**2. LPI Radar Waveform Design Method**

We presume the real radar signal *s*(*t*) is emitted by the transmitting antenna with gain *Gt* in the target direction. It can be scattered and intercepted by the target, which is equipped with a PIS. We denote *x*(*t*) = *y*(*t*) + *n*1(*t*) as the signal received by the receiving antenna of the radar with gain *Gr*, where *y*(*t*) = *αs*(*t*) ∗ *h*(*t*) is the target response, *h*(*t*) is the target impulse response, *α* = <*GtGrλ*2*L*<sup>1</sup> (4*π*) <sup>3</sup>*R*<sup>4</sup> is the energy attenuation coefficient, *<sup>λ</sup>* is the wavelength, *R* is the distance to the target, *L*<sup>1</sup> is the total radar path loss, and the symbol "∗" denotes convolution. Meanwhile, the signal intercepted by the receiving antenna of the PIS with gain *Gi* can be denoted as *z*(*t*) = *βs*(*t*) + *n*2(*t*), where *β* = <sup>&</sup>lt;*GtGiλ*2*L*<sup>2</sup> (4*π*) <sup>2</sup>*R*<sup>2</sup> , *<sup>L</sup>*<sup>2</sup> is the path loss between radar and target. We also assume that *h*(*t*), *n*1(*t*), and *n*2(*t*) are zero-mean Gaussian random processes.

For the convenience of analysis, we split the frequency interval W of the radar waveform into a large number of sufficiently small and disjointed frequency intervals F*<sup>k</sup>* = [ *fk*, *fk* + *δ f* ], so that for all *f* ∈ F*k*, we have *S*(*f*) ≈ *S*(*fk*), where *S*(*f*) is the frequency domain waveform of *s*(*t*). We denote **z***<sup>k</sup>* and **n**2,*<sup>k</sup>* as components of **z** and **n**2, with frequencies in F*k*, where **z** is the received signal vector of the PIS whose elements are the samplings of *z*(*t*), and **n**<sup>2</sup> is the corresponding background noise vector of the PIS. Based on sampling theory, we suppose the sampling frequency is 2*δ f* , and therefore the sample size is 2*δ f T*, where *T* denotes the duration of signals.

According to the expressions of the KLD between two Gaussian probability density functions (PDFs) and the entropy of a Gaussian random variable, the terms in Equation (1) can be calculated as

$$E\_{\mathbf{y}}[D(\mathbf{x}; \mathbf{n}\_{1}|\mathbf{y})] = T \int\_{\mathcal{W}} \frac{2a^{2}|S(f)|\_{2} \sigma\_{H}^{2}(f)}{TP\_{N\_{1}}(f)} \mathrm{d}f,\tag{2}$$

$$I(\mathbf{x}; \mathbf{y}) = T \int\_{\mathcal{W}} \ln \left[ 1 + \frac{2a^2 |S(f)|\_2 \sigma\_H^2(f)}{T P\_{\mathcal{N}\_1}(f)} \right] \mathrm{d}f,\tag{3}$$

where *σ*<sup>2</sup> *<sup>H</sup>*(*f*) is the variance of *H*(*f*), which is the Fourier transform of *h*(*t*), and *PN*<sup>1</sup> (*f*) is the one-sided power spectral density (PSD) of *n*1(*t*).

Thus, the KLD between **x** and **n**<sup>1</sup> can be written as

$$D(\mathbf{x}; \mathbf{n}\_1) = T \int\_{\mathcal{W}} \left\{ \frac{2a^2 |S(f)|\_2 \sigma\_H^2(f)}{T P\_{N\_1}(f)} - \ln \left[ 1 + \frac{2a^2 |S(f)|\_2 \sigma\_H^2(f)}{T P\_{N\_1}(f)} \right] \right\} \mathrm{d}f. \tag{4}$$

In the design of LPI radar waveforms, the resolution performance should also be considered, which is another quite important performance measure for radar. In this paper, we use autocorrelation to describe radar resolution performance, which is a more concise way of representing the ambiguity function. For time, the worse the autocorrelation is, the better the range resolution is. Correspondingly, for frequency, the worse the autocorrelation is, the better the velocity resolution is. Next, we will utilize joint entropy to describe the autocorrelation.

