How i got here: Contemplating some filters using gnuradio-companion with a simple flowgraph (simple enough to describe in words alone). Noticed the frequency response with a Rectangular filter was exactly the same as with a Hamming filter and also the response with a Kaiser filter (while varying Beta) seemed quite wrong.
The flowgraph: noise source -> throttle -> filter -> FFT
really basic. used the "convenience" blocks which are wrappers for firdes.
After quite a while of scratching my head at the odd results observed, then checking (and double-checking) Oppenheim[1999] and others, I wrote a little python to have a direct look at the window function coefficients:
#!/usr/bin/env python
from gnuradio import gr, audio
from math import pi
sample_rate = 192000
ntaps = 16
#channel_coeffs = gr.firdes.low_pass(1.0,sample_rate,50e3,4e3,gr.firdes.WIN_HAMMING,beta)
#print channel_coeffs
#channel_coeffs = gr.firdes.low_pass(1.0,10,1,1,gr.firdes.WIN_HAMMING,beta)
#print channel_coeffs
print "\n\nRectangular window function for {} samples\n".format(ntaps)
win_coeffs = gr.firdes.window(gr.firdes.WIN_RECTANGULAR,ntaps,0)
print win_coeffs
print "\n\nHamming window function for {} samples\n".format(ntaps)
win_coeffs = gr.firdes.window(gr.firdes.WIN_HAMMING,ntaps,0)
print win_coeffs
print "\n\nKaiser window function for {} samples\n".format(ntaps)
alpha = 1.0
print "Alpha = {}\n".format(alpha)
win_coeffs = gr.firdes.window(gr.firdes.WIN_KAISER,ntaps,alpha*pi)
print win_coeffs
alpha = 2.5
print "\nAlpha = {}\n".format(alpha)
win_coeffs = gr.firdes.window(gr.firdes.WIN_KAISER,ntaps,alpha*pi)
print win_coeffs
alpha = 8.0
print "\nAlpha = {}\n".format(alpha)
win_coeffs = gr.firdes.window(gr.firdes.WIN_KAISER,ntaps,alpha*pi)
print win_coeffs
alpha = 20.0
print "\nAlpha = {}\n".format(alpha)
win_coeffs = gr.firdes.window(gr.firdes.WIN_KAISER,ntaps,alpha*pi)
print win_coeffs
print "\nDone\n"
.. and here's the essential extract of output from unmodified v3.6.5.1 source:
Rectangular window function for 16 samples
(0.07999999821186066, 0.11976908892393112, 0.23219992220401764, 0.39785218238830566, 0.5880830883979797, 0.7699999809265137, 0.9121478199958801, 0.9899479150772095, 0.9899479150772095, 0.9121478199958801, 0.7699999809265137, 0.5880830883979797, 0.39785218238830566, 0.23219992220401764, 0.11976908892393112, 0.07999999821186066)
Hamming window function for 16 samples
(0.07999999821186066, 0.11976908892393112, 0.23219992220401764, 0.39785218238830566, 0.5880830883979797, 0.7699999809265137, 0.9121478199958801, 0.9899479150772095, 0.9899479150772095, 0.9121478199958801, 0.7699999809265137, 0.5880830883979797, 0.39785218238830566, 0.23219992220401764, 0.11976908892393112, 0.07999999821186066)
Kaiser window function for 16 samples
Alpha = 1.0
(1.0, 0.9949779510498047, 0.9800193309783936, 0.9554436802864075, 0.9217740297317505, 0.8797227740287781, 0.8301725387573242, 0.7741525173187256, 0.7128111124038696, 0.6473857760429382, 0.5791705250740051, 0.5094824433326721, 0.43962839245796204, 0.3708721101284027, 0.3044034540653229, 0.241310253739357)
... the Rectangular coefficients aren't right. And sure, it's really weird the coefficients are the same as for the Hamming window. But look at the Kaiser coefficients! (this was giving me an awful headache and bothering me to no end).
With a little help from octave and some quick cut-n-pastes, I was now contemplating graphs of window functions. The Kaiser window function didn't look right at all. It started a 1.0 and tapered toward zero. No starting near zero and tapering -up- toward 1.0 present. That can't be right, can it? Well, no, it can't. Hmm!
Glossing over the many hours it took from the start of this journey to its conclusion, I present the essential extract of output from v3.6.5.1 source modified by the diff attached to this message:
Rectangular window function for 16 samples
(1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
Hamming window function for 16 samples
(0.07999999821186066, 0.11976908892393112, 0.23219992220401764, 0.39785218238830566, 0.5880830883979797, 0.7699999809265137, 0.9121478199958801, 0.9899479150772095, 0.9899479150772095, 0.9121478199958801, 0.7699999809265137, 0.5880830883979797, 0.39785218238830566, 0.23219992220401764, 0.11976908892393112, 0.07999999821186066)
Kaiser window function for 16 samples
Alpha = 1.0
(0.6473857760429382, 0.7741525173187256, 0.8301725387573242, 0.8797227740287781, 0.9217740297317505, 0.9554436802864075, 0.9800193309783936, 0.9949779510498047, 1.0, 0.9949779510498047, 0.9800193309783936, 0.9554436802864075, 0.9217740297317505, 0.8797227740287781, 0.8301725387573242, 0.7741525173187256)
Alpha = 2.5
(0.2832930386066437, 0.47887080907821655, 0.5863785147666931, 0.6930355429649353, 0.7924286127090454, 0.8780854940414429, 0.9441056251525879, 0.9857622385025024, 1.0, 0.9857622385025024, 0.9441056251525879, 0.8780854940414429, 0.7924286127090454, 0.6930355429649353, 0.5863785147666931, 0.47887080907821655)
Alpha = 8.0
(0.01416657492518425, 0.083808533847332, 0.16599954664707184, 0.29149267077445984, 0.45759791135787964, 0.6461663842201233, 0.8243628740310669, 0.9529938697814941, 1.0, 0.9529938697814941, 0.8243628740310669, 0.6461663842201233, 0.45759791135787964, 0.29149267077445984, 0.16599954664707184, 0.083808533847332)
... there you have it, folks. I've heard it said "it's a jungle in there" (referring to the gnuradio codebase), to which I replied, "yeah and there are hidden stone walls, too". I'm glad to have had the opportunity to remove one of the stone walls.
[second attempt to post due to recent
gnuradio.org domain issue]