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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

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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.

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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:

Hz= F z 2z-11−2−( F c Q c − F c 2)z-1+1− F c Q c 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 Fc 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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