Conventional Ladder Filters

Given a reflecting termination on the right, the half-rate DWF chain of Fig.I.2 can be reduced further to the conventional ladder/lattice structure of Fig.I.3. Every delay on the upper rail is pushed to the right until they have all been worked around to the bottom rail. In the end, each bottom-rail delay becomes seconds instead of seconds. Such an operation is possible because of the termination at the right by an infinite (or zero) wave impedance. In the time-varying case, pushing a delay through a multiply results in a corresponding time advance of the multiplier coefficient. The time arguments of the reflection coefficients in the figure indicate the amount of the time shift for each section. Note that because of 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 series (or acyclic tree) of waveguide sections is not immediately possible because of the delay-free path along the top rail. In particular, the output cannot be fed back to the input.

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written by 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 Associate Professor (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 http://ccrma.stanford.edu/~jos/ for details.