IEEE Solid-States Circuits Magazine - Winter 2022 - 18

The way to do this is to take the FFT of
the phase-ordered sine wave, zero out
dc, the fundamental, and all of the frequency
bins you don't want to see, and
then take the inverse FFT (IFFT) to get
back to the time domain. (See " Manipulating
Harmonics " for more details.)
Since the harmonics are in a separate
plot, you can see how they might be
affecting the shape of your waveform.
An even better idea is to amplify
2,000
1,000
-1,000
-2,000
1,024
2,048
3,072
Phase-Ordered Sequence
FIGURE 3: The phase-ordered sequence for the data of Figure 1. Now you can see what's wrong!
4,096
the harmonics mathematically by
scaling the data for the second, third,
fourth, and so on but leaving the
fundamental as it is. Now when you
return to the time domain with the
IFFT, the effect of the distortion on
waveform shape is exaggerated and
should be more obvious.
A real-life application of this method
IT'S JUST A JUMP TO THE LEFT
The phase-ordering process shown in Figure 2 is ridiculously easy to do if your data record
meets the criteria for an unwindowed FFT listed in the text. First, recognize that the phase increments
from one sample to the next in the original data are
Dir#=2input fundamental frequency/sampling frequency
or
Dir#=
2numberofinput cycles in therecord/
number of samplesinthe record
So, we can calculate the phase of each sample relative to the start of the record. (The FFT
criteria guarantee that each of the samples will have a unique phase, and that they will evenly
cover the unit circle.)
Sample1phase =
Sample2phase
= T (modulo2 )ir
Sample3phase2 (modulo2 )
= # Tir
Sample 43# (modulo 2)
phase = Tir
... ands noo .
The modulo 2r operation is critical to rearranging the data. Think of it as wrapping the points
around the unit circle.
Finally, you use these phase numbers to sort the sample data. Or just plot the sample data versus
the phase numbers. Voila! Math savants in the audience will recognize that you don't need the 2 .r
The calculation could be done in cycles rather than radians, in which case, you'd use modulo 1.
can be seen in Figure 4. This figure
was supposed to show an example of
slew-rate limiting created by driving
a capacitor load with a weak source
follower. Asymmetrical drive, right?
But when the harmonics were exaggerated
(by a factor of 100× in this
case), I kept getting the waveform of
Figure 4, which certainly does not look
like a slew-rate problem. What's wrong?
It looks like backgate modulation, but
that can't be because the backgate
is connected to the source. Or so I
thought. An hour later... I discovered
that the backgate was floating! A tiny
schematic error had deceived me, but
the exaggerated harmonics revealed
the truth.
You may not be fascinated by my
heart-pounding tale of debugging a
simulation, but consider this: What
other technique could detect this
error? Subtracting the input from the

IEEE Solid-States Circuits Magazine - Winter 2022

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