The Direction of Arrival Location Deception Model Counter Duel Baseline Phase Interferometer Based on Frequency Diverse Array

With the emergence and development of the passive localization, the radiation source is more visible for the location system which endangers their survival. Therefore, there is an urgent demand for the radio frequency (RF) stealth technology. An effective method to realize RF stealth is location deception, therefore, for the passive localization system, this paper proposes a direction of arrival (DOA) location deception method using the frequency diverse array (FDA) against the dual baseline phase interferometer. Since the direction-finding of the dual baseline phase interferometer is based on the received signal with fixed frequency, the FDA signal has a deception effect on the interferometer owing to the introduction of the small frequency increment. Considering the influence of the frequency increment sequence on the deception effect, we derive the optimizations of the DOA location deception via the average location deviation for the sampling time in the case of no noise and noise, respectively. Besides, considering the time dependency of the beam, we investigate the average SNR (ASNR) and the corresponding CRLB to verify the proposed method. Numerical examples and simulations show that the proposed method can counter the interferometer by realizing location deception.

Introduction

In modern warfare, the concealment of reconnaissance equipment is very important. By achieving target localization without initiative emission of electromagnetic waves, the passive localization technology brings new ideas to traditional localization methods. Passive localization technology mainly uses the radiation source signals to extract parameters, and then uses these parameters to estimate the location of the radiation source, the main extracted parameters are: angle-of-arrival (AOA) [1], time-of-arrival (TOA) [2], time difference of arrival (TDOA) [3], frequency difference of arrival (FDOA) [4], received signal strength (RSS) [5], and phase difference [6]. However, with the emergence of the passive localization system, the radiation source was more visible for the location system, which might lead to the exposure of the true location of the radar and communication systems, and even endangered their survival. Therefore, the countermeasure technology needed to be developed.

Frequency diverse array (FDA) [7] is considered a feasible array countermeasure technology. As we know, in the far-field, the beam pattern of the FDA is range-angle-time-dependent owing to the introduction of the frequency increment [8, 9], however, the beam pattern of the traditional phased array is only angle-dependent, and therefore, the FDA increases the degree of freedom of range relative to the phased array. FDA thus attracted considerable attention in recent years since its range-dependent beam pattern. The current researches of the FDA mainly focus on decoupled range-angle beam forming [10–13], joint angle-range estimation [14–17], deceptive jamming, and range-dependent clutter suppression [18–23], receiver design [24–26], and the application in the communications [27]. However, most of the research above considered the case of the instantaneous time to ignore the time-variance property in the FDA beam pattern, but as we know, FDA’s work is a continuous process, and its time parameters cannot be fixed in the actual work process. Therefore, several papers have studied the influence of FDA beam time-varying and its suppression methods [28–34]. However, for FDA, the effect of the introduction of frequency increment on time and range are related, and the suppression of the time-varying characteristic will also affect the range-dependent characteristic of the FDA beam pattern. Therefore, the beam time-varying problem of the FDA has not been well solved, which seriously restricts the development of the FDA.

When turning to the countermeasures based on the FDA, the current research mainly focuses on two parts: 1) Low interception probability (LPI) transmit beamforming. Using different frequency increments, [35, 36], propose LPI beam forming based on FDA and FDA-MIMO, respectively. 2) Deceptive signal generation. For the amplitude-based reconnaissance, Antonik analyzes the S-shaped beam of the FDA, and thus proposes conjecture of virtual radiation source which may realize location deception [37, 38] analyzes the deception effect on the sum and difference beam reconnaissance; For the phase-based reconnaissance [39], proposes a cognitive active anti-jamming method based on the FDA phase center [40, 41], study the deception effect of uniform linear FDA (ULFDA) on the interferometer, and [42] further analyzes the influence of the nonlinear frequency increment on the deception effect in the noise environment. However, both [40–42] only analyze the deception effect of the FDA, the direction of arrival (DOA) location deception optimization by regulating frequency increment is not investigated.

Therefore, during the sampling time for the direction-finding process of the interferometer, and considering regulation of the frequency increment sequence on the deception effect, we propose a DOA location deception method according to the average location deviation. Our main contributions are summarized as follows.

1) The regulating ability of the frequency increment sequence on the deceptive FDA signal is investigated. The dual baseline phase interferometer measures the DOA by process the phase difference of the same signal received by different receivers, however, the phase difference of the FDA signal contains the range parameter owing to its frequency increment, and thus the interferometer cannot measure the indicated angle accurately, this means, while the suitable frequency increment sequence selected, the FDA signal is obviously deceptive.

