Waveform Design and Signal Processing Method of a Multifunctional Integrated System Based on a Frequency Diverse Array（in English）

1. Introduction

In the domain of radar technology, phased array radars have evolved to generate multiple dependent beams through various channels. This advancement addresses the drawbacks of slow scanning and low accuracy associated with traditional mechanically scanned radars. Consequently, these phased array radars enable the extraction of target angle, range, and velocity, enhancing the efficiency of target detection and information acquisition through rapid electronic scanning. They exhibit characteristics such as high precision, resolution, robust jamming countermeasures, multifunctionality, multitasking, high-reliability measurement, and real-time processing ability. These attributes find widespread applications in early warning and surveillance, detection and identification, searching, and tracking. However, a limitation arises as the transmit antenna beampattern is solely a function of angle, making it challenging to differentiate between two targets from the same direction. This constraint hampers the performance of the system in acquiring target and environmental information.

A recent innovative radar architecture termed the Frequency Diverse Array (FDA) is built upon phased array radars^[1], where a frequency increment $\Delta f$(much smaller than the transmit carrier frequency) is introduced across the transmit array elements, leading to an angle-range-time-dependent transmit beampattern^[2]. This augmentation expands the dimensions of signal identification and beamformer control for radar systems ^{1 )[3-8]}. However, the range and angle information are coupled in the transmit steering vector of the FDA, and the transmit beampattern is time-variant. To overcome this, the combination with the Multiple-Input-Multiple-Output (MIMO) technique facilitates the separation of transmit waveforms, providing additional Degrees of Freedom (DOFs)^[9]. Range information is derived from the time delay of the target echo, whereas the range information is encompassed in the transmit steering vector of the FDA-MIMO radar. This distinction enables the differentiation of overlapping targets in the transmit spatial domain, addressing the issue of range ambiguity^[8]. In light of these developments, the FDA-MIMO radar finds application in joint range-angle parameter estimation, target detection, range-ambiguous clutter suppression, mainlobe deceptive jammer suppression, unambiguous High-Resolution and Wide-Swath (HRWS) imaging, and integrated radar-communication. A visual representation of the dynamic timeline of FDA development is illustrated in Fig. 1, accompanied by a detailed summary of each application field.

Figure  1.  The dynamic timeline of FDA development

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(1) Joint range-angle parameter estimation: Compared with conventional phased arrays or MIMO radars, the transmit steering vector of the FDA-MIMO radar is range-angle-dependent. Consequently, employing suitable methods enables the achievement of range-angle-dependent 2D beamforming, allowing for the extraction of both range and angle information through beamforming. Algorithms for joint range-angle estimation have been extensively explored, leveraging techniques such as Maximum Likelihood (ML)^[9,10], MUltiple SIgnal Classification (MUSIC) and its improved methods^[11,12], and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) and its improved methods^[13,14]. These approaches facilitate the acquisition of the target angle, range, and Doppler of the target^[15]. Furthermore, studies have examined the estimation performance, considering metrics such as Cramér-Rao Bound (CRB), Root Mean Square Errors (RMSEs), and identifiability^[16]. Apart from parameter estimation methods based on a Uniform Linear Array (ULA), exploration extends to methods based on planar arrays^[17], conformal arrays^[18], and coprime arrays^[19]. In addition, a grid-less compressed sensing method for joint range and angle estimation is investigated in Ref. [20] by solving the 2D Atomic Norm Minimization problem. Moreover, a method based on sparse iterative optimization is utilized to obtain the angle, range, and amplitude of the target^[21].

(2) Target detection: The FDA-MIMO radar, with its enhanced signal processing dimension, enables the acquisition of clutter and target information, thereby improving target detection performance in nonhomogeneous environments, including cluttered surroundings. The exploration of target detection performance amid Gaussian white noise, with a known target range and interference-plus-noise covariance matrix, is initially presented in Ref. [22]. In the absence of a priori knowledge of the interference-plus-noise covariance matrix, Ref. [23] accomplishes the detection of true targets and suppression of false targets through a conic model. Additionally, a detection architecture is developed, leveraging different frequency increments for exploiting double pulses^[24]. Considering Gaussian colored noise, an adaptive detection method is formulated based on the Generalized Likelihood Ratio Test (GLRT) using optimization approaches^[25]. Moreover, the study delves into blind-Doppler target detection by utilizing the Doppler-Spreading effect in the mainlobe clutter environment^[26]. Ref. [27] employs the multi-pulse detector for target detection, considering the Swerling I and Swerling II models, whereas Ref. [28] introduces the Space-Range-Doppler Focus (SRDF) approach for moving target detection and classification.

(3) Range-ambiguous clutter suppression: The spatial-temporal-frequency joint processing procedure leverages the DOFs in the range-angle-Doppler 3D domains in the FDA-MIMO radar. In this context, simultaneous mitigation of clutter and deceptive jamming signals is achieved through subspace projection, utilizing space-time-range adaptive processing as outlined in Ref. [29]. Based on the Secondary Range Dependence Compensation (SRDC) principle, the range-ambiguous clutter can be separated and suppressed using FDA-Space-Time Adaptive Processing (STAP)^[30]. Range ambiguity is addressed in a planar FDA-MIMO radar in Ref. [31] through spatial frequency compensation and pre-STAP. In addition, an adaptive beamforming method is proposed to suppress clutter, compensating for range via auxiliary channels^[32]. In Ref. [33], discrimination of clutter ridges from different range regions in the 3D space is achieved by appropriately designing frequency increments in a bistatic FDA-MIMO STAP radar.

