So What's Up with Repeated Poles?
In the previous section, we found that repeated poles give rise to polynomial amplitude-envelopes multiplying the exponential decay due to the pole. On the other hand, two different poles can only yield a convolution (or sum) of two different exponential decays, with no polynomial envelope allowed. This is true no matter how closely the poles come together; the polynomial envelope can occur only when the poles merge exactly. This might violate one's intuitive expectation of a continuous change when passing from two closely spaced poles to a repeated pole.
To study this phenomenon further, consider the convolution of two
one-pole impulse-responses
and
:
The finite limits on the summation result from the fact that both
![$ h_1$](http://www.dsprelated.com/josimages_new/filters/img782.png)
![$ h_2$](http://www.dsprelated.com/josimages_new/filters/img503.png)
![$\displaystyle \sum_{m=0}^n r^m = \frac{1-r^{n+1}}{1-r}
$](http://www.dsprelated.com/josimages_new/filters/img783.png)
![$ \,$](http://www.dsprelated.com/josimages_new/filters/img94.png)
![$\displaystyle h(n) = p_2^n \frac{1-(p_1/p_2)^{n+1}}{1-(p_1/p_2)}
= \frac{p_2^{n+1}-p_1^{n+1}}{p_2-p_1}
= \frac{p_1^{n+1}-p_2^{n+1}}{p_1-p_2}.
$](http://www.dsprelated.com/josimages_new/filters/img784.png)
![$ p_1$](http://www.dsprelated.com/josimages_new/filters/img785.png)
![$ p_2$](http://www.dsprelated.com/josimages_new/filters/img786.png)
![$ \left\vert p_1\right\vert>\left\vert p_2\right\vert$](http://www.dsprelated.com/josimages_new/filters/img787.png)
![$ h(n)$](http://www.dsprelated.com/josimages_new/filters/img536.png)
![$ p_1^n$](http://www.dsprelated.com/josimages_new/filters/img788.png)
![$ n$](http://www.dsprelated.com/josimages_new/filters/img89.png)
![$ \left\vert p_2\right\vert>\left\vert p_1\right\vert$](http://www.dsprelated.com/josimages_new/filters/img789.png)
![$ p_2^n$](http://www.dsprelated.com/josimages_new/filters/img790.png)
Going back to Eq.(6.14), we have
![]() |
(7.15) |
Setting
![$ p_1=p_2=p$](http://www.dsprelated.com/josimages_new/filters/img792.png)
![]() |
(7.16) |
which is the first-order polynomial amplitude-envelope case for a repeated pole. We can see that the transition from ``two convolved exponentials'' to ``single exponential with a polynomial amplitude envelope'' is perfectly continuous, as we would expect.
We also see that the polynomial amplitude-envelopes fundamentally
arise from iterated convolutions. This corresponds to the
repeated poles being arranged in series, rather than in
parallel. The simplest case is when the repeated pole is at , in
which case its impulse response is a constant:
![$\displaystyle \frac{1}{1-z^{-1}} \eqsp
1 + z^{-1}+ z^{-2}+ \cdots \;\longleftrightarrow\; [1,1,1,\ldots]
$](http://www.dsprelated.com/josimages_new/filters/img795.png)
![$\displaystyle h_1(n)\eqsp \sum_{m=0}^n 1\cdot 1 \eqsp n+1
$](http://www.dsprelated.com/josimages_new/filters/img796.png)
![\begin{eqnarray*}
h_2(n)&=&\sum_{m=0}^n (m+1)\cdot 1 \eqsp \frac{(n+1)(n+2)}{2}\...
...+1)(m+2)}{2}\cdot 1\eqsp \frac{(n+1)(n+2)(n+3)}{3!}\\
&\cdots&
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img797.png)
Next Section:
Example 2
Previous Section:
Impulse Response of Repeated Poles