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Angular resampling converts a signal recorded at known times into values at equally spaced shaft angles. It is a change of sampling coordinate, used in order tracking and time-synchronous averaging. The signal must be accompanied by a reliable monotonic angle history over the interval being transformed.^[ref]

For positive speed $r(t)$ in revolutions per minute, an unwrapped reference angle isChoose ${\theta }_j={\theta }_0+j\Delta \theta$, find times $t_j$ satisfying $\theta (t_j)={\theta }_j$, and estimate $x(t_j)$ by interpolation from the original samples. Uniform time samples generally produce nonuniform angle samples; evaluating at the new $t_j$ is the resampling step. A modulo-$2\pi$ angle must first be unwrapped, rather than inverted as if each revolution restarted time.

An original acceleration model has $r(t)=600+600t$ revolutions per minute, with $t$ in seconds. Taking ${\theta }_0=0$ givesSampling every quarter revolution requires $10t_j+5t_j^2=j/4$, henceThe first quarter-turn intervals are about 0.02470 and 0.02411 seconds: angular spacing stays fixed while time spacing decreases. A third-order sinusoid becomes a fixed sinusoid of three cycles per revolution on the resampled coordinate.

With $N$ equally spaced samples per revolution, the angular Nyquist limit is $N/2$ cycles per revolution. Thus representing a maximum order $q_{\max }$ requires more than $2q_{\max }$ angular samples per turn, with practical margin and appropriate filtering. This does not repair aliasing already present in the original time record. Its sampling rate, anti-aliasing filter and the fastest physical signal frequency also matter.

A once-per-revolution tachometer constrains completed turns but does not directly measure every speed fluctuation within a turn. Inferring the intermediate angle introduces a model; a higher-resolution rotary encoder can supply more information. Stops, reversals, missing pulses and timing offsets need explicit handling. Resampling an uncertain angle history can produce a sharp-looking spectrum without making that angle history accurate.

Order tracking follows vibration or other signal components according to their ratio to a reference shaft's rotational frequency. That ratio is the rotational order:where $r$ is speed in revolutions per minute and $f,f_{\mathrm{rot}}$ are measured in hertz. A component at three cycles per reference revolution has order 3 even while its ordinary frequency changes. The reference shaft must be specified; changing the reference changes the order value.^[ref]

For an original example, a shaft accelerates from 600 to 1200 revolutions per minute. Its third-order component sweeps from 30 to 60 hertz, while its seventh-order component sweeps from 70 to 140 hertz. A Fourier transform over the entire time record spreads those components over frequency. An unrelated fixed 90-hertz tone instead retains its frequency but changes from order 9 to order 4.5.

Angular resampling expresses the record on a uniform shaft-angle grid. A speed-locked component $\cos (q\theta (t)+\varphi )$ then becomes $\cos (q\theta +\varphi )$, with fixed angular periodicity. An order spectrum or order-versus-speed map can separate that structure from a frequency-domain smear. MathWorks demonstrates this distinction with explicitly simulated helicopter-vibration data.^[ref]

In an apparatus, an accelerometer can record housing motion while a tachometer or rotary encoder records the reference. Tracking an order's amplitude and phase over a run-up can reveal where the machine's response grows, but a large amplitude need not identify the forcing mechanism by itself. Mechanical resonance, load changes and sensor placement can also change the response.

Time-synchronous averaging is related but answers a different question: it estimates a waveform repeatable over the chosen reference period. Order tracking can retain amplitude variation over speed and inspect selected components. Record speed-estimation assumptions, angular resolution and amplitude normalization with either analysis.

Robert S. Wagoner is a coauthor of experimental research using rotating accelerometers to detect damage in transmission gears. In the2015 report with David G. Lewicki and Nicholas A. Lambert, his affiliation is Ridgetop Group. The experiment compared embedded microelectromechanical system sensors with stationary housing sensors while an intentionally notched pinion was run toward tooth failure.^[ref]

A central signal-processing problem was separating repeatable shaft-related motion from unrelated vibration. The study reconstructed a rotation timing signal from a strong periodic component of the measured data and used it for time-synchronous averaging. Its comparison with a measured timing reference tested whether that reconstruction preserved the rotation-related waveform.^[ref]

The underlying averaging idea can be illustrated by records $a_j(\theta )=g(\theta )+{\eta }_j(\theta )$ aligned at the same shaft angle. Averaging $M$ revolutions preserves the repeatable waveform $g$; independent zero-mean noise with common variance ${\sigma }^2$ has mean-record variance ${\sigma }^2/M$. Timing error or changing operating conditions breaks those assumptions. This distinguishes detecting a repeatable vibration feature from proving that it uniquely identifies a tooth defect. The report found that fault indicators generally responded near actual tooth fracture, limiting the demonstrated early-warning interval.^[ref]

Time-synchronous averaging aligns repeated signal segments to a common reference phase and averages corresponding positions. In rotating machinery, a tachometer or rotary encoder provides a shaft reference. Components repeating at that reference phase remain coherent, while unrelated variations can cancel. MathWorks implements the operation using supplied pulse times and allows the averaging interval to span one or several rotations.^[ref]

For angle-aligned records $x_m(\theta )=s(\theta )+{\eta }_m(\theta )$, the average over $M$ revolutions isIf the noise terms are centered, independent between revolutions and each have variance ${\sigma }^2$ at that phase, the average's noise variance is ${\sigma }^2/M$. In an original example, noise standard deviation 4 signal units falls to 0.5 after 64 independent aligned records. Shared drift or other inter-record correlations invalidate that reduction rule.

Alignment is essential when speed changes. Equal time intervals after a trigger can represent different shaft angles in different revolutions. Angular resampling supplies a common angle grid before averaging. Errors in the reference phase also matter: a rotational order $q$ acquires phase error $q\delta \theta$ from a shaft-angle error $\delta \theta$, making high orders especially sensitive.

Cancellation is not proof that a component was unwanted noise. Consider a half-order component on a continuously increasing shaft angle, $x(\theta )=\sin (\theta /2)$. Consecutive one-revolution records satisfyAn even number of such records averages to zero, although the component is perfectly deterministic and repeats every two revolutions. Averaging over a two-revolution reference interval can retain it. The reference period therefore selects what the procedure regards as repeatable.

A useful bench acquisition records vibration and the reference pulses together, checks missing pulses and sensor timing, and compares individual aligned records with their average. Time-synchronous averaging estimates a repeatable waveform; it does not preserve every transient or every component merely because it is mechanically important.