Monopulse Parameter Estimation for FDA-MIMO Radar under Mainlobe Deception Jamming

Abstract

Multiple input multiple output with frequency diversity array (FDA-MIMO) radar has unique advantages in mainlobe deception jamming suppression and target location. However, if the training sample contains the target signal, it will lead to poor jamming suppression performance and large target measurement error. To deal with the problem, a method of coarse target location in the time domain is proposed based on the cumulative sampling analysis. Taking full advantages of the strongest correlation characteristic between the expected steering vector and the true target, the feature vector and feature value corresponding to the true target are found after feature decomposition. The time domain location of the target is roughly estimated during the cumulative sampling analysis from near to far. Then, a pure jamming training sample can be obtained by avoiding the location. Noise subspace projection algorithm is used to measure the angle and range of the target while suppressing mainlobe jamming. The simulation results show that the proposed method can roughly estimate the target location in the time domain when the mainlobe deception jamming completely covers the target. Compared with conventional methods, the performance of jamming suppression and target localization error are closer to the performance of ideal sampling.

1. Introduction

Compared with conventional phased array, a small frequency interval is applied across the adjacent elements of frequency diverse array (FDA) [1]. It provided an “S”-shaped beam which is related to range and angle [2]. FDA radar can control the beam to point at a different range from the same angle [3]. However, if the FDA radar only relies on a small carrier frequency difference to separate the signals received by each array, existing filters are difficult to achieve [4,5]. Considering the orthogonal waveform of multiple-input multiple-output (MIMO) is conducive to signal separation, it is one of the best ways for FDA practical application [6]. Multiple-input multiple-output with frequency diversity array (FDA-MIMO) radar combines the beam range-angle correlation of FDA [7] and the waveform orthogonal characteristics of MIMO radar [8]. It has unique advantages and potential application prospects [9] in mainlobe deception jamming suppression [10] and target location [7].

For mainlobe deception jamming suppression, the range-angle coupling characteristics of the beam can be used to identify the true and false targets in the transmitter–receiver two-dimensional domain to achieve jamming suppression [11]. Even if the angle of the target and the jamming are the same, as long as the ranges are different, no matter how many false target jamming are released, these jamming only correspond to the same position in the transmitter–receiver two-dimensional domain and are different from the true target position, so as to suppress the jamming, but the case that the training sample contains the target is not considered [12]. Blind source separation algorithm is also used to separate range deception jamming and target in different channels [13] and the method does not need the target location information, but the angle cannot be measured later, and the target channel needs to be further judged. Quadratic phase code (QPC) is used for transmitting in slow time dimension, and the jamming and target can be distinguished by decoding and main distance compensation. Especially, the method can move the jamming from the mainlobe to the sidelobe [14], but the method has poor pulse compression performance when the target signal has large Doppler frequency shift. For smeared spectrum [15,16] and intermittent sampling and forwarding jamming [17] in mainlobe, the jamming can be suppressed by combining the space-time two-domain algorithm or by controlling the transmitting and receiving beam based on the oblique projection algorithm; the jamming can also be well suppressed even if both mainlobe and sidelobe are mismatched [18]. For swarm jamming, two data-independent phase-only jamming suppression algorithms are proposed. Constant mode constraint and sidelobe level constraint are applied or constant mode decomposition is performed to obtain adaptive weights [19]. For range gate pull-off mainlobe jamming, a two-dimensional transceiver beamforming approach is proposed [20,21] to null out the jamming while maintaining the real target. However, the above methods are all based on ideal sampling and do not consider the problem that the mainlobe jamming suppression performance drops sharply when the training sample contains the true target.

For target location, the conventional method is to combine angle and delay parameter estimation [22,23], while the range and angle of the target for FDA-MIMO radar can be estimated [24] by making full use of the range-angle correlation beam. But the range-angle coupling of the transmitted beam will lead to multi-value problems. Therefore, the single-value accurate measurement of angle and range can be realized by decoupling beam [25].

A method of dividing an array into multiple subarrays working at different frequencies with non-zero frequency shifts is proposed [26]. It can obtain the range-angle-decoupled beampattern, but the array freedom is lost. Working at nonlinear frequency interval [27] or coprime frequency interval [28] for FDA-MIMO radar also can decouple the beam, but it is difficult to achieve in practice. A simple range-angle localization method is proposed by dual-pulse emission with different frequency intervals [29]. The radar transmits two pulses with zero and non-zero frequency intervals and the angle and range of targets are estimated directly from the beamforming output peaks. But the method has great loss of pulse accumulation. In addition, target parameter estimation based on phased array with FDA cooperation [30] and bistatic FDA-MIMO [31] are studied. Although accurate and single value measurement of target angle and range is possible, these methods require complex equipment and heavy computation. For meter-wave FDA-MIMO radar, the signal model and joint angle and range estimation in a low-elevation area are derived and a generalized multiple signal classification algorithm is proposed to estimate target location [32]. However, the above methods do not consider the mainlobe jamming and mainly study the target location method based on spatial spectrum estimation [33], which requires a large amount of calculations and a high signal-to-noise ratio (SNR). It may be not suitable in actual radar.

Therefore, considering the mainlobe dense false target jamming (DFTJ) and the true target will be contained by training sample sampling, a sum-difference beam localization method of FDA-MIMO radar is proposed in this paper. The main contributions of this paper are as follows:

• The equation of sum-difference monopulse ratio curves for FDA-MIMO radar target angle and range measurement are derived.
• A coarse target location method based on cumulative sampling analysis is proposed in the case that the mainlobe DFTJ completely overwhelms the true target.
• A mainlobe jamming suppression method based on noise subspace projection (NSP) is proposed. And it has almost exactly the same performance with minimum variance distortionless response (MVDR) in output signal-to-jamming-plus-noise ratio (SJNR).
• The true target position in the time domain can be avoided in sampling to obtain jamming samples and the location performance is better than conventional methods, which is similar to ideal sampling.

The rest of this paper is organized as follows: Section 2 provides the FDA-MIMO radar signal basic model, which combines the beam range-angle correlation and the waveform orthogonal characteristics. The basic angle and range measurement principles for sum-difference channels are presented in Section 3. A method of jamming suppression and sum-difference beam localization is proposed in Section 4. The results of target sum-difference beam location without jamming, target coarse location in the time domain based on cumulative sampling, and target location under the condition of mainlobe DFTJ are given and analyzed in Section 5. The sample cumulative sampling analysis is discussed in Section 6. Finally, the conclusions of this paper and future works are summarized in Section 7.

2. FDA-MIMO Radar Signal Basic Model

Suppose the array of FDA-MIMO radar is composed of $N$ ($N$ is even) elements with uniform spacing $d$ ($d$ is equal to half wavelength). The transmitted signal is a far-field narrow-band signal. The transmission and receiving of the signal used the same elements. And the angle to the array normal direction is $\theta$. The first array element is taken as the reference element. The range from the scattering point to the reference element is $r$. The signal model is shown in Figure 1.

