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Poisson Window
Definition:
determines the time constant 
:
denotes the sampling interval in seconds.
![\includegraphics[width=3.5in]{eps/poissonwindow}](http://www.dsprelated.com/josimages_new/sasp/img400.png)
The Poisson window is plotted in Fig.3.13. 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 h(n) (z/r)^{-n} \\
&=& H\left(\frac{z}{r}\right)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp/img403.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.3.14.
![\includegraphics[width=3.5in]{eps/zplane2}](http://www.dsprelated.com/josimages_new/sasp/img407.png)
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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Next: Hann-Poisson Window
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Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.
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