# Compressive Sampling with Multiple Bit Spread Spectrum-Based Data Hiding
## 1 Introduction
[PERSON] only explained CS techniques as a technique of inserting data with the property of being semi-fragile. [PERSON], in [3], proposed the CS method to compress the sinusoidal signal. [PERSON] investigated whether CS can be used to compress sinusoidal audio at a low speed because audio models like this have a high degree of spacing in the frequency domain. In his proposed method, [PERSON] performed CS techniques on single channels and multiple channels of audio signals with sinusoidal characteristics only. [PERSON] stated that the research he did was not to develop audio compression techniques and compare with existing compression techniques, but to find out how far CS was able to be applied in reducing the size of audio files so that the application applied, in this case, was for wireless sensor networks. [PERSON] could produce the smallest compression ratio by 5.4%. He applied spectral whitening first on the new audio, then applied the CS technique to the spectral results, so that he produced a tiny compression ratio with a good quality of the reconstruction results. [PERSON] in [4] proposed an insertion method using CS techniques by first thinning the host audio and watermark signals using the Walsh-Hadamard Transform (WHT), Discrete Cosine Transform (DCT), and Karhunen-Loeve Transform (TLC). Watermark extraction and the audio host are done by reconstruction \(L_{1}\) minimization. [PERSON] claimed that the technique could withstand MP3 attacks at the lowest rate of 64 kbps with an 11 bps watermark payload and the highest payload at 172 bps against additive noise attacks. However, [PERSON] used CS not for compression techniques, but as an insert technique. [PERSON] used MP3 attacks as compression to reduce the size of the audio signal after embedding a watermark. The watermarking applied to the compressed sensing domain was also proposed by [PERSON] in [5; 6]. [PERSON] used DWT-DCT as the host sparsity before he applied the CS acquisition and the watermark embedding procedure. In the other scenario, [PERSON] in [7] used CS to detect the embedded data by Least Significant Bit (LSB) and the DCT method. CS in [7]was used as a denoising filter to detect the hidden data. However, the embedding method in [5; 6; 7] worked on the image as the host. In [1], [PERSON] proposed a data hiding technique that was combined with CS synthetically. Suppose we define an over-complete dictionary \(\mathbf{A}\in\mathbb{R}^{p\times r}\), an uncompressed vector \(\mathbf{z}\in\mathbb{R}^{r\times 1}\), a watermark bit to be inserted as \(b\in\{-1,+1\}\), a watermark code sequence \(\mathbf{w}\in\mathbb{R}^{r\times 1}\), a compressed vector \(\mathbf{y}\in\mathbb{R}^{p\times 1}\), and \(\alpha\) as the gain control of the watermark, then we have:  
\[\mathbf{y}=\mathbf{A}(\mathbf{z}+\alpha b\mathbf{w}), \tag{1}\]  
[PERSON] inserted \(b\) as the additional operation to \(\mathbf{z}\) after multiplying by \(\alpha\mathbf{w}\). In this paper, we embed the watermark bits into the over-complete matrix \(\mathbf{A}\). Then, we multiply \(\mathbf{A}\) by the diagonal matrix from the singular values of the host audio. The data hiding technique proposed in this paper is multiple orthogonal codes based on Spread Spectrum (SS), as formerly introduced by [PERSON] in [8] in time domain embedding and continued by [PERSON] in DCT domain embedding in [9; 10]. We use the Hadamard code as the sequence for multiple bits of the watermark due to its best code performance [11]. The matrix \(\mathbf{A}\) consists of \(p\) Hadamard sequences that represent \(p\) groups of multiple bits. One of the signal sparsity techniques is a shrinkage technique on Singular Value Decomposition (SVD) output. This SVD technique truncates \(\mathbf{U}\), \(\mathbf{S}\), and \(\mathbf{V}\) with a specific rank as also described in [12; 13; 14; 15]. This shrinkage technique yields a more compressed signal as the CS output, but certainly decreases the quality of the reconstructed signal. In this paper, we decompose a host signal using SVD.