Technical Note · Sine Vibration Control
Digital Tracking Sampling in Sine Vibration Control
Why measuring a sine in cycles rather than seconds gives cleaner control, why tracking the analog sample clock without anti-alias filters corrupts the data, and how digital tracking delivers order-domain data that stay alias-free at every drive frequency.
Key takeaways
- A sine controller always knows its own drive frequency, so the natural way to measure a sine is in cycles, not seconds: a constant number of samples per cycle and an integer number of cycles per measurement. This is order-domain, or tracking, sampling.
- Estimators working on fixed-rate data see a non-integer and changing number of cycles per frame. Leakage and sweep smearing must be managed with window, bandwidth and frame-length choices, especially at low frequencies and fast sweep rates.
- Older controllers tracked by changing the analog sample clock. The anti-alias filter must then track the clock too, and because tracking analog filters add cost and limit dynamic range, it was often compromised or left out.
- Without that filter, everything above half the tracked sample rate folds into the measurement. Harmonics land exactly on the fundamental, fixture resonances sweep through it, and broadband noise piles up. It is worst at low frequencies.
- Digital tracking keeps a fixed, well-filtered high-rate front end, then filters and tracks the rate digitally. The data stay alias-free at every drive frequency, with the full dynamic range of the converter.
What a sine controller has to measure
A sine test controls the amplitude of one frequency: the drive frequency. On every control and response channel the controller must measure the amplitude and phase of the component at that frequency, and ignore everything else on the channel: harmonics from the shaker and fixture, fixture resonances excited by those harmonics, rattles, and broadband noise. Our note on Tracking Filters in Sine Vibration Control shows why that matters and how the measurement method changes the delivered test level.
This note goes one level deeper, to how the data are sampled before any filter or estimator sees them. The controller has one great advantage here: it generates the drive, so it knows the drive frequency and phase exactly, at every instant. The question is how best to use that knowledge.
Fixed-rate sampling: measuring in seconds
The simplest architecture, A in Figure 1, samples every channel at a fixed rate behind a fixed anti-alias filter, then estimates the fundamental with a tracking filter or a frame-based calculation whose settings are in hertz. The data are alias-free, because the anti-alias filter never has to move. But the measurement is out of step with the signal it is measuring:
- Frames hold a non-integer number of cycles. A 1 s frame holds 37.4 cycles at 37.4 Hz. The partial cycle causes leakage: energy from harmonics and other tones spreads into the fundamental, and windows that reduce leakage also widen the measurement and cost time (see FFT Fundamentals: Resolution, Windowing and Leakage).
- The frequency changes within a frame during a sweep. The fundamental smears over several frequency bins and harmonics smear twice as far. More resolution means a longer frame and more smearing.
- Bandwidth set in hertz does not scale with the signal. Harmonic spacing equals the drive frequency, so a filter that rejects harmonics well at 500 Hz may barely reject them at 5 Hz. Proportional-bandwidth filters help, but the frame or filter is still not synchronous with the cycle.
In Figure 2 the drive moves 4.6 Hz while a 1 s frame is captured. Read as a peak in a 1 Hz-resolution frame, the fundamental appears 14% low and the second harmonic 35% low, because their energy is spread over several bins. A fixed-rate estimator can recover the amplitude with a wider band, a shorter frame, or by demodulating against the known drive phase, but it then has to manage leakage, bandwidth and sweep rate itself, and each choice costs noise rejection or test time. Order-domain data remove that trade-off.
Order-domain sampling: measuring in cycles
Suppose instead that the data are sampled at a fixed number of samples per cycle of the drive, say 64, however the drive frequency changes. Then:
- Every frame holds a whole number of cycles, so the fundamental and every harmonic fall exactly on analysis lines. There is no leakage between them.
- A sweep looks like a steady signal: the fundamental is always order 1 and the second harmonic always order 2 (green in Figure 2).
