Time-Compressing Infrasonic Recordings to Discover New Sounds, by Clark
Analysis of Flywheel Recording Reveals Aliasing
While introducing the Optical Probe in Overview Part 3,
I mentioned that one way to record a rotating object is to
spot-illuminate it with a laser and record the reflected light. A
flywheel accelerating then coasting to a stop is an example (hear the
time-compressed soundfile on the Sound Example Page or just click here: FlywheelX165.wav [3.5 sec]). On the present page are some details about this
recording and its high-resolution spectral analysis, which reveals aspects of Data Converter performance. You may also want to check out my related demonstration of tones made by repeating noise motifs.
How Recording was Made. The photos below (Figure 1)
show the 6-inch diameter brass wheel and its aluminum frame; steel pins
project from the frame in the plane of the wheel. It's an old assembly,
probably from the 1950s or -60s, which belonged to my
late father. Most likely it was used to demonstrate angular momentum
and reaction wheel (gyroscope) effects. Each exposed length of the
steel axle has a hole, so a pull string can be used for acceleration; I
used my hand instead. I did not service the ball bearings; the massive,
well-balanced wheel probably performed better when the
bearings were in new, well-lubricated condition.
Photos of 6-inch diameter flywheel used in the recording. Left image
shows overall assembly. Right image is close-up showing 3/4-inch wide outer
surface (rim) of wheel, which was the target of laser beam.
A sketch of the setup for recording the flywheel is shown below (Figure 2). While the wheel's patina (tarnish) helped to limit specular reflection of the laser, the Optical Probe
was placed off-axis to pick up scattered light. Variations in light
intensity falling on the probe are due to random
differences in the wheel's surface texture and patina. Recording was done in a
darkened room. The laser pointer module was powered at constant current
using a Heathkit IP-28 regulated DC power supply. I chose a current
setting (i.e., laser intensity) that placed the average scattered light
intensity at about mid-scale in the recorded waveform. Note
that I used the 50-Hz low-pass filter setting in the Data Converter.
Figure 2. Drawing of setup for recording flywheel, showing the relative placement and orientation
of red laser module, wheel, and optical probe. Some recording channel
settings are also noted. Solid red line represents laser beam; dashed
red line shows path of scattered light from wheel to sensor chip.
Analytical Methods. A composite image of waveform (time domain) and spectrographic (frequency domain) plots of FlywheelX165.wav (3.5 sec) is shown in Figure 3
(below). The zoomed-in waveform at top left was obtained using Audacity
software. The rest of Figure 3 came from
Spectrogram 16 software. I highly recommend each of these excellent free software packages. Compare this spectrogram
with the one shown in Figure 6 of the Optical Probe overview page,
which used Digidesign's Sound Designer 2 software. Spectrogram 16 gives by far the better resolution and sensitivity.
The original time and frequency scales are shown in Figure 3, not the 165-fold time compressed scales for the WAVE file at "face value" (which you hear when you listen to FlywheelX165.wav [3.5 sec]). Spectrogram 16 software has a feature that proved helpful here: It allows correction for time expansion for users who analyze frequency-divided ultrasonic
signals recorded as WAVE files, for integer expansion factors of 2- to
64-fold. I specified 6-fold time expansion because 1000 times
1/165 is 6.06 (for a known 1-percent error), and simply re-labeled the
resulting "KHz" frequency scale as Hz and the "msec" time scale as sec.
Granted, it would be more convenient if the software could directly
re-scale time-compressed infrasonic signals, but I suppose ultrasonic bat
detectors are presently more popular than infrasonicons.
Waveform Results. Each
revolution of the wheel contributes a pattern of noise (a "motif") to the waveform.
This specific pattern repeats with increasing period as the wheel slows,
creating a periodic signal whose component frequencies (harmonics) decrease,
converging on 0 Hz when the wheel stalls. At maximum speed (immediately
after acceleration), the highlighted cycle of the zoomed-in waveform
(top left in Figure 3) indicates that one
revolution required 57 samples; this corresponds to 213 milliseconds
(sampling frequency was 268 Hz). Thus, maximum wheel speed was 4.70
revolutions per second (RPS). Of course this is the same as the
maximum fundamental frequency: 4.70 Hz. After this maximum, the wheel
coasted for 353 seconds, decelerating at a constant -0.0133 RPS per
second until stall.
Composite of recorded waveforms and spectrogram of rotating flywheel (FlywheelX165.wav [3.5 sec]
The scales are labeled with original, not time-compressed, time and
frequency. Time 0 is the top of the sound-file. The
zoomed-in waveform at top left shows about 4.5 cycles of the signal
when the wheel was at maximum speed; it was screen-captured from
software. One complete cycle is highlighted, represented by 57
samples. The rest of the figure (waveform overview, spectrogram, and
color scale) is from Spectrogram 16
Directly above the spectrogram in Figure 3 is a time-aligned overview of the waveform.
