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Steady State Signals

We can also speak of the transient response and steady-state response given any fixed sum of sinewaves as input, since a sum of sinewaves can be regarded as a ``steady-state signal'' (after it is switched on). Of course, any practical signal whatsoever, including a switch-on transient, can be represented by an infinite sum of sinusoids (just consider its inverse Fourier transform representation [83]). However, we're speaking in this section about practical engineering terms--not pure mathematics. Therefore, we'll consider a valid ``steady state input signal'' to be any fixed linear combination of a finite number of sinusoids (allowing dc as a zero-Hz sinusoid).6.6 Under this constraint, the turn-on transient cannot be represented by a finite sum of sinusoids. Generally speaking, any discontinuity in an input signal must be regarded as a transient, and transients will always disturb the steady-state operation of a filter, resulting in a transient response at the filter output.

In summary, a filter transient response is caused by suddenly switching on a filter input signal, or otherwise disturbing a steady-state input signal away from its steady-state form. After the transient response has died out, we see the steady-state response, provided that the input signal itself is a steady-state signal (a fixed linear combination of sinusoids) and the filter is LTI.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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