# On the Use of Structured Prior Models for Bayesian Compressive Sensing of Modulated Signals
## 2 Proposed Method
### Group Sparse Model for Modulated Signals
#### 2.3.2 Hierarchical Model  
In our case, the target spectrum exhibits modulations around the carrier frequency and its harmonics. Therefore, It is judicious to exploit the in-block quasi-symmetry of the spectrum in each active block \(b\) by jointly estimating coefficients \(x_{-q,b}\) and \(x_{q,b}\) belonging to symmetric frequency bins with respect to the \(b^{th}\) harmonic of the carrier frequency \(bf_{c}\) (see Figure 1). In fact, if a non-zero coefficient is related to a modulating signal, then there is a high probability that its symmetrical point with respect to the harmonic of the carrier frequency in this block is also nonzero with a quasi-similar magnitude. In that respect, we define the vector \(\mathbf{x}_{q,b}\in\mathbb{C}^{2}\), where  
\[\forall b\in\{1,\,\ldots,\,B\}\quad\forall q\in\{1,\,\ldots,\,Q\},\quad \mathbf{x}_{q,b}=[x_{-q,b},x_{q,b}] \tag{3}\]We further aim to exploit the similarities across active blocks. Our idea is to capture such dependencies by jointly estimating the coefficients in the same position with respect to the carrier harmonics. To do this, we stack the coefficients of all \(B\) groups at the same position and their symmetrical points with respect to the carrier harmonics and then build the following vectors of multi-block coefficients of size \(2B\)  
\[\forall q\in\{1,\,\ldots,\,Q\},\quad\mathbf{x}_{q}=\left[\mathbf{x}_{q,1}^{ \top},\ldots,\mathbf{x}_{q,B}^{\top}\right]^{\top} \tag{4}\]  
where \(\mathbf{x}_{q,1}^{\top}\) is the transpose of vector \(\mathbf{x}_{q,1}\). We also denote, by \(\mathbf{x}_{0}\), the vector of size \(B\) containing the coefficients of the \(B\) carrier harmonics. The sparsity of the spectrum and the dependencies between the Fourier coefficients are captured by assuming that \((\mathbf{x}_{q})_{1\leqslant q\leqslant Q}\) are realizations of a heavy tailed multivariate distribution. We consider the class of scale mixture of multivariate Gaussian (SMG)  
\[\forall q\in\{1,\,\ldots,\,Q\},\quad\mathbf{x}_{q}\sim\mathcal{CN}\big{(} \mathbf{0},\gamma_{q}\mathbf{\Sigma}\big{)} \tag{5}\]  
where \(\mathcal{CN}\), here, denotes the multivariate complex normal distribution (As the covariance matrix is real-valued, (5) means that the real and imaginary parts of \(\mathbf{x}_{q}\) follow the Gaussian distribution of zero mean and with covariance matrix \(\tau\gamma_{q}\mathbf{\Sigma}\)), \(\gamma_{q}\geqslant 0\) is the mixing parameter following some mixing distribution and \(\mathbf{\Sigma}=\text{Diag}(\mathbf{v})\bigotimes\mathbf{I}_{2}\in\mathbb{R}^{ 2B\times 2B}\) where \(\bigotimes\) denotes the Kronecker product, \(\mathbf{I}_{2}\) the identity matrix of size \(2\times 2\), and \(\mathbf{v}=[v_{1},\,\ldots,\,v_{B}]^{\top}\in(\mathbb{R}^{+})^{B}\). We further assume a Gaussian prior for the vector of carrier coefficients, i.e., \(\mathbf{x}_{0}\sim\mathcal{CN}(\mathbf{0},\tau\gamma_{0}\text{Diag}(\mathbf{v }))\), where \(\gamma_{0}>0\). It is noteworthy that, although we have considered a diagonal covariance matrix, coefficients of \(\mathbf{x}_{q}\) are uncorrelated but still dependent. Note that the block model (5), which is referred to as SMG-block, is a particular case of (A2) for which the variance of the Gaussian depends on the position of the Fourier coefficient, i.e., \(\lambda_{k}=v_{b}\gamma_{q}\). The variable \(v_{b}\) acts as a global indicator of the overall magnitude of harmonics in the active block \(b\), while the local variable \(\gamma_{q}\) controls the sparsity inside the blocks. In fact, if \(\gamma_{q}\) is too small, then the spectrum coefficients in the position \(q\) and those in their symmetrical positions with respect to the carrier harmonics through the different \(B\) active groups approach zero. Inversely, if \(\gamma_{q}\) is high, then all coefficients at positions \(q\) and \(-q\) around the carrier harmonic in any active block \(b\) with non-zero \(v_{b}\), they are also non-zero and then have a high probability to belong to modulations.