i\right)}(t)$ implies

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 $h^{\left(i\right)}(n)$ is causal with length $N^i=2^i(N-1)+1$ . By construction, then, ${\phi }^{\left(i\right)}(t)$ will be zero outside the interval $\left[0 , \frac{2^i(N-1)+1}{2^i}\right)$ . Assuming that the regularity conditions are satisfied so that $\lim_{i\to }i\to$∞ φ ( i ) t φ t , it follows that $\phi (t)$ must be zero outside the interval $\left[0 , N-1\right]$ . In this case we say that $\phi (t)$ has compact support . Finally, the wavelet scaling equation implies that, when $\phi (t)$ is compactly supported on $\left[0 , N-1\right]$ and $g(n)$ is length $N$ , $\psi (t)$ will also be compactly supported on the interval $\left[0 , N-1\right]$ .

Daubechies constructed a family of $H(z)$ with impulse response lengths $N\in \{4, 6, 8, 10, \dots \}$ which satisfy the regularity conditions. Moreover, her filters have the maximum possible number of zeros at $\omega =\pi$ , and thus are maximally regular (i.e., they yield the smoothest possible $\phi (t)$ 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 $\phi (t)$ , $\Phi (\Omega )$ , $\psi (t)$ , and $\Psi (\Omega )$ for various members of the Daubechies' wavelet system.

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