IEEE Solid-States Circuits Magazine - Winter 2022 - 19

different waveform. A little bit of
harmonic amplification can make
the distortion more obvious and
point directly at the source of the
corruption. Too much amplification
can produce an irrelevant waveshape
and suggest distortion mechanisms
that are red herrings. Think
of it like a martini: a tiny bit of vermouth
makes the drink easily recognizable;
too much makes it into
a different cocktail altogether...and
not a very good one.
While we're in disclaimer mode,
I should point out the limitation
of all techniques related to phaseordered
data. Perhaps this is obvious,
but phase ordering will obscure
any errors that are dependent on
the order in which the samples are
taken. This would include things
like sample acquisition problems,
leftover charge from sample to sample,
interchannel crosstalk in interleaved
ADCs, input stage kickback,
and so on. In short, it will scramble
anything that behaves like " intersymbol
interference " in ADCs.
What phase-ordered techniques
excel at is finding data out of place.
MANIPULATING HARMONICS
If you're not used to working with FFT output, it can be a bewildering
dump of complex data. Don't worry, Uncle Chris has got your back. Figure
S1 is a flowchart representation of the process that created Figure 4.
The first thing to do is to make the phase-ordered sine wave from your
ADC data. (See " It's Just a Jump to the Left. " ) Now when you do the FFT,
you won't have to go hunting for the fundamental and the harmonics.
They'll all be clustered around dc. (It is possible to do the sorting in the
frequency domain without the phase-ordering step, but you must adjust
the phase of each data point. This complex procedure is only recommended
for those desiring intellectual calisthenics.)
The next step is to " filter " the spectrum by editing the FFT of the phaseordered
waveform. Just zero out all of the frequency bins you don't want
Original Time
Sequence
(Real)
PhaseOrdered
Sequence
(Real)
3512
2927
2342
1757
1172
587
10
Sort
3513
2928
to
plot and then take the IFFT of what's left to get back to the time domain.
(The data points in an FFT output are referred to as " bins " because they
hold all of the energy in each tiny slice of the spectrum.) In Figure S1, for
example, the fundamental and the first four harmonics are kept. The harmonics-but
not the fundamental-are amplified by a scale factor, which
is equal to 100 in Figure 4. All of the other bins, including dc (the very
first bin), are zeroed out. Remember that the raw FFT output is " double
sided, " so that the last five bins in the record are not zeroed out either.
Also remember that the FFT data are complex, meaning that each data
point (except possibly dc) will have both a real and an imaginary part, so
deal with it appropriately.
That's it. Not so bad, right?
FFT Results
(Complex)
Fundamental
2nd Harmonic
3rd Harmonic
4th Harmonic
5th Harmonic
6th Harmonic
8th Harmonic
9th Harmonic
Filtered FFT
Results (Complex)
Fundamental
Phase-Ordered Sequence
With Scaled
Harmonics (Real)
11 DC 01
2
3
4
5
6
7
2nd Harmonic × Scale
3rd Harmonic × Scale
4th Harmonic × Scale
5th Harmonic × Scale
82 7th Harmonic 02
9
3512
2927
2342
1757
1172
587
3513
2928
4088
4089
4090
4091
4092
4093
4094
4095
4096
●● ●● ●
●● ●● ●
●● ●● ●
●● ●● ●
●● ●● ●
●● ●● ●
9th Harmonic
8th Harmonic
7th Harmonic
6th Harmonic
5th Harmonic
4th Harmonic
3rd Harmonic
2nd Harmonic
Fundamental
FFT
Edit
IFFT
1170
585
4096
3511
2926
2341
1756
1171
586
5th Harmonic × Scale
4th Harmonic × Scale
3rd Harmonic × Scale
2nd Harmonic × Scale
Fundamental
FIGURE S1: The flowchart of the process used to create the exaggerated harmonics of Figure 4.
1170
585
4096
3511
2926
2341
1756
1171
586
IEEE SOLID-STATE CIRCUITS MAGAZINE WINTER 2022
19

IEEE Solid-States Circuits Magazine - Winter 2022

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