Time-Angle Periodically Correlated Processes

Abstract

Cyclostationary processes have now become an essential mathematical representation of vibration and acoustical signals produced by rotating machines. However, to be applicable the approach requires the rotational speed of the machine to be constant, which imposes a limit to several applications. The object of this chapter is to introduce a new class of processes, coined time-angle periodically correlated, which extends second-order cyclostationary processes to varying regimes. Such processes are fully characterized by a time-angle autocorrelation function and its double Fourier transform, the frequency-order spectral correlation. The estimation of the latter quantity is briefly discussed and demonstrated on a real-world vibration signal captured during a run-up.

Appendices

Appendices

1.1 Proof of Eq. (17)

The key point is to carefully rewrite the interval of integration over $\tau$ as $\left[ {t\,-\,W/2;t\,+\,W/2} \right]$ when $W$ grows to infinity. Thus,

$$\begin{aligned} S_{2X} \left( {v,\,f} \right)&=\mathop {\lim }\limits _{W\rightarrow \infty } \frac{1}{W} \mathop \int \limits _{-W/2}^{W/2} {\mathop \int \limits _{t-W/2}^{t+W/2} {{\mathbb {E}}\left\{ {X(t)X(t-\tau )} \right\} } } \dot{\theta }(t)e^{-j2\pi v {\theta (t)}/\Theta }e^{-j2\pi f\tau }d\tau dt \nonumber \\&=\mathop {\lim }\limits _{W\rightarrow \infty } \frac{1}{W}\mathop \int \limits _{-W/2}^{W/2} {\mathop \int \limits _{-W/2}^{W/2} {{\mathbb {E}}\left\{ {X(t)X(u)} \right\} } } \dot{\theta }(t)e^{-j2\pi v {\theta (t)}/\Theta }e^{-j2\pi f(t-u)}dudt \nonumber \\&=\mathop {\lim }\limits _{W\rightarrow \infty } \frac{1}{W}{\mathbb {E}}\left\{ {\mathop \int \limits _{-W/2}^{W/2} {X(u)e^{j2\pi fu}du \mathop \int \limits _{-W/2}^{W/2} {X(t)} } \dot{\theta }(t)e^{-j2\pi v {\theta (t)}/\Theta }e^{-j2\pi ft}dt} \right\} \nonumber \\&=\mathop {\lim }\limits _{W\rightarrow \infty } \frac{1}{W}{\mathbb {E}}\left\{ {\mathcal{F}_W \left\{ {X(t)} \right\} ^{*}\mathcal{F}_W \left\{ {X(t)\dot{\theta }(t)e^{-j2\pi v {\theta (t)}/\Theta }} \right\} } \right\} \end{aligned}$$

(A1)

1.2 Symmetric Statistics

A symmetric version of the time-angle autocorrelation function is as follows

$$\begin{aligned} R_{2X} \left( {\theta ,\tau } \right) \mathop =\limits ^{def } {\mathbb {E}}\left\{ {X\left( {t\left( \theta \right) +\tau /2} \right) X\left( {t\left( \theta \right) -\tau /2} \right) } \right\} , \end{aligned}$$

(A2)

which has the advantage of returning an even function of $\tau$. This does not change definition (15) of the frequency-order spectral correlation, yet the counterpart of Eq. (17) requires an approximation. Indeed,

$$\begin{aligned} S_{2X} \left( {v,\,f} \right)&=\mathop {\lim }\limits _{W\rightarrow \infty } \frac{1}{W} \mathop \int \limits _{-W/2}^{W/2} {\mathop \int \limits _{-W+2|t|}^{W-2|t|} {{\mathbb {E}}\left\{ {X\left( {t+\frac{\tau }{2}} \right) X\left( {t-\frac{\tau }{2}} \right) } \right\} } } \dot{\theta }(t)e^{-j2\pi v \frac{\theta (t)}{\Theta }}e^{-j2\pi f\tau }d\tau dt \nonumber \\&=\mathop {\lim }\limits _{W\rightarrow \infty } \frac{1}{W} \mathop \int \limits _{-W/2}^{W/2} {\mathop \int \limits _{-W/2}^{W/2} {{\mathbb {E}}\left\{ {X(v)X(u)} \right\} } } \dot{\theta }\left( {\frac{u+v}{2}} \right) e^{-j2\pi \frac{v }{\Theta }\theta \left( {\frac{u+v}{2}} \right) }e^{-j2\pi f(v-u)}dvdu \end{aligned}$$

which cannot be factored into the product of two integrals, unless the speed variation is small enough as compared to the correlation length of the process so that

$$\begin{aligned} \dot{\theta }\left( {\frac{u+v}{2}} \right) e^{-j2\pi \frac{v }{\Theta }\theta \left( {\frac{u+v}{2}} \right) }\simeq \sqrt{\dot{\theta }\left( u \right) \dot{\theta }\left( v \right) }e^{-j2\pi \frac{v \left( {\theta \left( u \right) +\theta \left( v \right) } \right) }{2\Theta }} \end{aligned}$$

(A3)

over the whole domain where ${\mathbb {E}}\left\{ {X(v)X(u)} \right\}$ is significantly different from zero. Therefore,

$$\begin{aligned} S_{2X} \left( {v,\,f} \right) \simeq \mathop {\lim }\limits _{W\rightarrow \infty } \frac{1}{W}{\mathbb {E}}\left\{ {\mathcal{F}_W \left\{ {X(t)\dot{\theta }(t)^{\frac{1}{2}}e^{j\pi v {\theta (t)}/\Theta }} \right\} ^{*}\!\mathcal{F}_W \left\{ {X(t)\dot{\theta }(t)^{\frac{1}{2}}e^{-j2\pi v {\theta (t)}/\Theta }} \right\} } \right\} . \end{aligned}$$

(A4)

Because of the assumption required in (A3), one may prefer the exact asymmetric form (17) to the approximate symmetric form (A4).