## Hann-Poisson Window

**Definition:**

(4.35) |

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

### 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