One-Pole Transfer Functions

We can apply the same analysis to a one-pole transfer function. Let $p\in{\bf C}$ denote any real or complex number:

$\displaystyle H(z) \eqsp \frac{1}{1-pz^{-1}} \eqsp 1 + pz^{-1}+ pz^{-2}+ pz^{-3} + \cdots$

The convergence criterion is now $\vert pz^{-1}\vert<1$ , or $\vert z\vert>\vert p\vert$ . For the region of convergence to include the unit circle (our frequency axis), we must have $\vert p\vert<1$ , which is our usual stability criterion for a pole at

. The inverse z transform is then the causal decaying sampled exponential

$\displaystyle H(z) \;\longleftrightarrow\; h(n) = u(n)p^n$

Now consider the rewritten case:

$\begin{eqnarray*} \frac{1}{1-pz^{-1}} &=& \frac{-p^{-1}z}{1-p^{-1}z} \\ &=& -p^... ...cdots\right]\\ &\leftrightarrow& - u(-n-1)p^n,\quad n\in{\bf Z} \end{eqnarray*}$

where the inverse z transform is the inverse bilateral z transform. In this case, the convergence criterion is $\vert p^{-1}z\vert<1$ , or $\vert z\vert<\vert p\vert$ , and this region includes the unit circle when $\vert p\vert>1$ .

In summary, when the region-of-convergence of the z transform is assumed to include the unit circle of the plane, poles inside the unit circle correspond to stable, causal, decaying exponentials, while poles outside the unit circle correspond to anticausal exponentials that decay toward time $-\infty$ , and stop before time zero.

Figure 8.8 illustrates the two types of exponentials (causal and anticausal) that correspond to poles (inside and outside the unit circle) when the z transform region of convergence is defined to include the unit circle.

myFourFiguresToWidthpolesout11polesout21polesout12polesout220.52Left column: Causal exponential decay, pole at . Right column: Anticausal exponential decay, pole at . Top: Pole-zero diagram. Bottom: Corresponding impulse response, assuming the region of convergence includes the unit circle in the plane.

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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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Introduction to Digital Filters
    Pole-Zero Analysis
       Unstable Poles--Unit Circle Viewpoint
          One-Pole Transfer Functions