2) A DOA location method based on the regulation of the FDA frequency increment sequence is proposed. Using the regulating effect of the frequency increment sequence, and considering the sampling time of the dual baseline phase interferometer, the optimization of DOA location deception via the average location deviation is derived in the no noise and noise environment, respectively. Besides, considering the time-dependent beam of the FDA, the average SNR (ASNR) and the corresponding CRLB is derived to measure the superiority of the proposed method.

3) The improved particle swarm-immune optimization (PSO-IMMU) algorithm is applied. Since the periodicity of the received phase differs, the optimization problem is considered as a non-convex problem, therefore, the PSO-IMMU algorithm is used to get the optimal frequency increment sequence, in which the extra immune algorithm is used to solve the local optimality problem.

The remaining sections are organized as follows. The Difference of Arrival Location Deception Model Counter Dual Baseline Phase Interferometer Based on Frequency Diverse Array Section analyzes the countering effect of the FDA against the dual baseline phase interferometer. By analyzing the regulating ability of frequency increment sequence, and considering the sampling time, The Difference of Arrival location deception method based on Frequency Diverse Array Section formulates the DOA location deception optimization problem in the no noise and noise environment, respectively. Numerial Results Section gives the simulation results, and the conclusions are drawn in Conclusion Section.

The Direction of Arrival Location Deception Model Counter Dual Baseline Phase Interferometer Based on Frequency Diverse Array

Signal Model of Frequency Diverse Array Radar

Figure 1 shows a Q-element linear FDA with element spacing d, thus the array factor of the far-field target that locates at $\left(\theta ,R\right)$ can be given as

$A{F}_{FDA}={\displaystyle \sum _{q=1}^Q{A}_q\exp \left\{j2\pi f_q\left[t-\frac{R-\left(q-1\right)d\sin \theta }{c}\right]\right\}}q=\mathrm{1,2},\dots ,Q(1)$

where $c=3\times {10}^{8}m/s$ denotes the speed of light, and $f_q=f_0+\Delta f_q$ is the radiation frequency of the qth element, with $f_0$ and $\Delta f_q$ being the carrier frequency and the frequency increment of the qth element, respectively, note that we set $\Delta f_1=0$. Besides, $A_q$ denotes the signal radiation amplitude of the qth element, generally, we suppose $A_q=A=1$, thus we can learn

$\begin{array}{l}A{F}_{FDA}={\displaystyle \sum _{q=1}^Q\exp \left\{j2\pi \left(f_0+\Delta f_q\right)\left[t-\frac{R-\left(q-1\right)d\sin \theta }{c}\right]\right\}}\\ =\exp \left[j2\pi f_0\left(t-\frac{R}{c}\right)\right]{\displaystyle \sum _{q=1}^Q{A}_q\exp \left\{j2\pi \left[f_0\frac{\left(q-1\right)d\sin \theta }{c}+\Delta f_{q}t-\Delta f_q\frac{R}{c}+\Delta f_q\frac{\left(q-1\right)d\sin \theta }{c}\right]\right\}}\end{array}(2)$

FIGURE 1

FIGURE 1. Q—element linear Frequency Diverse Array.

With the assumption that $\Delta f_q\ll f_0$, (Eq. 2) can be approximately rewritten as

$A{F}_{FDA}\approx \exp \left[j2\pi f_0\left(t-\frac{R}{c}\right)\right]{\displaystyle \sum _{q=1}^Q\exp \left\{j2\pi \left[f_0\frac{\left(q-1\right)d\sin \theta }{c}+\Delta f_{q}t-\Delta f_q\frac{R}{c}\right]\right\}}(3)$

For the simplified expression, we define

$a_t\left(t\right)={\left[\begin{array}{cccc}1& e^{j2\pi \Delta f_{2}t}& \cdots & e^{j2\pi \Delta f_{Q}t}\end{array}\right]}^T$(4-1)

$a_R\left(R\right)={\left[\begin{array}{cccc}1& e^{-j2\pi \Delta f_{2}R/c}& \cdots & e^{-j2\pi \Delta f_{Q}R/c}\end{array}\right]}^T$(4-2)

$a_{\theta }\left(\theta \right)={\left[\begin{array}{cccc}1& e^{j2\pi f_{0}d\sin \theta /c}& \cdots & e^{j2\pi f_0\left(Q-1\right)d\sin \theta /c}\end{array}\right]}^T$(4-3)

$a_0\left(t,R\right)={\left[\begin{array}{cccc}{e}^{j2\pi f_0\left(t-R/c\right)}& e^{j2\pi f_0\left(t-R/c\right)}& \cdots & e^{j2\pi f_0\left(t-R/c\right)}\end{array}\right]}^T$(4-4)

Thus (Eq. 3) can be rewritten as

$A{F}_{FDA}\left(R,\theta ,t\right)=a_0^T\left(t,R\right)\cdot \left[a_t\left(t\right)\odot a_{\theta }\left(\theta \right)\odot a_R\left(R\right)\right](5)$

where ^T is the transpose operator, and $\odot$ is the Hadamard product operator.