The transmitting carrier frequency of the $m\mathrm{th}$ element channel is

$$f_m=f_0+(m-1)\Delta f\hspace{0.17em}m=1,2,\dots N$$

(1)

where $f_0$ is the carrier frequency, and $\Delta f$ is the adjacent element frequency interval.

Ignoring all kinds of losses, the signal transmitted by the $m\mathrm{th}$ element of FDA-MIMO radar can be expressed as

$$s_m(t;\theta ,r)=\sqrt{\frac{E}{N}}{\phi }_m\left(t\right)\exp (-\mathrm{j}2\pi f_m{\displaystyle ⟮}t-\frac{r-\left(m-1\right)d\sin \theta }{c}{\displaystyle ⟯})\hspace{0.17em}\hspace{0.17em}0\le t\le T$$

(2)

where $E$ is the total energy of the transmitted signal. ${\phi }_m\left(t\right)$ is the complex envelope of the transmitted signal for the $m\mathrm{th}$ element. $T$ is the radar pulse duration and $t$ is the time index within the radar pulse. $c$ is the speed of light. Different from FDA radars, the signals from different array elements of FDA-MIMO radars are orthogonal to each other. That is

$${\displaystyle {\int }_T{\phi }_m(t){\phi }_{{}_l}^{\ast }(t-\tau )e^{\mathrm{j}2\pi \Delta f(m-l)t}}dt=0\hspace{0.17em}\hspace{0.17em}l\ne m,\forall \tau$$

(3)

where ${( )}^{\ast }$ denotes the conjugate operator, and $\tau$ is an arbitrary time delay.

Each element of FDA-MIMO radar receives the signal transmitted by all elements, and then the received data are retaken. In this case, the virtual aperture characteristics of MIMO radar are obtained and N array elements are equivalent to $N^2$ element.

Regardless of the time factor, after matching filtering, the target signal received by the entire array can be expressed as

$$\mathit{X}(\theta ,r)=\xi {\displaystyle \sum _{m=1}^N{\displaystyle \sum _{n=1}^N\exp \left\{-\mathrm{j}2\pi (m-1)\Delta f\frac{2r}{c}+\mathrm{j}2\pi \left[\frac{(n-1)d\sin \theta }{\lambda }+\frac{(m-1)d\sin \theta }{\lambda }\right]\right\}}}$$

(4)

where $\xi$ is the amplitude information of the target.

Then, the received signal can be expressed as

$$\mathit{X}(\theta ,r)=\xi {\left[x_{11},x_{12},\cdots ,x_{1N},x_{21},\cdots x_{NN}\right]}^{\mathrm{T}}=\xi \mathit{b}(\theta )\otimes \mathit{a}(\theta ,r,\Delta f)=\xi \mathit{v}(\theta ,r,\Delta f)$$

(5)

where superscript ${}^{\mathrm{T}}$ represents transpose, $\otimes$ is Kronecker product, $\mathit{v}(\theta ,r,\Delta f)$ is array steering vector, $\mathit{a}(\theta ,r,\Delta f)$ and $\mathit{b}(\theta )$ represent the transmit and receive steering vectors, respectively. Specific expression is as follows.

$$\begin{array}{l}\mathit{a}(\theta ,r,\Delta f)={\mathit{a}}_{\theta }(\theta )\odot {\mathit{a}}_r(r,\Delta f)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\left[1,\exp \left\{\mathrm{j}2\pi \frac{d}{{\lambda }_{}}\sin \theta \right\},\cdots ,\exp \left\{\mathrm{j}2\pi \frac{d}{{\lambda }_{}}(N-1)\sin \theta \right\}\right]}^{\mathrm{T}}\odot \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} {\left[1,\exp \left\{-\mathrm{j}4\pi \frac{\Delta f_1}{c_{}}r\right\},\cdots ,\exp \left\{-\mathrm{j}4\pi \frac{\Delta f_N}{c_{}}r\right\}\right]}^{\mathrm{T}}\end{array}$$

(6)

$$\mathit{b}\left(\theta \right)={\left[1,\exp \left\{\mathrm{j}2\pi \frac{d}{{\lambda }_{}}\sin \theta \right\},\cdots ,\exp \left\{\mathrm{j}2\pi \frac{d}{\lambda }(N-1)\sin \theta \right\}\right]}^{\mathrm{T}}$$

(7)

where $\odot$ represents Hadamard product. For the convenience of expression, the steering vector $\mathit{v}(\theta ,r,\Delta f)$ is denoted as $\mathit{v}$, and the signal $\mathit{X}(\theta ,r)$ is denoted as $\mathit{X}$ in the following.

3. Target Location Basic Principles of Sum-Difference Three Channels

Due to the range-angle correlation of the FDA-MIMO radar beam, the sum-difference three channels are different from that of conventional phased array radar. Sum-difference three channels for conventional phased array radar refer to the sum channel (azimuth sum and pitch sum channel), azimuth difference channel (azimuth difference and pitch sum channel), and pitch difference channel (azimuth sum and pitch difference channel), while sum-difference three channels for the FDA-MIMO radar refer to the sum channel (angle sum and range sum channel), angle difference channel (angle difference and range sum channel), and range difference channel (angle sum and range difference channel). The target location of the sum-difference three channels for FDA-MIMO radar refers to angle measurement by the use of the sum channel and angle difference channel, and range measurement by the use of the sum channel and range difference channel. The target is located by integrating the information of angle and range measurement.

3.1. Basic Principles of Sum-Difference Beam Angle Measurement

Suppose a target’s location is $(\theta ,r)$ and the FDA-MIMO radar beam is directed to $({\theta }_0,r_0)$. The received array element-level data of the sum channel and angle difference channel are weighted by ${\mathit{W}}_{\Sigma -\Sigma }$ and ${\mathit{W}}_{\Delta -\Sigma }$, respectively. Where ${\mathit{W}}_{\Sigma -\Sigma }^{}=\mathit{v}({\theta }_0,r_0,\Delta f)$, ${\mathit{W}}_{\Delta -\Sigma }^{}=[w_{\theta \Delta }\odot b(\theta )]\otimes [a_{\theta }(\theta )\odot a_r(r,\Delta f)]$, $w_{\theta \Delta }={[1,1,\cdots ,-1,-1,\cdots -1]}^{\mathrm{H}}$, the sum beam can be expressed as

$$\begin{array}{l}{P}_{\Sigma -\Sigma }^{}(\theta ,r)={\mathit{W}}_{\Sigma -\Sigma }^{\mathrm{H}}\mathit{v}\\ =\frac{1-e^{\frac{\mathrm{j}2\pi Nd(\sin \theta -\sin {\theta }_0)}{\lambda }}}{1-e^{\frac{\mathrm{j}2\pi d(\sin \theta -\sin {\theta }_0)}{\lambda }}}•\frac{1-e^{\mathrm{j}2\pi N(\frac{d(\sin \theta -\sin {\theta }_0)}{\lambda }-\Delta f\frac{2(r-r_0)}{c})}}{1-e^{\mathrm{j}2\pi (\frac{d(\sin \theta -\sin {\theta }_0)}{\lambda }-\Delta f\frac{2(r-r_0)}{c})}}\end{array}$$

