Frequency-Hopping Frequency-Diverse MIMO Radar for Target Detection and Localization Under False-Target Jamming

A novel FH-LFM-FD-MIMO radar system is proposed to detect and localize true targets while suppressing false-target jamming signals. Linear frequency modulated (LFM) pulses are transmitted by a transmit array, with the carrier frequencies optimized to implement frequency-diverse multiple-input-multiple-output (FD-MIMO) and frequency-hopping (FH) schemes. The frequency offsets at all the transmit-array elements are represented as a frequency-diverse (FD) code, which is generated by combining a particle swarm optimization (PSO) algorithm and a rank-order-value (ROV) mapping to enhance the jamming suppression performance. The frequency-hopping (FH) scheme is implemented by applying specific FD codes in different pulse repetition intervals to suppress jamming signals in both frequency domain and spatial domain. The target echoes carrying matched FD code are detected at the receiver, while the false-target jamming signals carrying mismatched FD codes are suppressed. A spotlight response with low sidelobe level is achieved by optimizing the spotlight-range beamforming vector with a second PSO algorithm. After a target is detected, a two-dimensional minimum-variance distortionless response (2D-MVDR) is applied to pinpoint the target with higher precision. A binary integration detection scheme is proposed to enhance the detection performance under more severe jamming condition. The simulation results verify that the proposed FH-LFM-FD-MIMO radar system can increase the output signal-to-jamming-plus-noise ratio (SJNR) by more than 20 dB compared to conventional approaches, acquiring higher probability of detection under constant false alarm rate and enhancing the localization precision.


I. INTRODUCTION
False-target jamming is often used to divert the opponent from detecting the true target [1], [2], [3]. The jamming signal can be generated by using a false target generator (FTG) equipped with digital radio frequency memory (DRFM) [4], [5]. The FTG intercepts the radar signals and then re-transmits their replica at specific delay, mimicking targets at false positions [6], [7]. False-target jamming is typically countered by enhancing the echoes from true targets while suppressing the echoes from false targets [8], [9], [10].
The associate editor coordinating the review of this manuscript and approving it for publication was Fabrizio Santi .
An FTG requires at least one acting radar signal for duplication and mimicking [11], [12], [13], [14]. By casting some unique signature on each transmitted signal in different pulse repetition intervals (PRIs), the echoes from the true targets and the jamming signals can be differentiated. The signature can be cast in time, frequency or other domains. Pulse or waveform diversity is implemented in the time domain. In [15], the chirp rate of LFM signal was changed in each pulse repetition interval. In [16] and [17], different waveforms were transmitted in different PRIs, hence the jamming signals could be eliminated by waveform matching at the receiver.
Frequency hopping or frequency agility is implemented in the frequency domain. In [18], a frequency agile radar (FAR) randomly changed its carrier frequencies in each PRI. In [19] and [20], the carrier frequency and inter-pulse interval were varied from pulse to pulse, preventing the jammers from generating false signals at right carrier frequency. In [21], an extra phase modulation was imposed on each element and was changed from pulse to pulse, so that the true target and jammers can be separated in the spatial-frequency domain.
Spatial separation has been exploited to separate true targets from false ones. False targets can be generated to fulfill azimuth deception or mainlobe deception. An azimuth deception is usually achieved by transmitting signal to the sidelobe of the victim radar, making believe that a jammer appears in the mainlobe direction [22]. Sidelobe blanking [23] and sidelobe cancelation [24] were proposed to determine if the signal arrived from the mainlobe or sidelobe direction, followed by suppressing the jamming signals.
Data fusion based on a distributed multi-radar network could be used to recognize the scattering properties of targets and jammers. A typical target manifests specific bistatic scattering pattern, which is difficult to duplicate by a deceptive jammer [25]. In [26], the recognition task was reduced to a minimum-distance problem in terms of the target scattering signal and the jamming signal.
As for the mainlobe deception scheme, a false target can be generated in the same azimuth but different ranges from the target. A frequency diverse (FD) array can produce a spotlight beampattern by tuning frequency offsets across its array elements [27], [28], which is suitable for countermeasuring azimuth and mainlobe deception schemes. The diversity gain endowed provided by MIMO array and the range-azimuthdependent feature of FD array are combined to form an FD-MIMO array [29], [30], [31], which can spotlight the target at specific range-azimuth cell and suppress jamming signals from other cells.