To detect the far-field target at $\left({\theta }_0,R_0\right)$, we define the steering vector by

$w=a_0\left(0,-R_0\right)\odot a_R\left(-R_0\right)\odot a_{\theta }\left(-{\theta }_0\right)(6)$

Therefore, the unidirectional synthesized signal can be expressed as

$A{F}_{FDA}\left(R,\theta ,R_0,{\theta }_0,t\right)={\left[a_0\left(t,R\right)\odot w\right]}^T\cdot \left[a_t\left(t\right)\odot a_{\theta }\left(\theta \right)\odot a_R\left(R\right)\right](7)$

Actually, (Eq. 7) can be further expressed as

$A{F}_{FDA}\left(R,\theta ,R_0,{\theta }_0,t\right)=B{p}_{FDA}{e}^{j{\Psi }_{FDA}\left(R,\theta ,R_0,{\theta }_0,t\right)}(8)$

where

$B{p}_{FDA}=\left|A{F}_{FDA}\left(R,\theta ,R_0,{\theta }_0,t\right)\right|$(9-1)

${\Psi }_{FDA}\left(R,\theta ,R_0,{\theta }_0,t\right)=angle\left[A{F}_{FDA}\left(R,\theta ,R_0,{\theta }_0,t\right)\right]$(9-2)

with angle being the phase angle solving function.

Interferometer Direction Finding Principle

For the dual baseline phase interferometer shown in Figure 2, the receivers 1, 2, and 3 constitute a planar dual baseline phase interferometer direction-finding system. The receivers 1 and 2 form short baseline d₁ and the receivers 2 and 3 form long baseline d₂. Then there are,

${\varphi }_1={\phi }_1+2m\pi =\frac{2\pi d_1\sin \gamma }{\lambda },m=0,\pm 1,\pm 2\dots$(10-1)

${\varphi }_2={\phi }_2+2n\pi =\frac{2\pi d_2\sin \gamma }{\lambda },n=0,\pm 1,\pm 2\dots$(10-2)

where ${\phi }_1$ and ${\phi }_2$ are the phase differences obtained by the phase detector, ${\varphi }_1$ and ${\varphi }_2$ are the true values of the actual phase difference corresponding to the incident angle $\gamma$, and $\lambda$ is the incoming wavelength.

FIGURE 2

FIGURE 2. Dual baseline phase interferometer.

According to the design rules of the dual baseline phase interferometer, to ensure the short baseline being unambiguous, we set $d_1<\lambda /2$ that is, ${\varphi }_1={\phi }_1$. Then, (Eq. 10-1) can be rewritten as,

${\varphi }_1={\phi }_1=\frac{2\pi d_1\sin \gamma }{\lambda }$(11-1)

${\varphi }_2={\phi }_2+2n\pi =\frac{2\pi d_2\sin \gamma }{\lambda },n=0,\pm 1,\pm 2\dots$(11-2)

Supposing $p=d_2/d_1$ is ratio of the long and short baseline length, then ${\varphi }_2=p\times {\varphi }_1$. In this way, the blurred value n of the phase difference of the long baseline can be solved by the phase difference ${\phi }_1$ obtained from the short baseline, and the high-precision data measured from the long baseline can be obtained. The actual process of de-blurring is as follows: 1) For a known long baseline with a phase difference of ${\phi }_2$, a set of phase sequence of phase difference 2π is obtained from $n=0,\pm 1,\pm 2\dots$. 2) Finding the value closest to the $p\times {\varphi }_1$ is the actual exact value ${\varphi }_2$.3) The exact value ${\varphi }_2$ is the phase difference obtained from the long baseline with better accuracy than that from the short baseline, and finally, the incident angle $\gamma$ of the signal is derived. Therefore, the process of de-blurring can be expressed as,

$n_0=\underset{n}{\arg }\min \left(\left|{\phi }_2+2n\pi -p{\phi }_1\right|,n=0,\pm 1,\pm 2\dots \right)(12)$

Then, for the long baseline, the true value of the actual phase difference can be given by

${\varphi }_2={\phi }_2+2n_0\pi (13)$

As we know, there is a unique correspondence between the phase difference being de-blurred and the DOA. Generally, each de-blurring output can be used for direction finding, but the output accuracy of the long baseline is high. In order to simplify the calculation, we usually only use the output of long baseline to measure the DOA, that is,

$\widehat{\gamma }=\arcsin \left(\frac{c{\varphi }_2}{2\pi f_0{d}_2}\right)(14)$

Direction of Arrival Location Deception Model

Figure 3 gives the position relationship between the interferometer and the FDA radar, in this scenario, we set the far-field indicate angle as $\gamma$.

