# 9.1 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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