# Modulus Waveform Design Based on Manifold ADMM Idea in Dual-Function Radar-Communication System
## 4 Numeric Examples
### Radar Waveform Design and Analysis for Beampattern Matching  
Consider that the desired beampattern has three main lobes at \(\theta=-45^{\circ},0^{\circ},45^{\circ}\) corresponding to three targets \(Q=3\), their interval width is \(\Delta\theta=20^{\circ}\), and the penalty parameter of (16) is set as \(\rho=50\). Then, the desired beampattern can be formulated as  
\[d(\theta)= \begin{cases}1,&\theta\in\left[\theta_{q}-\frac{\Delta\theta}{2}, \theta_{q}+\frac{\Delta\theta}{2}\right],q=1,2,\ldots Q\\ 0,&\textit{others}\end{cases}  \tag{48}\]  
As discussed in [19], we also choose the squared-error cost function as the designing metric. Here, the variables \(\mathbf{h}^{(0)}\), \(\mathbf{\lambda}^{(0)}\) are initialized to be \(\mathbf{0}\), and \(\mathbf{z}^{(0)}\) is initialized to be a random vector. An illustration of the three main-lobes beampattern is shown in Figure 2, where we also provide results of WAPM and other prevalent ones. For comparison, the beampattern optimized by the semidefinite program (SDP) method in [22] and ADMM in [19] are also considered. Obviously, the beampattern obtained by WAPM is very close to the SDP-based beampattern, where it achieves the transmitting waveform in a direct manner rather than two-step way. Furthermore, compared with the main-lobe beamforming at \(0^{\circ}\), WAPM algorithm can generate a higher main lobe than the classical ADMM algorithm, also with a lower side lobe at \(20^{\circ}\). One of the indicators for measuring the constant modulus of a signal is the signal modulus. As can be seen from Figure 3, the waveform designed by the WAPM algorithm under a single beam has a strict constant modulus characteristic, while the waveform designed by the ADMM algorithm converges to the constant modulus. The minimum modulus value of the waveform designed by the ADMM algorithm is 0.9996, and the maximum modulus value is 1.0013. The superior performance of the WAPM algorithm in maintaining a constant modulus can be attributed to its inherent design, which emphasizes consistency and stability in waveform characteristics. Figure 2: Comparison of three main-lobes’ beampatterns in the case of \(\Delta\theta=20^{\circ}\). In the five-beam scenario, the detection is located at \(\theta=-60^{\circ}\), \(-30^{\circ}\), \(0^{\circ}\), \(30^{\circ}\), and \(60^{\circ}\), respectively. The five-beam waveform means that there are five main-lobe beams in the direction of the detection target. Figure 4 describes the five-beam pattern performance of the ADMM algorithm, WAPM algorithm and SDP algorithm, and the results are all normalized. As can be seen from Figure 4, the five beams formed by the above algorithms can form main-lobe beams similar to the standard waveform in the five directions of \(-60^{\circ}\), \(-30^{\circ}\), \(0^{\circ}\), \(30^{\circ}\) and \(60^{\circ}\). Unlike single-beam and three-beam, there are no side lobes between the main lobes of the five beams. The ADMM algorithm and the WAPM algorithm have the same main-lobe depression peak position at \(0^{\circ}\); that is, both are located at \(0^{\circ}\), and the SDP algorithm has a main-lobe depression peak position of \(1^{\circ}\) at \(0^{\circ}\). At the concave position, the normalized peak value of the waveform designed by the ADMM algorithm is 0.866, the normalized peak value of the waveform designed by the WAPM algorithm is 0.884, and the normalized peak value of the waveform designed by the SDP algorithm is 0.767. Figure 4: Comparison of five main-lobes beampattern in the case of \(\Delta\theta=20^{\circ}\). Figure 3: Performance comparison of ADMM and WAPM with the waveform modulus description. Furthermore, we also consider a main-lobe beampattern with a width \(\Delta\theta=60^{\circ}\), as shown in Figure 5. WAPM has a higher main lobe than ADMM at the \(-53^{\circ}\) and lower side lobe at the \(0^{\circ}\), which means some excellent beam-matching performance.