*2.1. Principles of Deceptive Jamming against SAR*

The principle of SAR deceptive jamming based on modulation–retransmission is presented in Figure 1 [17]. The jammer performs a serious of operations including the amplification, down-conversion, analog-to-digital conversion (A/D), and fast Fourier transform (FFT) on the intercepted radar radio frequency (RF) signal to obtain the frequency domain representation of the baseband, while the JSF is calculated based on the template and the SAR parameters including kinematic parameters (platform position, velocity, etc.), antenna parameters (antenna direction, beam pattern, etc.), and signal parameters (carrier frequency, PRI, etc.). Then, we multiply the two and perform an inverse fast Fourier transform (IFFT) to obtain the baseband of the jamming signal and finally perform the digital-to-analog conversion (D/A), up-conversion, gain control, and retransmission. By repeating the above steps for each pulse, a false image can be generated by the receiver. The template is an array of false scatters that depicts the electromagnetic characteristic of the fake scene artificially fabricated by the jammer.

**Figure 1.** Principle of synthetic aperture radar (SAR) deceptive jamming.

The JSF is the key to the generation of jamming signals. The following illustrates the basic idea for JSF in combination with the geometric model of SAR jamming. As shown in Figure 2, we assume that the SAR platform moves at a constant velocity *v* and the azimuth time *ta* = 0 when the plane of zero Doppler passes through the jammer. A Cartesian coordinate system with the location of the jammer as the origin is established in a two-dimensional slant range plane. The x-axis points to the range direction and the y-axis points to the azimuth direction. The shortest slant distance between the jammer and the SAR is *RJ*0, and the instantaneous slant distance is *RJ*(*ta*) at azimuth time *ta*. An arbitrary false point scatter *P* is generated by the jammer, the scattering coefficient of *P* is σ*<sup>P</sup>* and the location is (*x*, *y*) in the coordinate above. *RP*(*ta*) denotes the instantaneous slant distance between *P* and the jammer.

**Figure 2.** The geometric model of SAR deceptive jamming.

*Remote Sens.* **2020**, *12*, 53

In order to generate the fake point target *P* in the SAR image, the JSF at azimuth time *ta* is [17]

$$H\_P(f\_r, t\_a) = \sigma\_P \exp\left[-j2\pi(f\_r + f\_0)\frac{2\Delta R\_P(t\_a)}{c}\right] \tag{1}$$

where *fr* is the range frequency, *c* is the velocity of light, Δ*RP*(*ta*) is the difference between *RP*(*ta*) and *RJ*(*ta*):

$$
\Delta R\_P(t\_d) = R\_P(t\_d) - R\_f(t\_d) = \sqrt{\left(\mathbf{x} + R\_{\!/\!0}\right)^2 + \left(y - vt\_d\right)^2} - \sqrt{R\_{\!/\!0}^2 + \left(vt\_d\right)^2}.\tag{2}
$$

By calculating the JSF for each scatter in the template *T*, the JSF for the deception scene can be derived as follows:

$$H(f\_{r}, t\_{a}) = \sum\_{P \in \mathcal{T}} H\_{P}(f\_{r}, t\_{a}) = \sum\_{P \in \mathcal{T}} \sigma\_{P} \exp\left|-j2\pi (f\_{r} + f\_{0})\frac{2\Delta R\_{P}(t\_{a})}{c}\right|.\tag{3}$$

In the implementation of jamming, the jammer must calculate JSF *H*(*fr*, *ta*) and modulate the intercepted signal real-time in each PRI. The calculation of Equation (2) is time-consuming, therefore the method represented by Equation (3) cannot be used directly for large-scene deceptive jamming unless some improvements are made.

#### *2.2. TDFS-TS Algorithm*

### 2.2.1. Deceptive Jamming Based on TDFS

For space-borne SAR, we can assert that *RJ*<sup>0</sup> *x*, *RJ*<sup>0</sup> *y*, and *RJ*<sup>0</sup> *vta* throughout the synthetic aperture time. After the Taylor series expansion, Equation (2) can be approximated as follows [11]:

$$\begin{split} \Delta R\_P(t\_d) & \approx \left[ \mathbf{x} + R\_{f0} + \frac{y^2}{2\left(\mathbf{x} + R\_{f0}\right)} - \frac{yvt\_a}{\mathbf{x} + R\_{f0}} + \frac{\left(vt\_a\right)^2}{2\left(\mathbf{x} + R\_{f0}\right)} \right] - \left[R\_{f0} + \frac{\left(vt\_a\right)^2}{2R\_{f0}}\right] \\ & \approx \mathbf{x} + \frac{y^2}{2R\_{f0}} - \frac{yvt\_a}{R\_{f0}}. \end{split} \tag{4}$$

