The DFT - Synthesis Lectures on Signal Processing - page 149

which must be discriminated by performing a DFT on the composite signal (specific calls are given for
the examples discussed below).
Let's do several experiments in which there are two sinusoids in white noise. In the first experi-
ment, the sinusoids will have frequencies of 66 and 68 cycles and amplitudes of 1.0 each. These integral
frequencies coincide perfectly with FFT test correlators, so such frequencies are called “ on-bin .” Fre-
quencies that are not integers are called “ off-bin .” There is noise throughout the signal's spectrum that
contributes (undesirably) to all bins, helping to blur the magnitude distinctions between bins. The script
performs the same experiment for each of four different windows, namely, rectwin , kaiser , blackman , and
hamming . The experiment is to construct a test waveform having the two frequencies mixed with random
noise of standard deviation k , window the test waveform, and then compute the magnitude of the DFT.
This is repeated 30 times, the average taken of the DFT magnitudes, and the result plotted. Then the
next window is selected, the test waveform is constructed and evaluated 30 times, averaged, plotted, and
so forth. In this manner, a good idea can be obtained of the average performance in noise.
Our first call will be
LVxWindowingDisplay(1,66,68)
in which the first argument is the desired noise amplitude, the second argument is the first test frequency,
and the third argument is the second test frequency.
The result is shown in Fig. 3.40, plot (a). Here it can be seen that the rectwin window is by far
the best at separating the two test frequencies, which are marked with vertical dotted lines. The blackman
window, which has the widest central lobe, and the deepest skirt attenuation, is the poorest in this case,
with the kaiser ( 5 ) and hamming windows placing between the blackman and the rectwin . Note that the
two test frequencies in this case are “on-bin,” and hence are orthogonal to one another and therefore
cannot influence each other's DFT response.
Plot (b) of Fig. 3.40 shows the next case, close, nonintegral frequencies (66.5 and 68.6 ) . The call
used to create the plot was
LVxWindowingDisplay(1,66.5,68.6)
In this case, it is clear that no window is adequate to separate the two frequencies; there is simply
too much leakage from the off-bin test frequencies (and noise) into nearby (in fact, all) bins; the result is
a “washing-out” or “de-sharpening” of the DFT response to the two frequencies.
In the third case, we extend the spacing between the off-bin test frequencies to see if the situation
can be improved. The call used to create plot (c) of Fig. 3.40 was
LVxWindowingDisplay(1,66.5,70.7)
In this case, the rectwin is clearly the poorest performer, with kaiser ( 5 ) and hamming doing much
better.
• The conventional wisdom is that the rectwin is the poorest performer for general purposes, due to
its wideband leakage. The vast majority of frequencies in an unknown signal are likely to be off-bin
(a good assumption unless the contrary is known), and hence a nonrectangular window is likely to
be the better choice.
• When all frequencies being detected are orthogonal to each other, and each is “on-bin” for the DFT,
and noise levels are not excessively high, the rectangular window can perform well for frequency