Very narrow FFT window functions What is the flat-top window function that provides the narrowest possible lobe width?
I'm doing FFT analysis and I need the resulting main lobe of a sine wave to be as narrow as possible but avoiding scalloping loss. I ask for flat-top functions because these are best for scalloping reduction.
I don't mind sidelobes, even the rectangular window is good enough for me if it wasn't for its massive scalloping...
Right now I'm using the SFT3M window from this paper:
http://www.rssd.esa.int/SP/LISAPATHFINDER/docs/Data_Analysis/GH_FFT.pdf
 A: I experimented a bit more with windows and noticed that whenever a window starts and ends below zero, it creates a flat top.
I took familiar windows and scaled them like: $w'_j = factor \cdot (w_j - 1) + 1$ (or in a simpler way if the formula allows that).
The results, sorted by main lobe width, are:
Name               Formula                  PSLL   3dB BW  Ripple
-----------------  -----------------------  -----  ------  ------
Flattop_Welch      1.7081(4x(1-x)-1)+1      11.66  2.22    0.039
Flattop_Halfsine   1.6025(sin(0.5z)-1)+1    12.61  2.25    0.038
Flattop_Connes     1.3185((4x(1-x))^2-1)+1  15.38  2.36    0.033
Flattop_Barlett    1-1.3525|2x-1|           16.62  2.38    0.032
Flattop_Hann       1-1.7124 cos(z)          17.36  2.42    0.031
SFT3M <reference>  <see pdf>                44.2   2.92    0.022

[edit] New calculation with a ripple of exactly 0.1dB  (if you find that acceptable) and slightly narrower center lobe. (table above was for minimum ripple.)
Name               Formula                  PSLL [dB]  3dB BW [Bins]
-----------------  -----------------------  ---------  -------------
Flattop_Welch      1.6849(4x(1-x)-1)+1      12.00      2.17
Flattop_Halfsine   1.5813(sin(0.5z)-1)+1    12.97      2.20
Flattop_Connes     1.3023((4x(1-x))^2-1)+1  15.83      2.31
Flattop_Barlett    1-1.3384|2x-1|           17.03      2.32
Flattop_Hann       1-1.6580 cos(z)          17.88      2.36

[edit2] First table updated with more accurate calculations.
Images from the first table:






A: I designed a 2-term flat-top window (equivalent to what would be called SFT2; there is no distinction between F and M anymore because there is only one degree of freedom).
I optimized (actually, eyeballed) the parameter for minimal pass-band ripple.
$w_j = 1 - [1.7028 \pm 0.0002] \cos(z)$.
It has a peak side lobe level of PSLL $\approx$ 17dB
