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One-Pole Transfer Functions

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

$\displaystyle H(z) \eqsp \frac{1}{1-pz^{-1}} \eqsp 1 + pz^{-1}+ p^2z^{-2}+ p^3z^{-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 $ z=p$ . 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:

\frac{1}{1-pz^{-1}} &=& \frac{-p^{-1}z}{1-p^{-1}z} \\
&=& -p^{-1}z\left[1 + p^{-1}z + p^{-2}z^2 + p^{-3}z^3 + \cdots\right]\\
&=& -\left[p^{-1}z + p^{-2}z^2 + p^{-3}z^3 + p^{-4}z^4 + \cdots\right]\\
&\leftrightarrow& - u(-n-1)p^n,\quad n\in\mathbb{Z}

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 $ z$ 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 $ p=0.9$ . Right column: Anticausal exponential decay, pole at $ p=1/0.9$ . Top: Pole-zero diagram. Bottom: Corresponding impulse response, assuming the region of convergence includes the unit circle in the $ z$ plane.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2019-06-25 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University