Characterization of damped oscillations in noisy signals

This example is based on the paper by Nguyen Phong Dien, “Damping identification using the wavelet-based demodulation: application to gearbox signals”, Technische Mechanik 28/3-4, 324-333, 2008, which was brought to my attention by Donald Butts at Sikorsky Aircraft Corp. The damping of vibrations in a gear box is an important diagnostic, and traditional methods are ill-suited to the transient phenomenon.

Simulation is a good way to assess the capabilities of a method, since we can compare our a priori knowledge of the model signal and our ability (or not) to recover the information. A model signal was constructed by superposing noise and exponentially damped oscillations at 3 frequencies (at 250, 500 and 1250 Hz, respectively). The `ping’ was generated at time t=0.03s, and the resulting signal is shown below.

For the wavelet transform, the Morlet wavelet (locally periodic) is a good template for the information we are trying to recover. The norm of the transform is shown above in the time/frequency domain. (You can think of the color scale as showing the energy distribution.) The three oscillations are clearly distinguished, and their decreasing amplitudes can be plotted by taking horizontal sections at the relevant frequencies. If these amplitudes decrease exponentially as expected, their logarithmic plot from the time of the ping should be a straight line. The result is shown next:

In spite of the noise, a straight line can be fitted over each curve, between the time the oscillation is identified (it takes a couple of cycles to establish the existence of an oscillation) and the time when it has decayed to the level of the noise.

The challenge, as for a conversation at a rock concert, is that useful information can be overwhelmed by noise. Indeed, with noise amplitude 10 times as large as above, we get the following:

No features are recognizable! Note, however, that qualitative differences (between the middle third of the signal and the rest) are picked up by our eyes/brain, and it is an interesting challenge to replicate such ability in a mathematically defined algorithm. But here we can simply `ask to repeat’ and take advantage of the distinction between the noise (different each time) and the oscillations (repeatable each time we run the `experiment’ and ping the system), so that repeating the measurement will cancel out the noise and reinforce the scraps of signal. Indeed, averaging the above over 100 experiments and plotting the decaying amplitudes gives a clear result:

Again, straight lines can be fitted, and each decay exponent is measured as the slope of the line.

Conclusion

By carefully matching the analyzing wavelet (Morlet) to distinctive features of a signal (oscillations), we can extract valuable quantitative information from very noisy data. The simulation of such data allows the assessment of methods prior to the expensive implementation in the laboratory or production unit. Selecting the correct wavelet (template) facilitates selective hearing: it is not about more information, it is about useful information, about finding the needle in the haystack. Have fun!