## Poisson Window

The *Poisson window* (or more generically *exponential window*)
can be written

(4.33) |

where determines the time constant :

(4.34) |

where denotes the sampling interval in seconds.

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
:

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.

**Next Section:**

Hann-Poisson Window

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Bartlett (``Triangular'') Window