The spotlight pattern of an FD-MIMO array can suppress range-dependent jamming or interference more effectively than conventional MIMO arrays. In [41], FD technique was combined with space-time adaptive processing (STAP) to form an FD-STAP radar for detecting fast-moving target and increasing the signal-to-interference-plus-noise ratio (SINR). Various methods on FD-MIMO arrays were proposed to detect target under Gaussian-like interference. In [42], an adaptive detector for FD-MIMO radar was implemented on a generalized likelihood ratio test (GLRT) criterion. In [43], a detector based on two-step GLRT was proposed to operate at constant false alarm rate. In [44], structured and unstructured GLRT based on Bayesian inference was introduced. Detection schemes could also be based on Rao test or Wald test [45], [46]. In [47], an FD-MIMO array was proposed to identify moving targets immersed in clutter by exploiting the azimuth, range and Doppler information. In [48], a cognitive FD-MIMO radar was proposed to detect moving targets by adaptively maximizing the SINR. The FD-MIMO arrays were shown to achieve better target detection performance than FD, MIMO or phased arrays [49].
In terms of target localization, various novel methods based on MIMO radar have been proposed to estimate directionof-departure (DoD) and direction-of-arrival (DoA) of targets in recent years. In [50], a sparse L-shaped electromagnetic vector sensor (EMVS) MIMO radar was proposed for 2D-DoD and 2D-DoA estimation, and a two-step algorithm was designed to acquire unambiguous angle estimation with high accuracy. In [51], the conventional multiple signal classification (MUSIC) was combined with a scale discrete Fresnel transform (SDFnT) to form SDFnT-MUSIC, which was used to improve the accuracy of angle estimation. In [52], a max-MUSIC was proposed to estimate the DoA with high accuracy and without spatial aliasing, by constructing multiple subarrays from a sparse MIMO radar.
In contrast to DoD and DoA estimation aimed for azimuthangle estimation, FD-MIMO is capable of estimating both target range and azimuth angle. In [53], an FD-MIMO technique was proposed to extend the conventional MUSIC to the range-angle domain. In [54], a true target was localized with an FD-MIMO technique in the range-angle-Doppler domain, in the presence of false-target jamming signal. In [55], the MVDR spectrum for FD-MIMO was derived to perform joint range-angle estimation, with the simulated localization accuracy approaching the Cramér-Rao bound (CRB).
However, the pattern of an FD-MIMO array varies with time, and the transmitting waveforms are not orthogonal. Time-variant frequency offsets were proposed to generate stationary pattern [56], [57], [58], but was not successful [59], [60]. In [61], [62], and [63], multiple bandpass filters and mixers were utilized to remove time dependency from the pattern, which will be incorporated into the proposed radar system. In [64] and [65], multiple matched filters were applied to eliminate the time-variant term.
The orthogonality of waveforms achieved with conventional MIMO receiver was practically unattainable at all Doppler-delay instances [66]. It was impractical to extend the orthogonality assumption to some FD-MIMO applications [55], [67], [68]. Instead, waveforms with good auto-correlation and cross-correlation properties can be attained. In [69], [70], and [71], phase-coded sequences were optimized with cyclic algorithms to generate waveforms with good correlation properties. An alternative to achieve orthogonality is through proper choice of frequency offsets. The orthogonality condition in [72] is adopted in this work. LFM pulse of duration T is chosen as the baseband waveform, the frequency offset between any two of the transmit-array elements is a non-zero integer multiply of 1/T .
In this work, we propose an FH-LFM-FD-MIMO radar by applying a frequency-hopping (FH) scheme to an FD-MIMO array, to detect and localize true targets while suppressing false-target jamming signals. Fig.1 shows the block diagram of steps in the proposed approach. The radar configuration and baseband waveform are decided first. Then, the FD codebook is generated by combining a particle swarm optimization (PSO) algorithm and a rank-order-value (ROV) mapping. For each FD code in the codebook, a corresponding spotlight-range beamforming vector is obtained with a second PSO algorithm to create low-sidelobe spotlight response for target detection and localization.
By exploiting the disparity between target echo and jamming signals in frequency and spatial domains, the target echo can be extracted and the jamming signals are significantly suppressed. Each of the transmit-array element in the FH-LFM-FD-MIMO array emits signal at a specified carrier frequency, which is properly offset from the reference frequency via a frequency offset index (FOI), as illustrated in Fig.3. The frequency offset indices (FOIs) applied across the whole transmit array are stored in a vector, called frequency-diverse (FD) code. The frequencyhopping (FH) scheme is implemented by applying different FD codes in different pulse repetition intervals. The FD codebook, FH sequence and the baseband waveform need to be orchestrated at the transmit and receive arrays, as illustrated in Fig.2.
