The Continuous-Time Impulse

An impulse in continuous time must have ``zero width'' and unit area under it. One definition is

An impulse can be similarly defined as the limit of any integrable pulse shape which maintains unit area and approaches zero width at time 0. As a result, the impulse under every definition has the so-called sifting property under integration,

provided is continuous at . This is often taken as the defining property of an impulse, allowing it to be defined in terms of non-vanishing function limits such as

(Note, incidentally, that is in but not .)

An impulse is not a function in the usual sense, so it is called instead a distribution or generalized function [10,36]. (It is still commonly called a ``delta function'', however, despite the misnomer.)

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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.