 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.^[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.