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Surround Sound: Chamberlin Filters

Module by: Douglas L. Jones, Swaroop Appadwedula, Matthew Berry, Mark Haun, Jake Janovetz, Michael Kramer, Dima Moussa, Daniel Sachs, Brian Wade. E-mail the authors

Summary: 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 are necessary. Chamberlin implementations do not suffer from some stability problems that arise in direct-form implementations of 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 Equation 1:

H⁢z= F z 2⁢z-11−(2−( F c ⁢ Q c − F c 2))⁢z-1−1⁢z-2 H z F z 2 z -1 1 2 F c Q c F c 2 z -1 1 F c Q c z -2
(1)
where F⁢c F c determines the frequency where the filter peaks, and Q c ⁢1Q Q c 1 Q determines the rolloff. Q is defined as the positive ratio of the center frequency to the bandwidth. A derivation and more detailed explanation is given in Dattorro. The topology of the filter is shown in Figure 1. Note that the final feedback stage puts a pole just inside the unit circle on the real axis. For a response with smaller bandwidth, move the pole closer to the unit circle, but do not move it so far that the filter becomes unstable. Multiple second-order sections can be cascaded to yield a sharper rolloff.

Figure 1: Chamberlin Filter Topology
Figure 1 (chamberlin.png)

Figure 2 and Figure 3 show how the response of the filter varies with Q c Q c and F c F c .

Figure 2: Chamberlin filter responses for various Q c Q c ( F c =.3 F c .3 )
Figure 2 (chamberlinQc.png)
Figure 3: Chamberlin filter responses for various F c F c ( Q c =.8333 Q c .8333 )
Figure 3 (chamberlinFc.png)

Exercise

First, create a MATLAB script that takes two parameters, Q c Q c and F c F c , and plots the frequency response of a filter with a transfer function given in Equation 1. Then implement a Chamberlin filter on the DSP and compare its performance with that of your MATLAB simulation for the same values of Q c Q c and F c F c . What do you observe?

References

  1. J. Dattorro. (1996, September). Effect Design Part 1: Reverberator and Other Filters. Journal Audio Engineering Society, vol. 45, 660-684.
  2. Implementing IIR/FIR Filters with Motorola's DSP56000/SPS/DSP56001, Digital Signal Processors. [http://merchant.hibbertco.com/mtrlext/fs22/pdf-docs/motorola/apr7.rev2.pdf]. Motorola.

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