The jamming signals emitted from a false-target generator are presumed to be replica of acting FD codes. The target echoes carrying matched FD codes are detected at the receiver, as illustrated in Fig.4, while the jamming signals carrying mismatched FD codes are suppressed. In this work, the FD codes are designed by using a particle swarm optimization (PSO) method via a rank-order-value (ROV) mapping. The optimized FD codes are compiled to an FD codebook, in which each FD code is assigned an integer index. The sequential order of dispatching the FD codes to the arrays is recorded as an FH sequence. The generation of FD codebook is illustrated in the flow-chart of Fig.5.
A spotlight pattern on specific range-azimuth cell is achieved via the endowed properties of an FD-MIMO array. Phase adjustment at the receive-array elements is implemented by using a second PSO algorithm, as shown in Fig.6, to further focus the spotlight pattern. Finer range resolution is achieved by using linear frequency modulated (LFM) pulses than continuous-wave (CW) pulses. With the sharp spotlight pattern response of the proposed FH-LFM-FD-MIMO radar, the signal-to-jamming-plus-noise ratio (SJNR) is significantly increase, leading to much better detection probability under a constant false alarm rate. A binary integration detector approach is incorporated to suppress jamming signals that accidentally carry identical FD code to the target echo. A two-dimensional minimum-variance distortionless response (2D-MVDR) spectrum is used to pinpoint the target location at high precision within the target rangeazimuth cell.
The rest of this paper is organized as follows. The signal model of FH-LFM-FD-MIMO is presented in Section II, the design of FD codebook is presented in Section III, the beamforming approach to achieve spotlight response is given in Section IV, the detection and localization algorithms are presented in Section V, and simulation results are discussed in Section VI. Finally, some conclusions are drawn in Section VII.
In For convenience to the readers, all the symbols and definitions used in this work are listed in Table 1 in an alphabetical order (English symbols first, followed by Greek symbols), accompanied with the location of their first appearance.   Fig.2 shows the configuration of the proposed FH-LFM-FD-MIMO radar, where the co-located transmit array and receive array are composed of M and N elements, respectively, with the inter-element spacings of d t and d r , respectively. The FD codebook C is a composed of K FD codes,

II. SIGNAL MODEL
wherec k = [c k1 , c k2 , · · · , c kM ] t and the superscript t denotes a transpose operator. The FH sequence S is a sequence of integers between 1 and K . One FD code index is read from the FH sequence in each pulse repetition interval, and the carrier frequencies of the transmit-array elements are determined according to the FD code.
where f m and w aθ m are the carrier frequency and the weighting, respectively, of the mth transmit-array element, and s tb (t) is the baseband waveform of an LFM pulse, where E is the energy radiated from an antenna element in a PRI, B is the bandwidth of the LFM pulse, T is the pulse 62124 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   length, and rect(t/T ) is a window function, with The carrier frequency f m is given by where f 0 is the reference frequency, f = 1/T is frequency spacing [72], η m is an integer frequency offset index (FOI), with 0 ≤ η m ≤ M t , and M t is the maximum allowable FOI. The FOIs assigned to all the transmit-array elements in a pulse repetition interval are stored in an FD codeη = [η 1 , η 2 , · · · , η M ] t ∈ C. For example, ifη =c k , then f m = f 0 + c km f . To steer the beam of transmit array towards the azimuth of θ, a phase w aθm = e jk 0 d t (m−1) sin θ is imposed to the mth element, with m = 1, · · · , M , where k 0 = 2π/λ 0 is the wavenumber at the reference frequency f 0 . These phase terms are stored in a transmit-azimuth beamforming vector asw  jamming signals, the received signal at the nth receive-array element is given by where the qth signal is with q = 0 for the target echo and q = 1, · · · , Q for the jamming signals, where α q and f qm = f 0 + η qm f are the amplitude and carrier frequency, respectively, of the qth signal. The FD codes for the target echo and the qth jamming signal areη 0 andη q , respectively, withη 0 ,η q ∈ C. The roundtrip delay from the mth transmit-array element via the qth target to the nth receive-array element is where c is the speed of light. The true target is located at (r 0 , θ 0 ), and the qth jamming signal is designed to camouflage a false-target at (r q , θ q ). The Doppler frequency shift is neglected in this work. Assume that the total bandwidth of the received signal is much smaller than the carrier frequency, namely, B + M t f ≪ f 0 , (8) can be approximated aŝ The round-trip delay of the target to the first element of both co-located arrays is τ 0 = 2r 0 /c. Assume the jamming signals are generated with a false-target generator (FTG) located at (r j , θ j ), and the delay of the qth jammer is designed to be where τ j = 2r j /c is the time delay of the FTG to the first element of both co-located arrays, τ jq is the camouflage delay controlled by the FTG, and r q = τ q c/2. The received signal in (7) is demodulated to the baseband as where s bqn (t, θ) = s qn (t, θ)e −j2πf 0 t . The contribution in s bqn (t, θ) from the mth transmit-array element is extracted by mixing s bqn (t, θ) with a signal e −j2π η 0m ft , followed by a low-pass filter H LP (f ) = rect(f /B) and a matched filter where F and F −1 are Fourier transform and inverse Fourier transform, respectively, and is the Fourier transform of s tb (t). Thus, with the approximation in (10), (13) is reduced to is the code-matching response between elements m and m ′ . All the signals {s bqnm (t, θ, η 0m )} in (15), with q = 0, 1, · · · , Q, are stacked into an MN × 1 vector withv Eqn. (17) is rewritten asv q (t, θ,η 0 ) = α ′ qμ (r, θ,η 0 ; r q , θ q , η q ), with r = ct/2 and µ(r, θ,η; r ′ , θ ′ ,η ′ ) =b(θ ′ ) ⊗ā(r, θ,η; r ′ , θ ′ ,η ′ ) (19) where ⊗ denotes Kronecker product, (r, θ) indicates the range-azimuth cell of interest,η is the FD code at the receive array, which matches the FD code of the transmit array to suppress jamming signal, namely,η =η 0 , (r ′ , θ ′ ) indicates the range-azimuth cell of a true target or a false-target jamming signal that carries FD codeη is the receive-azimuth steering vector, and is the transmit-array steering vector, ⊙ denotes the Hadamard product,ā r (r ′ ,η) andā θ (θ ′ ) are the transmit-range steering vector and transmit-azimuth steering vector, respectively, with the explicit forms of w aθ (θ) is the transmit-azimuth beamforming vector defined in (6), withw aθ (θ) =ā θ (θ ) to point the beam to the azimuth θ of interest, andh(t,η =η 0 ,η ′ =η q ) is an M × M codematching matrix, with the mm ′ th element given in (16).
The received signals after demodulation can now be represented as Next, design a spotlight beamforming vector as wherew b (θ ) is the spotlight-azimuth beamforming vector, which collaborates withw aθ (θ ) to form a beam in the azimuth direction θ of interest, leading tow b (θ) =b(θ), andw ar (r,η 0 ) is the spotlight-range beamforming vector for detecting targets in a range cell centered at r of interest, having the explicit form where φ m with 1 ≤ m ≤ M are implicitly dependent on the FD codeη 0 ∈ C. The optimization of φ m 's will be elaborated in Section IV. The 2-norm of the spotlight beamforming vectorw(r, θ,η 0 ) is MN .

III. DESIGN OF FD CODEBOOK
Jamming suppression is achieved by applying frequencyhopping scheme to induce mismatch between FD codes of target echo and jamming signals. Fig.5 shows the flow-chart of the proposed procedure of selecting FD codes in the codebook. A total of K frequency-diverse (FD) codes are selected sequentially. The first FD code is generated by picking M candidates from the integer set I = {0, 1, · · · , M t }. Each candidate is randomly picked with equal probability. If a newly picked candidate repeats with previous ones, it will be discarded and repicked. The M candidates serve as frequency offset indices (FOIs). They are arranged in an ascending order asc 1 = [c 11 , c 12 , · · · , c 1M ] t , with c 11 < c 12 < · · · < c 1M and (c 1M − c 11 ) > 0.7 M t . The condition (c 1M − c 11 ) > 0.7M t is imposed to acquire a sufficiently narrow half-power beamwidth (HPBW) of the spotlight response in the range domain. By analog, a uniform linear array with constant frequency increment across all the elements has an HPBW in range domain inversely proportional to the frequency difference between the first and the last elements.
Next, define the maximum jamming suppression level of the kth FD codec k as where is the jamming suppression level ofc k against previous FD codec k ′ , with (32) and which is the integration between two overlapped spectra. When c km = c k ′ m ′ , (33) is reduced to The numerator and the denominator of (31) are proportional to the received energy if the FD codec k matches and mismatches, respectively. The kth FD code is selected among allc k 's that minimizes F 1 (c k ), namely,c A particle swarm optimization (PSO) algorithm with rank-order value (ROV) mapping [73] is applied to construct the FD codebook. Define the position vector of the pth particle as where M t is the maximum frequency offset index. Initially, each coordinate ofχ p is a real number randomly picked from [0, 1]. The ROV mapping converts a sequence of continuous variables to a permutation of integers. Each coordinate is mapped to the rank of its value, with the smallest coordinate mapped to 1, the next to 2, and so on. Explicitly, the pth particle is mapped to an ROV vector asχ ′ p = ROV{χ p }. Sinceχ ′ p is a permutation of integer from 1 to M t , the first M integers inχ ′ p are rearranged in ascending order to form an FD code, denoted byc k {χ ′ p }. The fitness function is F 1 (c k ) in (30), and the order of the real-valued elements inχ p is adjusted to minimize F 1 (c k ).
