Hann function

The Hann function, used to perform Hann smoothing, is named after the Austrian meteorologist Julius von Hann, is a window function given by: The Hann function, used to perform Hann smoothing, is named after the Austrian meteorologist Julius von Hann, is a window function given by: For digital signal processing, the function can be sampled symmetrically as: where the length of the window is N + 1 , {displaystyle N+1,} and N can be even or odd. (see Window function#Hann and Hamming windows) It is also known as the raised cosine window, Hann filter, von Hann window, etc. The Hann window is a linear combination of modulated rectangular windows: Using Euler's formula to expand the cosine term, we can write: whose Fourier transform is just: The Discrete-time Fourier transform (DTFT) of the N+1 length, time-shifted sequence is defined by a Fourier series, which also has a 3-term equivalent that is derived similarly to the Fourier transform derivation: For even values of N, the truncated sequence { w [ n ] ,   0 ≤ n ≤ N − 1 } {displaystyle {w, 0leq nleq N-1}} is a DFT-even (aka periodic) Hann window. Since the truncated sample has value zero, it is clear from the Fourier series definition that the DTFTs are equivalent. However, the approach followed above results in a significantly different-looking, but equivalent, 3-term expression: An N-length DFT of the window function samples the DTFT at frequencies f = k / N , {displaystyle f=k/N,} for integer values of k . {displaystyle k.} From the expression immediately above, it is easy to see that only 3 of the N DFT coefficients are non-zero. And from the other expression, it is apparent that all are real-valued. These properties are appealing for real-time applications that require both windowed and non-windowed (rectangularly windowed) transforms, because the windowed transforms can be efficiently derived from the non-windowed transforms by convolution.