We can apply the same analysis to a one-pole transfer function. Let denote any real or complex number:

The convergence criterion is now , or . For the region of convergence to include the unit circle (our frequency axis), we must have , which is our usual stability criterion for a pole at . The inverse

Now consider the rewritten case:

where the inverse *z* transform is the inverse *bilateral* *z* transform. In this
case, the convergence criterion is
, or
, and
this region includes the unit circle when
.

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
, 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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University