Beginning at 33 seconds into the sound-file (original time), the wheel
is accelerated to maximum speed during a 14-second interval. Both there
and in the subsequent 353-second deceleration period, signal amplitude
is inversely related to wheel speed. Amplitude at maximum speed is
-20.5 dBFS peak-to-peak; near the stall it is -15.1 dBFS peak-to-peak.
The 5.4-dB difference may be due to attenuation of high-frequency components according to the
channel's low-pass filter characteristic (3rd-order with 50-Hz corner).
By chance, the wheel stalled where the scattered light seen by the Optical Probe
was more intense than average. The resulting positive DC
offset sustains to the end of the recording (the channel is
DC-coupled), explaining the "click" one hears at the end of the
Spectrogram Results. The
spectrogram is dominated by a set of harmonics spaced 4.70 Hz apart at
maximum wheel speed (at 47 seconds). This frequency interval decreases
in proportion to speed as the wheel coasts. All of these hamonics
converge on 0 Hz as the wheel stalls (at 400 seconds). Amplitude
differences between the harmonics (see color scale at upper right in Figure 3) must be a function of the specific noise pattern that repeats with each
revolution. The time-compressed sound's timbre depends on
this pattern, which in turn is due to a random choice of the
circumferencial "track" that was spot-illuminated. A different track
would yeild a
somewhat different timbre. In a related experiment (click here), I demonstrate a
stochastic effect on timbre when randomly chosen noise motifs are
marked on the right side of the spectrogram, there are detectable 60-
and 120-Hz components independent of wheel movement, which is noise
from the mains AC service. Their amplitude is low compared to the
overall signal. It's not clear where this hum bled into the
channel. One candidate is the laser-module's power supply. In any case,
this artifact offers an internal standard to check the accuracy of the
Aliasing. A more troublesome artifact revealed in the
spectrogram is aliasing. Tracing the harmonic lines backward in time
from wheel-stall at 400 seconds, notice how the ones that reach the
Nyquist frequency (134 Hz, or one-half the 268-Hz sampling frequency)
"reflect" back into the lower frequency range. These reflections are
aliases, where the recorded frequency is the difference between the
sampling and input signal frequencies. Figure 4 (below) gives a diagram and equations describing the phenomenon. As noted in the Data Converter's technical page, complete elimination of aliasing would require a sharp cutoff (high-order) filter with a corner below the Nyquist frequency.
Interpretation of spectrogram of flywheel recording, showing how
aliases are created. Definitions of terms and basic equations are
listed to the left of the diagram. Recorded frequency equals input
frequency up to the Nyquist frequency; beyond that, aliases are
generated by the sampling process, and their frequency is the
difference between the sampling and input frequencies. A visual analogy
is available, for example, in videos of propeller-driven aircraft.
Propellers may appear to turn "in slow motion" due to aliasing,
stop if the propeller's RPM is an integer multiple of the video refresh
The Data Converter's
third-order low-pass filter was set at 50 Hz, thus as frequency increases past 100 Hz, the
roll-off should reach -18 dB per octave. Given this relatively low-order filter, some aliasing is
expected for a signal
source like this (rich in high harmonics). However, looking at Figure 3,
as one traces a recorded harmonic in reverse time as it rises past 100
Hz, reflects off the Nyquist limit, and drops to 68 Hz (the alias
of a 200-Hz input signal), the amplitude-indexed color does not change
as expected. With color increments on 3.33-dB steps (see scale at
upper right in Figure 3),
one would expect the low-pass filter to attenuate such a
harmonic through five shades of color as its frequency doubles (from 100 to 200 Hz). Could some of the input signal
have bypassed the filter, such as by coupling through the Data Converter's
power supply distribution network--even though I carefully tried to
prevent this, and the dynamic range of the 12-bit A-D converter is only
72 dB? I suppose this needs to be considered a possibility.
Role of Illumination Area. Besides the Data Converter's
50-Hz filter, another effective low-pass filter is due to the finite
width of the
laser-illuminated area ("spot") compared to the (speed-dependent)
length of wheel rim traversed per unit time. At 31 inches from the
laser module, the spot's diameter is about 2 mm (I'll assume
light intensity across spot is uniform). If the wheel has surface
features ("information") at all size scales that can differentially
scatter light (evidence that this may not be a good assumption is suggested at this link), there should be a corner frequency near 1125 Hz at
maximum wheel speed (4.70 RPS). Higher frequency
information would be progressively attenuated. For the recording
this filtering effect only becomes relevant when speed drops below 0.56
RPS, which occurs at the 358-second mark; there, 2 mm of wheel rim
(equal to width of laser spot) is traversed during each Nyquist period.
Notice in the spectrogram (Figure 3)
that loss of all detectable highest-frequency
information sets in at about 390 seconds; speed there is 0.14 RPS and
only 0.49 mm of rim was traversed per Nyquist period--only about 1/4th the
width of the laser spot.
Related content: Stochastic timbre differences in tones made by repeating noise motifs; Optical Probe overview and technical description.
Infrasonicon Home Page
Example Page Optical Probe: Overview Page; Technical Page
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