The best fitness of the pth particle is labeled as pbest, associated with the best position vectorχ pb . The best fitness among the whole swarm is labeled as gbest, associated with the global best position vectorχ gb . The velocity of the pth particle is updated at iteration d as [74] where the components ofβ 1 ,β 2 ∈ R M are random numbers from uniform distribution in [0, 1],χ are the position vector and velocity vector, respectively, of particle p at iteration d. The habit weight h w is linearly decreased from 0.9 to 0.4 with iterations, and ρ 1 = ρ 2 = 2 are empirical constants. The population size is set to P = 20 [40], and the velocity ceiling is set to V max = 1. The position of particle p is then updated asχ The PSO algorithm halts when the number of iterations reaches a pre-defined number D 1 . The gbest at the end is mapped to an ROV vector asχ ′ gb = ROV{χ gb }. The elements inχ ′ gb are rearranged in an ascending order to form an FD codec k .

IV. BEAMFORMING FOR SPOTLIGHT RESPONSE
The azimuth dependence of the spotlight response is controlled by the transmit-azimuth beamforming vector and the receive-azimuth beamforming vector. The range dependence of the spotlight response is achieved by orchestrating the frequency offsets across the transmit array with an FD code to form a transmit-range beamforming vector [32], [33], [34], [35], [36], [37], [38], [39], [40], as well as a spotlight-range beamforming vector, which will be optimized in this Section.
The spotlight response focused on a range-azimuth cell centered at (r s , θ s ) is defined as with whereμ(r s , θ s ,η 0 ; r, θ,η 0 ) is defined in (19). The normalized spotlight response is defined as g 0 (r, θ) = g(r s , θ s ,η 0 ; r, θ,η 0 ) with g(r s , θ s ,c k ; r, θ,c k ′ ) = G(r s , θ s ,c k ; r, θ,c k ′ ) G(r s , θ s ,c k ; r s , θ s ,c k ) The spotlight response means a single peak centered in the specified range-azimuth cell and very low sidelobe level in the other cells.
The auto-correlation function of the LFM waveform significantly reduces the sidelobe level, which will be further reduced by optimizing the spotlight-range beamforming vectorw ar (r,η 0 ). By substituting the spotlight beamforming vector in (25) into (39), we have where By choosingw b (θ s ) =b(θ s ), the azimuth response G b (θ ) is reduced to that of conventional MIMO arrays. In the response G 0a (r, θ), the transmit-azimuth beamforming vectorw aθ (θ) in (6) is used to optimize the azimuth response andw ar (r s ,η 0 ) in (26) is used to optimize the range response.   6 shows the flow-chart of optimizing the spotlightrange beamforming vectorw ar (r s ,η 0 ) in (26) with another PSO algorithm. Define the position vectorφ p of the pth particle asφ p = φ p1 , φ p2 , · · · , φ pM t (46) with the initial value of φ pm randomly picked from [0, 2π].
Next, specify a normalized response mask g mask (r, θ s ) as 10 log 10 g mask (r, θ s ) = 62128 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. and define a fitness function as F 2 (φ p ) = R min ≤r≤R max max 0, 10 log 10 g 0 (r, θ s ;φ p ) g mask (r, θ s ) (48) where g 0 (r, θ s ;φ p ) is defined in (41), pending onφ p , and the range cell of interest is R min ≤ r ≤ R max . The best fitness of the pth particle is labeled as pbest, associated with the position vectorφ pb . The best fitness among the whole swarm is labeled as gbest, associated with the position vectorφ gb . The velocity of the pth particle is updated at iteration d as [74] where the components ofβ 1 ,β 2 ∈ R M are random numbers from a uniform distribution in [0, 1],φ are the position vector and velocity vector, respectively, of particle p at iteration d. The habit weight h w is linearly decreased from 0.9 to 0.4 with iterations, and ρ 1 = ρ 2 = 2 are empirical constants. The population size is set to P = 20 [40], and the velocity ceiling V max is set to 20 % that of the maximum position coordinate. The position of particle p is then updated asφ The PSO algorithm halts when the number of iterations reaches a specified number D 2 . The average normalized response to the target over all FD codes is computed as and the average response to jamming signals is computed as

V. TARGET DETECTION AND LOCALIZATION
By applying the FH-LFM-FD-MIMO scheme to extract the signals that carry matched FD code and suppress the signals that carry mismatched FD codes, a target in a specific range-azimuth cell of size r × θ can be detected at high signal-to-jamming-plus-noise ratio (SJNR).