(4) Mainlobe deceptive jammer suppression: Leveraging the differences between true targets and mainlobe deceptive jammers (also known as false targets) in the range domain, an effective approach for suppressing mainlobe deceptive jammers is developed in the transmit side of the FDA-MIMO radar. Consequently, spatial processing methods have been extensively explored^[34-46], providing an overview in Ref. [40]. Existing suppression approaches are broadly categorized into three groups, encompassing data-dependent beamforming-based methods^[34-38], subspace projection-based methods^[43,44], and beampattern synthesis-based methods^[45,46]. In addition, a robust nonhomogeneous sample detection method is proposed in Ref. [41] to address the pseudo-random distribution of false targets. This method involves two steps: selecting nonhomogeneous samples and removing the target signal, followed by the suppression of false targets owing to range mismatch after data-dependent beamforming. In addressing array/system errors, a beampattern synthesis method is developed in Ref. [45], broadening the null regions and mainlobe of the 2D beampattern to suppress false targets by nulling in the beampattern. Furthermore, the Element-Pulse-Coding (EPC)-MIMO radar, based on the FDA-MIMO radar, is developed to suppress false targets^[46]. The authors’ group establishes the FDA experimental system, verifying the jammer suppression performance with real data^[8,40].

(5) Unambiguous High Resolution and Wide Swath (HRWS) imaging: Owing to the extra DOFs in the range domain after receive processing in the FDA-MIMO radar, echoes corresponding to different range ambiguity regions can be separated with an equivalent mainlobe “spatial flag”. This spatial flag can be utilized to solve the problem of range ambiguity in high-resolution and wide swath imaging. The application of FDA in Synthetic Aperture Radar (SAR) is initially investigated in Refs. [47,48]. Ref. [49] adopts non-uniformly increased frequencies, utilizing transmit-receive beamforming for imaging in FDA. In Ref. [50], HRWS imaging is achieved, resolving range ambiguity through the use of the Range Dependence Compensation (RDC) technique. To solve the problems of image co-registration and channel phase errors in a multichannel SAR-Ground Moving Target Indication (SAR-GMTI) system, Ref. [51] proposes an approach to estimate radial velocity in FDA-SAR, establishing a linear relationship between the interferometric phase among channels and the Doppler frequency. Furthermore, Ref. [52] employs a coding technique in both transmit channels and slow-time pulses. This technique allows for the extraction of desired echoes from the presumed range region while suppressing undesired echoes from other range ambiguous regions. In Ref. [53], echoes from the mainlobe corresponding to each range ambiguity region are independently extracted using a set of transmit filters, reducing the impact of residual range ambiguous echoes through azimuth compression. Consequently, this approach alleviates the contradiction between HRWS bands in SAR/GMTI systems.

(6) Integrated Radar-Communication: By encoding communication information into the transmit waveforms of FDA and selecting an appropriate frequency increment, the functions of communication information decoding and radar signal processing can be simultaneously achieved after dual-channel processing in the receiver^[3]. In Ref. [54], the communication signal is projected along the null radiation direction of the mainbeam of the radar system, and a Butler matrix is utilized to address cross-interference between the communication signal and radar target detection. In Ref. [55], a secondary communication function is developed, embedding communication information into the frequency increments of FDA-MIMO. Additionally, the bit error rate is analyzed to evaluate the probability of error. Linearly increased frequencies are employed in FDA for range-angle-dependent Quadrature Spatial Modulation (QSM) wireless communications in Ref. [56], where the in-phase and quadrature components of the transmit vector are transmitted using different frequencies. Furthermore, green secure communication is achieved in mmWave wireless communication with range-angle focusing QSM frequency-modulated diverse retrospective arrays. This approach shares the advantage of automatic self-tracking to alleviate the Channel State Information requirements of the transmitter/receiver^[57]. In Ref. [58], the frequency diverse array concept is applied to construct an Orthogonal Frequency-Division Multiplexing (OFDM) transmitter, enhancing the secrecy performance of information transmission compared with the conventional OFDM system.

Furthermore, FDA is utilized for target tracking^[59-62], Low Probability of Intercept (LPI)^[63,64], and other applications. The advantages and methods of FDA-MIMO radar in various applications are summarized in Tab. 1, and their relationships are illustrated in Fig. 2. However, current FDA studies focus on designing system parameters for specific functions, lacking flexibility in terms of different parameters and functions that operate independently. The functions of radar systems span a wide range of applications, including early warning detection, electromagnetic countermeasures, imaging, identification, and situational awareness. In complex electromagnetic environments, satisfying the multifunctional processing requirement using a single system with independent functions or simple stacking to multiple systems becomes challenging, necessitating a multifunctional integrated radar system. In recent years, techniques such as the AMDR (AN/SPY-6(V)) have been employed. This multifunctional carrier-borne active phased array radar caters to air defense and missile defense, comprising two radars—one utilizing the S-band and the other the X-band for horizon search, accurate tracking, and missile guidance. Additionally, DARPA conducted the “Concerto” project, developing a multifunctional radio frequency system that adapts and flexibly switches between electronic warfare, communication, and radar modes, exhibiting superior overall performance compared to independent systems. Consequently, multifunctional integrated radar technology has emerged as a crucial direction for radar application development, and signal processing methods compatible with multiple tasks under the same system should be explored urgently.

