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Conventional Ladder Filters

Given a reflecting termination on the right, the half-rate DWF chain of Fig.C.25 can be reduced further to the conventional ladder/lattice filter structure shown in Fig.C.26.

figure[htbp] \includegraphics{eps/DWFHRT}

To make a standard ladder/lattice filter, the sampling rate is cut in half (i.e., replace $ z^{-2}$ by $ z^{-1}$), and the scattering junctions are typically implemented in one-multiply form (§C.8.5) or normalized form (§C.8.6), etc. Conventionally, if the graph of the scattering junction is nonplanar, as it is for the one-multiply junction, the filter is called a lattice filter; it is called a ladder filter when the graph is planar, as it is for normalized and Kelly-Lochbaum scattering junctions. For all-pole transfer functions $ H(z)=F^{-}_0(z)/F^{+}_0(z)$, the Durbin recursion can be used to compute the reflection coefficients $ k_i$ from the desired transfer-function denominator polynomial coefficients [449]. To implement arbitrary transfer-function zeros, a linear combination of delay-element outputs is formed using weights that are called ``tap parameters'' [173,297].

To create Fig.C.26 from Fig.C.24, all delays along the top rail are pushed to the right until they have all been worked around to the bottom rail. In the end, each bottom-rail delay becomes $ 2T$ seconds instead of $ T$ seconds. Such an operation is possible because of the termination at the right by an infinite (or zero) wave impedance. Note that there is a progressive one-sample time advance from section to section. The time skews for the right-going (or left-going) traveling waves can be determined simply by considering how many missing (or extra) delays there are between that signal and the unshifted signals at the far left.

Due to the reflecting termination, conventional lattice filters cannot be extended to the right in any physically meaningful way. Also, creating network topologies more complex than a simple linear cascade (or acyclic tree) of waveguide sections is not immediately possible because of the delay-free path along the top rail. In particular, the output $ f^{{+}}_M$ cannot be fed back to the input $ f^{{+}}_0$. Nevertheless, as we have derived, there is an exact physical interpretation (with time skew) for the conventional ladder/lattice digital filter.

Previous: Half-Rate Ladder Waveguide Filters
Next: Power-Normalized Waveguide Filters

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See for details.


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