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 L {\displaystyle L} and amplitude 1 / L , {\displaystyle 1/L,} is given by: w 0 ( x ) ≜ { 1 L ( 1 2 + 1 2 cos ⁡ ( 2 π x L ) ) = 1 L cos 2 ⁡ ( π x L ) , | x | ≤ L / 2 0 , | x | > L / 2 } . {\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 L / N {\displaystyle L/N} and amplitude 1 {\displaystyle 1} ): w [ n ] = L ⋅ w 0 ( L N ( n − N / 2 ) ) = 1 2 [ 1 − cos ⁡ ( 2 π n N ) ] = sin 2 ⁡ ( π n N ) } , 0 ≤ n ≤ N , {\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 N + 1 {\displaystyle N+1} samples, and N {\displaystyle N} can be even or odd. It is also known as the raised cosine window, Hann filter, von Hann window, Hanning window, etc.