With the approximation above, the JSF—i.e., Equation (1)—can be rewritten as follows:

$$\begin{split} H\_{P}(f\_{r},t\_{d}) &= \sigma\_{P} \exp\left[-j2\pi(f\_{r}+f\_{0})\left(\frac{2\pi}{\mathcal{E}} + \frac{y^{2}}{cR\_{\parallel 0}} - \frac{2yvt\_{d}}{cR\_{\parallel 0}}\right)\right] \\ &= \sigma\_{P} \exp\left[-j2\pi(f\_{r}+f\_{0})\frac{2\pi}{\mathcal{E}}\right] \exp\left(j2\pi f\_{0}\frac{2yvt\_{d}}{cR\_{\parallel 0}}\right) \\ &\quad \exp\left(-j2\pi f\_{0}\frac{y^{2}}{cR\_{\parallel 0}}\right) \exp\left(j2\pi f\_{r}\frac{2yvt\_{d}-y^{2}}{cR\_{\parallel 0}}\right). \end{split} \tag{5}$$

where the third exponential term is independent of *fr* and *ta* and is equivalent to introducing a fixed phase that has no effect on the imaging; thus, it can be ignored. The fourth exponential term can also be omitted when *y* is small enough (details will be analyzed in the next subsection). Thus, Equation (5) can be simplified as follows:

$$H\_P(f\_{r\prime}, t\_a) = \sigma\_P \exp\left[-j2\pi(f\_r + f\_0)\frac{2\chi}{c}\right] \exp\left(j2\pi f\_0 \frac{2yvt\_a}{cR\_{f0}}\right). \tag{6}$$

Actually, at the broadside mode, the azimuth frequency modulation rate of the SAR echo signal from the location of the jammer is [22]

$$K\_a = -\frac{2f\_0 v^2}{cR\_{f0}}.\tag{7}$$

Combined with Equations (6) and (7), we can derive

$$H\_P(f\_r, t\_a) = \sigma\_P \exp\left[-j2\pi(f\_r + f\_0)\frac{2\pi}{c}\right] \exp\left(-j2\pi K\_a \frac{y}{\upsilon} t\_a\right). \tag{8}$$

This is equivalent to delaying the original echo signal 2*x*/*c* in fast-time to achieve position deception in the range dimension and a shifting frequency −*Kay*/*v* in slow-time. According to the frequency shifting property of the linear frequency modulation (LFM) signal, the frequency shifting is equivalent to time-delaying *y*/*v* in slow-time, which can implement position deception in the azimuth dimension as well.

If the scattering coefficient of the false point target which locates (*x*, *y*) in the template is σ(*x*, *y*), the JSF for the deception scene can be derived as follows:

$$\begin{split} H(f\_{r}, t\_{d}) &= \sum\_{\mathbf{x}} \sum\_{\mathbf{y}} \sigma(\mathbf{x}, \mathbf{y}) \exp[-j2\pi(f\_{r} + f\_{0})\frac{2\pi}{\mathfrak{c}}] \exp\Big(j2\pi f\_{0} \frac{2\mathbf{y}vt\_{\mathfrak{c}}}{\varepsilon\mathbb{R}\_{\parallel 0}}\Big) \\ &= \sum\_{\mathbf{y}} \exp\Big(j4\pi f\_{0} \frac{\mathbf{y}vt\_{\mathfrak{c}}}{\varepsilon\mathbb{R}\_{\parallel 0}}\Big) \sum\_{\mathbf{x}} \sigma(\mathbf{x}, \mathbf{y}) \exp[-j4\pi \left(f\_{r} + f\_{0}\right)\frac{\mathbf{x}}{\mathfrak{c}}]. \end{split} \tag{9}$$

The second summation term in Equation (9) is independent of azimuth time *ta* and only related to the relative position of each point in the false scene, which can be calculated off-line to reduce the real-time computational burden. This is the TDFS jamming algorithm, which has the advantages of simplicity and high computational efficiency.

