Quantum reflection

File:Window function and frequency response - Hann.svg

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

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

or

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

or, in terms of the haversine function,

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

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

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

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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