## 4 Numerical Examples
### Detection Performance  
Here, we illustrate the detection performance of the designed waveform and uniform waveform. We apply NUDFT, IAA, and OMP to verify the detection performance of the designed waveform, and the sparsity \(K_{s}=4.28\%\) among these algorithms. We apply the DFT, IAA, and OMP to verify the detection performance of the uniform waveform. In order to achieve the threshold of false alarm probability (Pfa), 30,000 Monte Carlo experiments are performed. Moreover, 500 independent Monte Carlo experiments are performed per SCNR in order to achieve the detection probability (Pd). In each Monte Carlo experiment, the radial velocity of the target is \(v_{t}=45\) km/h, the radial range is \(r_{t}=30\) m, the CNR is 30 dB, and the Pfa is 0.001.  
The results are shown in Figure 10. The figure illustrates that the Pd of the designed waveform is over 95% when SCNR \(>-25\) dB and Pfa = 0.001. It should be noted that compared to the IAA, the OMP and NUDFT have a 3 dB loss in the detection performance of the designed waveform when Pd = 90%. And the results also illustrate that the detection performance of the uniform waveform with DFT is optimal when SCNR \(<-25\) dB. However, we note that the Pd of the designed waveform is lower than that of the uniform waveform when the SCNR \(<-25\) dB. Specifically, for the IAA, the detection performance of the designed waveform has a performance loss of less than 1 dB compared to the uniform waveform. For the OMP and NUDFT, the detection performance loss reaches more than 3 dB, compared to the OMP and DFT with uniform waveform. This is due to the non-sparsity of the clutter, and the noise distribution has an impact on the reconstruction of the target information in the case of a low SCNR ratio, which leads to the decrease of Pd. Moreover, the decrease in the number of sensing symbols similarly decreases Pd.  
In the next experiment, we evaluate the detection performance of the designed waveform and the uniform waveform in the case of a high SCNR. The SCNR = 0 dB, and the other parameters are the same as in the last experiment. Since the Pd converges to 100% for SCNR \(>-25\) dB, we define the output of SCNR (SCNRout) to describe the intensity contrast relationship between the signal, clutter, and noise after the signal processing. The SCNRout can be defined as  
\[\text{SCNRout}=10 lg\frac{P_{t}}{mean(P_{ref})} \tag{25}\]  
where \(P_{t}\) is the power of the target, and \(P_{ref}\) is the power of the reference cell. The result is shown in Figure 11. The figure illustrates that the detection performance of the designed waveform using the IAA is better than that of applying the OMP and NUDFT, which is similar to the results of the last experiment. Different from the last experiment, the detection performance of the uniform waveform applying IAA is the best. Moreover, the  
Figure 10: Pd vs. SCNR for different waveforms and algorithms.  
detection performance of the designed waveform and the uniform waveform applying IAA is close. This is because, as the SCNR increases, the non-sparsity of the clutter and noise distribution has less influence on the target information. The information of the target is more easily brought out, making the sparsity of the signal more obvious. Similar to the conclusion in Figure 5, the sparser the signal is, the higher the probability that the signal can be reconstructed by the sparse recovery algorithm. And the IAA employs sparse prior knowledge so that the IAA is more robust to the reduction in the number of sensing symbols compared to the OMP, DFT, or NUDFT. In addition, we note that the SCNRout curve of the designed waveform applying NUDFT is not smooth. The reason for this is the non-uniformity of the designed waveform. According to the second and third experiments, we can see that the detection performance applying OMP decreases as the waveform sparsity decreases.