The Continuous-Time Impulse

The continuous-time impulse response was derived above as the inverse-Laplace transform of the transfer function. In this section, we look at how the impulse itself must be defined in the continuous-time case.

An impulse in continuous time may be loosely defined as any ``generalized function'' having ``zero width'' and unit area under it. A simple valid definition is

More generally, an impulse can be defined as the limit of any 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

An impulse is not a function in the usual sense, so it is called instead a distribution or generalized function [13,44]. (It is still commonly called a ``delta function'', however, despite the misnomer.)

---

Previous: Impulse Response
Next: Poles and Zeros

Order a Hardcopy of Introduction to Digital Filters

---

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.