LabPlot/DataAnalysis/FourierTransfromation

Fourier Transform

Available window functions are

Rectangular (uniform)

$w(n)=1$

Triangular

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

Triangular (Bartlett)

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

Triangular (Parzen)

$w(n)=1-{\frac {2}{N+1}}\left|n-{\frac {N-1}{2}}\right|$

Welch (parabolic)

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

Cosine

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

Bartlett-Hann

$w(n)=0.62-0.48\left|{\frac {n}{N-1}}-0.5\right|-0.38\cos \left({\frac {2\pi n}{N-1}}\right)$

Lanczos

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

Higher-order generalized cosine window functions:

Hann (raised cosine)

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

Hamming

$w(n)=0.54-0.46\cos \left({\frac {2\pi n}{N-1}}\right)$

Blackman

$w(n)=0.42-0.5\cos \left({\frac {2\pi n}{N-1}}\right)+0.08\cos \left({\frac {4\pi n}{N-1}}\right)$

Nuttall

$w(n)=a_{0}+a_{1}\cos \left({\frac {2\pi n}{N-1}}\right)+a_{2}\cos \left({\frac {4\pi n}{N-1}}\right)+a_{3}\cos \left({\frac {6\pi n}{N-1}}\right),a_{0}=0.355768,a_{1}=-0.487396,a_{2}=0.144232,a_{3}=-0.012604$

Blackman-Nuttall

$w(n)=a_{0}+a_{1}\cos \left({\frac {2\pi n}{N-1}}\right)+a_{2}\cos \left({\frac {4\pi n}{N-1}}\right)+a_{3}\cos \left({\frac {6\pi n}{N-1}}\right),a_{0}=0.3635819,a_{1}=-0.4891775,a_{2}=0.1365995,a_{3}=-0.0106411$

Blackman-Harris

$w(n)=a_{0}+a_{1}\cos \left({\frac {2\pi n}{N-1}}\right)+a_{2}\cos \left({\frac {4\pi n}{N-1}}\right)+a_{3}\cos \left({\frac {6\pi n}{N-1}}\right),a_{0}=0.35875,a_{1}=-0.48829,a_{2}=0.14128,a_{3}=-0.01168$

Flat top

$w(n)=a_{0}+a_{1}\cos \left({\frac {2\pi n}{N-1}}\right)+a_{2}\cos \left({\frac {4\pi n}{N-1}}\right)+a_{3}\cos \left({\frac {6\pi n}{N-1}}\right)+a_{4}\cos \left({\frac {8\pi n}{N-1}}\right),a_{0}=1,a_{1}=-1.93,a_{2}=1.29,a_{3}=-0.388,a_{4}=0.028$