Quantum reflection

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.

See also

• Apodization
• Raised cosine distribution
• Window function

References

• Template:Cite doi

External links

• Hann function at MathWorld

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