(8)

The angle difference beam can be expressed as

$$\begin{array}{l}{P}_{\Delta -\Sigma }^{}(\theta ,r)={\mathit{W}}_{\Delta -\Sigma }^{\mathrm{H}}\mathit{v}\\ =\frac{1-e^{\frac{\mathrm{j}\pi Nd(\sin \theta -\sin {\theta }_0)}{\lambda }}}{1-e^{\frac{\mathrm{j}\pi d(\sin \theta -\sin {\theta }_0)}{\lambda }}}•\frac{1-e^{\mathrm{j}\pi N(\frac{d(\sin \theta -\sin {\theta }_0)}{\lambda }-\Delta f\frac{2(r-r_0)}{c})}}{1-e^{\mathrm{j}\pi (\frac{d(\sin \theta -\sin {\theta }_0)}{\lambda }-\Delta f\frac{2(r-r_0)}{c})}}•(1-e^{\frac{\mathrm{j}\pi 2Nd(\sin \theta -\sin {\theta }_0)}{\lambda }-\mathrm{j}\pi N\Delta f\frac{2(r-r_0)}{c}})\end{array}$$

(9)

The ratio of the angle difference beam to the sum beam is

$$\frac{P_{\Delta -\Sigma }^{}}{P_{\Sigma -\Sigma }^{}}=-j\tan (\frac{\pi }{\lambda }Nd(\sin \theta -\sin {\theta }_0)+\pi N\Delta f\frac{r-r_0}{c})$$

(10)

The deviation angle and deviation range are $\Delta \theta =\theta -{\theta }_0$ and $\Delta r=r-r_0$, respectively. Considering that the target is generally located within the 3 dB mainlobe beam (angle dimension mainlobe is $\frac{50.7\lambda }{Nd}$, range dimension mainlobe is $\frac{c}{2N\Delta f}$), it can be easily detected, so $\Delta \theta$ and $\Delta r$ are both small. Then

$$\frac{P_{\Delta -\Sigma }^{}}{P_{\Sigma -\Sigma }^{}}\approx -\mathrm{j}\tan (\frac{\pi }{\lambda }Nd\Delta \theta \cos \theta +\pi N\Delta f\frac{\Delta r}{c})$$

(11)

As can be seen from Equation (10), due to the range-angle correlation of the beam, when the sum-difference beam is applied to measure for FDA-MIMO radar, the ratio of the sum-difference beam is coupled with the angle and range of the target. Therefore, in theory, the FDA-MIMO radar can measure not only the angle of the target but also the range by applying the sum-difference beam.

In Equation (11), when $N\Delta f\Delta r\ll c$, compared with the phase difference brought by $\frac{\pi }{\lambda }Nd\Delta \theta \cos \theta$, the phase difference brought by $\pi N\Delta f\frac{\Delta r}{c}$ is almost negligible. The angle of the target can be measured by applying the sum-difference beam weights of ${\mathit{W}}_{\Sigma -\Sigma }^{}$ and ${\mathit{W}}_{\Delta -\Sigma }^{}$. Equation (11) can be rewritten as

$$\frac{P_{\Delta -\Sigma }^{}}{P_{\Sigma -\Sigma }^{}}\hspace{0.17em}\approx -j\tan (\frac{\pi }{\lambda }Nd\Delta \theta \cos \theta )$$

(12)

It can be derived as

$$\Delta \theta =\frac{\lambda }{\pi Nd\cos \theta }{\tan }^{-1}\left[j\frac{P_{\Delta -\Sigma }^{}}{P_{\Sigma -\Sigma }^{}}\right]$$

(13)

The actual target angle is measured as

$$\theta ={\theta }_0+\Delta \theta$$

(14)

The angle dimension monopulse ratio curve is

$$f(\theta )=\mathrm{Im}(\frac{P_{\Delta -\Sigma }^{}}{P_{\Sigma -\Sigma }^{}})\approx -\frac{\pi }{\lambda }Nd\cos \theta \Delta \theta =k\Delta \theta$$

(15)

where the proportionality coefficient is $k=-\frac{\pi }{\lambda }Nd\cos \theta$.

The angle dimension received data ratio is

$$Y_{\theta }=\frac{{\mathit{W}}_{\Delta -\Sigma }^{\mathrm{H}}\mathit{X}}{{\mathit{W}}_{\Sigma -\Sigma }^{\mathrm{H}}\mathit{X}}$$

(16)

The angle of the target is estimated by comparing the ratio of the received data in the angle dimension obtained by Equation (16) with the monopulse ratio curve obtained by Equation (15).

3.2. Basic Principles of Sum-Difference Beam Range Measurement

The received range difference channel data are weighted by ${\mathit{W}}_{\Sigma -\Delta }^{}=b(\theta )\otimes [a_{\theta }(\theta )\odot (w_{r\Delta }\odot a_r(r,\Delta f))]$. Where $w_{r\Delta }={[-1,1,\cdots ,-1,1,\cdots ,-1,1]}^{\mathrm{H}}$. Then, the range difference beam is

$$P_{\Sigma -\Delta }^{}(\theta ,r)={\mathit{W}}_{\Sigma -\Delta }^{\mathrm{H}}\mathit{v}=\frac{1-{(-1)}^{N-1}{e}^{\frac{\mathrm{j}2\pi Nd(\sin \theta -\sin {\theta }_0)}{\lambda }}}{1+e^{\frac{\mathrm{j}2\pi d(\sin \theta -\sin {\theta }_0)}{\lambda }}}•\frac{1-{(-1)}^N{e}^{\mathrm{j}2\pi N(\frac{d(\sin \theta -\sin {\theta }_0)}{\lambda }-\Delta f\frac{2(r-r_0)}{c})}}{1+e^{\mathrm{j}2\pi (\frac{d(\sin \theta -\sin {\theta }_0)}{\lambda }-\Delta f\frac{2(r-r_0)}{c})}}$$

(17)

The ratio of the range difference beam to the sum beam is

$$\begin{array}{l}\frac{P_{\Sigma -\Delta }^{}}{P_{\Sigma -\Sigma }^{}}=\frac{1-{(-1)}^{N-1}{e}^{\frac{\mathrm{j}2\pi Nd(\sin \theta -\sin {\theta }_0)}{\lambda }}}{1+e^{\frac{\mathrm{j}2\pi d(\sin \theta -\sin {\theta }_0)}{\lambda }}}•\frac{1-e^{\frac{\mathrm{j}2\pi d(\sin \theta -\sin {\theta }_0)}{\lambda }}}{1-e^{\frac{\mathrm{j}2\pi Nd(\sin \theta -\sin {\theta }_0)}{\lambda }}}\\ •\frac{1-{(-1)}^N{e}^{\mathrm{j}2\pi N(\frac{d(\sin \theta -\sin {\theta }_0)}{\lambda }-\Delta f\frac{2(r-r_0)}{c})}}{1+e^{\mathrm{j}2\pi (\frac{d(\sin \theta -\sin {\theta }_0)}{\lambda }-\Delta f\frac{2(r-r_0)}{c})}}•\frac{1-e^{\mathrm{j}2\pi (\frac{d(\sin \theta -\sin {\theta }_0)}{\lambda }-\Delta f\frac{2(r-r_0)}{c})}}{1-e^{\mathrm{j}2\pi N(\frac{d(\sin \theta -\sin {\theta }_0)}{\lambda }-\Delta f\frac{2(r-r_0)}{c})}}\end{array}$$

