# Sinusoids

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Sinusoids

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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# Sinusoids

A sinusoid is any function of time having the following form:

where all variables are real numbers, and

The term ''peak amplitude'' is often shortened to ''amplitude,'' e.g., ''the amplitude of the sound was measured to be 5 Pascals.'' Strictly speaking, however, the ''amplitude'' of a signal is its instantaneous value at any time . The peak amplitude satisfies . The ''instantaneous magnitude'' or simply ''magnitude'' of a signal is given by , and the peak magnitude is the same thing as the peak amplitude.

Note that Hz is an abbreviation for Hertz which physically means ''cycles per second.'' You might also encounter the older (and clearer) notation ''c.p.s.'' for cycles per second.

Since is periodic with period , the phase is indistinguishable from the phase . As a result, we may restrict the range of to any length interval. When needed, we will choose

i.e., . You may also encounter the convention .

Subsections

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