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Introduction to Digital Filters
    Background Fundamentals
       A Sum of Sinusoids at the Same Frequency is Another Sinusoid at that Frequency
          Phasor Analysis: Factoring a Complex Sinusoid into Phasor Times Carrier

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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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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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