0.8 Filter banks and transmultiplexers  (Page 10/23)

where ${\theta }_0=\frac{\pi }{3}$ and ${\theta }_1=-\frac{\pi }{12}$ . The fact that the filter bank is associated with waveletsis precisely because ${\theta }_0+{\theta }_1=\frac{\pi }{4}$ . More generally, for a filter bank with filters of length $2K$ to be associated with wavelets, ${\sum }_{k=0}^{K-1}{\theta }_k=\frac{\pi }{4}$ . This is expected since for filters of length $2K$ to be associated with wavelets we have seen (from the Householder factorization)that there are $K-1$ parameters $v_k$ . Our second example belongs to a class of unitary filter banks called modulated filter banks , which is described in a following section. A Type 1 modulated filter bank with filters of length $N=2M$ and associated with a wavelet orthonormal basis is defined by

where $i\in \left\{0,,,...,,,M,-,1\right\}$ and $n\in \left\{0,,,...,,,2,M,-,1\right\}$ [link] , [link] . Consider a three-band example with length six filters.In this case, $K=2$ , and therefore one has one projection $P_1$ and the matrix $V_0$ . The projection is one-dimensional and given by the Householder parameter

The third example is another Type 1 modulated filter bank with $M=4$ and $N=8$ . The filters are given in [link] . $H_p\left(z\right)$ had the following factorization

where $P_1$ is a two-dimensional projection $P_1=v_1{v}_1^T+v_2{v}_2^T$ (notice the arbitrary choice of $v_1$ and $v_2$ ) given by

and

Notice that there are infinitely many choices of $v_1$ and $v_2$ that give rise to the same projection $P_1$ .

$M$ -band wavelet tight frames

In [link] , Theorem 7 , while discussing the properties of $M$ -band wavelet systems, we saw that the lowpass filter $h_0$ ( $h$ in the notation used there) must satisfy the linear constraint ${\displaystyle \sum _n{h}_0\left(n\right)=\sqrt{M}}$ . Otherwise, a scaling function with nonzero integral could not exit.It turns out that this is precisely the only condition that an FIR unitary filter bank has to satisfy in order for it to generatean $M$ -band wavelet system [link] , [link] . Indeed, if this linear constraint is not satisfied the filter bank does not generate a wavelet system. This single linear constraint (for unitary filter banks) also impliesthat ${\displaystyle \sum _h{h}_i\left(n\right)=0}$ for $i\in \left\{1,,,2,,,...,,,M,-,1\right\}$ (because of Eqn.  [link] ). We now give the precise result connecting FIR unitary filter banks and wavelet tight frames.

Theorem 46 Given an FIR unitary filter bank with ${\displaystyle \sum _n{h}_0\left(n\right)=\sqrt{M}}$ , there exists an unique, compactly supported, scaling function ${\psi }_0\left(t\right)\in L^2\left(\text{ℝ}\right)$ (with support in $[0,\frac{N-1}{M-1}]$ , assuming $h_0$ is supported in $[0,N-1]$ ) determined by the scaling recursion :

Define wavelets, ${\psi }_i\left(t\right)$ ,

and functions, ${\psi }_{i,j,k}\left(t\right)$ ,

Then $\left\{{\psi }_{i,j,k}\right\}$ forms a tight frame for $L^2\left(\text{ℝ}\right)$ . That is, for all $f\in L^2\left(\text{ℝ}\right)$

Also,

Remark: A similar result relates general FIR (not necessarily unitary) filter banks and $M$ -band wavelet frames [link] , [link] , [link] .

Starting with [link] , one can calculate the scaling function using either successive approximationor interpolation on the $M$ -adic rationals—i.e., exactly as in the two-band case in Chapter  [link] . Equation  [link] then gives the wavelets in terms of the scaling function. As in the two-band case, the functions ${\psi }_i\left(t\right)$ , so constructed, invariably turn out highly irregular and sometimes fractal. Thesolution, once again, is to require that several moments of the scaling function (or equivalently the moments of the scaling filter $h_0$ ) are zero. This motivates the definition of $K$ -regular $M$ -band scaling filters: A unitary scaling filter $h_0$ is said to be $K$ regular if its $\mathcal{Z}$ -transform can be written in the form