Time-synchronous averaging

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.