Hann-Poisson Window

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Spectral Audio Signal Processing

Spectrum Analysis Windows

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Matlab for the Hann-Poisson Window

The Fourier transform of a signal is a frequency-domain function — Click for https://ccrma.stanford.edu/~jos/mdft/

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A side lobe in the Fourier transform of a window function, such as the Chebyshev window, is any local maximum in the transform magnitude outside of the main lobe (response about dc). — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window_Side_Lobes.html

The Discrete Time Fourier Transform (DTFT) is the appropriate Fourier transform for discrete-time signals of arbitrary length. It can be obtained as the limit of a Discrete Fourier Transform (DFT) as its length goes to infinity. — Click for https://ccrma.stanford.edu/~jos/mdft/Discrete_Time_Fourier_Transform.html

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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The Fourier Transform (FT) gives the (complex) spectrum of a continuous-time signal of any length. — Click for https://ccrma.stanford.edu/~jos/mdft/Fourier_Transform_FT_Inverse.html

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Spectrum Analysis Windows

Poisson Window

Matlab for the Hann-Poisson Window

Hann-Poisson Window

Definition:

$\displaystyle w(n) = \frac{1}{2}\left[1 + \cos\left(\pi\frac{n}{\frac{M-1}{2}}\right)\right] e^{-\alpha\frac{\vert n\vert}{\frac{M-1}{2}}}$

(4.35)

**Figure 3.21:** Hann-Poisson window (upper plot, circles) and Fourier transform (lower plot). The upper plot also shows (using solid lines) the Hann and Poisson windows that are multiplied pointwise to produce the Hann-Poisson window.
$\includegraphics[width=\twidth]{eps/hannPoissonWindow}$

The Hann-Poisson window is, naturally enough, a Hann window times a Poisson window (exponential times raised cosine). It is plotted along with its DTFT in Fig.3.21.

The Hann-Poisson window has the very unusual feature among windows of having ``no side lobes'' in the sense that, for $\alpha\geq 2$ , the window-transform magnitude has negative slope for all positive frequencies [58], as shown in Fig.3.22. As a result, this window is valuable for ``hill climbing'' optimization methods such as Newton's method or any convex optimization methods. In other terms, of all windows we have seen so far, only the Hann-Poisson window has a convex transform magnitude to the left or right of the peak (Fig.3.21b).

**Figure 3.22:** Hann-Poisson Slope and Curvature
$\includegraphics[width=\twidth]{eps/hannPoissonSlope}$

Figure 3.23 also shows the slope and curvature of the Hann-Poisson window transform, but this time with $\alpha$ increased to 3. We see that higher $\alpha$ further smooths the side lobes, and even the curvature becomes uniformly positive over a broad center range.

**Figure 3.23:** Hann-Poisson magnitude, slope, and curvature, in the frequency domain, for larger $\alpha$ .
$\includegraphics[width=\twidth]{eps/hannPoissonSlope2}$

Subsections

Matlab for the Hann-Poisson Window

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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

Hann-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