FIGURE 3

FIGURE 3. The position relationship between the interferometer and the FDA radar.

Then, supposing the range from the first element of FDA radar to the receiver 1 of the interferometer is $R$, and the beam of FDA points at the far-field target with angle-range pair $\left({\theta }_0,R_0\right)$ while t = 0, then the phase differences obtained by the phase detector can be given by

${\phi }_{FDA1}=\mathrm{mod}\left\{angle\left[A{F}_{FDA}\left(R,\gamma ,R_0,{\theta }_0,t\right)\right]-angle\left[A{F}_{FDA}\left(R+d_1\sin \gamma ,\gamma ,R_0,{\theta }_0,t\right)\right]+2\pi ,2\pi \right\}$(15-1)

${\phi }_{FDA2}=\mathrm{mod}\left\{angle\left[A{F}_{FDA}\left(R,\gamma ,R_0,{\theta }_0,t\right)\right]-angle\left[A{F}_{FDA}\left(R+d_2\sin \gamma ,\gamma ,R_0,{\theta }_0,t\right)\right]+2\pi ,2\pi \right\}$(15-2)

According to the de-blurring process given in the Part 2.2, the output phase difference of the long baseline can be given by ${\varphi }_{FDA2}$, and thus the measured DOA can be derived as

${\widehat{\gamma }}_{FDA}=\arcsin \left(\frac{c{\varphi }_{FDA2}}{2\pi f_0{d}_2}\right)(16)$

From (Eq. 7), we learn that due to the extra frequency increment, the array factor of the FDA is coupled with the range and the indicate angle, and thus the output phase difference ${\varphi }_{FDA2}$ is not only related to the DOA, but also to the range, owing to which the measured DOA deviates from the actual DOA, then we have

$\Delta {\gamma }_{FDA}={\widehat{\gamma }}_{FDA}-\gamma (17)$

Besides, the measured location (x—intercept) is

${\widehat{x}}_{FDA}=R\sin \left({\widehat{\gamma }}_{FDA}\right)(18)$

Thus, the location deviation from the interferometer center to the array center can be given by

$\Delta x_{FDA}={\widehat{x}}_{FDA}-R\sin \gamma +\left(Q-1\right)d/2-d_2/2(19)$

Therefore, while the measured location is outside the array, we regard that the FDA can generate deceptive signals to counter the interferometer, that is,

$\left|\Delta x_{FDA}\right|>\left(Q-1\right)d/2(20)$

For the linear FDA with the uniform linear frequency increment (ULFDA),

$A{F}_{ULFDA}\left(R,\gamma ,R_0,{\theta }_0,t\right)=A{F}_{ULFDA}\left(R,\gamma ,R_0,{\theta }_0,t\right)|\Delta f_q=\left(q-1\right)\Delta f(21)$

Then, we have

$A{F}_{ULFDA}\left(R,\gamma ,R_0,{\theta }_0,t\right)=\sin \frac{Q\alpha }{2}/\sin \frac{\alpha }{2}$(22-1)

${\Psi }_{ULFDA}\left(R,\gamma ,R_0,{\theta }_0,t\right)=2\text{π}{f}_0\left[t-\left(R-R_0\right)/c\right]+\left(Q-1\right)\alpha /2$(22-2)

where $\alpha =2\text{π}{f}_{0}d\left(\sin \gamma -\sin {\theta }_0\right)/c+2\text{π}\Delta f\left[t-\left(R-R_0\right)/c\right]$. Then, we have

${\varphi }_{ULFDA1}={\Psi }_{ULFDA}\left(R,\gamma ,R_0,{\theta }_0,t\right)-{\Psi }_{ULFDA}\left(R+d_1\sin \gamma ,\gamma ,R_0,{\theta }_0,t\right)=2\text{π}\left[f_0+\frac{\left(Q-1\right)\Delta f}{2}\right]\frac{d_1\sin \gamma }{c}$(23-1)

