Conventional Ladder Filters

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

A digital filter can be visualized as a 'black box' that accepts a sequence of numbers and emits a new sequence of numbers. In digital audio signal processing applications, such number sequences usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects. — Click for https://ccrma.stanford.edu/~jos/filters/

A feedback signal creates a loop. That is, the term 'feedback' refers to when an output signal is 'fed back' to an input of the same system. — Click for http://en.wikipedia.org/wiki/Feedback

A waveguide restricts wave propagation to a particular subspace, which is usually a line. Vibrating strings, woodwinds, and transmission lines are examples of one-dimensional waveguides. — Click for http://en.wikipedia.org/wiki/Waveguide

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

Click for http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html

The wave impedance in a propagation medium is the force variable (such as pressure for acoustic waves) divided by the velocity variable. In ideal plane waves and traveling waves, the wave impedance is a real, positive number. For spherical waves in air, the wave impedance is different at each frequency. — Click for https://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

Click for https://ccrma.stanford.edu/~jos/filters/Computing_Reflection_Coefficients_Check.html

The transfer function is defined for LTI filters as the z transform of the filter output signal, divided by the z transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

Click for https://ccrma.stanford.edu/~jos/pasp/Kelly_Lochbaum_Scattering_Junctions.html

Click for https://ccrma.stanford.edu/~jos/pasp/Scattering_Impedance_Changes.html

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

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.

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

from the desired transfer-function denominator polynomial coefficients [452]. To implement arbitrary transfer-function zeros, a linear combination of delay-element outputs is formed using weights that are called ``tap parameters'' [174,299].

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

seconds instead of

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.

Conventional Ladder Filters

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University