- Filter bandwidth and averaging are set in cycles, so harmonic rejection is the same at 5 Hz as at 2,000 Hz, and response time scales naturally with frequency.
This is the same principle used in order tracking of rotating machinery, where data are sampled at a constant number of samples per revolution so that shaft-related components stay sharp through a run-up. In sine control the "shaft" is the drive oscillator, and its phase is known perfectly.
The question is how to get samples at a constant number per cycle. There are two ways: change the converter's clock, or sample at a fixed rate and change the rate digitally.
Analog tracking sample clocks
The older method, B in Figure 1, generates a constant-level tracking signal at the drive frequency, called a COLA signal (constant output level amplitude), and multiplies its frequency with a phase-locked loop to clock the converters at N samples per cycle. With N = 32, a 20 Hz drive is sampled at 640 Hz, and a 2,000 Hz drive at 64,000 Hz.
The weakness is the anti-alias filter. A sampled system can only represent frequencies below half its sample rate. Everything above it must be filtered out before the converter, or it folds back into the measurement band and cannot be separated afterwards. With a tracking clock, half the sample rate is only N/2 times the drive frequency, and it moves. The anti-alias filter would have to move with it, on every channel, by a factor of several hundred across a typical sweep.
Why the filter is often missing. A tracking analog anti-alias filter must be built for every input channel, track the clock over a very wide range, and match its phase from channel to channel. Practical tracking filters, such as switched-capacitor designs, add cost and complexity per channel, add clock feedthrough, and have much less dynamic range than the converter behind them. Some analog-tracking designs therefore omitted the filter, or fitted a single fixed filter set for the highest sample rate, which gives little protection at low drive frequencies.
What goes wrong without the anti-alias filter
With the clock locked at N samples per drive cycle, aliasing follows simple rules:
Three things follow, and each corrupts a sine test in a different way.
1. Harmonics land on the fundamental
Harmonics of the drive alias to other whole orders. Orders N ± 1, 2N ± 1 and so on land exactly on order 1. They are coherent with the drive, so they add to the measured fundamental in amplitude and phase. No tracking filter, however narrow, and no amount of averaging can remove them, because they sit at the same frequency as the quantity being measured.
Smooth distortion has little energy at the 31st harmonic. Impacts do: a loose fixture bolt, a rattling component or a bottoming suspension produces a sharp spike every cycle, with energy spread over many harmonics. Table 1 simulates a 1 g sine with 10% second and 5% third harmonics and a 5 g, 0.3 ms impact once per cycle. The error depends on where the samples fall relative to the impact, which drifts during a test, so the result is a range. (The lower limit is the same at 32 and 64 samples per cycle because it occurs when no sample lands on the impact.)
| Drive frequency | True fundamental | Analog tracking, 32 samples/cycle | Analog tracking, 64 samples/cycle | Digital tracking |
|---|---|---|---|---|
| 20 Hz | 1.04 g | -4% to +26% | -4% to +11% | within ±0.01% |
| 50 Hz | 1.10 g | -9% to +20% | -9% to +6% | within ±0.01% |
| 100 Hz | 1.19 g | -16% to +10% | -3% to +2% | within ±0.01% |
At 20 Hz with 32 samples per cycle, the measured fundamental can be off by -4% to +26%. The controller responds by moving the drive to correct an error that is not real. Doubling the samples per cycle helps but does not remove the problem.
2. Fixture resonances sweep through the fundamental
Response that is not harmonically related to the drive, such as a fixture resonance ringing at its own frequency after repeated impacts, electrical pickup, or vibration from nearby machinery, aliases to a frequency that moves as the drive sweeps. Each time that alias crosses the fundamental, the measurement beats. In Figure 3, a 0.3 g component at 1,850 Hz that is not related to the drive crosses the fundamental ten times between 10 and 100 Hz.
