Window function

the window function is an important term from digital signal processing. It concerns thereby a mathematical procedure, with which the cutout of a signal with a window function is weighted. Windows or Fensterung means to multiply the cutout (with finite <length> math< M> /math) of a signal by the selected window function. Frequently used window functions are the rectangle window, the of Hann window (also as Hanning windows or Hann windows well-known), the Hamming window, the Blackman window, the Bartlett window (also triangle window called) and the what window.

Application finds a window function above all if the available signal of a Frequenzanalyse is to be submitted by means of discrete Fourier transform. This is gleichbebeutend with the fact that the examined signal cutout repeats itself periodically. If the cutout without further computations is processed, then this corresponds the Fensterung with rectangle window. Here however the so-called leakage effect ( Leakage Effect) arises, with which the spectrum is distorted and “runs out”. This can be reduced by application of a suitable window function. The application of a window function provides for a same value at the edges of the window, and thus for a periodic continuation barness of the signal without jumps and breaks.

There are further different window functions of different complexity. With simple cost of computation the Hann window can be computed (also as Hanning windows well-known). The generally best, in addition, aufwändigste window is the emperor window. The selection of a suitable window function is application dependent, and by cost of computation and spectral characteristics of the window function is determined.

Table of contents

rectangle window

the rectangle function is in the entire window area 1 and outside of 0.

Rechteck-Fensterfunktion
Rectangle window function

Hamming window

function: <math> w (n) = 0 {,} 54 + 0 {,} 46 \ cdot \ cos \ left (\ frac {2 \ pi n} {M} \ right), \; n= \ frac {M} {2}, \ ldots, \ frac {M} {2}< /math>

is <math> M< /math> the window width and <math> n< /math> the current value of the input signal. This window function is designated after smelling pool of broadcasting corporations Hamming.

Hamming-Fensterfunktion
Hamming window function

of Hann window

function: <math> w (n) = \ frac {1} {2} \ left [1+ \ cos \ left (\ frac {2 \ pi n} {M} \ right) \ right], \; n= \ frac {M} {2}, \ ldots, \ frac {M} {2}< /math>

is <math> M< /math> the window width and <math> n< /math> the current value of the input signal.

The designation Hann window originates from the publication “Particular Pairs OF Windows.” of R. B. Blackman and John W. Tukey (New York publishes in “The Measurement OF power Spectra, From the POINT OF View OF Communications engineering”: Dover, 1959, pp. 98-99), which designated this after Julius of Hann. From this article also the wide-spread wrong designation “Hanning window” originates. There the use of the Hann window in verb form is called “hanning”, what is not no more formulated in this form nowadays.

Hanning-Fensterfunktion
Hanning window function

Blackman window

<math> w (n) =0,42 + 0.5 \ cdot \ cos \ frac {2 n \ pi} {M} + 0.08 \ cdot \ cos \ frac {4 n \ pi} {M}, \; n= \ frac {M} {2}, \ ldots, \ frac {M} {2}< /math>

Bartlett window

function: <math> w (n) =1 \ left|\ frac {2n-M} {M} \ right|, \; n=0, \ ldots, M< /math>

Bartlett-Fensterfunktion
Bartlett window function

what window

function: <math> w (n) =1 \ left [\ frac {2n-M} {M} \ right] ^2, \; n=0, \ ldots, M< /math>

is <math> M< /math> the window width and <math> n< /math> the current value of the input signal.

Welch-Fensterfunktion
What window function

comparison of the window functions

Fensterfunktionen überlagert
window functions overlays

comparison of the effects in the frequency range

window designation rel. Amplitude of the Nebenmaximums width of the major peak
rectangle - 13 railways 4 <math> \ pi< /math> /(M+1)
Bartlett - 25 railways 8 <math> \ pi< /math> /M
von Hann - 31 railways 8 <math> \ pi< /math> /M
Hamming - 41 railways 8 <math> \ pi< /math> /M
Blackman - 57 railways 12 <math> \ pi< /math> /M
 

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