# On the Use of Structured Prior Models for Bayesian Compressive Sensing of Modulated Signals
## 2 Proposed Method
### Structured Spike and Slab for Modulated Signals  
#### 2.4.1 Related Works  
Spike and slab models that take the structure of the signals into account can be built by imposing a dependency between the model parameters of elements sharing the same cluster. For example, in [54], the authors exploit the statistical structure of the wavelet coefficients of their data by setting the model parameters \(\beta_{k}\) of each wavelet coefficient according to the value of its parent in the wavelet tree, so that, if a parent coefficient is zero, its children coefficients are likely to also be zero. In [55], the authors consider relations between each coefficient of the signal and its neighbors by choosing \(\beta_{k}\) from a set a different candidates according to the number of non-zero neighboring coefficients, yielding a high probability of a non-zero coefficients if all neighbors are non-zero. Such a model is motivated by sparse signals, where non-zero elements tend to appear in groups rather than isolated points. In [56,57], the authors take into account not only the spatial structure, but also the time evolving structure though an hierarchical Gaussian process. Such a model is convenient for temporal signals, where non-zero elements are clustered in groups for each timestamp and that these groups can move and evolve in time. These works are an inspiring point of our model.  
In order to capture the structure form discussed above, the key idea is to set the model parameters \(\beta_{k},z_{k}\), and \(\lambda_{k}\) in (2.4.1), according to the position \(k\) of the Fourier coefficient \(x_{k}\) in the spectrum, In fact, if the coefficient \(x_{k}\in\mathcal{A}\), then it is located at some position \(q\) in an active block \(b,b\in\{1,\,\ldots,\,B\},q\in\{-Q,\,\ldots,\,Q\}\), and it follows that \(z_{k}=z_{q,b}\) and \(\lambda_{k}=\lambda_{q,b}\). Therefore, we can add constraints on model parameters for the block \(b\) to be either equal or close for symmetrical positions \(q\) and \(-q\) in order to assure the quasi-symmetry of the spectrum around a given carrier harmonic \(b.f_{c}\). The similarity of the support of non-zero coefficients between actives blocks can be achieved while using additional constraints on model parameters at the same or symmetrical positions, but belonging to different active blocks \(b\) and \(b^{\prime}\).  
In the following, we propose different spike-and-slab models that range from weak to strong levels of prior symmetry assumptions and discuss their advantages and disadvantages. A spike-and-slab model is said to have a strong prior assumption on a given symmetrical structure if the latter is underlined through the binary selection variable \(z_{q,b}\), while it has a weak prior assumption if it is only described through the prior probability of zero coefficient \(\beta_{q,b}\) and/or the slab component parameters. In that respect, we can say that spike-and-slab models [54; 55] are examples of models using a weak prior assumption, as they only take the structure of the target signals into account sh through the spike probability \(\beta_{k}\).  
It seems interesting to note that, dealing with the structure in the spike-and-slab model is quite different from the continuous model (5), where the dependency is introduced in terms of energy contribution through the mixing variables. Here, instead, we are interested in the symmetrical properties of the support of non-zero coefficients in the spectrum rather than their contribution in the overall spectrum energy. However, nothing prevents us introducing energy constraints by setting the variance of the slab model well. This will discussed in the following parts.