# Hann function

(Redirected from Hann window)
Hann function with B = 1.50 (left), and its frequency response (right)

The Hann function, named after the Austrian meteorologist Julius von Hann, is a discrete window function given by

$w(n)= 0.5\; \left(1 - \cos \left ( \frac{2 \pi n}{N-1} \right) \right)$

or

$w(n)= \sin^2 \left ( \frac{ \pi n}{N-1} \right)$

or, in terms of the haversine function,

$w(n)=\operatorname{haversin}\left(\frac {2 \pi n} {N-1} \right).$

## Spectrum

The Hann window is a linear combination of modulated rectangular windows $w_r = \mathbf{1}_{[0,N-1]}$. Thanks to the Euler formula

$w(n)= \frac{1}{2} \,w_r(n) -\frac{1}{4} e^{\mathrm{i}2\pi \frac{n}{N-1}} w_r(n) - \frac{1}{4}e^{-\mathrm{i}2\pi \frac{n}{N-1}} w_r(n)$

Thanks to the basic properties of the Fourier transform, its spectrum is

$\hat{w} (\omega) = \frac{1}{2} \hat{w}_r (\omega) - \frac{1}{4} \hat{w}_r \left(\omega + \frac{2\pi}{N-1}\right) - \frac{1}{4} \hat{w}_r \left(\omega - \frac{2\pi}{N-1}\right)$

with the spectrum of the rectangular window

$\hat{w}_r (\omega) = e^{-\mathrm{i} \omega \frac{N-1}{2}} \frac{\sin(N\omega/2)}{\sin(\omega/2)}$

(the modulation factor vanished if windows are time-shifted around 0)

## Name

Hann function is the original name, in honour of von Hann; however, the erroneous 'Hanning' function is also heard of on occasion, derived from the paper in which it was named, where 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.

## Use

The Hann function is typically used as a window function in digital signal processing to select a subset of a series of samples in order to perform a Fourier transform or other calculations.

i.e. (using continuous version to illustrate)

$S(\tau)= \int w(t+\tau)f(t) \, dt$

The advantage of the Hann window is very low aliasing, and the tradeoff is slightly decreased resolution (widening of the main lobe). If the Hann window is used to sample a signal in order to convert to frequency domain, it is complex to reconvert to the time domain without adding distortions.