# On the Use of Structured Prior Models for Bayesian Compressive Sensing of Modulated Signals
## 1 Introduction  
The condition monitoring (CM) of machines consists in observing the trend of physical quantities and identifying the health state of mechanical components in order to avoid system failures. Most CM systems in the industrial field are based on the analysis of the vibration signal, which is easy to acquire and provides complete information on the condition of the machines. In modern industries, online CM systems use cables with exorbitant costs and a huge footprint. CM systems that are based on wireless sensors, known as remote CM, are a new alternative and they offer easy installation and low power consumption [1]. However, remote CM based on the analysis of vibration signals has limitations. Indeed, the vibration signals are acquired at a high sampling frequency. That of gears, bearings, or reactor blades in aeronautics can reach tens of kilohertz, leading to an enormous amount of data. The challenge for remote monitoring is managing the transmission of these volumes of data.Therefore, there is a need to reduce the size of the samples that are acquired while preserving the useful information. In this context, an alternative to the sampling theory has recently emerged, which shows that data canbe recovered from far fewer measurements than what the Shannon-Nyquist theorem states. This new theory, coined compressive sensing (CS) introduced in [2,3], relies on the sparsity or compressibility of data. This technique was applied in several fields, such as the medical, communication, presence detection, electromagnetic radiation, and structural health monitoring fields [4, 5, 6, 7, 8, 9], respectively. The theory of CS is a rising branch of signal processing, whose primary goal is to achieve compression during the sensing procedure, and to reconstruct the signal using just a small set of randomly acquired samples. Therefore, this theory concerns not only the reconstruction (decompression) of the signal, but also its sensing. The CS approach uses the sparse characteristic of the observation for the reconstruction process. It results in a representation of the observation in a specific dictionary, which is the set of elementary signals-or atoms-used to decompose the signal. The fundamental idea of the CS is that the observation must be sparse (or compressible) enough in the corresponding dictionary to reconstruct all important information. The choice of the dictionary that sparsifies the observation is one of the basic problems in the CS approach [10]. The Fourier dictionary, known as an orthogonal basis, is the most and probably the first used dictionary for periodic signals. In fact, the spectrum of a multi-sine waveform can be represented by a complex-weighted spike train. This means that sinusoidal components constructing the signal of interest are sparse in the Fourier transform basis [11, 12]. Here, it is useful to report that this paper deals with the CS of smoothly modulated signals. These signals have a well known structure in the Fourier domain. The Fourier spectrum of such a signal only contains a family of harmonics (made up of carriers) around which sidebands appear. All the rest of the spectrum is zero or not significant. Such a representation is well sparse and offers an ideal basis for describing smoothly modulated signals. This is the reason why the Fourier basis will be exploited in the rest of this paper. The problem of CS reconstruction, in a particular dictionary, is a linear inverse problem that is described by an underdetermined system of equation [13]. Two categories of approaches are used to solve the CS reconstruction problem. In the first category, the reconstruction problem is modeled as a minimization problem searching for the sparsest signal that yields the observations. One popular sparsity promoting penalization function is the \(\ell^{0}\) pseudo-norm. The latter corresponds to the number of non-zero coefficients in the sparse domain.