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Chapters

See Also

Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Pole-Zero Analysis
       Unstable Poles--Unit Circle Viewpoint
          Geometric Series

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Geometric Series

The essence of the situation can be illustrated using a simple geometric series. Let $ R$ be any real (or complex) number. Then we have

$\displaystyle \frac{1}{1-R} \eqsp 1 + R + R^2 + R^3 + \cdots \quad < \infty$   when$\displaystyle \quad\vert R\vert<1. $

In other words, the geometric series $ 1 + R + R^2 + R^3 + \cdots$ is guaranteed to be summable when $ \vert R\vert<1$, and in that case, the sum is given by $ 1/(1-R)$. On the other hand, if $ \vert R\vert>1$, we can rewrite $ 1/(1-R)$ as $ -R^{-1}/(1-R^{-1})$ to obtain

$\displaystyle \frac{1}{1-R} \eqsp \frac{-R^{-1}}{1-R^{-1}} \eqsp -R^{-1}\left[1 + R^{-1} + R^{-2} + R^{-3} + \cdots \right] $

which is summable when $ \vert R\vert>1$. Thus, $ 1/(1-R)$ is a valid closed-form sum whether or not $ \vert R\vert$ is less than or greater than 1. When $ \vert R\vert<1$, it is the sum of the causal geometric series in powers of $ R$. When $ \vert R\vert>1$, it is the sum of the causal geometric series in powers of $ R^{-1}$, or, an anticausal geometric series in (negative) powers of $ R$.

Previous: Unstable Poles--Unit Circle Viewpoint
Next: One-Pole Transfer Functions

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About the Author: 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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