# On the Use of Structured Prior Models for Bayesian Compressive Sensing of Modulated Signals
## 3 Experimental Results
### Application to Gearbox Vibrations  
Here, we consider real data from CETIM (Centre technique des industries mecaniques, France). They consist of vibration signals, expressed in [m/s\({}^{2}\)], which were acquired at high sampling rate (\(f_{s}\) = 20 KHz) with an accelerometer mounted in the vicinity of a gearbox reducer. The latter comprises two gears meshing together under constant speed. More specifically, one signal is acquired per day, during 12 days. A progressive fault in one wheel appeared around day 9. In such a system, the meshing forces are the main source of vibration that result from the teeth contact of the two wheels. The meshing force is naturally modulated by the rotation of the two wheels due to gears imperfection and the systematic variation of stiffness along the teeth. These modulations are typically weak for healthy signals, but their numbers and amplitude grow as the fault occurs and progresses. The energy of sidebands around the meshing frequencies in the spectrum are thus a good indicator about the health of the gear. In our test bench, the normalized meshing frequency is around \(f_{c}\) = 0.0165. Given a sub-sampled vector of vibration measurements, the goal is to detect this fault by estimating the sidebands around the meshing frequencies. A standard practice in vibration-based condition monitoring is to track the evolution of some health indicators that tell us about the energy and the shape of the spectrum. We propose to use, as indicator, the root mean square of amplitude of the modulations that are induced by the faulty gear in the spectrum. Therefore, such an indicator tends to increase when a defect occurs. Because we are mainly interested in recovering the small sidebands rather than zeroing the background noise, we use the SMG-block model and run the proposed Gibbs sampler for 5000 iterations, where the first 3000 samples are discarded as a burn-in period. The proposed indicators are then computed on the amplitudes of the peaks that are associated with target modulations using the 2000 last samples generated by the Gibbs algorithm. Note that we do not have access to the noisy-free signal, the measured vibration signal may contain wide-band noise and frequency-sparse noise (interfering peaks from others sources that we do not know). Thus, it is difficult to quantitatively  
Figure 7: Zoom on spectra around the interfering component (left) and marginal density for coefficient related to the interfering component using models (from top to bottom): t-univ, H-univ, and SMG-block.  
define the reconstruction error. However, the proposed indicator reflects and quantifies the reconstruction accuracy of sidebands. Rather than computing the indicators on point estimates (since we do not have the noisy-free signal to compare with it), we compute it on all samples and evaluate credible intervals. Figure 8 shows the evolution from day one to 12 of this indicator for different compression rates.  
We can clearly see a progressive increase with the defect that is similar to the raw signal, even for low compression rates, which indicates a good recovery of sidebands from healthy case (small sidebands) to faulty cases (large sidebands), offering the possibility of monitoring gearboxes through compressive sensing.  
Figure 8: _Cont._