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Mathematics of the DFT
    Sinusoids and Exponentials
       Complex Sinusoids

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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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written by 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.


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