${\varphi }_{ULFDA2}=2\text{π}\left[f_0+\frac{\left(Q-1\right)\Delta f}{2}\right]\frac{d_2\sin \gamma }{c}$(23-2)

Then the measured DOA and location can be given by

${\widehat{\gamma }}_{ULFDA}=\arcsin \left[\sin \gamma +\frac{\left(Q-1\right)\Delta f}{2f_0}\sin \gamma \right]$(24-1)

${\widehat{x}}_{ULFDA}=R\sin \widehat{\gamma }=R\sin \gamma +\frac{\left(Q-1\right)\Delta fR\sin \gamma }{2f_0}$(24-2)

Thus, the location deviation is

$\Delta x_{ULFDA}=\frac{\left(Q-1\right)\Delta fR\sin \gamma }{2f_0}-\frac{d_2-\left(Q-1\right)d}{2}(25)$

Therefore, for the ULFDA, to ensure the deception effect, we have

$\Delta f>\frac{f_0{d}_2}{\left(Q-1\right)R\sin \gamma }\Vert \Delta f<\frac{f_0{d}_2-2\left(Q-1\right)f_{0}d}{\left(Q-1\right)R\sin \gamma }(26)$

The Direction of Arrival Location Deception Method Based on Frequency Diverse Array

The Basic Principle of Direction of Arrival Location Method

From the deception model, we learn that when the frequency increment between adjacent elements is varying, due to the nonlinear frequency increment, the phase differences are related to the range, the length of baseline, and t. For this case, by controlling the frequency increment sequence, we can control the DOA and location deviation to realize DOA location deception on the dual baseline phase interferometer. Therefore, we have

$\Delta x_{FDA}\left(\mathbf{\Delta f}\right)=R\sin \left[{\widehat{\gamma }}_{FDA}\left(\mathbf{\Delta f}\right)\right]-R\sin \gamma +\left(Q-1\right)d/2-d_2/2(27)$

where $\mathbf{\Delta f}=\left[\begin{array}{cccccc}\Delta f_1& \Delta f_2& \cdots & \Delta f_q& \cdots & \Delta f_Q\end{array}\right]$ denotes the frequency diverse sequence.

To investigate the regulation effect of the frequency diverse sequence on the location deception, by change the $\mathbf{\Delta f}$ randomly, the 10,000 Monte Carlo results are shown in Figure 4.

FIGURE 4

FIGURE 4. Regulation effect of Δf on the location deviation.

It can be seen that, with $\mathbf{\Delta f}$ changing randomly, the most measured location deviates slightly, but for the special $\mathbf{\Delta f}$, there is an obvious location deviation, this means, when we select the appreciate $\mathbf{\Delta f}$, the FDA have a good deception effect on the dual baseline phase interferometer.

The Direction of Arrival Location Deception Method Based on Frequency Diverse Array

The Monte Carlo results show that the Δf has the regulation ability on the deception effect, it means, by regulating $\mathbf{\Delta f}$, we can realize DOA location deception on the interferometer. Besides, in practice, the passive location system estimates DOA by processing the received signals in a period of time, which means the azimuth angle measured by the interferometer is actually the average over a period of time, not the instantaneous value at a certain time.

Therefore, we define the average DOA and location deviation, respectively,

$\Delta {\widehat{\gamma }}_{ave}={\displaystyle {\int }_{t_1}^{t_2}\Delta {\gamma }_{FDA}\left(t\right)dt}$(28-1)

$\Delta {\widehat{x}}_{ave}={\displaystyle {\int }_{t_1}^{t_2}\Delta x_{FDA}\left(t\right)dt}$(28-2)

where $t_1$ and $t_2$ denote the start and end time of the sampling, respectively.

As we know, to realize the deception on the dual baseline phase interferometer, we should ensure that the measured location deviates out of the FDA antenna, thus, we have

$\left|\Delta {\widehat{x}}_{ave}\right|>\left(Q-1\right)d/2(29)$

As the analysis of part 3.1, we can learn that when an appropriate $\mathbf{\Delta f}$ is selected, the location that is derived by the interferometer will deviate greatly, it means that the location deviation can be controlled by changing the $\mathbf{\Delta f}$. Therefore, in order to realize the (Eq. 29), we can adjust the $\mathbf{\Delta f}$. That is, there is a $\mathbf{\Delta f}$, s.t.,

$\left|\Delta {\widehat{x}}_{ave}\left(\mathbf{\Delta f}\right)\right|>\left(Q-1\right)d/2(30)$

From the above analysis, the DOA location deception problem on interferometer using FDA can be converted into the following optimization problems,

