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 , 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.