## 3 Experiment and Result Analysis  
### Experiment  
In the experiment, we use MATLAB2014b as an algorithm verification tool. Our process of compressing the signal is shown in Figure 6.  
The lossless compression system does not include 'data quantization', which is a step of a lossy compression system. First, the test signal is processed by DC level shifting. Then, the transform coding or the predictive coding is used to process the signal. Finally, LZW is used to further compress the signal. The calculation formula of DC level shifting is  
\[y=x-x_{\min}, \tag{3}\]  
where \(x\) is compressed data and \(x_{\min}\) is the minimum value of the compressed data. The experiment contains a total of three sets of data:  
Figure 5: Example of predictive decoding algorithm based on adaptive dictionary. Figure 6: The block diagram of a compression system. 1. The first set of data is a periodic signal that changes period and amplitude with time to verify the feasibility of a lossless compression system. 2. The second set of data is a periodic signal with a period T = 200 and a signal length of L = 10 K. It is used to verify the effect of block size on compression efficiency. The formula of the signal is \[y=80z+50\sin(\frac{2\pi}{50}x)+10\sin(\frac{2\pi}{20}x),\] (4) \[z=\left\{ \begin{array}{ll}\frac{8}{9}(x-200*n)&200n\leq x<200n+180\\ -8(x-200*(n+1))&200n+180\leq x<200n+200\end{array} \right.,\] (5) where \(n\) = 0, 1,..., 50. 3. The last set of data we selected is 10 different periodic signals, whose periods and amplitudes are different. Because the wavelet basis function has rich frequency domain information, we perform periodic extension of the wavelet basis function to obtain the above-mentioned 10 different sets of periodic signals. The detailed information of these 10 sets of periodic signals is shown in Table 1.  
In [10,11], the authors studied the data compression algorithm based on the two-dimensional lifting format wavelet transform (2-D LWT) and two-dimensional discrete cosine transform (2-D DCT) in order to improve the data compression ratio and the efficiency of the compression algorithm. The basic idea is to first convert the periodic signal to be compressed from one-dimensional space to two-dimensional space, and then perform two-dimensional wavelet decomposition and two-dimensional discrete cosine transform, which can improve compression efficiency. The process of two-dimensional expression of one-dimensional data is as follows:  
1. Segment the one-dimensional periodic signal along the period of the signal. 2. Arrange the cut signal into a two-dimensional signal. In Figure 7, the process of two-dimension expression of one-dimension data is shown. In the experiment, we chose several methods that currently have a good compression effect on periodic signals. They are two-dimensional discrete cosine transform (2-D DCT) [10], two-dimensional lifting wavelet transform (2-D LWT) [11] and differential pulse code modulation (DPCM) [13]. In addition, we used LZW to further compress the encoded output. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Wavelet** & **Period** & **Length** & **Digital Width (Bit)** \\ \hline bior2.4 & 320 & 102.4 K & 12 \\ coif3 & 272 & 74 K & 12 \\ db4 & 224 & 50 K & 12 \\ dmey & 200 & 40 K & 12 \\ haar & 256 & 65.5 K & 12 \\ mexihat & 256 & 65.5 K & 12 \\ meyer & 256 & 65.5 K & 12 \\ morlet & 256 & 65.5 K & 12 \\ rbio2.4 & 288 & 82.9 K & 12 \\ sym4 & 224 & 50 K & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental signal information. The workflow of these three methods is as follows:  
In reference [10], 2-D DCT works as follows:  
1. Segment 1-D signal along with multiples of the signal's period. 2. Arrange the signal into a 2-D signal. 3. Divide the 2-D signal into 8\(\times\)8 matrix blocks. 4. Perform 2D discrete cosine transform on 8\(\times\)8 matrix blocks. 5. Quantize the transformed output. In reference [11], 2-D LWT works as follows:  
1. Segment 1-D signal along with multiples of the signal's period. 2. Arrange the signal into a 2-D signal. 3.