The azimuth resolution of a conventional linear array is determined by the half-power beamwidth (HPBW) of the array as 0.88λ 0 /(Md t ) radian [75], where Md t is the aperture length of the linear array. The range resolution of the FH-LFM-FD-MIMO scheme is determined as The range resolution of conventional LFM radars is r = c/(2B) [76], which is derived from the auto-correlation function of the LFM pulse where sinc (x) = sin(πx)/(πx). The array factor in the range domain of an FD-MIMO array operating with an FD codeη is given by To obtain a rough estimation of range resolution, consider uniform linear frequency increment f across the transmit-array elements andw ar (r s ,η) = [1, 1, · · · , 1] t , the array factor is reduced to which implies a range resolution of r = c/(2M f ) [77]. By analog, the HPBW in range domain is determined by the frequency span over the array, which is approximated as (η max − η min ) f , with η max and η min the maximum and minimum frequency-offset indices, respectively, within the FD codeη. The range response of the FH-LFM-FD-MIMO radar is jointly determined by the auto-correlation function of the LFM signals and the response of FD-MIMO array [77].

A. TARGET DETECTION
The signal after spotlight beamforming, u q (t, θ,η 0 ) in (28), is a complex number marked in Fig.7. The jamming signals after spotlight beamforming and FH jamming suppression tend to cluster around the origin of the complex plane, while the target echo, u 0 (t, θ,η 0 ), is deposited far away from the cluster. A boundary line (threshold) can be drawn for hypothesis test to separate jamming signals and the target echo. The strongest jamming signal u jk is marked by ⊗. In a jammingfree environment, u jk = 0.
To detect targets at a specific range-azimuth cell centered at (r, θ), with azimuth resolution θ, range resolution r, and FD codeη 0 =c k ∈ C, a binary hypothesis test is formulated as  where t = 2r/c, and u 0 (t, θ,η 0 ) = α ′ 0w † (r, θ,η 0 ) ·μ(r, θ,η 0 ; r, θ,η 0 ) (58) as in (28), with q = 0. The strongest jamming signal against u 0 (t, θ,η 0 ) has the magnitude of and the same phase as u 0 . Hypothesis H 0 holds if there is only noise in the specified range-azimuth cell. Define the signal-to-noise ratio (SNR) at a single receive-array element as Without loss of generality, assume that the false-target jamming signals have the same strength as the true target echo. After FD code matching and spotlight beamforming, the signal-to-jamming-plus-noise ratio (SJNR) under the kth FD code is defined as where σ 2 n = MN σ 2 0 . The average SJNR over all the FD codes is computed as The spotlight response also results in higher SJNR, improving the detection performance. Previous FD methods [68], [71] located the false targets first and then synthesized a beampattern to suppress them in the spatial domain. Our approach suppresses the jamming signals by implementing a spotlight response with a distinctive FD code.
The binary test in (57) aims to decide if a target exists. Hypothesis H 0 holds if there is no target, while jamming and/or noise is present. Hypothesis H 1 holds if a target is present, possibly including jamming signals that carry matched FD code. Next, the boundary line between H 0 and H 1 in the u plane is determined by applying a log-likelihood ratio test (LLRT) as [78] ln where p(u|H 0 ) and p(u|H 1 ) are the probability density function (PDF) of u when target is absent and present, respectively, and ζ k is the threshold underη 0 =c k . With the FH scheme, the threshold ζ k depends on specific codec k in the FD codebook, although the LLRT takes the same form as in conventional radar detection [78]. Eqn. (63) can be reduced to Then, the false alarm rate is computed as where erf(x) is the error function [78]. The threshold is determined, under a given P fa , as The probability of detection is then computed as Fig.8(a) shows the normalized threshold, η k /(|u 0 | 2 /σ 2 n ), versus FD code index k, under false alarm rate of P fa = 10 −2 and 10 −4 , respectively, at SNR = 10 dB. The standard deviation of the normalized threshold is less than 15% of its average value, with either false alarm rate. Fig.8(b) shows the detection probability, P dk , at SNR = 0, −10, −15, −20 dB, respectively, under P fa = 10 −2 . Fig.9 shows the average detection probability over all FD codes, versus SNR, under P fa = 10 −2 , 10 −3 and 10 −4 , respectively. The detection probability of conventional radar without 62130 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  frequency-hopping (FH) scheme, under P fa = 10 −2 , is also shown for comparison. Without the FH scheme, the jamming signal may be mis-recognized as echo from a true target, namely, u jam = u t . Hence, the two PDFs p(u|H 1 ) and p(u|H 0 ) overlap with each other, making the detection probability equal to the false-alarm rate, regardless of SNR. Fig.10(a) shows the SJNR versus FD code index k under various SNRs. As the SNR increases, the average SJNR increases, compatible with the increase of detection probability shown in Fig.8(b), which implies higher SJNR leading to higher detection probability. Fig.10(b) shows the average SJNR versus SNR. At SNR = −5 dB, the SJNR achieved with the proposed approach is 15 dB, which is sufficiently high to make the detection probability close to 1, under P fa = 10 −2 , 10 −3 , 10 −4 , respectively, as shown in Fig.9.
