Repeated Poles

When poles are repeated, an interesting new phenomenon emerges. To see what's going on, let's consider two identical poles arranged in parallel and in series. In the parallel case, we have

In the series case, we get

Thus, two one-pole filters in parallel are equivalent to a new one-pole filter^7.8 (when the poles are identical), while the same two filters in series give a two-pole filter with a repeated pole. To accommodate both possibilities, the general partial fraction expansion must include the terms

for a pole having multiplicity 2.

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Subsections

• Dealing with Repeated Poles Analytically
• Example
• Impulse Response of Repeated Poles
• So What's Up with Repeated Poles?

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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.