Hann function

The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing or hanning. The function, with length ${\displaystyle L}$ and amplitude ${\displaystyle 1/L,}$ is given by:

${\displaystyle w_{0}(x)\triangleq \left\{{\begin{array}{ccl}{\tfrac {1}{L}}\left({\tfrac {1}{2}}+{\tfrac {1}{2}}\cos \left({\frac {2\pi x}{L}} ight) ight)={\tfrac {1}{L}}\cos ^{2}\left({\frac {\pi x}{L}} ight),\quad &\left|x ight|\leq L/2\\0,\quad &\left|x ight|>L/2\end{array}} ight\}.}$

For digital signal processing, the function is sampled symmetrically (with spacing ${\displaystyle L/N}$ and amplitude ${\displaystyle 1}$):

${\displaystyle \left.{\begin{aligned}w[n]=L\cdot w_{0}\left({\tfrac {L}{N}}(n-N/2) ight)&={\tfrac {1}{2}}\left[1-\cos \left({\tfrac {2\pi n}{N}} ight) ight]\\&=\sin ^{2}\left({\tfrac {\pi n}{N}} ight)\end{aligned}} ight\},\quad 0\leq n\leq N,}$

which is a sequence of ${\displaystyle N+1}$ samples, and ${\displaystyle N}$ can be even or odd. It is also known as the raised cosine window, Hann filter, von Hann window, Hanning window, etc.