The SJNR of conventional method saturates at 0 dB, as we assume the jamming signals can be comparable to the signal amplitude of the true target. On the other hand, the SJNR of the proposed approach is higher than 20 dB at high SNR. This level of improvement will be verified later in Fig.15(b) where the jamming signals at the center of the spotlight response are suppressed by more than 20 dB.  The false-alarm rate P fa in (67) is derived in one PRI, under the condition that the jamming signals and the target echo carry different FD codes. However, the FTG may happen to match the FD code of the transmit-array with probability p g . Hence, the false alarm rate and the detection probability are corrected as P ′ fa and P ′ d , respectively, with P ′ fa = P fa + p g P d (71) where the false-alarm rate induced by the jamming signal with mismatched FD code is P fa , and that induced by the jamming signal with matched FD code is p g P d . The probability of detecting a true target is modified by excluding the probability of detecting the jamming signal with matched FD code from P d . The probability p g depends on the design of FH sequence S and the strategy adopted by the FTG. If the FTG randomly selects an FD code among the K FD codes, we have p g = 1/K , which is typically much larger than P fa , leading to p g P d ≫ P fa in (71).
By implementing a frequency-hopping scheme, different FD codes are applied in different PRIs. It is possible that the FTG may accidentally match the FD code in some PRIs and induce strong jamming signal into the range-azimuth cell of interest. To increase the overall detection probability, binary hypothesis tests in (64) are conducted in L PRIs, with different FD codes orchestrated in different PRIs. A target is claimed if the H 1 hypothesis holds at least L 1 out of L tests. The overall detection probability is thus computed as Similarly, the overall false alarm rate is computed as Fig.11 shows the effect of L and L 1 on P dL and P faL , respectively. Fig.11(c) indicates that high value of P dL = 0.995 and fairly low value of P faL = 7×10 −4 can be achieved by choosing L = 8 and L 1 = 5. Fig.12 shows the values of P dL and P faL , with L 1 = L/2 + 1 and p g = 0.05, where p g is one half of its counterpart in Fig.11(c). It is observed that P dL = 0.998 and P faL = 2 × 10 −4 are achieved by choosing L = 6 and L 1 = 4, which implies higher detection probability and lower false alarm rate can be achieved within fewer PRIs if p g is smaller.
The range resolution r and the azimuth resolution θ are determined by the HPBWs of the response function in range VOLUME 11, 2023  and azimuth, respectively, which depend on the physical configurations of the transmit-array and the receive-array. It does not take full advantage of the information embedded in the received signals.
The two-dimensional MVDR spectrum exploits the covariance matrix of the received signals to enhance the estimation precision from ( r, θ) to (δr, δθ). A target is first detected within a range-azimuth cell of size r × θ, then the MVDR spectrum is computed to search for the fine target location, achieving precisions on the order of (δr, δθ). The computational cost is significantly reduced by carrying out fine search only in the range-azimuth cell of interest. Table 2 lists the parameters used in the simulations. The reference frequency and the waveform bandwidth are 10 GHz and 1 MHz, respectively [43], [44]. The maximum range is 150 km at the pulse repetition interval of 1 ms, without causing range ambiguity [80]. The waveform duration is T = 100µs [81]. The number of elements in both transmit and receive arrays is 17 [36], [40]. The number of FD codes is K = 10. Based on the parameters listed in Table 2, we have r = c/(2M t f ) = 30 m and θ = 0.88λ 0 /(Md t ) (rad)= 5.9 • . Fig.13 shows an optimal codebook of K = 10, designed with the method in Section III. The vertical lines in each FD code mark the chosen frequency offset indices. Table 3 lists the jamming suppression levels between different FD codes, where the value of −J (c k ,c k ′ ) in (31) is listed in the kth row and the k ′ th column. All the diagonal elements are equal to 0 dB and all the off-diagonal elements are larger than 30 dB, indicating that signals emitted under different FD codes will be suppressed by more than 30 dB with the proposed approach. Fig.14 shows the average normalized response in (51) and the average response to jamming signals in (52). The center of the spotlight is chosen as (r s , θ s ) = (50 km, 0 • ). Fig.14(a) shows a single peak at the center, with HPBW of 30 m in range and 6 • in azimuth. The first sidelobe is about −20 