(18)

Considering that the array element number used by FDA-MIMO radar is even, the above Equation can be further derived as

$$\frac{P_{\Sigma -\Delta }^{}}{P_{\Sigma -\Sigma }^{}}\hspace{0.17em}\approx \frac{{\pi }^2{d}^2\Delta {\theta }^2{\cos }^2\theta }{{\lambda }^2}-\frac{2{\pi }^{2}d\Delta \theta \Delta f\cos \theta }{c\lambda }\Delta r$$

(19)

According to Equation (19), when the angle deviation is very small, compared with the phase difference brought by $\frac{2{\pi }^{2}d\Delta \theta \Delta f\cos \theta }{c\lambda }$, the phase difference brought by $\frac{{\pi }^2{d}^2\Delta {\theta }^2{\cos }^2\theta }{{\lambda }^2}$ is almost negligible. Then, the sum beam weights ${\mathit{W}}_{\Sigma -\Sigma }^{}$ and ${\mathit{W}}_{\Sigma -\Delta }^{}$ can be used to measure the range of the target. Equation (19) can be rewritten as

$$\frac{P_{\Sigma -\Delta }^{}}{P_{\Sigma -\Sigma }^{}}\hspace{0.17em}\approx \frac{2\pi d\Delta \theta \Delta f\cos \theta }{c\lambda }\Delta r$$

(20)

It can be derived as

$$\Delta r=\frac{c\lambda }{2\pi d\Delta \theta \Delta f\cos \theta }{\tan }^{-1}\left[\frac{P_{\Sigma -\Delta }^{}}{P_{\Sigma -\Sigma }^{}}\right]$$

(21)

The true target range is measured as

$$r=r_0+\Delta r$$

(22)

The range dimension received data ratio is

$$f(r)=\mathrm{Im}(\frac{P_{\Sigma -\Delta }^{}}{P_{\Sigma -\Sigma }^{}})\approx -\frac{2{\pi }^{2}d\Delta \theta \Delta f\cos \theta }{c\lambda }\Delta r=k\Delta r$$

(23)

where the proportionality coefficient is $k=-\frac{2{\pi }^{2}d\Delta \theta \Delta f\cos \theta }{c\lambda }$.

The range dimension received data ratio is

$$Y_r=\frac{{\mathit{W}}_{\Sigma -\Delta }^{\mathrm{H}}\mathit{X}}{{\mathit{W}}_{\Sigma -\Sigma }^{\mathrm{H}}\mathit{X}}$$

(24)

The range of the target is estimated by comparing the ratio of the received data in the range dimension obtained by Equation (24) with the monopulse ratio curve obtained by Equation (23).

Finally, the angle and range information are integrated to locate the target.

4. A method of Mainlobe DFTJ Suppression and Target Sum-Difference Localization

4.1. Principle of Mainlobe DFTJ Suppression Based on NSP

For the sake of analysis, assume that there is a target located at $({\theta }_s,r_s)$, on the $l_s\mathrm{th}$ range gate. There is a jammer located at $({\theta }_j,r_j)$, which releases K DFTJ signals respectively at the $l_1\mathrm{th},l_2\mathrm{th},\cdots ,l_K\mathrm{th}$ range gates. Suppose that the false target on the $l_k\mathrm{th}$ range gate happens to cover the whole target completely. The jamming angle is exactly the same as the target angle and the range is different, that is, ${\theta }_s={\theta }_j$, $r_s\ne r_j$. Since these jamming signals are all released by the same jammer, the steering vectors of these K jamming are the same and are represented by ${\mathit{v}}_j({\theta }_j,r_j)$. The signals received by the whole array can be represented as

$$\mathit{x}={\mathit{x}}_s+{\mathit{x}}_j+\mathit{n}={\xi }_s{\mathit{v}}_s({\theta }_s,r_s)+{\displaystyle \sum _{k=1}^K{\xi }_{jk}{\mathit{v}}_j({\theta }_j,r_j)}+\mathit{n}$$

(25)

where ${\xi }_s$ and ${\xi }_{jk}$ represent the received amplitude information of the target and the $k\mathrm{th}$ jamming, respectively. ${\mathit{v}}_s({\theta }_s,r_s)$ is the target steering vector. $\mathit{n}$ is the Gaussian white noise whose mean is 0 and variance is ${\delta }_n^2$. For simplicity, the following target and jamming steering vectors are denoted by ${\mathit{v}}_s$ and ${\mathit{v}}_j$, respectively.

After data retaking, the signal received by the whole array can be expressed in the time domain as

$$\mathit{x}={\mathit{x}}_s+{\mathit{x}}_j+\mathit{n}={\displaystyle \sum _{p=1}^{N\times N}{\xi }_{sp}{l}_s}+{\displaystyle \sum _{p=1}^{N\times N}{\displaystyle \sum _{i=1}^K{\xi }_{jip}{l}_{ip}}}+\mathit{n}$$

(26)

where ${\mathit{x}}_s={\displaystyle \sum _{p=1}^{N\times N}{\xi }_{sp}{l}_s}$, $x_j={\displaystyle \sum _{p=1}^{N\times N}{\displaystyle \sum _{i=1}^K{\xi }_{jip}{l}_{ip}}}$. ${\xi }_{sp}$ represents the target signal amplitude received by the equivalent pth array element. ${\xi }_{jip}$ represents the amplitude of the ith jamming signal received by the equivalent pth array element. Considering that the mainlobe DFTJ energy is generally larger than the true target signal energy, for any $p\in (1,N^2)$, there is ${\xi }_{jip}>{\xi }_{sp}$.

After sampling the received signal, the jamming plus noise covariance matrix ${\mathit{R}}_{j+n}$ is obtained, which can be derived by feature decomposition as

$${\mathit{R}}_{j+n}={\displaystyle \sum _{k=1}^{N\times N}{\lambda }_k{\mathit{u}}_k{\mathit{u}}_k^{\mathsf{H}}}={\mathit{U}}_j{\mathit{\Sigma }}_j{\mathit{U}}_j^{\mathrm{H}}+{\mathit{U}}_n{\mathit{\Sigma }}_n{\mathit{U}}_n^{\mathrm{H}}$$

(27)

where ${\lambda }_k$ is the $k\mathrm{th}$ feature value. ${\mathit{u}}_k$ is the feature vector corresponding to the $k\mathrm{th}$ feature value. ${\mathit{U}}_j$ and ${\mathit{\Sigma }}_j$ are the diagonal matrix composed of jamming signal subspaces and their corresponding feature values, respectively. ${\mathit{U}}_n$ and ${\mathit{\Sigma }}_n$ are the diagonal matrices composed of noise subspaces and their corresponding feature values, respectively. Since these jamming signals are all released by the same jammer, the steering vectors of these jammings are the same and there is only one eigenvector obtained from the decomposition. And the feature values of the noise are equal. The feature values are ${\lambda }_1\ge {\lambda }_2=\cdots ={\lambda }_{N\times N}$ in order of magnitude. The feature vectors corresponding to different feature values are orthogonal, so the following set of standard vectors can be formed as

$$\left\{\begin{array}{c}{\mathit{u}}_k^{\mathsf{H}}{\mathit{u}}_m=0\\ {\mathit{u}}_k^{\mathsf{H}}{\mathit{u}}_k=1\end{array}\right.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k\ne m$$

(28)

and it has ${\displaystyle \sum _{k=1}^M{\mathit{u}}_k{\mathit{u}}_k^{\mathsf{H}}}=\mathit{I}$, where $\mathit{I}$ is the identity matrix.

