Sinusoids as Eigenfunctions of LTI Systems Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


A Sum of Sinusoids at the
Same Frequency is Another
Sinusoid at that Frequency

It is an important and fundamental fact that a sum of sinusoids at the same frequency, but different phase and amplitude, can always be expressed as a single sinusoid at that frequency with some resultant phase and amplitude. An important implication, for example, is that

sinusoids are eigenfunctions of linear time-invariant (LTI) systems.
That is, if a sinusoid is input to an LTI system, the output will be a sinusoid at the same frequency, but possibly altered in amplitude and phase. This follows because the output of every LTI system can be expressed as a linear combination of delayed copies of the input signal. In this appendix, we derive this important result for the general case of $ N$ sinusoids at the same frequency.



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[How to cite this work] [Order a printed hardcopy]

``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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