Definition:

$$w_P(n) = w_R(n)e^{- \alpha \frac{\vert n\vert}{ \frac{M-1}{2} }}$$

where $\alpha$ determines the time constant $\tau$:

$$\frac{\tau}{T} = \frac{M-1}{2\alpha}\quad\hbox{samples}$$

where $T$ denotes the sampling interval in seconds.

Figure 3.13: The Poisson (exponential) window.

The Poisson window is plotted in Fig.3.13. In the $z$ 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 $w_R$) applied to a causal signal $h(n)$ having $z$ transform $H(z)$:

$$\begin{eqnarray*} H_P(z) &=& \sum_{n=0}^\infty [w(n)h(n)] z^{-n} \\ &=& \sum_{... ..._{n=0}^\infty h(n) (z/r)^{-n} \\ &=& H\left(\frac{z}{r}\right) \end{eqnarray*}$$

The effect of this radial $z$-plane contraction is shown in Fig.3.14.

Figure 3.14: Radial contraction of the unit circle in the $z$ plane by the Poisson window.

The Poisson window can be useful 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.