The weighted vector based on NSP is

$${\mathit{W}}_n={\mathit{U}}_n\mathrm{pinv}({\mathit{U}}_n^{}){\mathit{v}}_s$$

(29)

where ${\mathit{U}}_n$ is the noise subspace, ‘pinv’ means finding a pseudo-inverse matrix.

Since the DFTJ steering vectors are the same, only one large feature value can be obtained after the feature decomposition of ${\mathit{R}}_{j+n}$, and the corresponding feature vector can form the jamming subspace ${\mathit{U}}_s$, whose orthogonal basis is ${\mathit{u}}_1$. The other feature values corresponding to $N\times N-1$ small feature values can form noise subspace ${\mathit{U}}_n$, whose orthogonal basis is $\left[{\mathit{u}}_2,{\mathit{u}}_3,\cdots ,{\mathit{u}}_{N\times N}\right]$. According to Equation (28), the jamming subspace and noise subspace are orthogonal.

${\mathit{U}}_n$ can be represented by the feature vector as

$${\mathit{U}}_n=\left[{\mathit{u}}_2,{\mathit{u}}_3,\cdots ,{\mathit{u}}_{N\times N}\right]$$

(30)

By substituting Equation (30) into Equation (29), the expression of the NSP weighted vector can be obtained as

$${\mathit{W}}_n={\mathit{U}}_n\mathrm{pinv}({\mathit{U}}_n^{}){\mathit{v}}_s={\mathit{U}}_n{({\mathit{U}}_n^{\mathrm{H}}{\mathit{U}}_n)}^{-1}{\mathit{U}}_n^{\mathrm{H}}{\mathit{v}}_s={\displaystyle \sum _{k=2}^{N\times N}{\mathit{u}}_k^{\mathsf{H}}{\mathit{v}}_s{\mathit{u}}_k}$$

(31)

Since the jamming steering vector ${\mathit{v}}_j$ and the corresponding feature vector ${\mathit{u}}_1$ with a large feature value span the same jamming subspace, the jamming steering vector is orthogonal to the noise subspace, that is ${\mathit{u}}_k^{\mathrm{H}}{\mathit{v}}_j=0(k\ge 1)$, so that the DFTJ released by the same jammer satisfies ${\mathit{W}}_n^{\mathsf{H}}{\mathit{v}}_j=0$.

Considering the range-angle correlation of the FDA-MIMO radar beam, if the signal is different from the angle or range of the jamming, the space spanned by the signal steering vector and the jamming steering vector generated by the jammer does not intersect [34]. Therefore, the target steering vector is not orthogonal to the noise subspace, which is ${\mathit{W}}_n^{\mathsf{H}}{\mathit{v}}_s\ne 0$.

After filtering, the output signal can be expressed as

$$\begin{array}{cc}\hfill y={\mathit{W}}_n^{\mathsf{H}}\mathit{x}& ={\displaystyle \sum _{k=2}^{N\times N}{\mathit{u}}_k^{\mathsf{H}}{\mathit{v}}_s{\mathit{u}}_{{}_k}^{\mathrm{H}}}\left({\xi }_s{\mathit{v}}_s+{\displaystyle \sum _{k=1}^{N\times N}{\xi }_{jk}}{\mathit{v}}_{jk}+\mathit{n}\right)\hfill \\ \hfill & ={\xi }_s{\mathit{W}}_n^{\mathsf{H}}{\mathit{v}}_s+{\mathit{W}}_n^{\mathsf{H}}\mathit{n}\hfill \end{array}$$

(32)

As can be seen from the above equation, the output signal only contains the target and noise, and the jamming is suppressed.

It should be pointed out here that different from the MVDR algorithm in [35], NSP algorithm is to project the adaptive weight vector to the noise subspace completely, which belongs to the subspace algorithm. And the NSP algorithm needs to know the number of sources to obtain noise subspace by decomposition, while the MVDR algorithm needs to accurately know the jamming plus noise covariance matrix. The application conditions of the two methods are different.

4.2. Target Location Method Based on Sample Cumulative Sampling Analysis

In practice, under the condition of mainlobe DFTJ, the target is often submerged by the jamming and the true target signal will inevitably be picked up when the jamming sampling is carried out. When the NSP algorithm is adopted for adaptive jamming suppression, the target will be eliminated and the output SJNR will decrease. Even in the case of a strong target signal, the output SJNR is lower than that of a weak target signal and the target location performance of FDA-MIMO radar is worse. Therefore, a target location method based on cumulative sample analysis under the condition of mainlobe DFTJ is proposed. The basic principle is as follows.

First, according to all the signals received by the array, the covariance matrix is constructed and the feature decomposition is carried out. The feature vectors are sorted according to the order from the largest to the smallest feature values.

$${\mathit{R}}_{\mathit{x}}=E[\mathit{x}{\mathit{x}}^{\mathrm{H}}]={\displaystyle \sum _{k=1}^{N\times N}{\lambda }_k{\mathit{u}}_k{\mathit{u}}_k^{\mathsf{H}}}={\lambda }_j{\mathit{u}}_j{\mathit{u}}_j^{\mathsf{H}}+{\lambda }_s{\mathit{u}}_s{\mathit{u}}_s^{\mathsf{H}}+{\displaystyle \sum _{k=3}^{N\times N}{\lambda }_k{\mathit{u}}_k{\mathit{u}}_k^{\mathsf{H}}}$$

(33)

where ${\mathit{u}}_j$ and ${\lambda }_j$ are the corresponding feature vectors and feature values of the jamming, respectively. ${\mathit{u}}_s$ and ${\lambda }_s$ are the corresponding feature vectors and feature values of the target, respectively, and ${\lambda }_j\ge {\lambda }_s\ge {\lambda }_3={\lambda }_4=\cdots ={\lambda }_{N^2}$.

