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

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

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:

- Example Sinusoids
- Why Sinusoids are Important
- In-Phase & Quadrature Sinusoidal Components
- Sinusoids at the Same Frequency
- Constructive and Destructive Interference
- Sinusoid Magnitude Spectra

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University