#### 2.2.2. Jamming Signal Analysis

Due to the approximation and simplification, the TDFS algorithm has high computational efficiency. However, the approximation and simplification will cause a decline in the image quality of the deceptive target at the same time. In this subsection, we will analyze the impact of the simplified operations above on the imaging results by comparing the difference between the jamming signal and real point target echo in the range-Doppler domain [23].

The echo signal of a real scatter point *P*(*x*, *y*) is represented as follows [22]:

$$\rho s\_P(t\_r, t\_a) = \sigma\_P w\_a(t\_a) w\_r \left[ t\_r - \frac{2R\_P(t\_a)}{c} \right] \exp\left[ -j4\pi f\_0 \frac{R\_P(t\_a)}{c} \right] \exp\left\{ j\pi K\_r \left[ t\_r - \frac{2R\_P(t\_a)}{c} \right]^2 \right\},\tag{10}$$

where *wa*(*ta*) represents the azimuth amplitude, *wr*(*tr*) is the SAR pulse complex envelope, and *Kr* is the frequency modulation rate of the SAR pulse.

The parabolic approximation of the instantaneous slant range *RP*(*ta*) by Taylor series is [22]

$$R\_P(t\_d) = \sqrt{\left(\mathbf{x} + R\_{l0}\right)^2 + \left(y - vt\_d\right)^2} = \mathbf{x} + R\_{l0} + \frac{\left(y - vt\_d\right)^2}{2\left(\mathbf{x} + R\_{l0}\right)}.\tag{11}$$

We can derive the echo signal expression (12) of the real target point *P*(*x*, *y*) in the range-Doppler domain by bringing Equation (11) into Equation (10) and using the principle of stationary phase (POSP) [22]:

$$\begin{aligned} S\_P(t\_{r\_I}f\_a) &= \sigma\_P \mathcal{W}\_a(f\_a) w\_I \left| t\_r - \frac{2\mathcal{R}^\circ(f\_a)}{c} \right| \\ \exp\left\{ j\pi K\_r \left| t\_r - \frac{2\mathcal{R}\_P(f\_a)}{c} \right|^2 - j\pi \frac{f\_a^2}{K\_a} - j2\pi f\_a \frac{y}{v} - j4\pi f\_0 \frac{x + R\_0}{c} \right\}, \end{aligned} \tag{12}$$

where *RP*(*fa*) is the range cell migration (RCM) curve of the target in the range-Doppler domain:

$$R\_P(f\_a) = \mathbf{x} + R\_{f0} + \frac{\{\mathbf{x} + R\_{f0}\} \mathbf{c}^2}{8\nu^2 f\_0^2} f\_{a\nu}^2 \tag{13}$$

and the azimuth frequency modulation rate is

$$K\_4 = -\frac{2v^2 f\_0}{c \{\mathbf{x} + R\_{I0}\}}.\tag{14}$$

The echo signal will focus on the coordinate (*x*, *y*) after range compression, RCM correction (RCMC), and azimuth compression.

For comparison, when the JSF is Equation (8), the jamming signal is expressed as follows:

$$\begin{split} s\_{I}(t\_{r}, t\_{d}) &= \sigma\_{P} w\_{d}(t\_{d}) w\_{I} \left| t\_{r} - \frac{2\mathcal{R}\_{I}(t\_{d})}{c} - \frac{2\chi}{c} \right| \\ \exp\left\{ -j4\pi f\_{0} \left[ \frac{\mathcal{R}\_{I}(t\_{d})}{c} + \frac{\chi}{c} - \frac{yvt\_{d}}{c\mathcal{R}\_{I0}} \right] \right\} \exp\left\{ j\pi K\_{r} \left[ t\_{r} - \frac{2\mathcal{R}\_{I}(t\_{d})}{c} - \frac{2\chi}{c} \right]^{2} \right\}. \end{split} \tag{15}$$