At each crossing the controller sees an amplitude that is too high or too low and corrects the drive the wrong way. The test record shows narrow notches and spikes, the same kind of signature as a real resonance, at drive frequencies that have nothing to do with the test article. This is a typical symptom of analog tracking without anti-alias filters: spurious features in the control and response plots, unnecessary aborts, and over- or under-test at those frequencies.
3. Broadband noise piles up at low frequencies
Without a filter, all the noise up to the bandwidth of the transducer and signal conditioning folds into the band from zero to half the sample rate. At low drive frequencies that band is only a few hundred hertz wide, so the noise density rises sharply (Figure 4). A tracking filter still rejects noise outside its passband, but there is now more noise inside it.
At a 10 Hz drive with 32 samples per cycle, the noise floor can be up to 21 dB higher than with proper filtering (white noise to 20 kHz), so up to 21 dB of usable dynamic range is lost exactly where low-level measurements are hardest: at the start of a sweep, on response channels, and at anti-resonances where the fundamental is small.
Digital tracking: alias-free at every drive frequency
Digital tracking, C in Figure 1, gets order-domain data without moving the analog sample clock:
- Sample at a fixed high rate behind a fixed analog anti-alias filter. Because the filter never moves, it can be designed for very high alias rejection and close phase match between channels.
- Filter and decimate digitally. Digital low-pass filters, whose cutoff follows the tracked rate, remove everything above the new Nyquist frequency before the rate is reduced. Digital filters are identical on every channel, do not drift, and can have stop bands far deeper than any tracking analog filter.
- Track the rate digitally. The data are brought to a constant number of samples per drive cycle using the exact drive frequency and phase, which the controller already knows because it generated them (see Real-Time Digital Resampling in Vibration Control).
- Estimate in cycles. The fundamental and harmonics are measured in the order domain, without leakage or sweep smearing.
The result combines the strengths of A and B and avoids their main weaknesses. The cost is computation and some filter delay, which the controller must account for in its loop. The front end is alias-free like a fixed-rate system, and the measurement is synchronous with the drive like a tracking system. In Figures 3 and 4 and Table 1, it is the green line and the right-hand column.
The frequency reference is better too. A phase-locked loop following an analog COLA signal needs several cycles to settle after a change in frequency and picks up noise on the COLA line as clock jitter. A digital frequency track computed from the drive generator itself has neither problem.
| A. Fixed rate, estimator in hertz | B. Analog tracking clock | C. Digital tracking | |
|---|---|---|---|
| Samples per cycle | Changes with frequency | Constant | Constant |
| Cycles per analysis frame | Non-integer, changes during a sweep | Integer | Integer |
| Leakage and sweep smearing | Present; grows with sweep rate | None for harmonics | None for harmonics |
| Anti-alias protection | Fixed filter, effective | Often none, or fixed | Fixed analog filter plus digital filter that follows the rate |
| Aliasing at low drive frequencies | No | Severe: everything above N/2 orders folds in | No |
| Dynamic range | Set by the converter | Reduced by folded noise | Set by the converter; >96 dB digital stop band |
| Data can be re-analyzed later | Yes | Only at the tracking ratio used | Yes, from the stored time history and HPFT channel |
Recording and re-analyzing sine data
A further benefit of digital tracking is that the raw data stay at a fixed sample rate. If the time histories are recorded along with the frequency track, the test can be re-analyzed later with different filter settings, a different number of cycles per estimate, or different channels, and the measurement is the same as those settings would have produced on the recorded signals. (The closed-loop drive itself is not re-run.) Data recorded through an analog tracking clock are tied to the tracking ratio used during the test, and any aliasing is permanently part of the record.
