# Compressive Sampling with Multiple Bit Spread Spectrum-Based Data Hiding
## 2 Sparse Singular Value and CS Technique  
The host signal in the form of a vector \(\mathbf{x}=[x_{1},x_{2},\cdots x_{L}]\in\mathbb{R}^{1\times L}\) is converted into a two-dimensional matrix \(\mathbf{X}\in\mathbb{R}^{M\times M}\) where \(L=M^{2}\). The conversion to a two-dimensional matrix \(\mathbf{X}\) is applied in such a way that it produces:  
\[\mathbf{X}=\begin{bmatrix}x_{1}&x_{M+1}&\cdots&x_{M(M-1)+1}\\ x_{2}&x_{M+2}&\cdots&x_{M(M-1)+2}\\ \vdots&\vdots&\ddots&\vdots\\ x_{M}&x_{2M}&\cdots&x_{M^{2}}\end{bmatrix}. \tag{2}\]  
The SVD process of \(\mathbf{X}\) obtains orthogonal matrices \(\mathbf{U}\in\mathbb{R}^{M\times M}\), \(\mathbf{S}\in\mathbb{R}^{M\times M}\), and \(\mathbf{V}\in\mathbb{R}^{M\times M}\), where the relationship is described as:  
\[\mathbf{X}=\mathbf{U}\mathbf{S}\mathbf{V}^{T}, \tag{3}\]  
where \(\mathbf{S}\) is a sparse diagonal matrix having \(M\) non-zero elements in the diagonal of the matrix as \(M\) singular values. For compression needs, \(\mathbf{U}\), \(\mathbf{S}\), and \(\mathbf{V}\) can be truncated or reduced to \(\mathbf{U}[1,..,M;1,..,r]\in\mathbb{R}^{M\times r}\), \(\mathbf{S_{r}}=\mathbf{S}[1,..,r;1,..,r]\in\mathbb{R}^{r\times r}\), and \(\mathbf{V_{r}}=\mathbf{V}[1,..,M;1,..,r]\in\mathbb{R}^{M\times r}\) with \(r<M\). Then, we apply CS acquisition \(\mathbf{S_{r}}\) as:  
\[\mathbf{Y}=\mathbf{A}\mathbf{S_{r}}, \tag{4}\]  
where \(\mathbf{A}\in\mathbb{R}^{p\times r}\) is an over-complete dictionary containing the SS-based encoded watermark and \(\mathbf{Y}\in\mathbb{R}^{p\times r}\) is an output of CS acquisition with a smaller size than \(\mathbf{S}\). The truncated matrix \(\mathbf{S_{r}}\) has the form of:  
\[\mathbf{S_{r}}=\begin{bmatrix}\sigma_{1}&0&\cdots&0\\ 0&\sigma_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\sigma_{r}\end{bmatrix}, \tag{5}\]  
where \(\sigma_{1},\sigma_{2},...,\sigma_{r}\) are the singular value elements. The matrix \(\mathbf{A}\) is described later in Section 3.1. Finally, we have three matrices to be transmitted, that is \(\mathbf{U_{r}}\), \(\mathbf{V_{r}}\), and \(\mathbf{Y}\). From this result, we can calculate the Compression Ratio (CR) as the comparison between the original signal length and the transmitted signal length as:  
\[\text{CR}=\frac{L_{X}}{L_{T}}=\frac{M^{2}}{2 Mr+pr}, \tag{6}\]  
where \(L_{X}\) is the element number of \(\mathbf{X}\), that is \(M^{2}\), and \(L_{T}\) is the total number of the transmitted elements \(\mathbf{U_{r}},\mathbf{Y}\), and \(\mathbf{V_{r}}\), i.e., \(2 Mr+pr\).  
We can calculate the reconstructed audio matrix with the same size as \(\mathbf{X}\) in the form of:  
\[\mathbf{X_{r}}=\mathbf{U_{r}}\widehat{\mathbf{S_{r}}}\mathbf{V_{r}}^{T}= \begin{bmatrix}\hat{x}_{1}&\hat{x}_{M+1}&\cdots&\hat{x}_{M(M-1)+1}\\ \hat{x}_{2}&\hat{x}_{M+2}&\cdots&\hat{x}_{M(M-1)+2}\\ \vdots&\vdots&\ddots&\vdots\\ \hat{x}_{M}&\hat{x}_{2M}&\cdots&\hat{x}_{M^{2}}\end{bmatrix}, \tag{7}\]  
where \(\mathbf{X_{r}}\in\mathbb{R}^{M\times M}\), but its element values are slightly different than \(\mathbf{X}\). The \(r\) value controls the signal quality and the signal compression ratio. If \(r\) is lower, then the compression ratio is higher, but the signal quality is worse. Finally, we can get \(\hat{\mathbf{x}}=[\hat{x}_{1},\hat{x}_{2},\cdots\hat{x}_{M^{2}}]\) as a reconstructed or decompressed version of the signal by converting two-dimensional matrix \(\mathbf{X_{r}}\) back to a vector or one-dimensional signal \(\hat{\mathbf{x}}\); thus, we can calculate the signal quality by comparing \(\mathbf{x}\) and \(\hat{\mathbf{x}}\).