# Design of an Underwater Acoustic Waveform and Integrated System for Communication and Detection
## 4 Simulations
### Simulations on Ambiguity Function and BER of CPM-LFM Signals in AWGN Channels  
A comparison of the ambiguity functions of LFM and CPM-LFM signals in AWGN channels is presented. The detection performance of the shared waveform is analyzed based on the velocity ambiguity function given in formulas (19) and (20), as well as the range ambiguity function given in formulas (21) and (22). The following parameters are used: \(B=1\) kHz, \(T_{p}=0.5\) s, \(h=0.5\), \(L=2\), binary CPM, LREC, and N = 50. The normalized ambiguity functions of the proposed and LFM waveforms are shown in Figure 13(a),b, respectively. This shows that the proposed waveform has a spike ambiguity function, while the LFM signal has a much broader ambiguity function. Both waveforms demonstrate good detection capabilities, indicating the feasibility of using the proposed waveform for detection. The velocity resolution for the proposed and LFM waveform are shown in Figure 13(c),d respectively. This shows that the resolution is similar, which is consistent with the theory. The corresponding distance resolution results are shown in Figure 13(e),f. This shows that the waveforms have the same main lobe width, with a single peak, so the distance resolution is similar. However, the proposed waveform has larger and more irregular side lobes compared to LFM. This can be attributed to the CPM waveform as indicated by (15). Figure 14: Ambiguity functions of the CPM-LFM and LFM waveforms. Figure 15 shows the distance resolution of the LFM waveform and proposed waveform for different numbers of data symbols. This shows that the side lobes increase with the number of symbols, which is consistent with (15). This is because increasing the number of symbols leads to the condition \(0\leq\tau\ll T_{s}\) being violated. Consequently, the effect of the second term in (15) increases, resulting in more side lobes. However, the proposed waveform still maintains the single-peak characteristic of LFM, and the main lobe width is the same, indicating similar distance resolution capabilities. In summary, the proposed waveform inherits the good detection performance of the LFM signal. They have the same velocity ambiguity function and velocity resolution, as well as similar distance ambiguity functions and distance resolution. Next, based on the bit error rate for CPM-LFM signals given in (11) in an AWGN channel derived in Subsection B of Section 2, the communication performance of the shared waveform signal is analyzed under ideal conditions. The impact of the modulation index \(h\), the number of modulation orders  
\(M\), and the correlation length \(L\) on the communication performance are considered. Figure 16 gives the simulation results of the impact of the modulation index \(h\) on the system's BER under the conditions of \(M=2\), \(L=2\), and LREC. Figure 17 simulates the impact of the number of modulation orders \(M\) on the system's BER under the conditions of \(L=2\), \(h=1/2\), and LREC. Figure 18 shows the impact of the correlation length \(L\) on the system's BER. In previous theoretical analyses, a larger correlation length L needs to be used in conjunction with a larger modulation index h to achieve better communication performance. Therefore, combinations of correlation lengths \(L=1\), \(2\), \(3\) and modulation indices \(h=0.5\), \(0.8\) were tested under the conditions of \(M=2\) and LREC. The following conclusions can be drawn from the results shown in Figures 16-18. According to the upper bound of the minimum Euclidean distance shown in Figure 4, an increase in the modulation index \(h\) results in an increase in the upper bound on the Euclidean distance up to a given index, after which it fluctuates. In addition, the bit error rate decreases as \(h\) increases. Figure 15: Distance resolution of the LFM and proposed waveforms. The upper bound on the Euclidean distance increases with the modulation order \(M\). Thus, the BER decreases with increasing \(M\).