IEEE Solid-States Circuits Magazine - Fall 2023 - 16

ALIASING AND PHASE
What happens to the phase when a tone-either fundamental or
harmonic-is aliased? It's actually, pretty simple: if the unsampled
tone is in an odd-numbered Nyquist zone (I, III, V, etc.), nothing happens.
The phase of what you see in the 0-to-Nyquist FFT data is what
you'd expect. If the unsampled tone is in an even-numbered zone
(II, IV, etc.), the sign of the phase is reversed (and the tone seems to
be moving backward in the spectrum as the
frequency is increased).
Figure S1 struggles valiantly to make these
(a)
(b)
sign changes intuitively appealing. With appropriate
hand-waving, the FFT axis can be
thought of as repeating infinitely in both the
positive and negative directions. Given limited
page space, however, we will represent it
as a closed circle instead. See Figure S1(a). In
Figure S1(b), a single tone is represented with
its two complex conjugate pieces shown in
red and black. As the red component passes
into the next Nyquist zone, its conjugal partner
takes its place, reversing the sign of the
phase in Figure S1(c). When the input tone
exceeds the sampling frequency, as in Figure
S1(d), the result is still an alias because
it appears (on reconstruction) to be a lowfrequency
tone, but the sign of the phase is
back to normal.
Technically, you should make note of phase
(c)
reversals in the harmonics as you chase the
relative phase of distortion products across
the spectrum. But don't sweat it. If you get it
wrong, you will find an abrupt discontinuity
in the phase-versus-frequency curve, which is
easily fixed.
Two more pointers to help you sort all
(d)
of this out:
■ The phase change at the Nyquist boundary
is a sign reversal, NOT a phase shift of π!
Think complex conjugates.
■ Much of this stuff becomes easier to get
your head around if you think of an FFT
as extending from -fs
/2 to +fs
FIGURE S1: The wheel of frequency. (a) The FFT horizontal axis can be thought of
as a ring. (b) An input tone in the first Nyquist zone. The phase number associated
with the red component is what you'd expect. The black component is red's complex
conjugate twin, so its phase has the sign reversed. (c) Once the tone is greater
than the Nyquist frequency (fs/2), it passes into the second Nyquist zone, and its
conjugate appears as an alias in the first Nyquist zone. This means that the sign
of the phase has become inverted. (d) Increasing the input frequency even further
brings the red component back into the first Nyquist zone as a low-frequency alias.
The sign of the phase is now back to normal. (fs refers to the sampling frequency,
and the blue dashed arrows indicate how the components move with increasing
input frequency.)
/2, rather
than 0 to +fs (where fs is the sampling
frequency). lf you're OK with positive
and negative frequencies, you can see
the resemblance between Fourier and
his big brother Laplace.
16
FALL 2023
IEEE SOLID-STATE CIRCUITS MAGAZINE
IEEE Solid-States Circuits Magazine - Fall 2023

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