11.1 Surround sound: chamberlin filters

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The Chamberlin filter topology can implement very narrow-band, low-pass filters. This module provides the Chamberlin filter transfer function, an illustration of the topology, and sample frequency responses for different choices of design parameters.

Introduction

Chamberlin filter topology is frequently used in music applications where very narrow-band, low-pass filters arenecessary. Chamberlin implementations do not suffer from some stability problems that arise in direct-form implementationsof very narrow-band responses. For more information about IIR/FIR filter design for DSPs, refer to the Motorola Application Note .

Filter topology

A Chamberlin filter is a simple two-pole IIR filter with the transfer function given in [link] :

$H(z)=\frac{{F}_{z}^{2}z^{-1}}{1-(2-{F}_{c}{Q}_{c}-{F}_{c}^{2})z^{-1}-1z^{-2}}$
where $F(c)$ determines the frequency where the filter peaks, and ${Q}_{c}(\frac{1}{Q})$ determines the rolloff. Q is defined as the positive ratio of the center frequency to thebandwidth. A derivation and more detailed explanation is given in Dattorro . The topology of the filter is shown in [link] . Note that the final feedback stage puts a pole just inside theunit circle on the real axis. For a response with smaller bandwidth, move the pole closer to the unit circle, but do notmove it so far that the filter becomes unstable. Multiple second-order sections can be cascaded to yield a sharperrolloff.

[link] and [link] show how the response of the filter varies with ${Q}_{c}$ and ${F}_{c}$ .

Exercise

First, create a MATLAB script that takes two parameters, ${Q}_{c}$ and ${F}_{c}$ , and plots the frequency response of a filter with a transfer function given in [link] . Then implement a Chamberlin filter on the DSP and compare itsperformance with that of your MATLAB simulation for the same values of ${Q}_{c}$ and ${F}_{c}$ . What do you observe?

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