*Communication* **Spatial Information-Theoretic Optimal LPI Radar Waveform Design**

**Jun Chen 1,\*, Jie Wang 1,\*, Yidong Zhang 1, Fei Wang <sup>2</sup> and Jianjiang Zhou <sup>2</sup>**


**Abstract:** In this paper, the design of low probability of intercept (LPI) radar waveforms considers not only the performance of passive interception systems (PISs), but also radar detection and resolution performance. Waveform design is an important considerations for the LPI ability of radar. Since information theory has a powerful performance-bound description ability from the perspective of information flow, LPI waveforms are designed in this paper within the constraints of the detection performance metrics of radar and PISs, both of which are measured by the Kullback–Leibler divergence, and the resolution performance metric, which is measured by joint entropy. The designed optimization model of LPI waveforms can be solved using the sequential quadratic programming (SQP) method. Simulation results verify that the designed LPI waveforms not only have satisfactory target-detecting and resolution performance, but also have a superior low interception performance against PISs.

**Keywords:** LPI; radar waveform; passive interception systems; Kullback–Leibler divergence; joint entropy

**1. Introduction**

Low probability of intercept (LPI) radar waveforms have been developed to combat passive interception systems (PISs) for several decades [1–3]. Common LPI waveforms include FM/PM signals, FSK/PSK signals, etc. [2,4–7], which utilize wideband modulations to spread the energy in a frequency. LPI radar waveform design is a primary means of affecting the interception performance of PISs, which is actually a compromise between radar performance (which contains detection and resolution performance) and the interception performance of the PIS. In this paper, the dispersion of waveform energy in a frequency can be implemented through the compromise between the optimal detection performance, resolution performance, and LPI performance of radar by adjusting their frequency amplitudes.

In frequency amplitude adjustment modeling, it is crucial to establish the performance metrics of both the radar and the PIS. For radar detection performance, besides metrics such as output signal-to-noise ratio (SNR), relative entropy, and mean square error, the method of maximizing mutual information has been widely used in optimal radar waveform design (see [8–11] and references therein). Zhu et al. [12] presented the Kullback–Leibler divergence (KLD) as more appropriate than mutual information to describe optimal radar detection performance. The KLD is defined as

$$D(\mathbf{x}; \mathbf{n}\_1) = E\_\mathbf{y} [D(\mathbf{x}; \mathbf{n}\_1|\mathbf{y})] - I(\mathbf{x}; \mathbf{y}), \tag{1}$$

where **x** = **y** + **n**<sup>1</sup> is the received radar signal, **y** is the target response, **n**<sup>1</sup> is the background radar noise, *E*(·) denotes capture expectation, *I*(·) denotes mutual information, and *D*(·) denotes the KLD. For radar resolution performance, the most classic and common metric

**Citation:** Chen, J.; Wang, J.; Zhang, Y.; Wang, F.; Zhou, J. Spatial Information-Theoretic Optimal LPI Radar Waveform Design. *Entropy* **2022**, *24*, 1515. https://doi.org/ 10.3390/e24111515

Academic Editors: Karagrigoriou Alexandros and Makrides Andreas

Received: 11 September 2022 Accepted: 22 October 2022 Published: 24 October 2022

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is the ambiguity function [13]. In this paper, we will design a simpler resolution metric for radar by using joint entropy. For PISs, the common interception performance metric includes the peak-to-average power ratio, time–bandwidth product, and so on (see [2,7,14] and references therein). Here, we regard the KLD, denoted by *D*(**z**; **n**2), between the intercept signal **z** and background noise **n**<sup>2</sup> of a PIS as the effective interception performance metric of a PIS for LPI radar waveform design. Thus, by maximizing the detection and resolution performance of radar and minimizing the interception performance of the PIS, an optimization problem of frequency amplitudes can be established and solved with the constraint of a fixed transmission power.
