Hann function

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 probability mass 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) = haversin[2πn/(N-1)].

Name

Hann function is the original name, in honour of von Hann; however, the erroneous 'Hanning' function is also heard of on occasion. 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.

See also

• Apodization
• Raised cosine distribution
• Window function

External links

• Hann function at MathWorld