Table  1.  Advantages and signal processing methods of multifunctional FDA-MIMO

Functions of radars	Problems	Advantages of FDA	Methods
Parameter estimation	Range ambiguity reduced by high pulse repetition frequency	Estimating joint range, angle, and range ambiguity number	Subspace-based methods (MUSIC and ESPRIT), ML, monopulse-based methods
Clutter suppression		Discriminating echoes corresponding to different range ambiguity regions	Secondary range compensation, STAP, space-time-range adaptive processing
HRWS-SAR imaging		Discriminating echoes corresponding to different range ambiguity regions	Range compensation, transmit channel and slow-time processing
Target detection	Insufficient samples, nonhomogeneous environment	Improving the detection performance in a nonhomogeneous environment	Design of adaptive detectors based on GLRT
Jammer suppression	Mainlobe deceptive jammers	Nulling the jammers in the joint transmit-receive spatial	Data-dependent beamforming, space projection, beampattern synthesis-based methods

 | Show Table

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Figure  2.  Relationships among different FDA-MIMO radar applications

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Based on the comprehensive review of FDA applications, this study delves into the multifunctional integrated waveform design and signal processing methods of FDA radar. The investigation encompasses FDA waveform design methods, highlighting the advantages of FDA in various tasks such as integrated parameter estimation and adaptive detection, integrated range ambiguity resolution and jammer suppression, and integrated imaging and moving target detection. Technical challenges are identified, and future development trends are outlined, offering valuable insights for the advancement and application of multifunctional integrated radar systems.

The paper is structured as follows: Section 2 provides the signal model and waveform design of FDA, analyzing waveform design methods, LPI properties, and receive processing methods. Sections 3, 4, and 5 explore integrated multidimensional parameter estimation and adaptive detection, integrated range ambiguity resolution and jammer suppression, and integrated imaging and moving target detection, respectively. Finally, Section 6 presents conclusions and discusses potential developments.

Notations: The conjugate, transpose, inverse, and conjugate-transpose operators are denoted by the symbols ${\left( \cdot \right)^{\text{T}}}$, ${\left( \cdot \right)^{\text{*}}}$, and ${\left( \cdot \right)^{\text{H}}}$ respectively. ${\left( \cdot \right)^{{{ - 1}}}}$ indicates the operation of inverse. $\odot$and $\otimes$ represent the Hadamard and Kronecker products, respectively. $\max \left( \cdot \right)$ and $\min \left( \cdot \right)$ indicate the operation of taking the maximum and minimum values, respectively. ${{\boldsymbol{I}}_M}$, ${{{{\textit{0}}}}_M}$, and ${{{{\textit{0}}}}_M}$ respectively denote the $M \times M$-dimensional identity matrix, the $M \times 1$-dimensional vector with all zero entries, and the $M \times 1$-dimensional vector with all elements being one. ${\mathbb{C}^M}$, ${\mathbb{R}^M}$, ${\mathbb{C}^{M \times M}}$, and ${\mathbb{H}^{M \times M}}$ are, respectively, the sets of $M \times 1$-dimensional vectors of complex numbers, $M \times 1$-dimensional vectors of real numbers, $M \times M$-dimensional complex matrices, and $M \times M$-dimensional Hermit matrices. The symbol $\left\| \cdot \right\|$ indicates the Euclidean norm. $\Re \left( \cdot \right)$ indicates the real part of a complex number. $\left[ {a,b} \right]$ indicates a closed interval with the starting point a and ending point b. $\mathcal{O}\left( \cdot \right)$ represents the computational complexity.

2. Signal Model and Waveform Design for FDA

2.1 Transmit waveform design

Consider a ULA consisting of Mantenna elements, the transmit signal of the m-th $\left( m = 1,2, \cdots ,M \right)$ element within the pulse duration is expressed as^[4]

where ${\text{rect}}\left( {\dfrac{t}{{{T_{\mathrm{p}}}}}} \right) = \left\{ \begin{gathered} 1,\left| t \right| \le \frac{{{T_{\mathrm{p}}}}}{2} \\ 0,\left| t \right| > \frac{{{T_{\mathrm{p}}}}}{2} \\ \end{gathered} \right.$ denotes the pulse function, ${T_{\mathrm{p}}}$ is the pulse duration, ${f_0}$ is the carrier frequency, E indicates the total transmit energy, and ${\varphi _m}\left( t \right)$ represents the baseband waveform of the m-th element. The baseband waveform can be characterized by two scenarios:

(1) In coherent FDA, the identical baseband waveform is transmitted by all elements^[4], It is assumed that the Linear Frequency Modulation (LFM) waveform is transmitted, i.e., ${\varphi _m}\left( t \right) = {\varphi _{{\text{LFM}}}}\left( t \right) = {{\mathrm{e}}^{{\mathrm{j}}\pi K{t^2}}}$, where K denotes the chirp rate, namely, $K = {B}/{{{T_{\mathrm{p}}}}}$, with B being the bandwidth.

