# Mathematics of the DFTSinusoids and ExponentialsSinusoids

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## Sinusoids

A sinusoid is any function having the following form:

where is the independent (real) variable, and the fixed parameters , , and are all real constants. In audio applications we typically have

An example is plotted in Fig.4.1.

The term peak amplitude'' is often shortened to amplitude,'' e.g., the amplitude of the tone 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.

The phase'' of a sinusoid normally means the initial phase'', but in some contexts it might mean instantaneous phase'', so be careful. Another term for initial phase is phase offset.

Note that Hz is an abbreviation for Hertz which physically means cycles per second. You might also encounter the notation cps (or c.p.s.'') for cycles per second (still in use by physicists and formerly used by engineers as well).

Since the sine function is periodic with period , the initial phase is indistinguishable from . 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 .

Note that the radian frequency is equal to the time derivative of the instantaneous phase of the sinusoid:

This is also how the instantaneous frequency is defined when the phase is time varying. Let

denote the instantaneous phase of a sinusoid with a time-varying phase-offset . Then the instantaneous frequency is again given by the time derivative of the instantaneous phase:

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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 (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.