Suppose the samplings of the transmitted waveform **s** = [*s*(1),*s*(2), ··· ,*s*(*L*)] *<sup>T</sup>* come from the normal distribution with mean 0 and variance *σ*<sup>2</sup> *<sup>l</sup>* . Then, the entropy of sample *s*(*l*) can be computed as

$$\text{Entropy}[s(l)] = \frac{1}{2}[\log 2\pi + 1] + \frac{1}{2}\log \sigma\_l^2. \tag{5}$$

Since the joint probability density function of these samples is *p*(**s**) = (2*π*) *<sup>L</sup>*|**R**<sup>|</sup> − 1 2 exp- −1 <sup>2</sup> **<sup>s</sup>***T*|**R**<sup>|</sup> <sup>−</sup>1**s** , the joint entropy of these samples can be calculated as

$$\text{Entropy}[\mathbf{s}] = \frac{L}{2}[\log 2\pi + 1] + \frac{1}{2}\log|\mathbf{R}|\,\text{s}\tag{6}$$

where **R** is the sample covariance matrix of transmitted waveform **s**, which can be estimated as **R**ˆ = *ri*,*j <sup>L</sup>*,*<sup>L</sup> i*=1,*j*=1 , *ri*,*<sup>j</sup>* = <sup>1</sup> *L L*−|*i*−*j*| ∑ *l*=1 *s*(*l*)*s*(*l*+|*i* − *j*|), and **R** is a Toeplitz and symmetric matrix. If the designed waveform has a perfect resolution performance, then the samples are independent. It means that the joint entropy is equal to the sum of the entropy of each sample, that is: Entropy[**s**] <sup>=</sup> *<sup>L</sup>* ∑ Entropy[*s*(*l*)]. In fact, the designed waveform cannot have

*l*=1 a perfect resolution performance. The joint entropy and the sum of the entropy of each sample have a relationship as follows: Entropy[**s**] <sup>≤</sup> *<sup>L</sup>* ∑ *l*=1 Entropy[*s*(*l*)]. Therefore, we use the difference between the joint entropy and the sum of the entropy of each sample as the metric of resolution performance, which can be expressed as

$$\Delta \mathbf{E}\_{\sf s} = \sum\_{l=1}^{L} \text{Entropy}[s(l)] - \text{Entropy}[\mathbf{s}] = \frac{1}{2} \sum\_{l=1}^{L} \log \sigma\_{l}^{2} - \frac{1}{2} \log |\mathbf{R}| \,\tag{7}$$

where ΔE*<sup>s</sup>* ≥ 0.

From Equation (7), we find that the smaller the value of the metric, the better the resolution performance of the transmitted waveform, and when the transmitted waveform is white Gaussian noise, the value of the metric ΔE*<sup>s</sup>* is equal to zero, which means the white Gaussian noise has a perfect resolution performance. In the design of LPI waveforms, we need to minimize ΔE*s*. For solving it conveniently, since **R** is a Toeplitz and symmetric matrix, we can simplify the metric ΔE*<sup>s</sup>* to a convex function, which can be expressed as

$$\widehat{\rm \Delta E}\_{s} = \frac{\sum\_{j=2}^{\mathcal{E}} r\_{1,j}}{(c-1)P\_{\mathcal{S}}} = \frac{\sum\_{j=2}^{\mathcal{E}} \sum\_{l=1}^{L-|1-j|} s(l)s(l+|1-j|)}{(c-1)LP\_{\mathcal{S}}},\tag{8}$$

where *c* is the constraint number of the time delay or Doppler shift, which can be set according to the practical application, since we do not need to constrain the autocorrelation for each time delay and Doppler shift, and *PS* is the average power spectral density of the transmitted waveform, which is used for the purpose of normalization, that is, the upper bound of<sup>Δ</sup>E*<sup>s</sup>* is equal to one.

For frequency, resolution performance has the same computational procedure. We only need to substitute *s*(*l*) with *S*(*fl*), and we denote the metric of resolution performance for frequency as<sup>Δ</sup>E*s*.

It is a common view that white noise is the best LPI waveform. The closer the distance between intercept signal **z** and background noise **n**2, the more difficult it is to detect and recognize the intercept signal. The KLD has been confirmed to be a powerful and accurate tool to measure the information of multivariate data, with lesser complexity and superior performance among the existing distance measures, such as <sup>L</sup>1, Bhattacharyya distance, Hellinger distance, *f*− divergence, etc. [15–17]. Therefore, here we use KLD as the PDF distance measure, which is denoted as *D*(**z**; **n**2).