(2) In orthogonal FDA (i.e., FDA-MIMO) radar, the orthogonal waveforms are transmitted for different elements^[4], satisfying the relationship of $\displaystyle\int_o^{{T_{\mathrm{p}}}} {{\varphi _m}\left( t \right)\varphi _l^*\left( {t - \tau } \right){{\mathrm{e}}^{{\mathrm{j}}2\pi \Delta f\left( {m - l} \right)t}}{\mathrm{d}}t} = 0, l \ne m,\forall \tau$, with $\tau$ being an arbitrary time delay.

To measure the performance of waveform design, the multidimensional ambiguity functions of FDA concerning angle, distance, and Doppler dimensions are defined as^[65]

where ${\theta _0}$ denotes the angle of the target, $\theta$ denotes the angle of receiving, and $x\left( {\theta ,t} \right)$ is the transmission signal at angles $\theta$ and time t, i.e., $x\left( {\theta ,t} \right) = \displaystyle\sum\nolimits_{m = 1}^M {{{\mathrm{e}}^{{\mathrm{j}}2\pi {\textstyle\frac{d}{{{\lambda _0}}}}\left( {m - 1} \right)\sin \left( \theta \right)}}{s_m}\left( t \right)}$. To proceed, $\chi \left( {\tau ,\theta ,{\theta _0},{f_d}} \right)$ can be further expressed as

where d represents inter-element spacing (half the wavelength), and ${\lambda _0} = {{\mathrm{c}} \mathord{\left/ {\vphantom {c {{f_0}}}} \right. } {{f_0}}}$ represents the wavelength. The ambiguity function is a multidimensional function related to range, angle, and Doppler. When the Doppler is zero, Fig. 3 provides the angle-angle profile and angle-range profile of the FDA multidimensional ambiguity function, respectively. As depicted in Fig. 3(a), there is a high gain on the diagonal between the target angle and the receiving angle, ensuring uniform coverage throughout the entire spatial domain. Furthermore, Fig. 3(b) illustrates high received energy nearby, with low sidelobe characteristics in the range dimension. However, its mainlobe in the range dimension is large, resulting in poor resolution. Notably, the cumulative bandwidth corresponding to the frequency-time modulation waveform decreases, leading to a deterioration of the range resolution of coherent FDA radar. In response to this challenge, Ref. [66] proposes a waveform design method for coherent FDA radar, i.e., “tangent frequency modulation + spatial encoding”. The signal expression of the transmitted signal at $\theta$ is

Figure  3.  Multi-dimensional ambiguity function of FDA

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where ${\boldsymbol{a}} \left( \theta \right) = {\left[ {1, \cdots ,{{\mathrm{e}}^{{\mathrm{j}}2\pi \left( {m - 1} \right) {\textstyle\frac{{d\sin \left( \theta \right)}}{{{\lambda _0}}}}}}, \cdots ,{{\mathrm{e}}^{{\mathrm{j}}2\pi \left( {M - 1} \right) {\textstyle\frac{{d\sin \left( \theta \right)}}{{{\lambda _0}}}}}}} \right]^{\text{T}}} \in {\mathbb{C}^M}$represents the transmit steering vector, ${\boldsymbol{b}} = {\left[ {{b_1},{b_2}, \cdots ,{b_m}, \cdots ,{b_M}} \right]^{\text{T}}} \in {\mathbb{C}^M}$ represents the encoded phase vector, ${b_m}$ denotes the encoded phase of the $m{\text{ - th}}$ transmit element, and ${\boldsymbol{s}}\left( t \right) = {\left[ {{s_1}\left( t \right),{s_2}\left( t \right), \cdots ,{s_m}\left( t \right), \cdots ,{s_M}\left( t \right)} \right]^{\text{T}}} \in {\mathbb{C}^M}$ is the transmitted waveform vector, where the expression of ${s_m}\left( t \right)$ has been given in Eq. (1). Notice that $\sqrt { {E}/{M}}$ is omitted in the equation. In Ref. [66], the introduction of non-LFM waveform and spatial phase coding effectively reduces range sidelobes and enhances range resolution, respectively.

