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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): 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 $
