Two-Pole Partial Fraction Expansion
Note that every real two-pole resonator can be broken up into a sum of two complex one-pole resonators:
where
![$ g_1$](http://www.dsprelated.com/josimages_new/filters/img284.png)
![$ g_2$](http://www.dsprelated.com/josimages_new/filters/img285.png)
![$ \left\vert p\right\vert\to1$](http://www.dsprelated.com/josimages_new/filters/img1416.png)
![$ 0\ll\angle p \ll \pi$](http://www.dsprelated.com/josimages_new/filters/img1417.png)
![$ \angle p =\pi/2$](http://www.dsprelated.com/josimages_new/filters/img1418.png)
![]() |
To show Eq.(B.7) is always true, let's solve in general for
and
given
and
. Recombining the right-hand side
over a common denominator and equating numerators gives
![$\displaystyle g = g_1 - g_1 \overline{p}z^{-1}+ g_2 - g_2 pz^{-1}
$](http://www.dsprelated.com/josimages_new/filters/img1420.png)
![\begin{eqnarray*}
g_1+g_2 &=& g\\
g_1 \overline{p} + g_2 p &=& 0.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img1421.png)
The solution is easily found to be
![\begin{eqnarray*}
g_1 &=& g \frac{p}{2\mbox{im}\left\{p\right\}}\\
g_2 &=& -g \frac{\overline{p}}{2\mbox{im}\left\{p\right\}}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img1422.png)
where we have assumed
im, as necessary to have a
resonator in the first place.
Breaking up the two-pole real resonator into a parallel sum of two complex one-pole resonators is a simple example of a partial fraction expansion (PFE) (discussed more fully in §6.8).
Note that the inverse z transform of a sum of one-pole transfer
functions can be easily written down by inspection. In particular,
the impulse response of the PFE of the two-pole resonator (see
Eq.(B.7)) is clearly
![$\displaystyle h(n) = g_1 p^n + g_2 \overline{p}^n,\qquad n=0,1,2,\ldots
$](http://www.dsprelated.com/josimages_new/filters/img1424.png)
![$ h(n)$](http://www.dsprelated.com/josimages_new/filters/img536.png)
![$ g_2=\overline{g}_1$](http://www.dsprelated.com/josimages_new/filters/img1425.png)
![$ \left\vert p\right\vert=1$](http://www.dsprelated.com/josimages_new/filters/img1426.png)
![$ h(n)$](http://www.dsprelated.com/josimages_new/filters/img536.png)
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Exercise
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Resonator Bandwidth in Terms of Pole Radius