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A sinusoid is any function having the following form:
is the independent (real) variable, and the fixed parameters
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
is indistinguishable from . As a result,
we may restrict the range of to any length interval.
When needed, we will choose
. 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
denote the instantaneous phase of a sinusoid with a time-varying
. Then the instantaneous frequency is again
given by the time derivative of the instantaneous phase:
Previous: Sinusoids and ExponentialsNext: 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/