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Complex Sinusoids

Recall Euler's Identity,

$\displaystyle e^{j\theta} = \cos(\theta) + j\sin(\theta).
$

Multiplying this equation by $ A \geq 0$ and setting $ \theta = \omega t
+
\phi$ , where $ t$ is time in seconds, $ \omega$ is radian frequency, and $ \phi$ is a phase offset, we obtain what we call the complex sinusoid:

$\displaystyle s(t) \isdef A e^{j(\omega t+\phi)} = A \cos(\omega t+\phi) + jA\sin(\omega t+\phi)
$

Thus, a complex sinusoid consists of an ``in-phase'' component for its real part, and a ``phase-quadrature'' component for its imaginary part. Since $ \sin^2(\theta) + \cos^2(\theta) = 1$ , we have

$\displaystyle \left\vert s(t)\right\vert \isdef \sqrt{\mbox{re}^2\left\{s(t)\right\} + \mbox{im}^2\left\{s(t)\right\}} \equiv A.
$

That is, the complex sinusoid has a constant modulus (i.e., a constant complex magnitude). (The symbol ``$ \equiv$ '' means ``identically equal to,'' i.e., for all $ t$ .) The instantaneous phase of the complex sinusoid is

$\displaystyle \angle s(t) = \omega t+\phi.
$

The derivative of the instantaneous phase of the complex sinusoid gives its instantaneous frequency

$\displaystyle \frac{d}{dt}\angle s(t) = \omega = 2\pi f.
$



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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 © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA