One-Pole Transfer Functions
We can apply the same analysis to a one-pole transfer function.
Let
denote any real or complex number:
![$\displaystyle H(z) \eqsp \frac{1}{1-pz^{-1}} \eqsp 1 + pz^{-1}+ pz^{-2}+ pz^{-3} + \cdots
$](http://www.dsprelated.com/josimages_new/filters/img1087.png)
![$ \vert pz^{-1}\vert<1$](http://www.dsprelated.com/josimages_new/filters/img1088.png)
![$ \vert z\vert>\vert p\vert$](http://www.dsprelated.com/josimages_new/filters/img1089.png)
![$ \vert p\vert<1$](http://www.dsprelated.com/josimages_new/filters/img778.png)
![$ z=p$](http://www.dsprelated.com/josimages_new/filters/img1090.png)
![$\displaystyle H(z) \;\longleftrightarrow\; h(n) = u(n)p^n
$](http://www.dsprelated.com/josimages_new/filters/img1091.png)
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*}](http://www.dsprelated.com/josimages_new/filters/img1092.png)
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.
Next Section:
Direct-Form I
Previous Section:
Geometric Series