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 formula_1 and amplitude formula_2 is given by: For digital signal processing, the function is sampled symmetrically (with spacing formula_4 and amplitude formula_5): which is a sequence of formula_7 samples, and formula_8 can be even or odd. (see ) It is also known as the raised cosine window, Hann filter, von Hann window, etc. Fourier transform. The Fourier transform of formula_9 is given by: Transforming each term: Discrete transforms. The Discrete-time Fourier transform  (DTFT) of the formula_7 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: The truncated sequence formula_14 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 formula_16 for integer values of formula_17 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. Name. The function is named in honor of von Hann, who used the three-term weighted average smoothing technique on meteorological data. However, the term "Hanning" function is also conventionally used, derived from the paper in  which the term "hanning a signal" was used to mean applying the Hann window to it. The confusion arose from the similar Hamming function, named after Richard Hamming.