0.8 Filter banks and transmultiplexers  (Page 11/23)

for maximal possible $K$ . By default, every unitary scaling filter $h_0$ is one-regular (because ${\sum }_n{h}_0\left(n\right)=\sqrt{M}$ - see [link] , Theorem  [link] for equivalent characterizations of $K$ -regularity). Each of the $K$ -identical factors in Eqn.  [link] adds an extra linear constraint on $h_0$ (actually, it is one linear constraint on each of the $M$ polyphase subsequences of $h_0$ - see [link] ).

There is no simple relationship between the smoothness of the scaling function and $K$ -regularity. However, the smoothness of the maximally regular scaling filter, $h_0$ , with fixed filter length $N$ , tends to be an increasing function of $N$ . Perhaps one can argue that $K$ -regularity is an important concept independent of the smoothness of the associated wavelet system. $K$ -regularity implies that the moments of the wavelets vanish up to order $K-1$ , and therefore, functions can be better approximated by usingjust the scaling function and its translates at a given scale. Formulae exist for $M$ -band maximally regular $K$ -regular scaling filters (i.e., only the sequence $h_0$ ) [link] . Using the Householder parameterization, one can then design the remaining filters in the filter bank.

The linear constraints on $h_0$ that constitute $K$ -regularity become nonexplicit nonlinear constraints on theHouseholder parameterization of the associated filter bank. However, one-regularity can be explicitly incorporated and thisgives a parameterization of all $M$ -band compactly supportedwavelet tight frames. To see, this consider the following two factorizations of $H_p\left(z\right)$ of a unitary filter bank.

and

Since $H_p\left(1\right)=V_0$ and $H_p^T\left(1\right)=W_0$ , $V_0=W_0^T$ . The first column of $W_0$ is the unit vector ${[H_{0,0}\left(1\right),H_{0,1}\left(1\right),...,H_{0,M-1}\left(1\right)]}^T$ . Therefore,

But since ${\displaystyle \sum _n{h}_0\left(n\right)=H_0\left(1\right)=\sqrt{M}}$ ,

Therefore, for all $k$ , $H_{0,k}\left(1\right)=\frac{1}{\sqrt{M}}$ . Hence, the first row of $V_0$ is $[1/\sqrt{M},1/\sqrt{M},...,1/\sqrt{M}]$ . In other words, a unitary filter bank gives rise to a WTF iff the first rowof $V_0$ in the Householder parameterization is the vector with allentries $1/\sqrt{M}$ .

Alternatively, consider the Given's factorization of $H_p\left(z\right)$ for a two-channel unitary filter bank.

Since for a WTF we require

we have $\text{Θ}={\sum }_{k=0}^{K-1}{\theta }_k=\frac{\pi }{4}$ . This is the condition for the lattice parameterization tobe associated with wavelets.

Modulated filter banks

Filter bank design typically entails optimization of the filter coefficients to maximize some goodness measure subject to the perfect reconstruction constraint.Being a constrained (or unconstrained for parameterized unitary filter bank design) nonlinear programming problem, numerical optimization leads to localminima, with the problem exacerbated when there are a large number of filter coefficients. To alleviate this problem one can try to impose structuralconstraints on the filters. For example, if [link] is the desired ideal response, one can impose the constraint that all analysis (synthesis)filters are obtained by modulation of a single “prototype” analysis (synthesis) filter. This is the basic idea behind modulated filter banks [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] . In what follows, we only consider the case where the number of filtersis equal to the downsampling factor; i.e., $L=M$ .