Similarly, the parabolic approximation and POSP is used to obtain the expression of the jamming signal in the range-Doppler domain:

$$\begin{split} S\_{\boldsymbol{f}}(t\_{\boldsymbol{r}}, f\_{\boldsymbol{a}}) &= \sigma\_{\boldsymbol{P}} \mathcal{W}\_{\boldsymbol{a}}(f\_{\boldsymbol{a}}) w\_{\boldsymbol{r}} \Big[ t\_{\boldsymbol{r}} - \frac{2 \mathcal{R}\_{\boldsymbol{f}\boldsymbol{P}}(f\_{\boldsymbol{a}})}{\boldsymbol{c}} \Big] \\ \exp\Big\{ j\pi \mathcal{K}\_{\boldsymbol{r}} \Big[ t\_{\boldsymbol{r}} - \frac{2 \mathcal{R}\_{\boldsymbol{f}\boldsymbol{P}}(f\_{\boldsymbol{a}})}{\boldsymbol{c}} \Big]^{2} - j\pi \frac{f\_{\boldsymbol{a}}^{2}}{\mathcal{K}\_{\boldsymbol{f}\boldsymbol{a}}} - j2\pi f\_{\boldsymbol{a}} \frac{\mathcal{Y}}{\boldsymbol{v}} - j4\pi f\_{0} \frac{\boldsymbol{x} + \mathcal{R}\_{\boldsymbol{f}\boldsymbol{0}}}{\boldsymbol{c}} + j2\pi f\_{0} \frac{\mathcal{Y}^{2}}{\mathcal{K}\_{\boldsymbol{f}\boldsymbol{0}}} \Big\}, \end{split} \tag{16}$$

where the RCM of fake target is

$$R\_{IP}(f\_a) = \mathbf{x} + R\rho\_0 + \frac{R\_{f0}c^2}{8v^2f\_0^2}f\_a^2 - \frac{yc}{2vf\_0}f\_a + \frac{y^2}{2R\_{f0}},\tag{17}$$

and the azimuth frequency modulate rate of jamming signal is

$$K\_{la} = -\frac{2v^2 f\_0}{cR\_{f0}}.\tag{18}$$

It can be found that there are differences between the real target echo signal and the jamming signal in the RCM curve and azimuth frequency modulation rate, ignoring the phase terms unrelated to pulse compression. The effects of these differences on the imaging results and the corresponding effective region of deceptive jamming are analyzed in detail below.

First, it is obvious that the jamming signal introduces the azimuth frequency modulation rate error, which will cause a mismatch of the azimuth matched filter and finally lead to the main lobe broadening of the azimuth pulse compression result. The azimuth frequency modulation rate error is

$$
\Delta K\_{\rm d} = K\_{\rm ld} - K\_{\rm d} = -\frac{2v^2 f\_0 \mathbf{x}}{c R\_{\rm l0} (\mathbf{x} + R\_{\rm l0})}.\tag{19}
$$

The effect of Δ*Ka* on the main lobe broadening can be measured by the quadratic phase error (QPE); the expression of QPE is as follows [22]:

$$\text{QPE} = \pi \Delta \mathcal{K}\_{\mathsf{d}} \left( \frac{T}{2} \right)^2 = -\frac{2\pi v^2 f\_{\mathsf{f}0} \text{x}}{c R\_{\mathsf{f}0} \text{(x} + R\_{\mathsf{f}0} \text{)}} \left( \frac{L}{2v} \right)^2,\tag{20}$$

where *T* = *L*/*v* is the synthetic aperture time and *L* represents the synthetic aperture length.

For a typical Kaiser window with β = 2.5, if the broadening is required to be less than 2%, 5%, and 10%, the corresponding QPE absolute value should be less than 0.27π, 0.41π, and 0.55π [22]. Here, we define the azimuth QPE factor ε; when the condition |QPE| ≤ επ is required, the range coordinate *x* should satisfy

$$|\mathbf{x}| \le \frac{2\varepsilon c R\_{\!/\!0\!\!0} \left(\mathbf{x} + R\_{\!\!\!\!0\!\!0} \right)}{f\_0 L^2} \approx 2\varepsilon \frac{c}{f\_0} \left(\frac{R\_{\!\!\!\!0\!\!0}}{L} \right)^2. \tag{21}$$

Second, in the range-Doppler domain, the RCM error of the jamming signal is

$$
\Delta R\_P(f\_a) = R\_{IP}(f\_a) - R\_P(f\_a) = -\frac{\text{xc}^2}{8v^2 f\_0^2} f\_a^2 - \frac{yc}{2vf\_0} f\_a + \frac{y^2}{2R\_{f0}}.\tag{22}
$$