Then, under ideal conditions, the true target is located in the mainlobe of FDA-MIMO radar. Compared with the jamming, the range and angle in the expected target steering vector are closer to the range and angle in the true target signal. And the jamming energy is generally greater than the true target, so the correlation between the expected target steering vector ${\mathit{v}}_{s0}$ and the feature vector corresponding to the true target should be the strongest. This feature can be used to determine whether there is a true target in the received signal, and to return the corresponding feature vector of the target and its position in the sequence of feature vectors.

$$[\beta ,\rho ,{\mathit{u}}_l]=\arg \underset{\begin{array}{c}{\mathit{u}}_m\\ 1\le m\le N^2\end{array}}{\max }{\left| {\mathit{u}}_m^{\mathrm{H}}{\mathit{v}}_{s0} \right|}^2$$

(34)

where ${\mathit{u}}_l$ and $\rho$ are the feature vector with the strongest correlation with the desired target steering vector and their positions in the feature vector ordering, respectively. $\beta$ is the correlation coefficient.

The threshold is set as $\eta$. If the correlation coefficient is larger than the threshold, it is denoted as ${\mathrm{H}}_1$. If the correlation coefficient is smaller than the threshold, it is denoted as ${\mathrm{H}}_0$. The specific expression is as follows.

$$\left\{\begin{array}{l}{\mathrm{H}}_1:\beta \ge \eta \hspace{0.17em}\\ {\mathrm{H}}_0:\beta <\eta \hspace{0.17em}\end{array}\right.$$

(35)

It should be pointed out here that if the decision is no target, that is ${\mathrm{H}}_0$, the selected data can be sampled according to Equation (36) and denoted as an adaptive weight vector to suppress jamming. If it is determined to have a target, that is ${\mathrm{H}}_1$, it is necessary to analyze the approximate location of the target in the time domain based on cumulative sampling and then selectively sample the received signal of the array. The specific diagram is shown in Figure 2.

First, a sampling threshold $\partial$ is defined to select data larger than the threshold, effectively reducing the sample size and calculation amount. It should be pointed out that, considering the low signal energy, some noise energy may be higher than the set threshold, so the data whose number of consecutive sampling points higher than the threshold are greater than the number of sampling points corresponding to 2/3 signal pulse width can be selected as the cumulative sampling sample. Generally, the data meeting the conditions are mostly DFTJ when DFTJ completely overwhelms the target after pulse compression, which is denoted as $x_{j1},x_{j2},\cdots ,x_{jk},\cdots ,x_{jK}$. Wherein,

$${\mathit{x}}_{jk}\ge \partial \hspace{0.17em}\hspace{0.17em}k\in [1:K]$$

(36)

Then, the cumulative sampling analysis is carried out for the data larger than the threshold from near to far. The first cumulative sampling data are $x_{j1}$, and the second cumulative sampling data are $x_{j1}+x_{j2}$. So the kth cumulative sampling data are

$${\mathit{x}}_k={\displaystyle \sum _{p=1}^k{\mathit{x}}_{jp}}\hspace{0.17em}\hspace{0.17em}k\in [1:K]$$

(37)

The kth sampling covariance matrix is constructed and the feature decomposition is carried out. The feature vectors are arranged in order from large to small according to the feature values, as shown in Equation (38). According to the position $\rho$ of ${\mathit{u}}_l$ obtained above, the sum of correlation ${\mu }_k$ between the two feature vectors and the expected steering vector before the kth sampling is calculated. This is shown in Equation (39).

$${\mathit{R}}_{Xk}=E[{\mathit{X}}_k{\mathit{X}}_k{}^{\mathrm{H}}]={\displaystyle \sum _{m=1}^{\rho }{\lambda }_m{\mathit{u}}_m{\mathit{u}}_m^{\mathsf{H}}}+{\displaystyle \sum _{m=\rho +1}^{N\times N}{\lambda }_m{\mathit{u}}_m{\mathit{u}}_m^{\mathsf{H}}}$$

(38)

$${\mu }_k={\displaystyle \sum _{m=1}^{\rho }{\left| {\mathit{u}}_m^{\mathrm{H}}{\mathit{v}}_{s0} \right|}^2}$$

(39)

Since the true target is just covered by the false target located on the $l_k\mathrm{th}$ range gate, the target will just be included in the $k\mathrm{th}$ cumulative sampling. The correlation between the $\rho \mathrm{th}$ feature vector obtained after feature decomposition and the expected target steering vector will be enhanced. And then the obtained $u_k$ will be much larger than the $u_{k-1}$ obtained in the previous cumulative sampling. Hence, the target location decision threshold $\varsigma$ can be set to compare the magnitude of adjacent correlation coefficients difference $\Delta u$ and $\varsigma$.

$$\Delta u_{k-1}=\left|{\mu }_k-{\mu }_{k-1}\right|\ge \varsigma$$

(40)

If Equation (40) is satisfied, it can be roughly determined that the target is located on the $l_k\mathrm{th}$ range gate. When constructing the adaptive mainlobe jamming suppression weight vector, the $l_k\mathrm{th}$ range gate can be avoided and sampled in the selected data. If Equation (40) is not satisfied, the $(k+1)\mathrm{th}$ sampling shall be conducted until all the data higher than the threshold $\partial$ are collected cumulatively. If $\Delta u$ obtained by cumulative sampling to all data greater than the threshold $\partial$ still does not satisfy Equation (40), it can be considered that the data higher than the threshold $\partial$ do not contain target signals. When constructing an adaptive weight vector, it can be sampled in these data.

Finally, the adaptive weight vector of sum-difference three channels is constructed to locate the target while suppressing the jamming. Sample the data larger than the detection threshold $\partial$ to obtain data $\widehat{X}$ by avoiding the target position. And then the noise subspace ${\mathit{U}}_n^{}$ is obtained by feature decomposition according to Equation (27).

Sum channel adaptive weight can be expressed as

$$W_{\mathrm{A}-\Sigma -\Sigma }={\mathit{U}}_n\mathrm{pinv}({\mathit{U}}_n^{})W_{\Sigma -\Sigma }^{}$$

(41)

Angle difference channel adaptive weight can be expressed as

$$W_{\mathrm{A}-\Delta -\Sigma }={\mathit{U}}_n\mathrm{pinv}({\mathit{U}}_n^{})W_{\Delta -\Sigma }^{}$$

(42)

Range difference channel adaptive weight can be expressed as

$$W_{\mathrm{A}-\Sigma -\Delta }={\mathit{U}}_n\mathrm{pinv}({\mathit{U}}_n^{})W_{\Sigma -\Delta }^{}$$

(43)

The adaptive angle and range monopulse ratio under the condition of mainlobe DFTJ can be expressed as

$$\left\{\begin{array}{l}{f}_{\mathrm{A}}(\theta )=\mathrm{Im}(\frac{W_{\mathrm{A}-\mathsf{\Delta }-\mathsf{\Sigma }}^{\mathrm{H}}\mathit{v}}{W_{\mathrm{A}-\mathsf{\Sigma }-\mathsf{\Sigma }}^{\mathrm{H}}\mathit{v}})\\ f_{\mathrm{A}}(r)=\mathrm{Im}(\frac{W_{\mathrm{A}-\mathsf{\Sigma }-\mathsf{\Delta }}^{\mathrm{H}}\mathit{v}}{W_{\mathrm{A}-\mathsf{\Sigma }-\mathsf{\Sigma }}^{\mathrm{H}}\mathit{v}})\end{array}\right.$$

