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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 $ 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:

\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 $ 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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