Hann function

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

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

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

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

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