(44)

The ratio of angle and range dimension received data can be expressed as

$$\left\{\begin{array}{l}{Y}_{\mathrm{A}-\theta }=\frac{W_{\mathrm{A}-\mathsf{\Delta }-\mathsf{\Sigma }}^{\mathrm{H}}\mathit{X}}{W_{\mathrm{A}-\mathsf{\Sigma }-\mathsf{\Sigma }}^{\mathrm{H}}\mathit{X}}\\ Y_{\mathrm{A}-r}=\frac{W_{\mathrm{A}-\mathsf{\Sigma }-\mathsf{\Delta }}^{\mathrm{H}}\mathit{X}}{W_{\mathrm{A}-\mathsf{\Sigma }-\mathsf{\Sigma }}^{\mathrm{H}}\mathit{X}}\end{array}\right.$$

(45)

Similar to the target location method of FDA-MIMO radar under the condition of no jamming, the target location under the condition of mainlobe DFTJ is also estimated by comparing the ratio of angle and range received data obtained from Equation (45) with the monopulse ratio curve of angle and range obtained from Equation (44).

The specific process steps of the whole method are as follows.

(1)

All data received including the target are sampled and a covariance matrix ${\mathit{R}}_x$ is constructed. Feature decomposition is carried out and feature vectors are arranged in order from largest to smallest according to the feature values, as shown in Equation (33).

(2)

According to Equation (34), the position $\rho$ and correlation coefficient $\beta$ of the feature value corresponding to the feature vector with the strongest correlation with the expected target steering vector are obtained.

(3)

Set the sampling threshold $\partial$ and select the data ${\mathit{x}}_{jk}$ higher than the threshold.

(4)

Set the threshold $\eta$ of whether the target is present or not and determine whether there is a target according to Equation (35). If there is no target, an adaptive weight vector can be obtained by sampling $\widehat{X}$ in the selected data.

(5)

According to Equation (37), the sampling data ${\mathit{X}}_k$ are obtained by the $k\mathrm{th}$ time cumulative sampling from near to far.

(6)

According to Equation (38), the $k\mathrm{th}$ sampling covariance matrix ${\mathit{R}}_{Xk}$ is constructed and the feature decomposition is carried out. The feature vectors are arranged in order from the largest to the smallest according to the feature values.

(7)

The sum of the correlation coefficients ${\mu }_k$ is calculated between the first $\rho$ feature vectors and the expected steering vector according to Equation (39).

(8)

The target position decision threshold $\varsigma$ is set to compare whether the difference of the correlation coefficients $\Delta u$ obtained by two adjacent samples meets Equation (40). If yes, the target can be roughly judged to be located on the $l_k\mathrm{th}$ range gate and data $\widehat{X}$ can be obtained by avoiding the $l_k\mathrm{th}$ range gate to sample in the selected data. If not, conduct the $(k+1)\mathrm{th}$ sampling and turn to step (5) until the cumulative sampling reaches all the data greater than the threshold $\partial$.

(9)

If the cumulative sampling data in all do not satisfy Equation (40), it can be considered that the data higher than the threshold do not contain the target signal. When constructing the adaptive weight vector, data $\widehat{X}$ can be obtained by sampling in these data.

(10)

The covariance matrix is constructed from the sampled data $\widehat{X}$ in steps (3), (8), or (9). And the noise subspace ${\mathit{U}}_n$ is obtained by feature decomposition according to Equation (27).

(11)

Finally, according to Equations (41)–(43), the adaptive weights of the sum-difference three channels are obtained, respectively. Then, the angle and range of the target are estimated by comparing the angle and range ratio of the received data obtained by Equation (45) with the monopulse ratio curve obtained by Equation (44).

It is worth mentioning that the target location method based on sample cumulative sampling is not based on the time delay of the target signal in the time domain. Taking advantage of the range correlation characteristics of FDA-MIMO radar beam, the method uses the sum-difference beam to measure target range, which is similar to the angle measurement. The pulse width affects the number of sampling points. Under the condition of constant sampling rate, the wider the pulse, the more sampling points above the threshold will be, and the larger the calculation amount will be, but the target location performance of the proposed method is not influenced.

5. Simulation Results and Analysis

5.1. Simulation Analysis of Target Sum-Difference Location without Jamming

The main parameters of FDA-MIMO radar are set as follows in

Table 1. FDA-MIMO radar main parameters.

$\mathit{N}$	$\mathit{d}$	${\mathit{f}}_0$	$\mathit{\Delta }\mathit{f}$	Bandwidth	PRI	Snapshots	Size of Range Gate
8	$\lambda /2$	3 GHz	1 kHz	75 kHz	20 kHz	2000	1 km

. Assume the target is located at $(5°,400\mathrm{km})$ and SNR is 5 dB. The snapshots refer to the total number of samples from 320 km to 520 km on the range dimension, so the target is located on the 80th range gate. The sum-difference three data are used to locate the target and are shown in Figure 3.

As can be seen from Figure 3, the location of the target can be found well in the sum channel and SNR is 20 dB. Meanwhile, no target can be found in the angle difference and range difference channel, which is similar to the data receiving effect of conventional phased array radar sum-difference three channels.

When the sum-difference beam is applied to measure the angle and range of the target, the angle and range sum-difference beam and ratio curves are obtained, as shown in Figure 4 and Figure 5. It can be seen that no matter the angle dimension or range dimension, the difference beam forms a zero trap of about −32 dB at the location of the target. Therefore, the ratio curve formed is close to a straight line near the angle or range of the target, with good angle and range measurement performance.

SNR is changed and other parameters remain unchanged. For a total of 5000 times, simulations are carried out by Monte Carlo. The range measurement unit is km and the angle measurement unit is degree. Measurement error is calculated in the above units and angle and range measurement mean square error is shown in Figure 6.

As can be seen from Figure 6, with the increase in SNR, both the target angle and range measurement errors show a decreasing trend. But the angle measurement decreases to convergence significantly faster than the range measurement. And the angle measurement error is much smaller than the range measurement error, mainly because the mainlobe width of the angle dimension of FDA-MIMO radar is one order of magnitude lower than that of the range.

5.2. Simulation Analysis of Target Coarse Location in Time Domain Based on Cumulative Sampling

The target and jamming parameters are shown in

Table 2. Target and jamming parameters.

Target Location	SNR	Target Range Gate Location	Jammer Location	Number of Jamming	Jamming Range Gates Location	JNR
$(5°,400\mathrm{km})$	5 dB	80th	$(5^{\circ },500\mathrm{km})$	9	20th 40th 60th 80th 100th 120th 140th 160th 180th	All are 30 dB

. FDA-MIMO radar parameters are consistent with Section 5.1.

The data before filtering are shown in Figure 7. It can be found that no matter before or after pulse compression, the true target located on the 80th range gate is completely submerged by the false target jamming (the 4th jamming on the left) and it is unable to measure the angle and range.

In this paper, the cumulative sampling method is proposed to estimate the target location in the time domain. Then, in the follow-up sampling of jamming signals, the time domain location of the target is avoided, and more pure jamming samples are extracted to improve the jamming suppression effect and target location performance.