$\begin{array}{l}\mathbf{\Delta }\widehat{\mathbf{f}}=\arg \max \left\{\left|\Delta {\widehat{x}}_{ave}\left(\mathbf{\Delta f}\right)\right|\right\}\\ s.t.\mathrm{\Delta }{f}_{\min }\le \Delta f_q\le \Delta f_{\max }\\ \left|\Delta {\widehat{x}}_{ave}\left(\mathbf{\Delta f}\right)\right|-\left(Q-1\right)d/2>0\end{array}(31)$

Converting the optimization problem to a minimizing optimization problems, we have

$\begin{array}{l}\mathbf{\Delta }\widehat{\mathbf{f}}=\arg \min \left\{-\left|\Delta {\widehat{x}}_{ave}\left(\mathbf{\Delta f}\right)\right|\right\}\\ s.t.\Delta f_{\min }\le \Delta f_q\le \Delta f_{\max }\\ \left|\Delta {\widehat{x}}_{ave}\left(\mathbf{\Delta f}\right)\right|-\left(Q-1\right)d/2>0\end{array}(32)$

Let $f\left(\mathbf{\Delta f}\right)=\left|\Delta {\widehat{x}}_{ave}\left(\mathbf{\Delta f}\right)\right|$, (32) be further rewritten to

$\begin{array}{l}\mathbf{\Delta }\widehat{\mathbf{f}}=\arg \min \left\{-f\left(\mathbf{\Delta f}\right)+r{P}^2\left(\mathbf{\Delta f}\right)\right\}\\ s.t.\Delta f_{\min }\le \Delta f_q\le \Delta f_{\max }\end{array}(33)$

where $P\left(\mathbf{\Delta f}\right)=\max \left(0,-f\left(\mathbf{\Delta f}\right)+\left(Q-1\right)d/2\right)$, r is the penalty coefficient.

However, in practice, there will be noise in the process, so the noise should be also taken into consideration. The RMSE is used to solve this problem, that is,

$RMS{E}_{\widehat{\gamma }}=RMS{E}_{\Delta {\widehat{\gamma }}_{ave}}=\sqrt{E\left[{\left(\Delta {\widehat{\gamma }}_{ave}-\mathbf{0}\right)}^2\right]}$(34-1)

$RMS{E}_{\Delta {\widehat{x}}_{ave}}=\sqrt{E\left[{\left(\Delta {\widehat{x}}_{ave}-\text{0}\right)}^2\right]}$(34-2)

Therefore, in the noise environment, the DOA location deception problem on interferometer using FDA can be expressed as

$\begin{array}{l}\mathbf{\Delta }\widehat{\mathbf{f}}=\arg \min \left\{-\left|RMS{E}_{\Delta {\widehat{x}}_{ave}}\left(\mathbf{\Delta f}\right)\right|\right\}\\ s.t.\Delta f_{\min }\le \Delta f_q\le \Delta f_{\max }\\ \left|RMS{E}_{\Delta {\widehat{x}}_{ave}}\left(\mathbf{\Delta f}\right)\right|-\left(Q-1\right)d/2>0\end{array}(35)$

Let $f_1\left(\mathbf{\Delta f}\right)=\left|\Delta {\widehat{x}}_{ave}\left(\mathbf{\Delta f}\right)\right|$, (Eq. 35) can be further rewritten to

$\begin{array}{l}\mathbf{\Delta }\widehat{\mathbf{f}}=\arg \min \left\{-f_{\mathbf{1}}\left(\mathbf{\Delta f}\right)+r_{\mathbf{1}}{P}_1^2\left(\mathbf{\Delta f}\right)\right\}\\ s.t.\Delta f_{\min }\le \Delta f_q\le \Delta f_{\max }\end{array}(36)$

where $P_{\mathbf{1}}\left(\mathbf{\Delta f}\right)=\max \left(0,-f_{\mathbf{1}}\left(\mathbf{\Delta f}\right)+\left(Q-1\right)d/2\right)$, $r_1$ is the penalty coefficient.