Common misconceptions
| Misconception | Reality |
|---|---|
| "Tracking the sample clock to the drive is the most accurate way to measure a sine." | Only if every channel has an anti-alias filter that tracks the clock too. Without one, anything above half the sample rate folds into the measurement, and some of it lands exactly on the fundamental. |
| "A tracking filter removes aliases." | Not when the alias falls inside its passband. Aliased harmonics of the drive land on whole orders, including order 1, where no tracking filter can separate them from the real fundamental. |
| "Averaging longer will clean it up." | Aliased harmonics are coherent with the drive, so they do not average away. They bias both amplitude and phase. |
| "Aliasing only matters for high-frequency tests." | The opposite. With an analog tracking clock, the sample rate is lowest when the drive frequency is lowest, so aliasing and noise folding are worst at the start of a low-frequency sweep. |
| "Fixed-rate sampling is good enough for sine." | It is alias-free, but frames hold a non-integer, changing number of cycles, so leakage and sweep smearing have to be traded against bandwidth and test time (Figure 2). Digital tracking keeps the fixed-rate front end and adds measurement in cycles. |
Quick reference
| Quantity | Formula or rule | Example |
|---|---|---|
| Sample rate, tracking at N samples/cycle | fs = N · f | 32 × 20 Hz = 640 Hz |
| Highest frequency that does not alias | fs/2 = (N/2) · f | 16 orders: 320 Hz at a 20 Hz drive |
| Where a tone at fx appears | |fx − k · fs|, nearest whole k | 1,850 Hz at 640 Hz → 70 Hz |
| Harmonics that alias onto the fundamental | Orders kN ± 1 | 32 samples/cycle: orders 31, 33, 63, 65 … |
| Drive frequencies where a tone hits the fundamental | f = fx / (kN ± 1) | 1,850 Hz, N = 32: 59.7, 56.1, 29.4, 28.5 Hz … |
| Noise-floor increase, no anti-alias filter | 10 · log10(Bnoise / (fs/2)) | 20 kHz noise, 10 Hz drive, N = 32 (160 Hz Nyquist) → 21 dB |
| Drive change during a frame, log sweep | f · ln2 · R / 60 per second | 4 oct/min at 100 Hz → 4.6 Hz/s |
How Panther helps
- Digital frequency tracking: Panther Sine tracks digitally through an internally generated High Precision Frequency Track (HPFT) channel, directly coupled to the digital output and input subsystems and typically generated at the system's highest sample rate. This is far more precise than the older analog tracking devices that relied on COLA signals to adjust the sampling rate.
- Alias-free front end: analog anti-aliasing is fixed (524 kHz) with more than 105 dB of alias attenuation, ahead of simultaneous 24-bit sampling at up to 262,144 samples/s.
- Digital filtering and decimation: frequency-range reduction uses on-board FPGA and DSP digital filters with more than 96 dB of stop-band attenuation at Nyquist, so data stay alias-free as the frequency range is reduced.
- Digital tracking filters: Panther Sine measures with high-performance digital tracking filters (see Tracking Filters in Sine Vibration Control).
- Re-analysis from stream files: when STX stream files are recorded, the HPFT channel is stored with the time histories, so Sine Analysis can post-process a test later from the recording. A separate Panther analyzer recording the same test can have the HPFT channel copied from the sine test under sub-system control, or synthesized with the PEM tool from a recorded COLA or COLA-proportional signal, such as a Tee'd control channel.
- Order domain for rotating machinery: Panther RMA converts data to the order domain frame by frame, referenced to its dedicated 1 MHz hardware tachometer input (requires Spectral Dynamics tachometer hardware).
References and further reading
- A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed., Pearson (sampling, aliasing and sample-rate conversion).
- R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing, Prentice Hall, 1983.
- K. R. Fyfe and E. D. S. Munck, "Analysis of Computed Order Tracking," Mechanical Systems and Signal Processing 11(2), 1997.
- F. J. Harris, "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform," Proceedings of the IEEE 66(1), 1978.
- C. M. Harris and A. G. Piersol (eds.), Harris' Shock and Vibration Handbook, 5th ed., McGraw-Hill.
- Spectral Dynamics Technical Library: Tracking Filters in Sine Vibration Control; FFT Fundamentals: Resolution, Windowing and Leakage; Dynamic Range Considerations in Vibration Testing; Rotating Machinery Analysis.