First, the received signals are fully sampled. The feature values are obtained by feature decomposition and sorted from largest to smallest, as shown in Figure 8a. The threshold $\eta =30$ whether there is a target that is defined and the correlation coefficient between the feature vector and the expected target steering vector is calculated, as shown in Figure 8b.

As can be seen from Figure 8a, there is only one large characteristic value, which is mainly caused by the DFTJ released by the jammer. It is consistent with the theoretical analysis. It can be seen from Figure 8b that the feature vector corresponding to the second feature value has the strongest correlation with the expected target steering vector. Its correlation coefficient is about 62, which is much higher than other correlation coefficients. It is also not the feature vector corresponding to the first feature value (corresponding to the mainlobe DFTJ), which also indicates that the second feature value and its feature vector are highly corresponding to the target. And the correlation coefficient is higher than the set threshold $\eta$, so it is determined that the received signal contains the target.

The threshold $\varsigma =30$ of target location decision is delimited and cumulative sampling is carried out from near to far. Then, covariance matrix is constructed and feature decomposition is carried out. According to the conclusion obtained in Figure 8b, the sum of the correlation coefficients between the first two feature vectors and the expected steering vector was calculated as the correlation coefficient obtained by this cumulative sampling. And then the difference of the correlation coefficients obtained from two adjacent cumulative samples was calculated, as shown in Figure 9. It is found that when cumulative sampling reaches the 4th false target jamming, the difference of correlation coefficients is greater than the set threshold. So it can be roughly determined that the true target is located at the 4th cumulative sampling location, that is, the range gate $l_4=80$ where ${\mathit{x}}_{j4}$ is located and is consistent with the simulation setting.

5.3. Simulation Analysis of Target Sum-Difference Location

5.3.1. Comparisons Jamming Suppression in the Time Domain

Based on the true target location in the time domain determined in Section 5.2, the sampling of this location is avoided to make the jamming samples purer. Then, the NSP algorithm is applied to suppress the jamming and locate the target. Compared with full jamming sampling +MVDR method (FJS+MVDR), ideal sampling (ideal jamming plus noise covariance matrix) +MVDR method (IS+MVDR), and ideal sampling +NSP method (IS+NSP), the obtained sum channel data are shown in Figure 10. The SNR was changed, Monte Carlo simulation is performed 5000 times, and the output SJNR are obtained, as shown in Figure 11.

As can be seen from Figure 10, when full jamming sampling is used for jamming suppression, mainlobe DFTJ suppression is not clean and there is still much energy left because of the sample containing the target. However, both the IS+MVDR/NSP method and the proposed method can well suppress jamming and improve the output SJNR, which is equivalent to about 20 dB. As shown in Figure 11, with the increase in the input SNR, the output SJNR of the proposed method is almost consistent with the IS+MVDR/NSP method. Especially in the case of high SNR, the output SJNR is much higher than that of the full jamming sampling method.

5.3.2. Comparisons of Sum-Difference Beam and Monopulse Ratio Curve

The sum-difference beam and sum-difference curve are shown in Figure 12, Figure 13 and Figure 14.

As can be seen from Figure 12, Figure 13 and Figure 14, compared with the FJS+MVDR method, the sum beam of the proposed method has no distortion in angle and range dimension. And the difference beam can form a null trap at the angle and range of jamming. Compared with the IS+MVDR/NSP method, the angle dimension null trap is shallower, about −15 dB, and the range dimension null trap is similar, about −47 dB. Moreover, the sum-difference curves formed by the proposed method in angle and range dimension are closer to the IS+MVDR/NSP method, while the FJS+MVDR algorithm directly distorts the sum-difference curve because the jamming is not suppressed cleanly. The error of angle and range measurement are larger theoretically.

5.3.3. Comparisons of Output SJNR and Measurement Error

The SNR was changed, and Monte Carlo simulation is performed 5000 times, the probability of false alarm is 10⁻⁶. The target probability of detection (Pd) and the error of angle and range measurement are obtained, as shown in Figure 15 and Figure 16, respectively.

As shown in Figure 15, with the increase in the input SNR, the Pd of the proposed method is almost consistent with the IS+MVDR/NSP method. And the Pd approaches 100% faster than the FJS+MVDR method. As can be seen from Figure 16, both the angle and range measurement errors of the proposed method and the IS+MVDR/NSP method decrease with the increased SNR. The proposed method tends to be the IS+MVDR algorithm, which has the same target measurement effect at high SNR. However, the measurement error in the method of FJS+MVDR at high SNR is larger than that at low SNR and the target location effect is the worst. In addition, the effect of IS+MVDR and IS+NSP is almost exactly the same whether it is output SJNR, angle and range sum-difference ratio curve and measurement error, which also proves the effectiveness of the NSP algorithm for jamming suppression and measurement.

6. Discussions for Sample Cumulative Sampling Analysis

In order to test the coarse location method in the time domain based on sample cumulative sampling analysis under the condition of partial or complete stagger of DFTJ and true target in the time domain, the location of the target on the range gate is changed to the 99th and 130th range gates. The other parameters are seen in Section 5.3. The simulation diagram is shown in Figure 17 and Figure 18.

If the target is on the 99th range gate, it will partially overlap with false target jamming on the 100th range gate. The true target, together with the 5th jamming from the left, meet the condition above the threshold $\partial =3\mathrm{dB}$. They are treated as a signal, as shown in Figure 17a. When sample cumulative sampling analysis is conducted, their location will be regarded as the location of the target, which is the location of the 5th cumulative sampling, as shown in Figure 17b.

If the target is located at the 130th range gate, it is completely staggered from all jamming and they are larger than the threshold, as shown in Figure 18a. At this time, during the sample cumulative sampling analysis, the target is considered as the 7th jamming, as shown in Figure 18b.

On the basis of identifying the true target location in the time domain, the pure jamming sample can be obtained by avoiding location sampling.

If the threshold is greater than the energy of the target signal, then the selected data sample will not contain the target information, as shown in Figure 19a. If the sample cumulative sampling analysis is carried out, all the selected data are not greater than the threshold $\varsigma =30$, as shown in Figure 19b. It can be assumed that the selected data do not contain the target, which is consistent with reality.

7. Conclusions

In this paper, considering the mainlobe DFTJ and the true target will be included in the training sample, a sum-difference localization method of FDA-MIMO radar is proposed. Based on the derived monopulse-ratio curve expression of angle and range measurement, a coarse target location method in the time domain based on sample cumulative sampling analysis is proposed to sample away from the target and obtain a cleaner jamming sample. Then, the NSP algorithm is used in the sum-difference beam to suppress the mainlobe jamming and accurately measure the angle and range of the target at the same time. The simulations verify that the effectiveness of the proposed method tends to be IS+MVDR/NSP algorithm, which has greater performance than the FJS+MVDR in mainlobe jamming suppression and angle and range measurement.

Nevertheless, the multi-target case is not considered in this paper, and the jamming suppression and localization method under the multi-target case are supposed to be further studied in the future.