# On the Use of Structured Prior Models for Bayesian Compressive Sensing of Modulated Signals
## 3 Experimental Results
### Simulated Example
#### 3.1.3 Results  
For the different models, we run the Gibbs algorithms for 6000 iterations, with the first 4000 iterations as burn-in. The last 2000 samples are used to empirically compute point estimates (maximum a posteriori). We begin with comparing the performance of the different models for different levels of Gaussian noise and compression rates. The noise level is given in terms of SNR. The reconstruction quality is evaluated in terms of normalized mean square error (NMSE) in Figure 3. The latter is defined as \(\text{NMSE}_{\text{e}}\)\(\frac{\|\mathbf{x}-\mathbf{\hat{x}}\|_{2}^{2}}{\|\mathbf{x}\|_{2}^{2}}\), in which \(\mathbf{x}\) is the real signal, \(\mathbf{\hat{x}}\) is its estimate, and \(\|.\|_{2}\) stands for the \(\ell^{2}\)-norm. A quick view of the NMSE plots shows that the proposed models (expect S&S-3A and expect S&S-3B) always improve the reconstruction error as compared to their univariate versions. We first discuss the results for continuous models then for spike-and-slab models and, finally, we compare both of them. In the one hand, for continuous models, H-univ and t-univ lead approximately to the same reconstruction error, while the continuous SMG-block achieves almost the best results among all models. The contribution of the support structure in continuous models is mainly noticed for increasing noise levels. Using the SMG-block instead of H-univ and t-univ helps to decrease the error for 5% for low noise levels until 20% for higher noise levels. In order to evaluate the reconstruction error visually, we plot the reconstructed spectra using the H-univ, t-univ, and SMG-block in Figure 4 and in Figure 5 for the lowest noise level (13.97 dB) and a compression rate equal to 0.9 and for a high noise level (0 dB) and a compression rate equals 0.85, respectively  
We can see that, for very low noise levels, all of the continuous models recover the modulations, but the SMG-block better shrinks background noise, since we observe some peaks that are related to noise in the reconstructed spectrum by the univariate models. For the high noise level, although the NMSE values are close for continuous models, t-univ and H-univ fail to reconstruct modulations with small amplitudes, while SMG-block enhances the reconstruction of these small sidebands. Figure 3: Normalized mean square error (NMSE) (in logarithmic scale) for different Gaussian noise levels and compression rates. Figure 4: Reconstructed spectra using (from top to bottom) t-univ, H-univ, and scale mixture of multivariate Gaussian (SMG)-block for compression rate 0.9 and noise level 13.97 dB. Figure 5: Reconstructed spectra using (from top to bottom) t-univ, H-univ, and SMG-block for compression rate 0.85 and noise level 0 dB. On the other hand, the addition of structure in the spike-and-slab model spectacularly improves the numerical results. In particular, we can see that S&S-1A and S&S-2A achieve a good improvement in the reconstruction error when compared to the S&S-univ. This means that adding the weak or strong assumption of the perfect symmetry of the support around the carrier frequencies is very advantageous, especially for low noise levels. The contribution of the assumption on support symmetry is less noticed for high noise levels, where the reconstitution error is only slightly improved when compared to S&S-univ. The use of additional constraints on the energy for S&S-1B and S&S-2B contributes to further enhance the results especially in high noise levels. In Figure 6, we display the reconstructed spectra using the S&S-1A and S&S-1B for initial SNR 1.93 dB and a compression rate equals 0.7. We can see that the addition of energy constraints in S&S-1B improves the recovering of the small sidebands as compared to S&S-1A. Figure 6: Reconstructed spectra using S&S-1A (top) and S&S-1B (bottom) for noise level 1.93 dB and compression rate 0.7. We can also note that a weak and strong assumption about support in-block symmetry (with the weak assumption about the support inter-block similarity) yields close results.