# Compressive Sampling with Multiple Bit Spread Spectrum-Based Data Hiding
## 3 Data Hiding Model
### Data and Dictionary Detection
Thus, (21) is:  
\[K_{ij}=\left|\mathbf{y}_{i_{1}}\mathbf{p}_{j}^{\mathsf{T}}\right|. \tag{23}\]  
Substituting (20) into (23) results in:  
\[K_{ij}=\left|\mathbf{p}_{t_{i}}\mathbf{S}_{\mathbf{r}}\mathbf{p}_{j}^{\mathsf{ T}}\right|. \tag{24}\]  
Assume that \(t_{i}=j\); thus, \(\mathbf{p}_{t_{i}}=\mathbf{p}_{j}\), then (24) is an autocorrelation as:  
\[K_{a}=\left|\mathbf{p}_{j}\mathbf{S}_{\mathbf{r}}\mathbf{p}_{j}^{\mathsf{T}} \right|. \tag{25}\]  
Assume that \(\mathbf{p}_{j}\) consists of such elements as:  
\[\mathbf{p}_{j}=\left[p_{j_{1}}\quad p_{j_{2}}\quad\cdots\quad p_{j_{r}}\right]; \tag{26}\]  
therefore, (24) becomes:  
\[K_{a}=\left| \begin{bmatrix}p_{j_{1}}\\ p_{j_{2}}\\ \vdots\\ p_{j_{r}}\end{bmatrix} ^{\mathsf{T}} \begin{bmatrix}\sigma_{1}&0&\cdots&0\\ 0&\sigma_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\sigma_{r}\end{bmatrix}\begin{bmatrix}p_{j_{2}}\\ \vdots\\ p_{j_{r}}\end{bmatrix} \right|. \tag{27}\]  
By a matrix multiplication operation, (27) is described as:  
\[K_{a}=\left|p_{j_{1}}^{2}\sigma_{1}+p_{j_{2}}^{2}\sigma_{2}+\cdots+p_{j_{r}}^ {2}\sigma_{r}\right|=\left|\sum_{i=1}^{r}p_{j_{i}}^{2}\sigma_{i}\right|. \tag{28}\]  
Since \(\sigma_{i}>0\) and \(p_{j_{i}}^{2}\gg 0\) for all \(j\) and all \(i\), then (28) becomes:  
\[K_{a}=\sum_{i=1}^{r}p_{j_{i}}^{2}\sigma_{i}\gg 0. \tag{29}\]  
If \(\mathbf{p}_{t_{i}}=\mathbf{p}_{k}\) and \(\mathbf{p}_{t_{i}}\
eq\mathbf{p}_{j}\), then (24) is a cross-correlation as:  
\[ \begin{split} K_{c}&=\left|\mathbf{p}_{k}\mathbf{S}_{ \mathbf{r}}\mathbf{p}_{j}^{\mathsf{T}}\right|\\ &=\left|\begin{bmatrix}p_{k_{1}}\\ p_{k_{2}}\\ \vdots\\ p_{k_{r}}\end{bmatrix}^{\mathsf{T}}\begin{bmatrix}\sigma_{1}&0&\cdots&0\\ 0&\sigma_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\sigma_{r}\end{bmatrix}\begin{bmatrix}p_{j_{1}}\\ p_{j_{2}}\\ \vdots\\ p_{j_{r}}\end{bmatrix}\right|\\ &=\left|p_{k_{1}}p_{j_{1}}\sigma_{1}+p_{k_{2}}p_{j_{2}}\sigma_{2}+\cdots+p_{ k_{r}}p_{j_{r}}\sigma_{r}\right|\\ &=\left|\sum_{i=1}^{r}p_{k_{i}}p_{j_{i}}\sigma_{i}\right|.\end{split}  \tag{30}\]  
Since \(\mathbf{p}_{k}\) is mutually orthogonal with \(\mathbf{p}_{j}\), it is confirmed that \(K_{a}\) is comparable to \(K_{c}\) with the following inequality:  
\[K_{a}\gg K_{c}, \tag{32}\]which means that the autocorrelation of the same Hadamard sequence is still much higher than the cross-correlation of the different Hadamard sequence on the singular value intervention. It confirms that the Hadamard sequence can be detected successfully; thus, from (22), \(t_{i}\) is detected for \(t_{i}=\{t_{1},t_{2},...,t_{p}\}\), then we can obtain the associated watermark bits \(\hat{\mathbf{w}}_{t_{i}}=\{\hat{\mathbf{w}}_{t_{1}},\hat{\mathbf{w}}_{t_{2}},...,\hat{\mathbf{w}}_{t_{p}}\}\) and also all Hadamard sequences \(\hat{\mathbf{p}}_{t_{i}}=\{\hat{\mathbf{p}}_{t_{1}},\hat{\mathbf{p}}_{t_{2}},...\hat{\mathbf{p}}_{t_{p}}\}\), which form \(\widehat{\mathbf{A}}\) using (13) and (14) as:  
\[\widehat{\mathbf{A}}=\frac{1}{p} \begin{bmatrix}\hat{\mathbf{p}}_{t_{1}}\\ \hat{\mathbf{p}}_{t_{2}}\\ \vdots\\ \hat{\mathbf{p}}_{t_{p}}\end{bmatrix} , \tag{33}\]  
where \(p\) is the row number of \(\widehat{\mathbf{A}}\). This procedure assures that there is no dictionary needed to detect the hidden data and also to reconstruct the signal, since the associated watermark bits \(\hat{\mathbf{w}}_{t_{i}}\) are detected. Thus we can calculate the Bit Error Rate (BER) as a robustness parameter. The following equation is a BER formula:  
\[\text{BER}=\frac{\sum_{i=1}^{L_{w}}|\hat{w}_{i}-w_{i}|}{L_{w}}, \tag{34}\]  
where \(w_{i}\) is the original watermark bit, \(\hat{w}_{i}\) is the detected watermark bit, and \(L_{w}\) is the total number of watermark bits.