The Hann function, named after the Austrian meteorologist Julius von Hann

Julius von Hann

Julius Ferdinand von Hann was an Austrian meteorologist. He is seen as the father of modern meteorology.-Biography:...

, is a discrete probability mass function

Probability mass function

In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...

 given by


or


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

Richard Hamming

Richard Wesley Hamming was an American mathematician whose work had many implications for computer science and telecommunications...

.

Use

The Hann function is typically used as a window function

Window function

In signal processing, a window function is a mathematical function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation...

 in digital signal processing

Digital signal processing

Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...

 to select a subset of a series of samples in order to perform a Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

 or other calculations.

i.e. (using continuous version to illustrate)

The advantage of the Hann window is very low aliasing

Aliasing

In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...

, 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.

External links

• Hann function at MathWorld
  MathWorld
  MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...