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Data Tapering Windows
Data tapering windows are used to reduce spectral leakage in Fourier spectra.
The following fixed shape data tapering windows are available in PeakLab:
· None
· Welch
· Bisquare
· Bartlett
· cs2 Hann
· Tukey-Hann
· cs2 Hamming
· Bartlett Mod
· cs3 Nuttall C3
· cs3 Blackman
· cs3 BHarris 3
· cs3 Nuttall C1
· cs3 BlckmnExct
· cs3 BHarris min
· cs3 Nuttall min
· cs4 Nuttall C5
· cs4 BHarris 4
· cs4 Nuttall C3
· cs4 Nuttall C1
· cs4 BHarris min
· cs4 Nuttall min
The following adjustable shape data tapering windows are available in PeakLab:
· Beta
· csx maxRolloff
· Kaiser-Bessel
· VanderMaas
· Chebyshev
· Chebyshev Appr
· Slepian DPSS
· Gaussian
· Tapered-Cosine
Welch
1.0-(((n-1)-2*i)/(n-1)*((n-1)-2*i)/(n-1)), i=0..n-1
Bisquare
(1.0-(abs(i-0.5*n+0.5))^2/(0.5*n-0.5)^2)^2, i=0..n-1
Bartlett
2*i/(n-1), i=0..(n-1)/2
2-2*i/(n-1), i=(n-1)/2+1..n-1
cs2 Hann
0.5-0.5*cos(2*Pi*i/(n-1)), i=0..n-1
Tukey-Hann
Dirichlet(theta)=sin((2*Pi+0.5)*theta)/(2*Pi*sin(0.5*theta))
0.25*Dirichlet(2*(i/(n-1))-1.0-0.5)+(1.0-2.0*0.25)*Dirichlet(2*(i/(n-1))-1.0)+0.25*Dirichlet(2*(i/(n-1))-1.0+0.5),
i=0..n-1
cs2 Hamming
0.53836-0.46164*cos(2*Pi*i/(n-1)), i=0..n-1
Bartlett Mod
(sin(Pi*(2*(i/(n-1))-1)))^2/(2*Pi*sin((i/(n-1))-0.5))^2, i=0..n-1
cs3 Nuttall C3
0.375-0.5*cos(2*Pi*i/(n-1))+0.125*cos(4*Pi*i/(n-1)), i=0..n-1
cs3 Blackman
0.42-0.5*cos(2*Pi*i/(n-1))+0.08*cos(4*Pi*i/(n-1)), i=0..n-1
cs3 BHarris 3
0.44959-0.49364*cos(2*Pi*i/(n-1))+0.05677*cos(4*Pi*i/(n-1)), i=0..n-1
cs3 Nuttall C1
0.40897-0.5*cos(2*Pi*i/(n-1))+0.09103*cos(4*Pi*i/(n-1)), i=0..n-1
cs3 BlckmnExct
0.42659071367153912296-0.49656061908856405847*cos(2*Pi*i/(n-1))+0.076848667239896818573*cos(4*Pi*i/(n-1)), i=0..n-1
cs3 BHarris min
0.42323-0.49755*cos(2*Pi*i/(n-1))+0.07922*cos(4*Pi*i/(n-1)), i=0..n-1
cs3 Nuttall min
0.4243801-0.4973406*cos(2*Pi*i/(n-1))+0.0782793*cos(4*Pi*i/(n-1)), i=0..n-1
cs4 Nuttall C5
0.3125-0.46875*cos(2*Pi*i/(n-1))+0.1875*cos(4*Pi*i/(n-1))-0.03125*cos(6*Pi*i/(n-1)), i=0..n-1
cs4 BHarris 4
0.40217-0.49703*cos(2*Pi*i/(n-1))+0.09892*cos(4*Pi*i/(n-1))-0.00188*cos(6*Pi*i/(n-1)), i=0..n-1
cs4 Nuttall C3
0.338946-0.481973*cos(2*Pi*i/(n-1))+0.161054*cos(4*Pi*i/(n-1))-0.018027*cos(6*Pi*i/(n-1)), i=0..n-1
cs4 Nuttall C1
0.355768-0.487396*cos(2*Pi*i/(n-1))+0.144232*cos(4*Pi*i/(n-1))-0.012604*cos(6*Pi*i/(n-1)), i=0..n-1
cs4 BHarris min
0.35875-0.48829*cos(2*Pi*i/(n-1))+0.14128*cos(4*Pi*i/(n-1))-0.01168*cos(6*Pi*i/(n-1)), i=0..n-1
cs4 Nuttall min
0.3635819-0.4891775*cos(2*Pi*i/(n-1))+0.1365995*cos(4*Pi*i/(n-1))-0.0106411*cos(6*Pi*i/(n-1)), i=0..n-1
Beta
alpha=main lobe width
(4*(1-i/(n-1))*i/(n-1))^(-3.218913776512187+2.760793796409310*alpha), i=0..n-1
csx maxRolloff
alpha=main lobe width
This window produce the maximum rolloff characteristics.
(abs(0.5*(1-cos(2*Pi*i/(n-1)))))^(alpha-1), i=0..n-1
Kaiser-Bessel
alpha=main lobe width, I0 is Modified Bessel
I0(sqrt((alpha*alpha-1.0)/0.10132118361)*sqrt(i*(2*((n-1)/2)-i))/((n-1)/2))/I0(sqrt((alpha*alpha-1.0)/0.10132118361)),
i=0..n-1
VanderMaas
alpha=main lobe width, I1 is Modified Bessel
0.5*I1(2*(0.5*Pi*sqrt(4*alpha*alpha-1))*sqrt((i/(n-1))*(1-(i/(n-1)))))/(sqrt((i/(n-1))*(1-(i/(n-1))))*I1(0.5*Pi*sqrt(4*alpha*alpha-1))),
i=0..n-1
Chebyshev
This window produces the minimum sidelobe leakage, but offers zero rolloff. It is constructed in the frequency domain and inverted to produce the time domain data. For more information, see program 5.2 in IEEE Programs for Digital Signal Processing, IEEE Press, 1979. Unlike the alpha in the other adjustable windows, the Chebyshev adjustable parameter sets the sidelobe level in -dB. PeakLab offers the Chebyshev window from -30dB to -150dB.
Chebyshev Appr
This is an PeakLab approximation to the Chebshev window that is valid between -50 and -130 dB. The main application is in in the Fourier Spectrum of Unevenly Sampled Data option.
Slepian DPSS
This is the first of the Slepian discrete prolate spheroidal sequences, or eigentapers, with alpha equal to approximately the main lobe width. For more information, see Jonathan Lees and Jeffrey Park, "Multiple Taper Spectral Analysis", Computers and Geosciences, v21, p199, 1995.
Gaussian
alpha=main lobe width
exp(-4*ln(2)*((i-((n-1)/2))/((n-1)*((2.453274901281656+alpha*0.3202556336455866)/(1.0+alpha*(2.348619671799226)))))^2),
i=0..n-1
Tapered-Cosine
taper ranges from Hann with alpha=0.5 to rectangular as alpha approaches 0
if(i>(int)(alpha*n+0.5) and i<(int)((1.0-alpha)*n-0.5)), 1.0, 0.5-0.5*cos((Pi/alpha/(n-1))*(if(i>0.5*n,
n-i-1, i)))), i=0..n-1
The following reference may be of interest: Albert H. Nuttall, "Some Windows with Very Good Sidelobe Behavior", IEEE Trans. ASSP, v29-1, Feb. 1981.
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