Order tracking (signal processing)

Computed order tracking[1] is a resampling technique based on interpolation.
The procedure begins by estimating the time instants ${\displaystyle T_{k}}$ ${\displaystyle (k=1:K)}$ corresponding to integer rotations of the shaft (i.e. angle equal to ${\displaystyle 2\pi k}$). Then an angular rotation vector is defined:

${\displaystyle \alpha _{i}=2\cdot \pi {\frac {iK}{N}}}$

accordingly to the desired angular resolution:

${\displaystyle \Delta \alpha ={\frac {K}{N}}}$

A corresponding vector of time instants is obtained by means of a first interpolation step

${\displaystyle t(i\Delta \alpha )={\text{interpolation}}(\{2\pi k,T_{k}\},\alpha _{i})}$

A second interpolation step is then applied to obtain the angular resampled signal ${\displaystyle x(i\Delta \alpha )}$ from the original time domain signal ${\displaystyle x(j\Delta t)}$:

${\displaystyle x(i\Delta \alpha )={\text{interpolation}}(\{x(j\Delta t),j\Delta t\},t(i\Delta \alpha ))}$

Vold-Kalman[2] filter is a particular formulation of Kalman filter, able to estimate both instantaneous speed and amplitude of a series of harmonics of the shaft rotational velocity.

Order tracking transforms are mathematical transforms which perform in a single step both the order tracking (synchronization of the signal domain with the reference shaft) and the Fourier transform to assess amplitude and phase of each order of the so obtained spectrum. With such transforms it is possible to directly assess the amplitude of a synchronous, sub-synchronous or super-synchronous shaft-locked harmonics, without an additional resampling step.

The most recent formulation of such transforms is the Velocity Synchronous Discrete Fourier Transform,[3] defined as follows:

${\displaystyle X(\Omega )={\frac {\Delta t}{\Theta }}\sum _{n=1}^{N}x(n\Delta t)e^{-j\Omega \theta (n\Delta t)}\omega (n\Delta t)}$

where ${\displaystyle \Omega }$ is the order of the harmonics to be estimated, ${\displaystyle \Theta }$ is the total angular rotation of the shaft in the acquisition window, ${\displaystyle \theta }$ and ${\displaystyle \omega }$ are respectively the instantaneous angular rotation and velocity of the reference shaft.