In fact, the DOFs of the FDA-MIMO in the range domain depend on the separation of the transmitted waveforms at the receiver. The orthogonality of the waveforms between the transmit channels significantly influences the performance of FDA-MIMO in applications related to target detection and estimation, resolving ambiguity, interference/clutter suppression, and imaging^[8]. Although much of the literature related to FDA-MIMO assumes that the transmitted waveforms are completely orthogonal, in practical scenarios, achieving perfect orthogonality in temporal and Doppler domains is challenging. The performance of FDA-MIMO is consequently impacted by the degree of orthogonality. To address this challenge, the traditional MIMO orthogonal waveform designing method can be used to design orthogonal waveforms under constraints such as constant modulus^[67], peak-to-average ratio (PAR)^[68], and similarity^[69]. To improve the performance of FDA-MIMO radar applications, including detection, countermeasures, and imaging, common criteria for orthogonal waveform design include minimizing the weighted integral sidelobe level^[70,71], maximizing the Signal to Interference plus Noise Ratio (SINR)^[72,73], and minimizing the Mean Square Error criterion. Minimizing the weighted integral sidelobe level is particularly conducive to reducing residuals after waveform separation and improving the performance of FDA-MIMO radar in resolving ambiguity. For the discrete waveform set $\left\{ {{{\boldsymbol{s}}_m}\left( p \right)} \right\}_{m = 1,p = 1}^{M,P}$, the weighted integral sidelobe level ${{\text{P}}_{{\text{WISL}}}}$can be expressed as^[74]

where the first part of the equation is the sidelobe of the autocorrelation function, the second part denotes the cross-correlation function, P denotes the number of polyphase codes, which refers to the sub-pulse of each pulse, ${{\boldsymbol{J}}}_{k} = \left[\begin{array}{l}\stackrel{k个\text{0}}{\overbrace{0\cdots 0 }}1 \;\;{{\bf{0}}}\\ \qquad\quad\;\;\ddots \\ \;\; \cdots\quad\quad 1\\ \quad\quad {\bf{0}}\end{array}\right] \in {\mathbb{C}}^{P\times P}$ represents the shift matrix, and $\lambda \in \left( {0,1} \right)$ is the weighting factor.

For MIMO orthogonal waveforms, orthogonal signals can be achieved not only through code division methods but also through frequency-division orthogonal methods^[75,76], incorporating transmit power range-angle coupling^[77]. However, a key distinction from FDA-MIMO lies in the transmission of the same baseband waveforms, such as linear and non-LFM signals. In general, the frequency spacing $\Delta f$ between adjacent channels is maintained greater than or equal to the bandwidth to ensure the orthogonality of baseband waveforms.

2.2 Low probability of interception property and waveform design

The transmit beampattern of coherent FDA exhibits the angle-range-time-dependent property^[4], and the expression for the transmit beampattern of coherent FDA in a pulse radar system is given by^[4,8]

where ${R_0}$ denotes range.

In Fig. 4(a), the transmit beampattern of FDA displays an S-shaped property, distributing energy in all directions. At any point in the far field, it periodically encounters the mainlobe and sidelobe of the beampattern over time^[4]. Compared with traditional phased array radars, the energy of FDA at the target area can be reduced by employing wide transmitting antenna beams. As depicted in Fig. 4(a)–Fig. 4(c), the spatial coverage is determined by ${T_{\mathrm{p}}}$ and $\Delta f$. For a fixed $\Delta f$, the full spatial coverage is achieved when ${T_{\mathrm{p}}} = {1}/{{\Delta f}}$, and only limited spatial coverage can be achieved when ${T_{\mathrm{p}}} < {1}/{{\Delta f}}$. Moreover, the range domain profile is illustrated in Fig. 4(d), and for the same pulse duration, the width of the mainlobe in the range domain is determined by $\Delta f$, namely, ${{2{\mathrm{c}}}}/({{M\Delta f}})$^[8].

Figure  4.  Analysis on the spatial coverage of the transmit beampattern for coherent FDA

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For the FDA-MIMO radar, owing to the transmitted orthogonal waveforms, its beampattern aligns with traditional MIMO radar, lacking directionality in its transmit beampattern, and uniform energy is obtained in the spatial and temporal domain. Furthermore, the S-shaped distribution property cannot be achieved^[8]. The design of the receiving matched filter for FDA-MIMO radar will be detailed in the next section.

For the LPI waveform of FDA-MIMO radar, under the constraint of constant energy on each antenna, combined with the constraint of the output Signal-to-Clutter-plus-Noise Ratio (SCNR), the LPI waveform is designed by minimizing the energy of the target area (range-azimuth). The optimization problem is formulated as^[63]

where ${\boldsymbol{W}} \in {\mathbb{C}^{M \times K}}$ represents the array of transmit beams, K denotes the number of orthogonal waveforms, $\lambda$ denotes the SCNR threshold of the target, x is the receiving filter, and $P({\boldsymbol{W}})$ is the energy of the transmitting signals at the target area. The first constraint ensures that the output SCNR surpasses a predetermined threshold. The second constraint dictates that the energy on each antenna remains constant. Moreover, an iterative algorithm is employed to achieve this optimization. Initially, the variable W is randomly initialized, and x is determined using the generalized Rayleigh quotient. Subsequently, W is computed based on the auxiliary variable and alternating direction multiplier method until convergence is reached. This iterative process yields the optimized LPI waveform, ensuring adherence to both constraints.

