*3.2. LPI Waveform Design Considering Radar Detection and Resolution Performance and PIS Interception Performance*

Since the LPI waveforms (which are designed to minimize the interception performance of a PIS with a certain loss of radar detection performance) do not have good range and velocity resolution, we also need to further design LPI radar waveforms to satisfy a given requirement of radar resolution performance. In this subsection, we further optimize the LPI radar waveforms, which are designed in *experiment 2* of Section 3.1, to meet the requirements of radar detection and resolution performance simultaneously. Under the background of colored Gaussian noise, whose PSD is displayed in Figure 2, the designed LPI waveforms are shown in Figure 4 with different values of resolution constraints *ν*<sup>1</sup> and *ν*2, and a given radar detection performance constraint *γ* = 0.0001. In these designs, the constrained normalized Doppler shifts are in the interval [−2, 2], and the constrained normalized time delays are in the interval [−0.1575, 0.1575], the length of which can be set according to the actual demands, which is the reflection of parameter *c* in Equation (8). As shown in Figure 4, the power of these optimized LPI waveforms has been put at the frequencies where C(*f*) has a local extremum to implement the maximization of the radar detection performance. In Figure 4, we can also find that the smaller the value of the resolution performance constraints *ν*<sup>1</sup> and *ν*2, that is, the higher the requirement level of radar resolution performance, the more the power is put at the frequencies where C(*f*) has a local extremum. This is because it is meant to satisfy the higher requirement of resolution performance by sacrificing more LPI performance, that is to put more power at the frequencies where C(*f*) has a local extremum, under a certain requirement *γ* = 0.0001 of radar detection performance. There are the same conclusions when the constrained normalized Doppler shifts and normalized time delays are extended to the intervals [−3, 3] and [−0.2362, 0.2362], respectively, as Figure 5 shows.

**Figure 4.** Optimized LPI radar waveforms under the constraint of radar detection and resolution performance (colored Gaussian noise, *PN*<sup>1</sup> = *PN*2, **Ps** = 5 kw, *γ* = 0.0001; constrained normalized Doppler shifts are in [−2, 2]; constrained normalized time delays are in [−0.1575, 0.1575]).

**Figure 5.** Optimized LPI radar waveforms under the constraint of radar detection and resolution performance (colored Gaussian noise, *PN*<sup>1</sup> = *PN*2, **Ps** = 5 kw, *γ* = 0.0001; constrained normalized Doppler shifts are in [−3, 3]; constrained normalized time delays are in [−0.2362, 0.2362]).

The one-dimensional zero-delay and zero-Doppler cuts of the ambiguity function of these optimized LPI radar waveforms are shown in Figures 6 and 7 for different intervals of constrained normalized Doppler shifts and normalized time delays. Figure 6a displays the one-dimensional zero-delay cuts of the ambiguity function of those optimized LPI radar waveforms, whose constrained normalized Doppler shifts are in the interval [−2, 2]. From Figure 6a, we find that the peak values of each sidelobe become smaller and smaller with the fall in the value of resolution performance metrics *ν*<sup>1</sup> and *ν*<sup>2</sup> in the constrained interval [−2, 2], which means velocity resolution can be improved effectively under the constraint of resolution performance. From Figure 6a, we can find that the first sidelobe can be suppressed to −20 dB below the maximum of the mainlobe with the resolution performance constraint *<sup>ν</sup>*<sup>1</sup> <sup>=</sup> *<sup>ν</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup>−7, while the first sidelobe is <sup>−</sup>4.2 dB without a resolution performance constraint. Figure 7a shows the one-dimensional zero-delay cuts of the ambiguity function, whose constrained normalized Doppler shifts are in the interval [−3, 3]. In the same way, we can find the sidelobes have better suppression, with reduction of the values of resolution performance metrics *ν*1, and *ν*2, in a wider range [−3, 3] of normalized Doppler shift. From Figure 7a, we can see that the first sidelobe can be suppressed to −22 dB below the maximum of the mainlobe with the resolution performance constraint *<sup>ν</sup>*<sup>1</sup> <sup>=</sup> *<sup>ν</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup>−7, while the first sidelobe is <sup>−</sup>4.3 dB without resolution performance constraint. Figures 6b and 7b give the one-dimensional zero-Doppler cuts of the ambiguity function of those optimized LPI radar waveforms with the constrained intervals [−0.1575, 0.1575] and [−0.2362, 0.2362] of normalized time delay, respectively. In Figure 6b, we see that the sidelobes can acquire better suppression with reduction of the values of the resolution performance metrics *ν*<sup>1</sup> and *ν*<sup>2</sup> in the constrained normalized time delay interval [−0.1575, 0.1575], which means the range resolution of radar can be effectively improved under the constraint of resolution performance. From Figure 6b, we see that the first sidelobe can be suppressed from −51 dB to −62 dB below the maximum of the mainlobe with the resolution performance constraint *<sup>ν</sup>*<sup>1</sup> <sup>=</sup> *<sup>ν</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup>−7. In Figure 7b, we have the same result in a wider constrained normalized time delay interval [−0.2362, 0.2362]. Thus, we can draw the conclusion that the LPI radar waveforms can be designed to effectively satisfy the given requirements of radar detection and resolution performance, which can also be verified in Figures 8–10. These three figures furnish the ambiguity functions with different radar detection performance constraints *γ* = 0.0001 (Figures 8 and 9) and *γ* = 0.0005 (Figure 10), and different constrained intervals for radar resolution performance (Figures 8 and 10: normalized Doppler shifts [−2, 2], normalized time delays [−0.1575, 0.1575]; Figure 9: normalized Doppler shifts [−3, 3], normalized time delays [−0.2362, 0.2362]). In each figure, different constraint levels of resolution performance have been simulated, and we can see that the sidelobe in the constraint interval has been effectively suppressed with *<sup>ν</sup>*<sup>1</sup> <sup>=</sup> *<sup>ν</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup>−<sup>7</sup> compared to other subfigures, which can be suppressed in more Doppler shifts and time delays by increasing the value of parameter *c* in Equation (8).

