Poisson Window in System Identification

In the z-plane, the Poisson window has the effect of contracting the spectrum toward zero inside 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 $H(z)$ :

$$\begin{eqnarray*} H_P(z) &=& \sum_{n=0}^\infty [w(n)h(n)] z^{-n} \\ &=& \sum_{n=0}^\infty \left[h(n) e^{- \frac{ \alpha n}{ M/2 }}\right] z^{-n} \qquad\hbox{(let $r\mathrel{\stackrel{\Delta}{=}}e^{\frac{ \alpha}{ M/2 }}$)}\\ &=& \sum_{n=0}^\infty h(n) z^{-n} r^{-n} = \sum_{n=0}^\infty h(n) (zr)^{-n} \\ &=& H( zr ) \\ \end{eqnarray*}$$

• Unit-circle response moved to $\vert z\vert=1/r < 1$
• Marginally stable poles now decay as $r^{-n}=e^{-\alpha n/(M/2)}$

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.