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Sinusoids

A sinusoid is any function having the following form:

$\displaystyle x(t) = A \sin(\omega t + \phi)
$

where $ t$ is the independent (real) variable, and the fixed parameters $ A$ , $ \omega$ , and $ \phi$ are all real constants. In audio applications we typically have

\begin{eqnarray*}
A &=& \mbox{peak amplitude (nonnegative)} \\
\omega &=& \mbox{radian frequency (rad/sec)} \\
&=& 2\pi f \; \mbox{($f$\ in Hz)}\\
t &=& \mbox{time (sec)} \\
f &=& \mbox{frequency (Hz)} \\
\phi &=& \mbox{initial phase (radians)}\\
\omega t + \phi &=& \mbox{instantaneous phase (radians).}
\end{eqnarray*}

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 $ x$ is its instantaneous value $ x(t)$ at any time $ t$ . The peak amplitude $ A$ satisfies $ \left\vert x(t)\right\vert\leq A$ . The ``instantaneous magnitude'' or simply ``magnitude'' of a signal $ x(t)$ is given by $ \vert x(t)\vert$ , 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 $ 2\pi $ , the initial phase $ \phi \pm 2\pi$ is indistinguishable from $ \phi$ . As a result, we may restrict the range of $ \phi$ to any length $ 2\pi $ interval. When needed, we will choose

$\displaystyle -\pi \leq \phi < \pi,
$

i.e., $ \phi\in[-\pi,\pi)$ . You may also encounter the convention $ \phi\in[0,2\pi)$ .

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

$\displaystyle \frac{d}{dt} (\omega t + \phi) = \omega
$

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

$\displaystyle \theta(t) \isdef \omega t + \phi(t)
$

denote the instantaneous phase of a sinusoid with a time-varying phase-offset $ \phi(t)$ . Then the instantaneous frequency is again given by the time derivative of the instantaneous phase:

$\displaystyle \frac{d}{dt} [\omega t + \phi(t)] = \omega + \frac{d}{dt} \phi(t)
$



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``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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA