# On the Use of Structured Prior Models for Bayesian Compressive Sensing of Modulated Signals
## 1 Introduction
The solution to such an optimization problem is known to be computationally intractable or NP-Hard, so it is more realistic to solve a computationally tractable alternative. Greedy iterative methods attempt to find approximations to this minimization problems [14] like matching pursuit [15], orthogonal matching pursuit [16], and compressive sampling matching pursuit [17]. These algorithms are known to have low computational complexity, but they have the drawback of providing a large reconstruction error. Another attempt is based on convex optimization that relaxes the \(\ell^{0}\) loss by using the \(\ell^{1}\)-norm [18], then solving the problem using the basis pursuit [19], linear programming [20], gradient projection sparse reconstruction [21], or shrinkage and selection operators [22].The second category covers the Bayesian methods. Unlike the first category, the Bayesian approaches treats unknown signal and model parameters as random variables by assigning them probability distributions and provides an entire posterior distribution [23,24].The latter can be used not only to compute point estimates, but also to provide additional information, such as credible regions. It has also shown that Bayesian methods outperform certain deterministic methods [25]. Other than the sparsity of the signal, structures on the sparse patterns of the signal have also been used as an additional prior in numerous works, in which the signal is described as a finite set of sparse vectors [26]. Those vectors share a common cluster (zero or non-zero coefficients) in the chosen dictionary, and this allows for considering the signal as the so-called block sparse structured signal [27]. The block sparse structure naturally arises for example when dealing with multi-band signal [28,29], measurement of gene expression [30], or magnetoencephalography signal [31]. Taking the block sparse structure in the signal into account reduces the number of required compressed measurement samples and enhancesthe recovery efficiency [32].Taking advantage of the signal structure, Bayesian approaches have been proposed to treat signals with block-sparse structure [33; 34]. This paper concerns the block sparse Bayesian CS framework for the vibration data recovery in the context of mechanical condition monitoring. Vibration-based Condition monitoring is the process of observing the evolution of the vibrations that are generated by a mechanical component, aiming to identify a significant change that can be correlated with the health state of that component. The reason why CS is beneficial in such applications is that vibration data analysis is usually performed while using a relatively high sampling frequency. This results in a huge amount of data to be stored and transmitted. For example, a health monitoring system installed on a Vestas V47 wind turbine measures 2 terabytes per month [35]. In addition, such acquisition systems require high power consumption which can significantly affect the autonomy of the condition monitoring system. The latter point can be problematic in many applications, such as offshore wind turbines and airplanes remote monitoring. The use of CS for industrial signals is quite recent. Some works have been done on the condition monitoring of machine tools, structures [36; 37], bearings, and gears [38; 39; 40] using traditional CS reconstruction algorithms. Few works have attached importance to the use of specific structures of the signal [39; 41]. A gearbox is the monitored component addressed in this paper, being frequently used in mechanical systems to transmit and convert speed and torque into different part of the power transmission chain. The vibration signal that is generated by such a component is well studied in the literature and known from a signal point of view [42]. The CS of the gearbox vibration signal has already been addressed in previous works, since it is sparse in Fourier domain, Wigner-Ville distribution [38], and wavelet domain [43].