The KLD between **z***<sup>m</sup> <sup>k</sup>* and **<sup>n</sup>***<sup>m</sup>* 2,*k*, which are samples of **z***<sup>k</sup>* and **n**2,*<sup>k</sup>* in each frequency *fm* ∈ F*<sup>k</sup>* can be written as *D* - **z***m <sup>k</sup>* ; **<sup>n</sup>***<sup>m</sup>* 2,*k* <sup>=</sup> *<sup>β</sup>*<sup>2</sup>|*S*(*fk* )<sup>|</sup> 2 *PN*<sup>2</sup> (*fk* ) , where *PN*<sup>2</sup> (*f*) is the one-sided power spectral density (PSD) of *n*2(*t*), that is a function of SNR. The lower SNR a waveform possesses in each frequency, the harder it is for a PIS to intercept it. This agrees with our common knowledge and experience. In order to solve the following optimization problem smoothly, we take the natural exponential function of *D* - **z***m <sup>k</sup>* ; **<sup>n</sup>***<sup>m</sup>* 2,*k* , which can maintain the monotonicity near |*S*(*fk*)| 2 . It can be denoted as *D*\* - **z***m <sup>k</sup>* ; **<sup>n</sup>***<sup>m</sup>* 2,*k* <sup>=</sup> exp *<sup>β</sup>*<sup>2</sup>|*S*(*fk* )<sup>|</sup> 2 *PN*<sup>2</sup> (*fk* ) . Thus, the modified KLD between component **z***<sup>k</sup>* and **n**2,*<sup>k</sup>* is

$$D(\mathbf{z}\_k; \mathbf{n}\_{2,k}) = 2\delta f \ T\bar{D}\left(\mathbf{z}\_k^m; \mathbf{n}\_{2,k}^m\right) = 2\delta f T \exp\left\{\frac{\beta^2 \left|S(f\_k)\right|^2}{P\_{\mathcal{N}\_2}(f\_k)}\right\}.\tag{9}$$

When *δ f* → 0, the modified KLD between **z** and **n**<sup>2</sup> can be obtained as

$$D(\mathbf{z}; \mathbf{n}\_2) = 2T \int\_{\mathcal{W}} \exp\left\{ \frac{\beta^2 \left| S(f) \right|^2}{P\_{\mathcal{N}\_2}(f)} \right\} \mathrm{d}f. \tag{10}$$

Since the KLDs *D*(**x**; **n**1) and *D*(**z**; **n**2) can be used to measure the detection performance of radar and a PIS respectively, and <sup>Δ</sup>E*<sup>s</sup>* and <sup>Δ</sup>E*<sup>s</sup>* can be used to measure the resolution performance of radar, we can design an LPI radar waveform which not only has superior target detecting and resolution performance, but also has superior LPI performance against PIS interception based on these four metrics. The optimization problem of LPI radar waveform design can be straightforwardly described as **<sup>s</sup>** <sup>=</sup> *arg*{max**s***D*(**x**; **<sup>n</sup>**1), min**<sup>s</sup>** <sup>Δ</sup>E*s*, min**<sup>s</sup>** <sup>Δ</sup>E*s*, min**s***D*(**z**; **<sup>n</sup>**2)}, under the constraint that the average transmitted power is fixed, denoted by + +*S*(*f*) + + 2d*f* = **Ps** .

W LPI radar waveform design is a trade-off between the performance of radar and of PISs, which is to maximize the detection and resolution performance of radar and minimize the interception performance of PISs. In fact, the primary task of the emitted waveform is to accomplish target detection and resolution. The designed radar waveform should be considered for its LPI capability under the condition of meeting radar performance. Therefore, we minimize the KLD *<sup>D</sup>*(**z**; **<sup>n</sup>**2) of the PIS in the situation that the KLD *<sup>D</sup>*(**x**; **<sup>n</sup>**1), <sup>Δ</sup>E*s*, and<sup>Δ</sup>E*<sup>s</sup>* of radar make some concessions, which can be expressed as

$$\begin{array}{c} \mathsf{s}^{\star} = \arg \left\{ \min\_{\mathsf{s}} \boldsymbol{D}(\mathsf{z} \; ; \mathsf{n}\_{2}) \right\} \\ \mathsf{s} \text{ s.t. } \boldsymbol{D}(\mathsf{x}; \mathsf{n}\_{1}) \geq \gamma\_{\mathsf{s}} \quad \widehat{\mathsf{AE}\_{\mathsf{s}}} \leq \nu\_{1} \quad \widehat{\mathsf{AE}\_{\mathsf{s}}} \leq \nu\_{2\mathsf{s}} \quad \int\_{\mathcal{W}} |\mathsf{S}(f)|\_{2} \mathsf{d}f = \mathsf{P}\_{\mathsf{s}\prime} \end{array} \tag{11}$$

