IEEE Solid-States Circuits Magazine - Fall 2023 - 14

so zero phase means the frequency
component is at its maximum at
the start of the record.) Other than
simulations, you generally have no
synchronization between the input
waveform and the start of the digital
record. Thus, phases are completely
random. The phase is determined the
instant you push the button to record
the time data for the FFT input.
[Aside: Don't get confused here.
There often is synchronization between
the frequency of the signal and the
frequency of the sampling clock. In
fact, this is the very best way to do an
FFT. But there is no synchronization
between the input signal and the start
of the data record.]
So, the phase number for the fundamental
means absolutely nothing.
However, the relative phase rela20
40
60
80
-60
-40
-20
512 1,024 1,536 2,048
FFT Bin (Frequency)
(a)
180
270
90
-270
-180
-90
0 512 1,024 1,536 2,048
FFT Bin (Frequency)
(b)
180
270
90
-270
-180
-90
0 512 1,024 1,536 2,048
FFT Bin (Frequency)
(c)
2,560 3,072 3,584 4,096
2,560 3,072 3,584 4,096
2,560 3,072 3,584 4,096
tionship between the fundamental
and each of the harmonics is a gold
mine. OK, maybe it's more like a coal
mine, but it is a clue. Since the harmonics
are integer multiples of the
fundamental, their phases at the
zero-phase point of the fundamental
are uniquely defined and do not
depend on where the data record is
started. These " relative " phase relationships
carry information on how
the harmonics were created and
what singularities they encountered
since creation.
To examine the phase relation of
the harmonics to the fundamental,
the normal plots of phase versus frequency
like Figure 2(b) or (c) are not
much help. The data are visible, but
far from intuitive. Polar plots make
much more sense here because the
phase relationships are shown as
angles... which is what they are.
When we take the log magnitude
(in decibels) and phase data for
each point of the FFT and move it
to a polar plot, we get Figure 3(a). (If
you can't get past your rectilinear
fixation, consider that polar plots
are identical to a Cartesian plot of
imaginary versus real components,
as in Figure 1.) The frequency information
gets lost when you do a plot
like this, but when you're testing
something, you already know what
the frequencies are.
There are several points about
the plots in Figure 3 that may not be
immediately obvious:
■ Plotting the magnitude in decibels
is important to compress the
range of the data, just as it is in
the normal FFT plot, such as Figure
2(a). Otherwise, you won't see
anything except the fundamental.
■ If you plot the same range of magnitude
that you would in a normal
FFT plot, the noise floor will look
like a donut on the polar axes, and
it will waste a lot of valuable real estate
on the figure. For example, the
magnitude data for Figure 3 extend
from 0 down to -120 dB. But nobody
cares what the bottom of the
noise floor looks like, so the lower
limit on the radial axis for all plots
in Figure 3 is -90 dB. Choosing a
lower limit somewhere in the noise
floor yields a satisfying " star burst "
graphic rather than a " donut. "
FIGURE 2: (a) The normal view of FFT data shows only magnitude. No phase info at all.
(b) Phase data in their raw form are not much help. There's really no point in plotting them.
(c) Some instruments will plot only the phase of major spectral components, which is helpful,
but still doesn't add much insight.
14
FALL 2023
IEEE SOLID-STATE CIRCUITS MAGAZINE
■ FFT data points travel in pairs, just
like the complex singularities of
the Laplace transform. They have
to if they are going to represent
real quantities. So each component
(except dc) of the first Nyquist
zone has a complex conjugate twin
in the second Nyquist zone with
the same magnitude but phase
of the opposite sign. You can see
this in Figure 2(c) and in any of the
plots in Figure 3. For example, the
fundamental in Figure 3(a), marked
with " 1, " is at about 150°. If you
look at 210° (-150°) you will find its
twin. So you need to decide which
of the two vectors to track. If you
plot only the first Nyquist zone of
Phase (°)
Phase (°)
Magnitude (dB)

IEEE Solid-States Circuits Magazine - Fall 2023

Table of Contents for the Digital Edition of IEEE Solid-States Circuits Magazine - Fall 2023