Disk Algebra Bases

Abstract

This chapter considers the approximation of the transfer function of a linear system in terms of a causal filter bank. Assume that $f\left( {{\rm{e}}^{{\rm{i}}\theta } } \right)$ with $\theta \in [-\pi,\pi)$ is the transfer function of an arbitrary discrete-time linear system ${\cal L}$, and let $\Phi = \left\{ {\varphi _k } \right\}_{k = 1}^\infty$ be a set of transfer functions of an orthonormal filterbank. It is assumed that f as well as all $\varphi_{k}$ are elements of a certain Banach algebra ${\cal B}$ which characterizes the system theoretical properties of ${\cal L}$ and of the filterbank $\Phi$. Moreover, since f as well as $\{\varphi_{k}\}^{\infty}_{k=1}$ should represent causal systems, these transfer functions have to belong to the causal subspace ${\cal B}_ + \,{\rm{of}}\,{\cal B}$. Then, it is desirable to obtain an approximation of f in this filterbank of the form