The last term in Equation (22) can be omitted because *RJ*<sup>0</sup> *y*. The residual RCM introduced by Δ*RP*(*fa*) in broadside mode is represented as follows:

$$\text{RCM}\_{\text{res}} = \left| \Delta R\_P \left( \frac{B\_d}{2} \right) - \Delta R\_P \left( -\frac{B\_d}{2} \right) \right| = \frac{B\_d c}{2vf\_0} \left| y \right|. \tag{23}$$

where *Ba* is the azimuth Doppler bandwidth and can be expressed as [22]

$$B\_{\rm d} = \frac{L}{\upsilon} \middle| \mathbf{K}\_{\rm l} \underline{\mathbf{l}} = \frac{2 \upsilon f\_{\rm 0} L}{c R\_{\rm 0}}.\tag{24}$$

Combined with Equations (23) and (24), we can simplify the expression of residual RCM:

$$\text{RCM}\_{\text{res}} = \frac{L}{R\_{J0}} |y| \,\tag{25}$$

The residual RCM will result in the main lobe broadening in both range and azimuth dimensions and the extent of broadening can be measured by the ratio of the residual RCM to the range resolution. The range resolution ρ*<sup>r</sup>* ≈ *c*/2*B*, where *B* is the signal bandwidth. Then, we define the residual RCM factor η; if we require RCMres ≤ ηρ*r*, the azimuth coordinate *y* should satisfy

$$\left|y\right| \le \eta \frac{\rho\_r R\_{l0}}{L} = \eta \frac{cR\_{l0}}{2BL}.\tag{26}$$

According to [22], the residual RCM should be no more than 0.5 of the range cell; therefore, η should be no more than 0.5.

In addition, due to the difference between the false and real point targets in the instantaneous slant range history, the Doppler center frequency of the jamming signal shifts, and the azimuth main lobe broadening and ghost targets are introduced. This phenomenon limits the effective azimuth scale as well; according to [11], the azimuth coordinate *y* should satisfy

$$\left|y\right| \le \frac{cR\_{f0}}{2vf\_0} \left(\text{PRF} - \frac{v}{D}\right) - \frac{L}{2},\tag{27}$$

where PRF is the pulse repetition frequency and *D* is the antenna aperture in the azimuth direction.

Equations (21) and (26) describe the effective region of the range and azimuth directions of the TDFS algorithm with the specified azimuth QPE factor ε and residual RCM factor η. The jamming signals representing the false target located beyond the region will not achieve the desired deception. Equation (27) described the inherent limitations of SAR deceptive jamming in the azimuth direction, which is beyond the scope of this article. In the following discussion, we suppose that the size of the template in the azimuth dimension meets the requirement of Equation (27). When the typical C-band space-borne SAR parameters (see Table 1 in Section 3) are set as an example, we can calculate the <sup>e</sup>ffective region: <sup>|</sup>*x*<sup>|</sup> <sup>≤</sup> 1.25 km and  *y* <sup>≤</sup> 0.75 km with <sup>ε</sup> <sup>=</sup> 0.25 and <sup>η</sup> <sup>=</sup> 0.5; i.e., a rectangular area of 2.5 km × 1.5 km.


**Table 1.** The setting of SAR parameters in simulations.

2.2.3. Template Segmentation

The analysis in the previous subsection shows that the effective region of the TDFS algorithm is limited. In order to achieve deceptive jamming in a larger scene, we divide the jamming scene template into several blocks and apply a time-delay and frequency-shift in each block to calculate the partial JSF, which will be summed to get the JSF on the whole template. As shown in Figure 3, the template consisting of *m* × *n* point scatters is divided into *M* × *N* blocks; each block contains *U* × *V* point scatters, namely *U* = *m*/*M* and *V* = *n*/*N*. If the range interval between each point is Δ*x* and the azimuth interval is Δ*y*, *U* and *V* should satisfy the following conditions according to the limitation of the effective region Equations (21) and (26) with the required ε and η:

$$
\Delta U \le \eta \frac{cR\_{\rm min}}{BL\Delta y}, \quad V \le 4\varepsilon \frac{c}{f\_0 \Delta x} \left(\frac{R\_{\rm min}}{L}\right)^2,\tag{28}
$$

where *Rmin* is the minimum value of the shortest slant range of all point scatters, which can be approximated by *RJ*0.

