Hann function

It is a window function used to perform Hann smoothing.

is given by: For digital signal processing, the function is sampled symmetrically (with spacing

is given by: Using Euler's formula to expand the cosine term in

we can write: which is a linear combination of modulated rectangular windows: Transforming each term: The Discrete-time Fourier transform (DTFT) of the

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

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

From the expression immediately above, it is easy to see that only 3 of the N DFT coefficients are non-zero.

[5][c][d] The function is named in honor of von Hann, who used the three-term weighted average smoothing technique on meteorological data.

[6][2] However, the term Hanning function is also conventionally used,[7] derived from the paper in which the term hanning a signal was used to mean applying the Hann window to it.