# On the Use of Structured Prior Models for Bayesian Compressive Sensing of Modulated Signals
## 4 Discussion and Conclusions  
This paper concerns Bayesian CS of periodically modulated signals. This class of signals is in mechanical vibrations that are generated by rotating machines, and their analysis carries useful information regarding the health state of the component. The particularity of periodically modulated signals is that in addition to their sparsity in the Fourier basis, their spectrum exhibits two forms of symmetry. First, their spectrum is quasi-symmetrical around the carrier frequency. Second, the support of non-zero coefficients is almost identical around the different carrier harmonics. The main contribution of the paper is to consider the particular spectral structure of these signals in the design of prior models. More specifically, it proposes extensions of two families of hierarchical priors, namely continuous scale mixture of Gaussians and spike-and-slab model, to take the form of the target signal spectrum into account. First, a block-sparse model is proposed by jointly estimating coefficients being in similar or symmetrical positions relatively to the carrier harmonics. This form of dependency encourages such coefficients to contribute in an identical manner to the overall energy of the block to which they belong. Second, structured spike-and-slab models are built by letting the binary variable dependent on the position in the spectrum. This is ensured by introducing a Gaussian latent variable to decide for the prior probability of being non-zero. In that respect, different strength beliefs about the symmetrical properties of the spectrum lead to different hierarchical models. In fact, a strong assumption on the similarity of the support around the carrier harmonics leads us to set the same  
Figure 8: Indicators computed on the raw signals (blue) and on obtained samples (boxplots). The red line shows the median. binary selection variable for all coefficients sharing symmetrical or same positions, while a weak assumption only introduces dependency through the prior probability of being non-zeros. This form of dependency promotes quasi-symmetrical and identical spectrum support. The proposed methods have been evaluated on both simulated signals and real gearbox vibration signal. It was shown that, for such signals, structured priors achieve better results when compared to their univariate versions. In particular, the continuous block-sparse model outperforms all of the tested priors for all the compression rates and noise levels. Moreover, the discrete structured priors give more accurate results when combined with the energy constraint. Overall, the consideration of the intrinsic structure of the spectrum support is a very successful strategy and has shown its advantage for the recovering of mechanical signals for diagnosis purposes. However, it is worth to note that the considered models are only adapted to mechanical signals under the hypothesis that they are sparse in the Fourier domain. This hypothesis can be, for example, altered if the speed variation or the spectral leakage are high as pointed in Section 2.1.1. Furthermore, some mechanical signals, such as bearings, may exhibit cyclostationarity in their second order moments, rather than in the Fourier spectrum. This means that they are not perfectly sparse in the Fourier domain and alternative representations should be considered. As a further investigation, we can study the performance of such models in other applications, where we also encounter modulated signals (e.g., telecommunication, electricity, optics). The investigation of cases where the carrier frequency and the number of its harmonics are unknown may be very attractive for other real applications as an extension of our work. Methodology, Y.M.,Y.H. and D.A. ; Software, Y.M. ; Formal Analysis, Y.M. ; Writing--original draft, Y.H., A.A. and Y.M. ; Writing--review & editing, Y.H., D.A. and M.E.-B. ; Visualization, Y.M. and A.A.; Supervision/ Project Administration, M.E.-B. All authors have read and agreed to the published version of the manuscript.