Infrasonicon icon. Click here for home page.INFRASONICON: Time-Compressing Infrasonic Recordings to Discover New Sounds, by Clark Huckaby

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.

Figure 1: Photos of flywheel used in recording
Figure 1. 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
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.

Figure 3: Composite of recorded waveforms and spectrogram of rotating flywheel
Figure 3. 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 Audacity 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 software.

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 time-compressed sound-file.

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 wheel 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 repeated periodically.

As 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 frequency scale.

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.

Figure 4: Interpretation of aliases visible in Figure 3
Figure 4. 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, or even stop if the propeller's RPM is an integer multiple of the video refresh rate.

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 considered here, 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.

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