**Figure 3.** Jamming scene template segmentation diagram.

The scattering coefficients of point scatters in blockpq can be expressed as a matrix **T***pq*:

$$\mathbf{T}\_{pq} = \begin{bmatrix} \sigma\_{11} & \sigma\_{12} & \cdots & \sigma\_{1V} \\ \sigma\_{21} & \sigma\_{22} & \cdots & \sigma\_{2V} \\ \vdots & \vdots & \ddots & \vdots \\ \sigma\_{l11} & \sigma\_{l2} & \cdots & \sigma\_{lIV} \end{bmatrix} \tag{29}$$

Suppose the coordinate of the blockpq geometric center is *xq*, *yp* , where *p* = 1, 2, ··· , *M* and *q* = 1, 2, ··· , *N*. According to Equations (1) and (2), the JSF on the blockpq geometry center *Hcpq*(*fr*, *ta*) is represented as follows:

$$\mathcal{H}c\_{\mathcal{V}\mathbb{I}}(f\_{\mathbb{I}},t\_{\mathbb{d}}) = \exp\left\{-\frac{j4\pi\left(f\_{\mathbb{I}}+f\_{\mathbb{I}}\right)}{c}\left[\sqrt{\left(\mathbf{x}\_{\mathbb{q}}+\mathbf{R}\_{\mathbb{I}0}\right)^{2}+\left(y\_{\mathbb{P}}-vt\_{\mathbb{d}}\right)^{2}}-\sqrt{R\_{\mathbb{I}0}^{2}+\left(vt\_{\mathbb{d}}\right)^{2}}\right]\right\}.\tag{30}$$

The jamming signal generated by *Hcpq*(*fr*, *ta*) can generate a well-focused point target in the center of blockpq after imaging, which is equivalent to moving the jammer to the geometric center of blockpq. Then, we use the TDFS algorithm to generate the JSF on blockpq:

$$\begin{split} H\_{pq}(f\_r, t\_a) &= H c\_{pq}(f\_r, t\_a) \sum\_{k=1}^{lI} \exp\left[\frac{i4\pi f\_0 v t\_a \Delta y}{c\left(k\_{fl0} + \mathbf{x}\_q\right)} \Big(k - \frac{lI + 1}{2}\Big)\right] \\ &\sum\_{l=1}^{V} \sigma\_{kl} \exp\left[-\frac{j4\pi (f\_0 + f\_r) \Delta x}{c} \Big(l - \frac{V + 1}{2}\Big)\right], \end{split} \tag{31}$$

where σ*kl* is the element of row *k* and column *l* in the matrix **T***pq*. Finally, the JSF on the whole template can be derived by summing JSF on all blocks:

$$H(f\_{r\prime}, t\_a) = \sum\_{p=1}^{M} \sum\_{q=1}^{N} H\_{pq}(f\_{r\prime}, t\_a). \tag{32}$$

Since the number of blocks is very small compared to the total number of scatters in the template, Equation (30) has a limited effect on the computing load. In addition, the second summation term of Equation (31) is independent of slow time *ta* and can be calculated offline. The template segmentation method can solve the problem that the effective region of the TDFS algorithm is limited and achieve the purpose of the rapid generation of large scene deceptive jamming signals.

#### 2.2.4. Correction Algorithm in Squint Mode

The analyses above are based on the broadside mode with the squint angle θ = 0. In order to extend the broadside jamming algorithm to squint mode, this subsection will discuss the effect of the squint angle on the jamming result and the corresponding correction method. It should be pointed out that the parabolic approximation of the instantaneous slant range in Equation (4) will no longer be applicable under the condition of a large squint angle, so the jamming method of this article is limited to the small squint angle and the medium aperture length SAR.

First, the Doppler center frequency of the echo signal *fac* = 2*v f*<sup>0</sup> sin θ/*c*, the presence of the squint angle will result in the non-zero Doppler center frequency. At this time, the azimuth signal can be regarded as the non-baseband signal, and the frequency modulation rate error will cause the position offset besides main lobe broadening in pulse compression [22]. According to the pulse compression principle [22], the azimuth main lobe position offset of scatter with coordinate (*x*, *y*) is

$$y\_{ofs}(x,y) = -\frac{\Delta \mathcal{K}\_a}{\mathcal{K}\_a} t\_{ac}v = x \tan \theta,\tag{33}$$

where *tac* = −*RJ*<sup>0</sup> tan θ/*v* is the pulse center time of the jamming signal in the azimuth dimension.

