Allpass Filters

The allpass filter is an important building block for digital audio signal processing systems. It is called ``allpass'' because all frequencies are ``passed'' in the same sense as in ``lowpass'', ``highpass'', and ``bandpass'' filters. In other words, the amplitude response of an allpass filter is 1 at each frequency, while the phase response (which determines the delay versus frequency) can be arbitrary.

In practice, a filter is often said to be allpass if the amplitude response is any nonzero constant. However, in this book, the term ``allpass'' refers to unity gain at each frequency.

In this section, we will first make an allpass filter by cascading a feedback comb-filter with a feedforward comb-filter. This structure, known as the Schroeder allpass, is used extensively in the fields of artificial reverberation and digital audio effects. Next we will look at creating allpass filters by nesting them; allpass filters are nested by replacing delay elements (which are allpass filters themselves) with arbitrary allpass filters. Finally, we will consider the general case, and characterize the set of all single-input, single-output allpass filters. The general case, including multi-input, multi-output (MIMO) allpass filters, is treated in [460, Appendix D].

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Subsections

• Allpass from Two Combs
• Nested Allpass Filters
• More General Allpass Filters
• Example Allpass Filters
• Gerzon Nested MIMO Allpass

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written by 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.