## 2 Message Signal Modulation  
This paper modulates the phases of the communication symbols directly on the complex exponential function [32]:  
\[\mathbf{p}_{l}=exp\{j2\pi\mathbf{k}_{l}\} \tag{3}\]  
where \(\mathbf{k}_{l}\in[-0.5,0.5]\) is the \(l\)-the element of chaos sequence. We can see that chaos phase modulation sequences are complex sequences with randomly-distributed phases, and their envelopes are constant. We use the multi-segment piecewise linear mapping [25; 47] to generate the desired chaos sequence \(\mathbf{k}\). It is an iteration procedure that can be programmatically expressed as follows:  
1. Initialize the first element of the sequence \(\mathbf{k}_{0}\) with a random value, known as \"seed\" in computer science;
2. If \(\mathbf{k}_{l}>1/2\), repeat the computation \(\mathbf{k}_{l}:=\mathbf{k}_{l}-|2\tilde{\xi}\mathbf{k}_{l}|\) with \(\tilde{\xi}\in[0,1]\) until the constraint is satisfied, otherwise, go to the next step;
3. If \(\mathbf{k}_{l}<-1/2\), repeat the computation \(\mathbf{k}_{l}:=\mathbf{k}_{l}+|2\tilde{\xi}\mathbf{k}_{l}|\) until the constraint is satisfied, otherwise, go to the next step;
4. Compute the next element \(\mathbf{k}_{l+1}\): \[\mathbf{k}_{l+1}= \begin{cases}2(N-\tilde{\xi})\mathbf{k}_{l}+(N-0.5)&-1/2\leqslant\bm {k}_{l}<(1-N)/(2N)\\ 2(N-\tilde{\xi})\mathbf{k}_{l}+(N-1.5)&(1-N)/(2N)\leqslant\mathbf{k}_{l}<(2-N)/(2N)\\ &\vdots\\ 2(N-\tilde{\xi})\mathbf{k}_{l}+(-N+0.5)&(N-1)/(2N)\leqslant\mathbf{k}_{l}<1/2\end{cases} \] (4)
5. If \(l=L-1\), task completed, otherwise go back to the second step. Steps (2)-(4) above correct iteratively \(\mathbf{k}_{l}\) to keep it in the given interval with the help of a step coefficient \(\tilde{\xi}\in[0,1]\). \(L\) is the length of sequence \(\mathbf{k}\). Step (4) segments the interval \([-0.5,0.5]\) into \(2N\) pieces (\(N\in\mathds{N}^{+}\)). Now we can create the chaos spreading spectrum symbol \(\mathbf{u}(t)\):  
\[\mathbf{u}(t)=\sum_{l=0}^{L-1}\mathbf{d}(t)\times g(t,l,\mathbf{p}_{l}) \tag{5}\]with  
\[\mathbf{d}(t)=A\cdot\frac{e^{i2\pi f_{0}t}+e^{-i2\pi f_{0}t}}{2} \tag{6}\] \[g(t,l,\mathbf{p}_{l})= \begin{cases}\mathbf{p}_{l}&\text{if }t\in(\frac{T \cdot l}{L},\frac{T\cdot(l+1)}{L}]\\ 0&\text{otherwise}\end{cases}  \tag{7}\]  
where \(\mathbf{d}(t)\) is the carrier wave, \(A\) is amplitude, \(f_{0}\) is the frequency of carrier wave, \(g\) is a shaping function, \(T=T_{c}\times L\) is the time length of symbol and \(T_{c}\) is the time length of chip. In this work, cosine signal is used as the carrier wave. We can write the fourier transform of \(\mathbf{u}(t)\) as follows:  
\[\mathcal{F}[\mathbf{u}(t)] =\sum_{l=0}^{L-1}\int_{-\infty}^{+\infty}\mathbf{d}(t)\times g(t,l,\bm {p}_{l})\times e^{-i2\pi ft}dt \tag{8}\] \[=\sum_{l=0}^{L-1}\int_{\frac{T\cdot l}{L}}^{\frac{T\cdot(l+1)}{L} }\mathbf{d}(t)\times e^{i2\pi(\mathbf{k}_{l}-ft)}dt \tag{9}\]  
The equation above implies that chaos phase modulation associates the frequency spectrum of \(\mathbf{u}(t)\) with the chaos sequence \(\mathbf{k}\). According to the chaos theory, chaos sequences are sensitive to the initial conditions, even if they are approximately equal. Consequently, correlators will have a low output unless they are highly synchronized with the inputs. That allows to create a large number of highly orthogonal symbol candidates by changing the seeds. Figure 2a plots the normalized autocorrelation of a chaos phase modulation signal measured by using a classical matched filter (see Equation (2)). The correlation values of main and side lobes are marked. We can see clearly that their ratio is around \(4.1\times\), whereas, as shown in Figure 2b, it is hard to distinguish the main and side lobes on the cross-correlation curves, and the maximum cross-correlation coefficient is only \(0.217\), demonstrating that chaos phase modulation signals have nice orthogonality. For the purpose of textual communication, we map the 8-bit ASCII table to chaos phase modulation signals.