2.3 Received signal processing

For the non-adaptive beampattern of coherent FDA radar, achieving matched filtering at any point in the spatial domain requires devising a matched filter for all directions. The matched filter is determined by range and time and can be expressed as^[78]

The matched filtering function of the coherent FDA radar system depends on the baseband waveform and the transmit beampattern. Matched filtering in the fast-time domain and transmit beamforming can be simultaneously achieved. Fig. 5 illustrates two theoretical equivalent receiving structures for coherent FDA radar, where signal processing can be performed in the transmit beam domain since the transmit beampattern can be equivalently synthesized at the receiver^[78]. This involves constructing M orthogonal matched filtering functions in the transmit beam domain, with parameter ${\theta _m} = {\sin ^{ - 1}}\left( {({{2\left( {i - 1} \right)}}/{M}) - 1} \right)$. In the structure shown in Fig. 5(a), the corresponding echo vector in the transmit beam domain is only related to the receive steering vector. In the structure depicted in Fig. 5(b), the corresponding echo vector in the transmit beam domain is only related to the receive steering vector. In the structure depicted in Fig. 5(b), the corresponding echo vector in the transmit beam domain is determined by the virtual transmit-receive steering vector. Furthermore, a signal processor based on the Minimum Variance Distortionless Response (MVDR) criterion can be devised^[78].

Figure  5.  Basic structures of coherent FDA receiver

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In practical interference and clutter suppression, there are performance differences between the two processing structures owing to variations in the interference-plus-noise ratio (INR)/CNR. When the angle of the angle-time-dependent matched filtering is the same as the angle of the receiving beamforming, the maximum matched output in the target direction can be obtained^[78].

For FDA-MIMO radar, a matched filter is designed to eliminate the time-variation factor ${{\mathrm{e}}^{{\mathrm{j}}2\pi \Delta f\left( {m - 1} \right)t}}$ after mixing. Fig. 6 illustrates two matched filters corresponding to the $n{\text{-th}}$ receiving antenna, with the processing procedure being consistent for each receiving antenna. The first processing structure involves mixing the signal of each channel with $\Delta f$ and subsequently matching it with the transmitted waveform^[40]. Conversely, in the second structure, the matched filter, determined by $\Delta f$, is directly designed for processing^[25].

Figure  6.  Processing procedurse of receive matched filtering in FDA-MIMO radar

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In particular, the signal vector determined by the first structure can be expressed as^[40]

where ${\beta _1}$ denotes the complex amplitude of the target, ${\boldsymbol{r}} \in {\mathbb{C}^{MN \times 1}}$ is the output vector of matched filtering, and ${\boldsymbol{a}}\left( {{R_0},{\theta _0}} \right) \in {\mathbb{C}^{M \times 1}}$ and ${\boldsymbol{b}}\left( {{\theta _0}} \right) \in {\mathbb{C}^{N \times 1}}$ represent the transmit and receive steering vectors, respectively, which have the forms of

The output vector obtained by the second structure can be expressed as^[25]

where ${\boldsymbol{b}}\left( {{\theta _0}} \right)$ and ${\boldsymbol{d}}\left( {{\theta _0}} \right)$ denote the receive and transmit steering vector, respectively, ${\boldsymbol{R}} \in {\mathbb{C}^{M \times M}}$ represents the output array of matched filtering, and ${{\Delta }}\tau$ is the difference between the actual delay ${\tau _0}$ of the target within each range bin and the sampling ${t^\star }$^[25], i.e., ${{\Delta }}\tau = {t^\star } - {\tau _0}$, ${\boldsymbol{a}}(\Delta \tau )$ is the component of transmit steering vector related to range.

The range of the target is obtained after processing on the receiver. The two receiving structures are equivalent in the presence of Gaussian white noise.

3. Integrated Target Detection and Parameter Estimation

3.1 Model of integrated detection and estimation

Typically, target detection and parameter estimation are distinct processes. After confirming the presence of the target in the mainlobe through spatial processing with a single pulse, parameter estimation methods are employed to determine its location. Although spatial processing using a single pulse can confirm the existence of the target and provide angle information, existing approaches fall short of simultaneously obtaining range and angle information through spatial processing^[79]. In contrast, FDA-MIMO radar excels in obtaining both range and angle information simultaneously, allowing for multidimensional parameter estimation and target detection. Addressing the challenge of acquiring unknown parameters while detecting targets necessitates constructing a joint framework for target detection and parameter estimation in clutter and non-uniform color environments.

For adaptive target detection based on FDA-MIMO radar in a Gaussian background (clutter or white Gaussian noise), consider a set of training samples ${{\boldsymbol{z}}_k}(k = 1,2,\cdots,K )$, where $K \ge MN$ denotes the number of samples, and ${\boldsymbol{z}} \in {{\mathbb{C}}^{MN}}$ represents the echo vector from the unit to be detected after matching filtering. The detection problem can be formulated as a binary hypothesis testing problem^[25]:

where ${{\mathrm{H}}_{\text{0}}}$ denotes the assumption that the target does not exist, ${{\mathrm{H}}_1}$ represents that the target exists in the unit to be detected, n and ${{\boldsymbol{n}}_k}\text{～} \mathcal{C}\mathcal{N}(0,{\boldsymbol{Q}})$ indicate the noise component, ${\boldsymbol{Q}} \in {\mathbb{H}^{MN \times MN}}$ denotes the interference-plus-noise covariance matrix or the clutter covariance matrix, ${\boldsymbol{s}}\left( {{\theta _0},\Delta \tau } \right) = {\boldsymbol{b}}\left( {{\theta _0}} \right) \otimes \left\{ {\left[ {{{\boldsymbol{R}}^{\text{T}}}{\boldsymbol{d}}\left( {{\theta _0}} \right)} \right] \odot {\boldsymbol{a}}({{\Delta }}\tau )} \right\} \in {\mathbb{C}^{MN}}$ represents the joint transceiver steering vector after matching filtering, and K indicates the number of samples.