Eqs 33, 36 give the closed form of optimization to realize the DOA location deception problem on the interferometer in the case of no noise and noise, respectively. However, since the phase is periodic, the optimizations are non-convex problems, thus, the particle swarm immune optimization (PSO-IMMU) algorithm is used to solve this problem. According to the particle swarm optimization (PSO), the position and speed of the particle update with the following equation,

${\mathbf{v}}_i^{\left(j+1\right)}=\omega \cdot {\mathbf{v}}_i^{\left(j\right)}+c_1\left(\Delta f_{\max }-\Delta f_{\min }\right)\left[{\mathbf{P}}_{\mathbf{i}}-\mathbf{\Delta }{\mathbf{f}}_i^{\left(j\right)}\right]+c_2\left(\Delta f_{\max }-\Delta f_{\min }\right)\left[{\mathbf{P}}_{\mathbf{g}}-\mathbf{\Delta }{\mathbf{f}}_i^{\left(j\right)}\right]$(37-1)

$\mathbf{\Delta }{\mathbf{f}}_i^{\left(j+1\right)}=\mathbf{\Delta }{\mathbf{f}}_i^{\left(j\right)}+{\mathbf{v}}_i^{\left(j+1\right)}$(37-2)

where P_i and P_g denote the local optimal solutions and global optimal solutions, respectively. c₁ andc₂ are constant learning factors, and ω denotes the inertia weight. Thus, the flowchart of the optimization is given in Figure 5.

FIGURE 5

FIGURE 5. Flowchart of the optimization.

In summary, the output of the PSO-IMMU is the optimal frequency increment sequence, based on which, the FDA can realize DOA location deception on the dual baseline phase interferometer.

Model Verification and Performance Analysis

Signal to Noise Ratio Analysis

As we know, the beam of the FDA is time-dependent. This means, in the constant noise environment, the SNR of the FDA signal is varying with time, and thus we derive the instantaneous by

$ISN{R}_{FDA}\left(t\right)=10\log \left[\frac{B{p}_{FDA}^2\left(t\right)}{P_w}\right](38)$

where P_w is the average noise power. During the sampling time, the average SNR (ASNR) is

$ASN{R}_{FDA}=\frac{1}{t_2-t_1}{\displaystyle {\int }_{t_1}^{t_2}ISN{R}_{FDA}\left(t\right)dt}(39)$

Actually, for the PA signal, the SNR is constant, that is,

$SN{R}_{PA}=ISN{R}_{PA}\left(t\right)=ASN{R}_{PA}=10\log \left(\frac{P_{PA}}{P_w}\right)(40)$

where,

$P_{PA}={\left|A{F}_{PA}\right|}^2={\left|A{F}_{FDA}\left(t\right)\right|}_{\Delta f_{1:Q}=0}^2=\max \left[B{p}_{FDA}^2\left(t\right)\right](41)$

Therefore, we can learn that the ASNR of the PA is always higher than that of the FDA.

Cramer-Rao Lower Bound Analysis

To analyzes the proposed method in the noise environment, the CRLB is important. Let $A={\displaystyle {\int }_{t_1}^{t_2}AFdt}$, we derive the sample of the interferometer by,

$\mathbf{x}\left[n\right]=\mathbf{s}\left[n\right]+\mathbf{w}\left[n\right]=\left|A\right|\exp \left[j\left(-\varphi \frac{d_n}{d_{N-1}}+\psi \right)\right]+\mathbf{w}\left[n\right]n=\mathrm{0,1},.\dots ,N-1(42)$

where d₀ = 0.

Similar to [42], the CRLB for the dual baseline phase interferometer is

$CRL{B}_{\gamma }=\frac{\mathbf{3}{c}^2}{\mathbf{8}{\pi }^2{f}_0^2{d}_2^2{\cos }^2\gamma \left[3\left(1+\frac{d_1^2}{d_2^2}\right)-{\left(1+\frac{d_1}{d_2}\right)}^2\right]ASN{R}_{FDA}}$(43-1)

$CRL{B}_x=\frac{3R^2{c}^2}{8{\pi }^2{f}_0^2{d}_2^2\left[3\left(1+\frac{d_1^2}{d_2^2}\right)-{\left(1+\frac{d_1}{d_2}\right)}^2\right]ASN{R}_{FDA}}$(43-2)

In general, for the signals of the different array, the higher the CRLB is, the better the deception performance is.

Numerical Results

To verify the proposed method, we divide the numerical simulation into 3 parts: 1) The deception effect of the FDA counter the dual baseline phase interferometer. 2) The optimal deception method without noise. 3) The optimal deception method in the noise environment. Here the FDA radar and dual baseline phase interferometer is considered, the main simulation parameters are listed in Table 1.

TABLE 1

TABLE 1. Main simulation parameters.

Example 1: The deception effect of the FDA counter the dual baseline phase interferometer: In this example, we analyze different parameters the influence of different parameters on the deception effect, and three array structures are considered: 1) Phased array 2) FDA with frequency increment $\Delta f\cdot \left[\begin{array}{cccccc}0& 1& \dots & q-1& \dots & Q-1\end{array}\right]$ (named ULFDA) 3) FDA with frequency increment $\Delta f\cdot \left[\begin{array}{cccccc}0& \log \left(2\right)& \dots & \log \left(q\right)& \dots & \log \left(Q\right)\end{array}\right]$ (named LogFDA). Figure 6 shows the DOA deception effect of the FDA. From Figures 6A–E, we can learn that while fixing the time, the deception effect of the FDA is positively correlated with $\Delta f$, Q, $R$, and $\gamma$, while negatively correlated with $f_0$. From Figure 6F, we can learn that while other parameters are fixed, the ULFDA can realize deception to counter the dual baseline phase interferometer, besides, we can also learn that with the introduction of nonlinear frequency increment, the LogFDA signal has a time-varying deception ability.

FIGURE 6

FIGURE 6. The DOA deception effect of the FDA (A)x-intercept deviation with f₀(B)x-intercept deviation with Δf(C)x-intercept deviation with Q (D)x-intercept deviation with R(E)x-intercept deviation with γ(F)x-intercept deviation with t.

Example 2: The optimal deception method without noise. Supposing $t_1$ and $t_2$ are 0 and 1 ms, respectively, the optimal frequency increment sequence is determined according to (Eq. 33), then naming the FDA with optimal frequency increment sequence OFDA, we study the DOA deception effect of OFDA on the interferometer as shown in Figure 7. Figure 7 compares the x-intercept deviation of OFDA with that of PA and ULFDA, we can learn that in the sampling time, the OFDA can achieve a larger average x-intercept deviation than ULFDA, though the instantaneous x-intercept deviations of FDA in some certain times may not better than that of FDA. However, as the interferometer actually measures the average location over the sampling time, the proposed method in the case of no noise performs a better DOA location deception effect than ULFDA.

FIGURE 7

FIGURE 7. The Difference of Arrival deception effect of the OFDA.

Example 3: The optimal deception method in the noise environment. With the Gaussian white noise assumption, and setting the sampling time of the interferometer as 1ms, we derive the optimal frequency increment sequence according to (Eq. 36). Generally, assuming that the array structures are all in the same noise environment, and taking the SNR of PA as reference, Figure 8 gives the variation of deception effect with SNR, we can see that compared with PA and ULFDA, the proposed OFDA can achieve better deception effect on the interferometer than ULFDA. Then, Figure 9 gives the variation of location error with SNR, we can see the proposed OFDA has better DOA location deception effect. However, compared with the ULFDA, the OFDA perform better while the SNR is high, but for the case that the SNR is low, there is little difference between the OFDA and the ULFDA. To explain this phenomenon, Figure 10 gives the beampattern of the three array structures, we can see that compared with the other two structures, there are many times when the energy is zero in the beampattern of FDA, which means in those times, the ISNR of FDA will be infinitesimal, and the interferometer can only receive the noise, and that will make the average estimation of x-intercept inaccurate, especially when the SNR is low. In summary, the proposed method performs a better DOA location deception effect in the case of noise, especially when the SNR is high.

FIGURE 8

FIGURE 8. The variation of deception effect with SNR (A) RMSE of DOA (B) RMSE of location.

FIGURE 9

FIGURE 9. The variation of location error with SNR.

FIGURE 10

FIGURE 10. The beam pattern of three array structures.

Conclusion

In this paper, to counter the dual baseline phase interferometer, we proposed a DOA location deception method based on the FDA with non-linear frequency increment. The dual baseline phase interferometer measures the DOA by processing the phase difference of the same signal received by different receivers, however, the phase difference of the FDA signal contains the range parameter owing to its frequency increment, and thus the interferometer cannot measure the indicated angle accurately. Therefore, by analyzing the regulating ability of frequency increment sequence on the deception effect, considering the sampling time, we formulate the DOA location deception optimization problem in the no noise and noise environment, respectively. Considering the periodicity of the phase, the PSO-IMMU algorithm is used to solve this non-convex problem. The numerical simulations show that the proposed method can realize good deception effect on the dual baseline phase interferometer, however, since there are multiple zero energy points in the FDA signal during sampling time, the superiority of the proposed method is not more obvious than the ULFDA in the high noise environment, the higher the SNR is, the better the deception effect of the proposed method is. In summary, the proposed method is a good countermeasure for the interferometry reconnaissance.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author Contributions

JG: Conceptualization, Methodology, Simulation, Writing the original draft. JX: Validation, review and editing, Supervision. CC: Formal analysis, review and editing. BW: Supervision, Visualization.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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