# A Robust Method Based on Deep Learning for Compressive Spectrum Sensing
## 3 The Proposed CSS Method
### Beista-Net Architecture  
BEISTA-Net is introduced, which is a novel deep learning framework designed for wideband spectrum signal reconstruction. The framework incorporates BSF to enhance reconstruction performance. The key idea behind this approach is to map the iterative steps of ISTA onto a deep network framework, where each stage corresponds to one iteration of the traditional ISTA algorithm. At each stage, the network employs one-dimensional convolutional kernels of varying sizes to extract the BSF from wideband spectrum signals. Additionally, the CA mechanism is integrated at each stage to further enhance the extraction of BSF. As shown in Figure 3, BEISTA-Net consists of three components, with arrows indicating the direction of data flow. The components include the \(\textbf{R}^{(k)}\) module, \(\textbf{Z}^{(k)}\) module, and \(\textbf{X}^{(k)}\) module. The \(\textbf{R}^{(k)}\) module generates the immediate reconstruction result [17]. The \(\textbf{Z}^{(k)}\) module performs a soft-thresholding process that removes less important parts from \(\textbf{R}^{(k)}\). The \(\textbf{X}^{(k)}\) module extracts and enhances the BSF from \(\textbf{Z}^{(k)}\) to obtain accurate reconstruction results. Figure 3: The proposed BEISTA-Net is composed of three components: the gradient update module \(\textbf{R}^{(k)}\), the non-linear transformation module \(\textbf{Z}^{(k)}\), and the feature enhancement module \(\textbf{X}^{(k)}\). SDM denotes forward sparse domain mapping, while ISDM denotes inverse sparse domain mapping. BSFE denotes block sparse feature extraction, and IBSFE denotes inverse block sparse feature extraction. \(\mathbf{R}^{(k)}\) **Module:** This module corresponds to Equation (4) and is used to generate the instantaneous reconstruction result \(\mathbf{R}^{(k)}\). Note that \(\mathbf{\Phi}^{\top}(\mathbf{\Phi}\mathbf{X}^{(k-1)}-\mathbf{Y})\) is actually the gradient of the data-fidelity term \(\frac{1}{2}\parallel\mathbf{\Phi}\mathbf{X}-\mathbf{Y}\parallel_{2}^{2}\). To improve the reconstruction performance and increase its capacity, we permit the step size \(\rho\) to vary during iterations. The input to this layer is \(\mathbf{X}^{(k-1)}\), and the output is defined as  
\[\mathbf{R}^{(k)}=\mathbf{X}^{(k-1)}-\rho^{(k)}\mathbf{\Phi}^{\top}(\mathbf{ \Phi}\mathbf{X}^{(k-1)}-\mathbf{Y}). \tag{7}\]  
\(\mathbf{Z}^{(k)}\) **Module:** This module addresses the problem in Equation (5) using a neural network approach. BEISTA-Net employs a non-linear transformation function \(f(\cdot)\) to transform \(\mathbf{R}^{(k)}\) into the sparse domain, rather than directly performing subsequent operations in a fixed sparse domain. The parameters of this transformation function are learnable. We use a combination of two symmetric linear convolution operators, with ReLU units added in between, to implement proximal mapping [29; 30]. Additionally, a residual component is introduced to facilitate direct information transmission. The output of this module can be represented as  
\[\mathbf{Z}^{(k)}=\alpha^{(k)}\tilde{f}^{(k)}\Big{(}soft\Big{(}f^{(k)}\Big{(} \mathbf{R}^{(k)}\Big{)},\theta^{(k)}\Big{)}\Big{)}+(1-\alpha^{(k)})\mathbf{R} ^{(k)}, \tag{8}\]  
where \(\alpha^{(k)}\) represents the ratio of the residual component to the computational component, with a range of 0 to 1, and \(soft\Big{(}z,\theta^{(k)}\Big{)}\) is the soft-thresholding function, which can be expressed as \(soft\Big{(}z,\theta^{(k)}\Big{)}=\text{sign}(z)\text{ReLU}\Big{(}|z|-\theta^{(k )}\Big{)}\). \(\mathbf{X}^{(k)}\) **Module:** The network expansion diagram of the \(\mathbf{X}^{(k)}\) module is shown in Figure 4. First, this module extracts the BSF of the wideband spectrum signals by the BSFE-Block. Then, it enhances these features by the CA-Block. Finally, it maps the enhanced signals back to the original dimensions by the IBSFE-Block. The CA-Block transforms the captured features into corresponding coefficients and multiplies them with their original data to obtain the enhanced signals.