Applying the Neyman-Pearson criterion, the Likelihood Ratio Test of the binary hypothesis testing problem is the optimal detector. In practice, as ${\beta _1}$, ${\theta _0}$, $\Delta \tau$, and Q are unknown, detectors can be designed based on the GLRT criterion as

where $u = \sin {{\theta _0}}$ and $\delta = 2\Delta f\Delta \tau$. $\mathcal{A}$ and $\mathcal{B}$ denote the uncertain sets where u and are located, respectively, namely, $\left[ { - 1,1} \right]$ and $\left[ { - {{\Delta f} \mathord{\left/ {\vphantom {{\Delta f} B}} \right. } B},{{\Delta f} \mathord{\left/ {\vphantom {{\Delta f} B}} \right. } B}} \right]$. $g\left( {{\boldsymbol{z}},{{\boldsymbol{z}}_1}, \cdots ,{{\boldsymbol{z}}_K}\mid {\beta _1},u,\delta ,{\boldsymbol{Q}};{{\mathrm{H}}_1}} \right)$ and $g\left( {\boldsymbol{z}},\;{{\boldsymbol{z}}_1},\; \cdots ,\;{{\boldsymbol{z}}_K} \mid {\boldsymbol{Q}};{{\mathrm{H}}_0} \right)$ represent the Probability Density Function (PDF) under the assumption of ${{\mathrm{H}}_{\text{1}}}$ and ${{\mathrm{H}}_{\text{0}}}$, respectively. After conducting calculations, the GLRT detector can be expressed as^[25]

where $g\left( {u,\delta } \right) = \dfrac{{{{\left| {{{\boldsymbol{s}}^{\text{H}}}\left( {u,\delta } \right){{\boldsymbol{S}}^{ - 1}}{\boldsymbol{z}}} \right|}^2}}}{{{{\boldsymbol{s}}^{\text{H}}}\left( {u,\delta } \right){{\boldsymbol{S}}^{ - 1}}{\boldsymbol{s}}\left( {u,\delta } \right)}}$ and ${\boldsymbol{S}} = \displaystyle\sum\nolimits_{k = 1}^K {{{\boldsymbol{z}}_k}{\boldsymbol{z}}_k^{\text{H}}}$. Additionally, assuming that Q is known, a two-step GLRT detector, also known as Adaptive Matched Filter (AMF), can be constructed as

where $g\left( {{\boldsymbol{z}}\mid {\beta _1},u,\delta ,{\boldsymbol{Q}};{{\mathrm{H}}_1}} \right)$ and $g\left( {{\boldsymbol{z}}\mid {\boldsymbol{Q}};{{\mathrm{H}}_0}} \right)$represent the PDF under the assumptions of ${{\mathrm{H}}_{\text{1}}}$ and ${{\mathrm{H}}_{\text{0}}}$, respectively. After assuming that the covariance Q is known, Q can be replaced with the sample covariance matrix S. The final expression of the AMF detector is

In practice, based on the expression of $g\left( {u,\delta } \right)$, ${{{\varLambda }}_{{\text{AMF}}}}$ can be rewritten as

where ${{\boldsymbol{w}}_0}\left( {u,\delta } \right) = {\left[ {{{\boldsymbol{s}}^{\text{H}}}\left( {u,\delta } \right){{\boldsymbol{S}}^{ - 1}}{\boldsymbol{s}}\left( {u,\delta } \right)} \right]^{ - \frac{1}{2}}}{{\boldsymbol{S}}^{ - 1}}{\boldsymbol{s}}\left( {u,\delta } \right) \in {\mathbb{C}^{MN}}$ denotes the weight vector. The above equation is equivalent to estimating the target angle and distance under the ML criterion.

3.2 Approximate optimization method

In Ref. [9], the estimation of range and angle for problem (16) is accomplished through a 2D search method. However, this method, although accurate, involves the use of discrete intervals in both angle and range dimensions, resulting in high computational complexity. To address this, approximate optimization methods^[10,25] can be employed to obtain angle and range estimates. Corresponding adaptive detectors can then be designed using the following techniques:

(1) Coordinate Descent (CD) method: This method transforms the 2D search problem into two 1D subproblems. It fixes one of the two variables, either angle or range, and searches within the value region of the other unknown variable. After obtaining the result, it fixes the other variable and repeats until convergence. The discrete forms of $\mathcal{A}$ and $\mathcal{B}$ corresponding to the search intervals of angle and range are respectively expressed as^[10,25]

where ${N_u}$ and ${N_\delta }$ represent the discrete number of points in the range and angle dimensions, respectively. The estimated values $( {{{\hat u}_{{\text{CD}}}},{{\hat \delta }_{{\text{CD}}}}} )$ of angle and range can be obtained through iterations. The GLRT and AMF detectors can be expressed as