On the other hand, when *fac* - 0, the RCM error will cause the RCM curve to shift along the range dimension in addition to introducing the residual RCM, causing main lobe broadening. According to Equation (22), the offset in the range dimension of fake point *P*(*x*, *y*) is

$$
\Delta \chi\_{ofs}(\mathbf{x}, \mathbf{y}) = \Delta R\_P(f\_{\mathbf{a}\mathbf{c}}) \approx -\frac{\chi}{2} \sin^2 \theta - y \sin \theta. \tag{34}
$$

The residual RCM will increase at the same time:

$$\text{RCM}\_{\text{res}} = \left| \Delta R\_P \left( f\_{\text{ac}} + \frac{B\_d}{2} \right) - \Delta R\_P \left( f\_{\text{ac}} - \frac{B\_d}{2} \right) \right| = \left| \frac{B\_d c}{2vf\_0} y + \frac{B\_d c \sin \theta}{4vf\_0} x \right|. \tag{35}$$

In order to ensure the image quality of the fake scene, the size of the blocks should be reduced. However, the RCMres increment is not obvious when the squint angle is small. Thus, it can be ignored in this paper.

According to the analysis above, the main effect of the squint angle is the location offset of fake targets in the deceptive image, and the offset depends on the coordinates in the template. This effect will cause the distortion of the jamming image. Therefore, the coordinates of each scatter in the template should be corrected. For the false scatter with coordinate (*x*, *y*), the corrected coordinate (*xc*, *yc*) can be represented as follows:

$$\begin{cases} \mathbf{x}\_{\varepsilon} = \mathbf{x} - \mathbf{x}\_{ofs}(\mathbf{x}, \mathbf{y}) = \mathbf{x} + \frac{\mathbf{x}}{2} \sin^{2}\theta + \mathbf{y} \sin\theta, \\\ y\_{\varepsilon} = \mathbf{y} - y\_{ofs}(\mathbf{x}, \mathbf{y}) = \mathbf{y} - \mathbf{x} \tan\theta. \end{cases} \tag{36}$$

Correspondingly, Equation (31) will be modified as follows:

$$\begin{split} \mathcal{H}\_{pq}(f\_{\boldsymbol{r}},t\_{\boldsymbol{a}}) &= \mathcal{H}c\_{pq}(f\_{\boldsymbol{r}},t\_{\boldsymbol{a}}) \sum\_{\begin{subarray}{c} k=1 \ l=1 \\ k=1 \end{subarray}}^{\mathcal{U}} \sum\_{l=1}^{V} \sigma\_{kl} \exp\left\{ -\frac{j4\pi(f\_{\boldsymbol{r}}+f\_{\boldsymbol{0}})\Delta x}{\varepsilon} \left[ \left(1 + \frac{\sin^{2}\theta}{2}\right) \Big(l - \frac{V+1}{2}\right) \right. \\ &+ \Delta y \sin\theta \Big(k - \frac{\mathcal{U}+1}{2}\big) \Big] \exp\left\{ \frac{j4\pi f\_{\boldsymbol{r}}\omega\_{a}}{\varepsilon \left(k\_{\boldsymbol{r}}y + x\_{\boldsymbol{q}}\right)} \Big[ -\Delta x \cdot \tan\theta \Big(l - \frac{V+1}{2}\big) + \Delta y \Big(k - \frac{\mathcal{U}+1}{2}\big) \Big] \right\}. \end{split} \tag{37}$$

The range and azimuth-related terms in Equation (37) are separable, so Equation (37) can be rewritten as matrix operations to increase the calculation speed. Here, we define the time-delay matrixes **Hr**1, **Hr**<sup>2</sup> and frequency-shift matrixes **Ha***q*1, **Ha***q*2:

