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
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© 2009 Springer-Verlag Berlin Heidelberg
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Pohl, V., Boche, H. (2009). Disk Algebra Bases. In: Advanced Topics in System and Signal Theory. Foundations in Signal Processing, Communications and Networking, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03639-2_7
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DOI: https://doi.org/10.1007/978-3-642-03639-2_7
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03638-5
Online ISBN: 978-3-642-03639-2
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