# Compressive Sampling with Multiple Bit Spread Spectrum-Based Data Hiding
## 3 Data Hiding Model
### Feasible Parameters  
In this paper, there is more than one work to do in the signal processing environment. The first work is to encode the watermark into the secure Hadamard code. The second work is to make the host audio be a sparse signal. The third work is to hide the coded watermark into the sparse signal by CS acquisition. Thus, there are two objects for performance analysis, the detected watermark and the reconstructed audio from the detected sparse signal. From the embedded watermark relative to the length of the host audio, we can calculate the watermark payload, as described in (17). We can also calculate the CR of the sparse technique and CS performance as described in (6) from the host audio length relative to the coded and compressed audio. Mathematically, we can simply determine the trade-off parameters between the watermark payload and the CR as presented in (17) and (6), respectively. In (17) and (6), there are the three same parameters affecting the payload and the CR, \(M\), \(r\), and \(p\), where \(M\) is the square root of the host audio length or the row/column number of the diagonal matrix (\(\mathbf{S}\)), \(r\) is the row/column number of the truncated diagonal matrix (\(\mathbf{S_{r}}\)), and \(p\) is the sample number of the compressed signal or the row number of the output of CS acquisition (\(\mathbf{Y}\)). First, we can see that \(p\), \(r\), and \(M^{2}\) have different positions in (17) and (6). In (17), the positions of \(p\) and \(r\) are in the numerator, which means the decrease of \(p\) and \(r\) causes a lower payload. In (6), the positions of \(p\) and \(r\) are in the denumerator, which means the decrease of \(p\) and \(r\) causes a higher CR. Parameter \(M^{2}\) also has a different position. This case certainly is a trade-off between payload and CR, for which we can find the moderate value of \(p\) and \(r\) to produce a high payload and high CR. The relation between the three parameters \(p\), \(r\), and \(M\) is such that \(p\leq r<M\). Referring to (6), the above relation causes the denumerator \(pr\ll 2 Mr\) if \(M\) has a high value; thus:  
\[\text{CR}\approx\frac{M^{2}}{2 Mr}=\frac{M}{2r}. \tag{46}\]  
Note that CR for compression must be more than one; thus, \(M/2r>1\) or \(r<M/2\). This means that the minimum truncation for compression is applied at half of the diagonal matrix \(\mathbf{S}\in\mathbb{R}^{M\times M}\), obtaining \(\mathbf{S_{r}}\in\mathbb{R}^{\frac{M}{2}\times\frac{M}{2}}\). Consequently, the relation of the three parameters becomes:  
\[p\leq r<\frac{M}{2}. \tag{47}\]  
Thus, we can exploit those three parameters in the above relation. Next, we find possible \(p\) and \(r\) values such that (17) reaches the maximum payload. The positions of parameters \(p\) and \(r\) are in the numerator of (17); thus, \(r\) should be set to the maximum value or \(M/2\) in order to obtain the maximum payload and \(p\) should be set to approximately \(r\). Certainly, setting \(r\) to the maximum value or \(M/2\) obtains the minimum CR, then we have to be careful setting the \(r\) parameter since it controls the trade-off between \(C\) and CR. Due to its position, the \(p\) parameter should be to the maximum value for reaching the maximum payload. The maximum value of \(p\) is \(r\). If \(p=r\), then CS acquisition, as described in (4), produces an output with the same size as the input of CS. This condition is still acceptable when CR from (6) is more than one. CS acquisition still contributes to the watermarking process. Figure 2(a) displays the payload versus CR with \(M\in\{34,66,98,\cdots,482\}\) and \(r\in\{0.01M,0.02M,\cdots,0.5M\}\). All possibilities of the \(r\) and \(M\) combination with the restriction (47) are plotted as the magenta dots in Figure 2(b). Blue dots in Figure 2(a) mean the mapping between the payload using Equation (17) and CR using Equation (6) where \(p=r\), whereas magenta plus signs mean the mapping between the payload and CR where \(p=1\). The red vertical dotted line means the minimum CR or one.