= Angular resampling
<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.<Order Analysis of a Vibration Signal in MATLAB documentation>[RPM-order-map method, resampling at constant phase increments]{ref}
For positive speed $r(t)$ in revolutions per minute, an unwrapped reference angle is
$$
\theta(t)=\theta_0+2\pi\int_0^t\frac{r(u)}{60}\,du.
$$
Choose $\theta_j=\theta_0+j\Delta\theta$, 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 $\theta_0=0$ gives
$$
\theta(t)=2\pi(10t+5t^2).
$$
Sampling every quarter revolution requires $10t_j+5t_j^2=j/4$, hence
$$
t_j=-1+\sqrt{1+j/20}.
$$
The 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.
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