Poisson Window
The Poisson window (or more generically exponential window) can be written
![]() |
(4.33) |
where


![]() |
(4.34) |
where

The Poisson window is plotted in Fig.3.19. In the
plane, the Poisson window has the effect of radially contracting
the unit circle. Consider an infinitely long Poisson window
(no truncation by a rectangular window
) applied to a causal
signal
having
transform
:
![\begin{eqnarray*}
H_P(z) &=& \sum_{n=0}^\infty [w(n)h(n)] z^{-n} \\
&=& \sum_{n=0}^\infty \left[h(n) e^{- \frac{ \alpha n}{ M/2 }}\right] z^{-n}
\qquad\hbox{(let $r\isdef e^{-\frac{\alpha}{ M/2 }}$)}\\
&=& \sum_{n=0}^\infty h(n) z^{-n} r^{n}
= \sum_{n=0}^\infty h(n) (z/r)^{-n} \\
&=& H\left(\frac{z}{r}\right)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img450.png)
Thus, the unit-circle response is moved to
. This means, for
example, that marginally stable poles in
now decay as
in
.
The effect of this radial
-plane contraction is shown in Fig.3.20.
The Poisson window can be useful for impulse-response modeling by poles and/or zeros (``system identification''). In such applications, the window length is best chosen to include substantially all of the impulse-response data.
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Hann-Poisson Window
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Bartlett (``Triangular'') Window