**Hr**<sup>1</sup> = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ exp# <sup>−</sup> *<sup>j</sup>*4π(*fr*<sup>+</sup> *<sup>f</sup>*0)Δ*<sup>x</sup> c* 1 + sin<sup>2</sup> <sup>θ</sup> 2 <sup>1</sup> <sup>−</sup> *<sup>V</sup>*+<sup>1</sup> 2 \$ exp# <sup>−</sup> *<sup>j</sup>*4π(*fr*<sup>+</sup> *<sup>f</sup>*0)Δ*<sup>x</sup> c* 1 + sin<sup>2</sup> <sup>θ</sup> 2 <sup>2</sup> <sup>−</sup> *<sup>V</sup>*+<sup>1</sup> 2 \$ . . . exp# <sup>−</sup> *<sup>j</sup>*4π(*fr*<sup>+</sup> *<sup>f</sup>*0)Δ*<sup>x</sup> c* 1 + sin<sup>2</sup> <sup>θ</sup> 2 *<sup>V</sup>* <sup>−</sup> *<sup>V</sup>*+<sup>1</sup> 2 \$ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , (38) **Hr**<sup>2</sup> = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ exp# <sup>−</sup> *<sup>j</sup>*4π(*fr*<sup>+</sup> *<sup>f</sup>*0)Δ*<sup>y</sup>* sin <sup>θ</sup> *c* <sup>1</sup> <sup>−</sup> *<sup>U</sup>*+<sup>1</sup> 2 \$ exp# <sup>−</sup> *<sup>j</sup>*4π(*fr*<sup>+</sup> *<sup>f</sup>*0)Δ*<sup>y</sup>* sin <sup>θ</sup> *c* <sup>2</sup> <sup>−</sup> *<sup>U</sup>*+<sup>1</sup> 2 \$ . . . exp# <sup>−</sup> *<sup>j</sup>*4π(*fr*<sup>+</sup> *<sup>f</sup>*0)Δ*<sup>y</sup>* sin <sup>θ</sup> *c <sup>U</sup>* <sup>−</sup> *<sup>U</sup>*+<sup>1</sup> 2 \$ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , (39) **Ha***q*<sup>1</sup> = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ exp# <sup>−</sup> *<sup>j</sup>*4π*f*0*vta*Δ*<sup>x</sup>* tan <sup>θ</sup> *c*(*RJ*0+*xq*) <sup>1</sup> <sup>−</sup> *<sup>V</sup>*+<sup>1</sup> 2 \$ exp# <sup>−</sup> *<sup>j</sup>*4π*f*0*vta*Δ*<sup>x</sup>* tan <sup>θ</sup> *c*(*RJ*0+*xq*) <sup>2</sup> <sup>−</sup> *<sup>V</sup>*+<sup>1</sup> 2 \$ . . . exp# <sup>−</sup> *<sup>j</sup>*4π*f*0*vta*Δ*<sup>x</sup>* tan <sup>θ</sup> *c*(*RJ*0+*xq*) *<sup>V</sup>* <sup>−</sup> *<sup>V</sup>*+<sup>1</sup> 2 \$ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , (40) **Ha***q*<sup>2</sup> = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ exp# *j*4π*f*0*vta*Δ*y c*(*RJ*0+*xq*) <sup>1</sup> <sup>−</sup> *<sup>U</sup>*+<sup>1</sup> 2 \$ exp# *j*4π*f*0*vta*Δ*y c*(*RJ*0+*xq*) <sup>2</sup> <sup>−</sup> *<sup>U</sup>*+<sup>1</sup> 2 \$ . . . exp# *j*4π*f*0*vta*Δ*y c*(*RJ*0+*xq*) *<sup>U</sup>* <sup>−</sup> *<sup>U</sup>*+<sup>1</sup> 2 \$ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ . (41)

Equation (37) is rewritten as follows:

$$H\_{pq}(f\_r, t\_a) = H c\_{pq}(f\_r, t\_a) \Big[ \left( \mathbf{H} \mathbf{r}\_2 \circ \mathbf{H} \mathbf{a}\_{q2} \right)^T \mathbf{T}\_{pq} \Big( \mathbf{H} \mathbf{r}\_1 \circ \mathbf{H} \mathbf{a}\_{q1} \Big) \Big] \tag{42}$$

where (·) <sup>T</sup> represents the matrix transposition and the operator ◦ represents the Hadamard product of the matrix.

We can get the JSF on the entire scene by superimposing the JSF on all blocks, which is the same as Equation (32). The time-delay matrixes **Hr**<sup>1</sup> and **Hr**<sup>2</sup> are independent of the azimuth time *ta*, which can be calculated offline in advance to improve the real-time processing speed.
