Delay-Line Damping Filter Design

Let denote the desired reverberation time at radian frequency , and let denote the transfer function of the lowpass filter to be placed in series with the th delay line which is samples long. The problem we consider now is how to design these filters to yield the desired reverberation time. We will specify an ideal amplitude response for based on the desired reverberation time at each frequency, and then use conventional filter-design methods to obtain a low-order approximation to this ideal specification.

In accordance with Eq.(3.6), the lowpass filter in series with a length delay line should approximate

which implies

Taking of both sides gives

This is the same formula derived by Jot [217] using a somewhat different approach.

Now that we have specified the ideal delay-line filter in terms of its amplitude response in dB, any number of filter-design methods can be used to find a low-order which provides a good approximation to satisfying Eq.(3.9). Examples include the functions invfreqz and stmcb in Matlab. Since the variation in reverberation time is typically very smooth with respect to , the filters can be very low order.

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Subsections

• First-Order Delay-Filter Design
• Orthogonalized First-Order Delay-Filter Design
• Multiband Delay-Filter Design

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.