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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:

**Subsections**

**Previous:** Sinusoids and Exponentials**Next:** Example Sinusoids

**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.