In order to verify the superiority of LPI performance of the designed radar waveforms, we calculate the performance degradations *δsnr* for radar and PIS in different constraint parameters and compare the optimized waveforms with a common LPI radar waveform (Frank, P1–P4). In Table 1, we find that there is a huge gap in performance degradation between the radar and the PIS for the optimized waveforms (radar detection constraint *<sup>γ</sup>* <sup>=</sup> 0.0001; resolution performance constraint *<sup>ν</sup>*<sup>1</sup> <sup>=</sup> *<sup>ν</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup>−7; optimized waveform 1: constrained normalized Doppler shifts are in [−2, 2], constrained normalized time delays are in [−0.1575, 0.1575]; optimized waveform 2: constrained normalized Doppler shifts are in [−3, 3], constrained normalized time delays are in [−0.2362, 0.2362]). The performance degradation of radar is 2.6dB, while the performance degradation of the PIS is 11.2 dB. As there are such huge gaps, the optimized waveforms can achieve a superior LPI performance. Compared with the common LPI radar waveforms, our designed waveforms still have a significant LPI superiority, as shown in Table 1. For each compared LPI radar waveform (Frank, P1–P4), we can find the SNR loss of radar for optimized waveforms is approximately equal to that of the compared waveform, as the second row shows. However, the SNR losses of the PIS are different between compared waveforms and the optimized waveforms. As the third row shows, the SNR loss of the PIS of each compared waveform is nearly 3 dB less than that of the optimized waveforms. Therefore, we can conclude that our designed waveforms can maximize the performance degradations of PISs when they meet the requirements of radar detection and resolution performance.

**Figure 6.** (**a**) One-dimensional zero-delay cuts of the ambiguity function of optimized LPI radar waveforms; (**b**) One-dimensional zero-Doppler cuts of the ambiguity function of optimized LPI radar waveforms; (colored Gaussian noise, *PN*<sup>1</sup> = *PN*2, **Ps** = 5 kw, *γ* = 0.0001; constrained normalized Doppler shifts are in [−2, 2]; constrained normalized time delays are in [−0.1575, 0.1575]).

**Table 1.** SNR losses for the common low probability of intercept (LPI) radar waveforms and optimized waveforms (radar detection constraint *γ* = 0.0001; resolution performance constraint *<sup>ν</sup>*<sup>1</sup> <sup>=</sup> *<sup>ν</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup>−7; optimized waveform 1: constrained normalized Doppler shifts are in [−2, 2], constrained normalized time delays are in [−0.1575, 0.1575]; optimized waveform 2: constrained normalized Doppler shifts are in [−3, 3], constrained normalized time delays are in [−0.2362, 0.2362]).


**Figure 7.** (**a**) One-dimensional zero-delay cuts of the ambiguity function of optimized LPI radar waveforms; (**b**) One-dimensional zero-Doppler cuts of the ambiguity function of optimized LPI radar waveforms; (colored Gaussian noise, *PN*<sup>1</sup> = *PN*2, **Ps** = 5 kw, *γ* = 0.0001; constrained normalized Doppler shifts are in [−3, 3]; constrained normalized time delays are in [−0.2362, 0.2362]).

**Figure 8.** Ambiguity function of optimized LPI radar waveforms for different resolution performance constraints *ν*<sup>1</sup> and *ν*<sup>2</sup> (colored Gaussian noise, *PN*<sup>1</sup> = *PN*2, **Ps** = 5 kw, *γ* = 0.0001; constrained normalized Doppler shifts are in [−2, 2]; constrained normalized time delays are in [−0.1575, 0.1575]).

**Figure 9.** Ambiguity function of optimized LPI radar waveforms for different resolution performance constraints *ν*<sup>1</sup> and *ν*<sup>2</sup> (colored Gaussian noise, *PN*<sup>1</sup> = *PN*2, **Ps** = 5 kw, *γ* = 0.0001; constrained normalized Doppler shifts are in [−3, 3]; constrained normalized time delays are in [−0.2362, 0.2362]).

**Figure 10.** Ambiguity function of optimized LPI radar waveforms for different resolution performance constraints *ν*<sup>1</sup> and *ν*<sup>2</sup> (colored Gaussian noise, *PN*<sup>1</sup> = *PN*2, **Ps** = 5 kw, *γ* = 0.0005; constrained normalized Doppler shifts are in [−2, 2]; constrained normalized time delays are in [−0.1575, 0.1575]).
