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Phasor Analysis: Factoring a Complex Sinusoid into Phasor Times Carrier

The heart of the preceding proof was the algebraic manipulation

$\displaystyle \sum_{i=1}^N A_i e^{j(\omega t + \phi_i)} = e^{j\omega t} \sum_{i=1}^N A_i e^{j\phi_i}.
$

The carrier term $ e^{j\omega t}$ ``factors out'' of the sum. Inside the sum, each sinusoid is represented by a complex constant $ A_i e^{j\phi_i}$ , known as the phasor associated with that sinusoid.

For an arbitrary sinusoid having amplitude $ A$ , phase $ \phi$ , and radian frequency $ \omega$ , we have

$\displaystyle A\cos(\omega t + \phi) =$   re$\displaystyle \left\{(A e^{j\phi}) e^{j\omega t}\right\}.
$

Thus, a sinusoid is determined by its frequency $ \omega$ (which specifies the carrier term) and its phasor $ {\cal A}\isdef A e^{j\phi}$ , a complex constant. Phasor analysis is discussed further in [84].


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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