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 ${\displaystyle L}$ and amplitude ${\displaystyle 1/L,}$ is given by:

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

which is a sequence of ${\displaystyle N+1}$ samples, and ${\displaystyle N}$ can be even or odd. It is also known as the raised cosine window, Hann filter, von Hann window, Hanning window, etc.

The Fourier transform of ${\displaystyle w_{0}(x)}$ is given by:

Using Euler's formula to expand the cosine term in ${\displaystyle w_{0}(x),}$ we can write:

which is a linear combination of modulated rectangular windows:

Transforming each term:

The discrete-time Fourier transform (DTFT) of the ${\displaystyle N+1}$ 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: