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Introduction to Digital Filters
    The Simplest Lowpass Filter
       An Easier Way
          Phasor Notation

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Phasor Notation

The complex amplitude $ {\cal A}\isdef A e^{j\phi}$ is also defined as the phasor associated with any sinusoid having amplitude $ A$ and phase $ \phi$. The term ``phasor'' is more general than ``complex amplitude'', however, because it also applies to the corresponding real sinusoid given by the real part of Equations (1.9-1.10). In other words, the real sinusoids $ A\cos(\omega t+\phi)$ and $ A\cos(\omega nT+\phi)$ may be expressed as

\begin{eqnarray*} A\cos(\omega t+\phi) &\isdef & \mbox{re}\left\{A e^{j(\omega t... ...hi)}\right\} = \mbox{re}\left\{{\cal A}e^{j\omega nT}\right\}\\ \end{eqnarray*}

and $ {\cal A}$ is the associated phasor in each case. Thus, we say that the phasor representation of $ A\cos(\omega t+\phi)$ is $ {\cal A}\isdef A e^{j\phi}$. Phasor analysis is often used to analyze linear time-invariant systems such as analog electrical circuits.

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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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