dB. Fig.14(b) shows the average response to jamming signals, of which the maximum is −21.7 dB. Fig.15 shows the range profiles of average normalized response in (51) and average response to jamming signals  in (52), respectively, at θ s = 0 • . The responses in [40] is also shown for comparison. Fig.15(a) shows that the proposed approach achieves the mainlobe width of 30 m, comparable to that in [40], while significantly reducing the sidelobe level from about −10 dB in [40] to lower than −20 dB. Fig.15(b) shows that the jamming response is significantly suppressed by more than 20 dB with the proposed approach. In comparison, the jamming response with conventional approach is indistinguishable from the target response shown in Fig.15(a). Fig.16 shows the two-dimensional MVDR spectrum in the target range-azimuth cell centered at θ s = 0 • and r s = 50 km, as well as two profiles of spectrum at θ s = 0 • and r s = 50 km, respectively. The target is pinpointed to a small area of δr × δθ, much smaller than r × θ. Fig.17 shows the ratios of δr/ r and δθ/ θ, respectively, versus SNR. Both ratios depend linearly on the inverse of SNR, consistent with (90) and (91). Note that the range resolution r and the azimuth resolution θ are determined by the length of antenna arrays and the FD code. Fig.17(a) shows that δθ is about 40 times smaller than θ, even at SNR = −20 dB. Similar improvement on precision was observed in the literature of FDA [53], [55]. In [55], the number of transmit-array elements is 10, leading to θ = 10 • . The simulation results in Fig.10 of [55] shows that (δθ) 2 = 0.5 at SNR = −20 dB, which implies δθ/ θ ≃ 0.07. In [53], the number of transmit-array elements is 8, leading to θ = 12.5 • . The simulation results in Fig.4 of [53] shows that (δθ) 2 = 0.7 at SNR = −20 dB, which implies δθ/ θ ≃ 0.07.

VI. SIMULATIONS AND DISCUSSIONS
Since the frequency diverse scheme only affects the range resolution, the azimuth properties of FD-MIMO array are the same as those of MIMO array of the same physical length. The CRBs on estimating the direction-of-arrival

A. HIGHLIGHT OF CONTRIBUTIONS
The main contributions of this work are summarized as follows.
1) We propose for the first time a novel FH-LFM-FD-MIMO radar system to counter false-target jamming.
The proposed system inherits the merits of frequency hopping (FH) scheme and FD-MIMO array to suppress jamming signals in both frequency and spatial domains.
The proposed system proves more effective than conventional countermeasures by simulations.
2) The frequency-diverse codebook for implementing the FH scheme is generated by combining a particle swarm optimization (PSO) algorithm and a rank-order-value (ROV) mapping, effectively suppressing jamming signals carrying different FD codes. 3) By adopting linear frequency modulation (LFM) waveform in an FD-MIMO array, spotlight response with low sidelobe level is achieved. 4) The spotlight-range beamforming vector is fine-tuned with a second PSO algorithm to further reduce the sidelobe level of spotlight response. 5) A detection scheme is proposed by applying frequencyhopping (FH) scheme to the FD codes in a sequence of PRIs to increase the overall detection probability, under constant false-alarm rate. Binary integra- tion is adopted to counter more severe jamming pattern. 6) A two-dimensional MVDR spectrum is applied on the target range-azimuth cell to further enhance the localization precision.

VII. CONCLUSION
A novel FH-LFM-FD-MIMO radar system is proposed to counter false-target jamming. The combination of frequency-hopping scheme and spotlight response of the LFM-FD-MIMO array can suppress jamming signals more effectively than conventional countermeasures. Simulations verify that the jamming signals at the center of the spotlight response are suppressed by more than 20 dB compared to conventional approaches. The detection probability is improved by raising the signal-to-noise-plus-jamming ratio after processing the received signals. A two-step localization scheme is proposed by first detecting a target in specific range-azimuth cell with the spotlight response of the proposed scheme, then applying a two-dimensional MVDR spectrum to pinpoint the target at a much finer resolution restricted by the SNR of the received signals at each receivearray element. His research interests include different ideas, theories, and methods on various electromagnetic phenomena and possible applications, including remote sensing, radar signal processing, antennas, phased arrays, propagation, scattering, and communications. For more information visit the link (http://cc.ee.ntu.edu.tw/∼jfkiang/selected_publications.html).