where *γ* is the value of *D*(**x**; **n**1) required to meet radar detection performance, and *ν*<sup>1</sup> and *<sup>ν</sup>*<sup>2</sup> are the values of <sup>Δ</sup>E*<sup>s</sup>* and <sup>Δ</sup>E*<sup>s</sup>* needed to meet the radar resolution performance for range and velocity, respectively. All of them can be set to various values in different competing scenarios.

In order to achieve the optimal solution of Equation (11), the discrete form of the optimization problem first needs to be obtained, which can be written as

$$\begin{split} \mathsf{s}^{\star} &= \arg \left\{ \min\_{\mathsf{s}} 2Tb \sum\_{l=1}^{L} e^{\frac{\beta^{2} |S(f\_{l})|^{2}}{P\_{N\_{2}}(f\_{l})}} \right\}, \\ & \left\{ Tb \sum\_{l=1}^{L} \left\{ \frac{2\kappa^{2} |S(f\_{l})|\_{2} r\_{H}^{2}(f\_{l})}{T p\_{N\_{1}}(f\_{l})} - \ln \left[ 1 + \frac{2\kappa^{2} |S(f\_{l})|\_{2} r\_{H}^{2}(f\_{l})}{T p\_{N\_{1}}(f\_{l})} \right] \right\} \right\} \geq \gamma \\ & \text{s.t.} \left\{ \frac{\sum\_{l=1}^{L} \sum\_{l=1}^{|I|} s(|I|s(I+|1-j)|)}{(c-1) L P\_{\mathbb{S}}} \leq \nu\_{1}, \frac{\sum\_{l=1}^{L} \sum\_{l=1}^{|I-j|} s(f\_{l}) S(f\_{l+|1-j|})}{(c-1) L P\_{\mathbb{S}}} \leq \nu\_{2}, \\ & \quad \quad \quad \quad \quad \quad \frac{b \sum\_{l=1}^{L} |S(f\_{l})|^{2}}{l} = \mathbf{P\_{\mathsf{s}}} \\ & \quad \quad \quad \quad 0 \leq \Big\vert S(f\_{l}) \right\vert\_{2} \leq \frac{\mathbf{P\_{\mathsf{s}}}}{b}, l = 1, \cdots, l. \end{split} \tag{12}$$

where *fl*, *l* = 1, ··· , *L* are the uniform partition points in the frequency interval W and *b* = *fl*<sup>+</sup><sup>1</sup> − *fl*.

When *γ* = 0, *ν*<sup>1</sup> = 1, and *ν*<sup>2</sup> = 1, that is, the LPI waveform is optimized without regard to radar detection and resolution performance, the optimized waveform—considering only the interception performance of the PIS—is <sup>|</sup>*S*(*f*)|<sup>2</sup> <sup>=</sup> **Ps**/<sup>W</sup> , under the assumption that the background noise **n**<sup>2</sup> is a white Gaussian noise, whose PSD *PN*<sup>2</sup> (*f*) is a constant. Thus, in this case we can draw a conclusion that for PISs, it is most difficult to intercept a radar waveform whose power is distributed equally within the whole bandwidth W. When the radar resolution and interception performance of the PIS are not considered, and only the radar detection performance is considered, the radar detection performance metric *D*(**x**; **n**1) is maximized, and the optimized waveform concentrates all waveform energies at the frequency where *σ*<sup>2</sup> *<sup>H</sup>*(*fl*)/*PN*<sup>1</sup> (*fl*) achieves the maximum value. The optimized LPI waveform in Equation (12) should be a compromise between one optimized solution, i.e., energy is distributed equally in all frequencies, considering only the interception performance of the PIS, and another optimized solution, i.e., energy is concentrated at a certain frequency, considering only radar detection performance.

An efficient method for solving nonlinear constrained optimization problems is the combination of the interior point and sequential quadratic programming (SQP) methods. Here, we use the algorithm proposed by Byrd [18] to solve the optimization problem in Equation (12), which jointly utilizes trust regions to ensure the robustness of iterations.
