Hann function

The Hann function of length  used to perform Hann smoothing, is named after the Austrian meteorologist Julius von Hann, is a window function given by:

   

For digital signal processing, the function can be sampled symmetrically as:

where the length of the window is  and N can be even or odd.  (see Window function#Hann and Hamming windows)  It is also known as the raised cosine window, Hann filter, von Hann window, etc.

Fourier transform 

The Hann window is a linear combination of modulated rectangular windows:

Using Euler's formula to expand the cosine term, we can write:

whose Fourier transform is just:

Discrete transforms 

The Discrete-time Fourier transform (DTFT) of the 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:

For even values of N, 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  for integer values of   From the expression immediately above, it is easy to see that only 3 of the N DFT coefficients are non-zero.  And from the other expression, it is apparent that all are real-valued.  These properties are appealing for real-time applications that require both windowed and non-windowed (rectangularly windowed) transforms, because the windowed transforms can be efficiently derived from the non-windowed transforms by convolution.

Name 

The function is named in honour of von Hann, who used the three-term weighted average smoothing technique on meteorological data.  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.

See also
 Window function
 Apodization
 Raised cosine distribution
 Raised-cosine filter

References

External links 
 Hann function at MathWorld

Category:Signal processing