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Chapters

Spectral Audio Signal Processing
    Spectrum Analysis Windows
       Poisson Window
          Definition:

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Definition:

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

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

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

where $ T$ denotes the sampling interval in seconds.

Figure 3.10: The Poisson (exponential) window.
\includegraphics[width=3.5in]{eps/poissonwindow}

The Poisson window is plotted in Fig.3.10. 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.3.11.

Figure 3.11: Radial contraction of the unit circle in the $ z$ plane by the Poisson window.
\includegraphics[width=3.5in]{eps/zplane2}

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.

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written by Julius Orion Smith III
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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