IEEE Solid-States Circuits Magazine - Winter 2022 - 17

The only thing that Figure 1(b) does
well is deplete the ink cartridges in
your printer. If you zoom in on the
raw data, you are rewarded with
something that looks like a bunch
of random dots. Except for very lowfrequency
inputs, the sampling is too
sparse to see anything useful. If you
slow down the input so that you can
actually detect a problem in the time
data, all of the distortion and many of
the other defects disappear.
What you need is another way
to see the high-frequency time-do -
main data.
Let's Do the Time Warp Again
OK,
this first trick depends critically
on coherent sampling, where
the ADC signal input and sampling
clock are phase locked together.
This is the preferred setup for ADC
testing, but if, for some reason, you
are not doing this, and you are windowing
your data before calculating
the spectrum, this stunt won't
work correctly without substantial
extra effort. Also, we'll consider only
single-tone inputs. Multitone inputs
aren't much fun here.
Although we'll be looking at the
time domain, data captured for
FFT purposes are perfect. You probably
remember that the criteria for
unwindowed FFTs are these:
■ The number of samples must be a
power of 2, e.g., 210
= 1,024 samples
(required for the FFT algorithm).
■ The record must contain an integral
number of signal cycles, exactly
(the Fourier transform assumes
the input record is periodic).
■ The signal frequency and the
sampling frequency must be " mutually
prime, " meaning that they
have no common factors.
This last constraint is important
here because it guarantees that each
data point samples a unique phase
of the input signal. If you get this
wrong, you'll find that you are sampling
the same points over and over,
and your spectrum looks like a comb
with most of the teeth missing. That
is, there are many empty frequency
bins. If you get it right, each sample
represents a unique phase point of
the input fundamental, and that's
important for what happens next.
If you're a bit rusty on all of this FFT
stuff, Figure 2 might help. Figure 2(a)
represents samples of a sine wave as
points around a circle. In this example,
the sampling rate is high, i.e., near
Nyquist, so that there are only three
samples per cycle, and it takes about
three cycles of the input sine wave to
acquire eight data points. The green
numbers indicate the sequence in
which the samples are taken. The time
record for these same data is shown
in Figure 2(b) as the green line, which
is unintelligible.
Our first trick is to sort the data
by phase. (Please do not confuse this
" phase " with any phase data in the
FFT output. In this case, we mean
" phase of the input sine wave at the
time of each sample. " ) Instead of
plotting the data points in the order
in which they are taken, we plot them
in order of phase, as indicated by the
blue letters in Figure 2(a). When we do
that, we get a picture that looks a lot
more like a sine wave in Figure 2(b)
(the blue trace). You'll remember that
this is essentially what an oscilloscope
does when displaying a highfrequency
input: the waveform looks
highly oversampled, but it's not.
Nevertheless, the data are the real
deal. They're not interpolated or estimated;
they're just presented in a different
order that is more intelligible
to the human eye.
This rearrangement of the data
is labeled " phase-ordered sequence "
in the plots because I couldn't think
of a better name. I used to refer to it
as the " unscrambled " waveform, but
technically it is a scrambled version
of the real-time sequence. You might
call it a " unit cycle, " but then it would
be confused with what clowns ride in
the circus. Maybe you could call it an
" adolescent vector " because it's just
going through a phase. Anyway, the
floor is open for nominations.
If we use this " unscrambling "
trick on the data of Figure 1, we get
the result shown in Figure 3. (Please
see " It's Just a Jump to the Left "
for more details on the procedure.)
Notice anything? The defects that
were hidden in both the FFT and the
raw time-domain data now stand
out like a fish on a bicycle. Yes, this
synthetic example is a bit hokey,
but it does illustrate the revealing
power of the phase-ordered sine
wave. The eye instantly detects anything
out of place.
Attack of the 50-Foot Harmonics
The phase-ordered waveform can
help you figure out what's causing
spurious harmonics too, but it may
need a little help. Unless you've got
a massive distortion problem, the
effect on the sine wave can be hard
or impossible to see. Even at -40 dBc,
the harmonic amplitude is only 1%
of the fundamental. It's going to be
subtle. There are two ways to deal
with this.
The first way is to plot the harmonics
separately from the fundamental.
7
2
d
5
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f
(a)
3
g
h
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(b)
FIGURE 2: (a) A phase diagram for sampling a sine wave. Numbers indicate the order in
which samples are taken. Letters indicate the order in which the data are rearranged to
yield a phase-ordered sequence. (b) Original data sequence (green) and phase-ordered data
sequence (blue).
IEEE SOLID-STATE CIRCUITS MAGAZINE WINTER 2022
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IEEE Solid-States Circuits Magazine - Winter 2022

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