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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}

Matlab for the Hann-Poisson Window 

function [w,h,p] = hannpoisson(M,alpha)
%HANNPOISSON  - Length M Hann-Poisson window
Mo2 = (M-1)/2; n=(-Mo2:Mo2)';
scl = alpha / Mo2;
p = exp(-scl*abs(n));
scl2 = pi / Mo2;
h = 0.5*(1+cos(scl2*n));
w = p.*h;

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Poisson Window