Hann function information

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

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\}.

^[a]

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

\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 $N+1$ samples, and $N$ 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.^[2]^[3]

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Hann function information

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