Poisson Window

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Spectral Audio Signal Processing

Spectrum Analysis Windows

Matlab for the Bartlett Window:

Hann-Poisson Window

The impulse signal is the shortest pulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). The impulse is typically denoted by the Greek delta symbol δ and is often called the 'Dirac delta function', after Paul Dirac who pioneered its use (in addition to being a lead contributor in the development of Quantum Mechanics). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

Techniques for system identification are concerned with estimating the coefficients of a filter given observations of its input and output signals. — Click for https://ccrma.stanford.edu/~jos/mdft/FIR_System_Identification.html

The impulse response of a system is its output signal in response to the impulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

A causal signal is any signal that is zero prior to time zero. Thus, if x(n) denotes the signal amplitude at time (sample) n, the signal x is said to be causal if x(n)=0 for all n < 0. — Click for https://ccrma.stanford.edu/~jos/filters/Causal_Recursive_Filters.html

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The sampling interval (or sampling period) of a discrete-time signal is defined as the reciprocal of the sampling rate. Since the sampling rate is normally in units of samples per second (Hz), the sampling interval is normally in units of seconds. See also: sampling rate, sampling theorem — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theorem.html

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An exponential is any function of the form Ae^-t/τ, where t is the independent variable (often denoting time in seconds), A is an arbitrary scale factor, τ is called the time constant, and e is Euler's number (2.718281828459...). Many systems in nature exhibit exponential decay over time, because an exponential decay is produced whenever the rate of decay is proportional to how much is left. — Click for https://ccrma.stanford.edu/~jos/mdft/Exponentials.html

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Spectral Audio Signal Processing

Spectrum Analysis Windows

Matlab for the Bartlett Window:

Hann-Poisson Window

Poisson Window

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

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

(4.33)

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

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

(4.34)

where

denotes the sampling interval in seconds.

**Figure 3.19:** The Poisson (exponential) window.
$\includegraphics[width=3.5in]{eps/poissonwindow}$

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 :

$\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 $r\isdef e^{-\frac{\alpha}{ M/2 }}$)}\\ &=& \sum_{n=0}^\infty h(n) z^{-n} r^{n} = \sum_{n=0}^\infty h(n) (z/r)^{-n} \\ &=& H\left(\frac{z}{r}\right) \end{eqnarray*}$

Thus, the unit-circle response is moved to $\vert z\vert=1/r$ . This means, for example, that marginally stable poles in now decay as $r^{-n}=e^{-\alpha n/(M/2)}$ in .

The effect of this radial -plane contraction is shown in Fig.3.20.

**Figure 3.20:** Radial contraction of the unit circle in the plane by the Poisson window.
$\includegraphics[width=3.5in]{eps/zplane2}$

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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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Poisson Window

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University