(2) Linearized Array Manifold (LAM) method: This method unfolds the true steering vector of the target, i.e., ${{\boldsymbol{s}}_a}\left( {\Delta {\boldsymbol{\theta }}} \right) = {\boldsymbol{s}}\left( {{u_0},{\delta _0}} \right) + {{\boldsymbol{s}}_u}\Delta u + {{\boldsymbol{s}}_\delta }\Delta \delta$, where ${{\boldsymbol{s}}_u}$ and ${{\boldsymbol{s}}_\delta }$ denote the derivative of ${\boldsymbol{s}}\left( {u,\delta } \right)$ at ${u_0}$ and ${\delta _0}$, and $\Delta {\boldsymbol{\theta }} = {\left[ {\Delta u,\Delta \delta } \right]^{\text{T}}}$ represents the deviation vector between the angle and range. Eq. (16) can then be equivalent to

where $\mathcal{C}=\left[-1,1\right]\times \left[-\Delta f/B,\Delta f/B\right]$. Furthermore, ${\beta _1}$ and $\Delta {\boldsymbol{\theta }}$ can be jointly optimized through iteration, assuming that the result obtained in step h regarding ${\beta _1}$ is

After substituting ${\beta _1}^{\left( h \right)}$ into Eq. (20), the optimization function w.r.t. $\Delta {\boldsymbol{\theta }}$ can be constructed as

where ${\boldsymbol{\tilde r}} = {{\boldsymbol{S}}^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}\left( {{\boldsymbol{r}} - {\beta _1}^{\left( h \right)}{\boldsymbol{s}}\left( {{u_0},{\delta _0}} \right)} \right)$ and ${\boldsymbol{\hat s}} = {{\boldsymbol{S}}^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}} \cdot\left[ {{\beta _1}^{\left( h \right)}{{\boldsymbol{s}}_u},{\beta _1}^{\left( h \right)}{{\boldsymbol{s}}_\delta }} \right]$. The deviation vector $\Delta {{\boldsymbol{\theta }}^*} = {\left[ {\Delta {u^*},\Delta {\delta ^*}} \right]^{\text{T}}}$ and the angle and range of the target can be obtained by iterating, i.e., $\left( {{\hat u}_{{\text{LAM}}}}, {{\hat \delta }_{{\text{LAM}}}} \right) = \left( {{u_0} + \Delta {u^*},{\delta _0} + \Delta {\delta ^*}} \right)$. Afterward, ${{{\varLambda }}_{{\text{GLRT-LAM}}}}$ and ${{{\varLambda }}_{{\text{AMF-LAM}}}}$ can be obtained.

The computational complexities of the 2D ML, CD, and LAM are $\mathcal{O}\left( {{N_u}\left( {{M^2}{N^2} + {N_\delta }{M^2}} \right)} \right)$, $\mathcal{O} \left( {{N_u}{M^2}{N^2} + {N_{{\mathrm{it}},{\mathrm{CD}}}}\left( {{N_u} + {N_\delta }} \right){M^2}} \right)$, and $\mathcal{O} \left( {K{M^2}{N^2}} \right)$, respectively, where ${N_{{\mathrm{it}},{\mathrm{CD}}}}$ represents the number of cycles for the CD algorithm, which satisfies ${N_u}{N_\delta } > {N_{{\mathrm{it}},{\mathrm{CD}}}}\left( {{N_u} + {N_\delta }} \right)$. The proposed optimization method significantly reduces the computational complexity resulting from a 2D search.

Fig. 7 shows the results of integrated range-angle multidimensional parameter estimation and adaptive target detection based on FDA-MIMO radar under a Gaussian noise background. The simulation parameters include $M = N = 4$, and the false alarm rate is ${10^{ - 4}}$. Fig. 7(a) and Fig. 7(b) present the RMSE results for angle and range estimation, respectively. It is observed that as the input SINR increases, the RMSE of the estimated results of the proposed CD and LAM methods decreases. Additionally, when the SINR is greater than around 12 dB, the RMSE approaches their respective CRBs. Fig. 7(c) illustrates the result of adaptive detection. It is observed that in the presence of mismatch errors, the AMF outperforms the GLRT, namely, a higher probability of detection is achieved with an identical input Signal-to-Noise Ratio (SNR). In addition, the detection performance of the sub-optimal methods approximates that of the benchmark detectors, and much better than the mismatched detectors.

Figure  7.  Integrated detection and estimation results in FDA-MIMO radar

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4. Integrated Ambiguity Resolution and Jammer Suppression

4.1 Principle of range ambiguity resolution with FDA-MIMO radar

In an FDA-MIMO radar system, the transmit spatial frequency of the target can be obtained from its transmit steering vector as

Assuming that the target has range ambiguity, its range can be expressed as

where ${r_0}$ denotes the principal range in a range non-ambiguity period, p is the index of range ambiguous, ${R_{\mathrm{u}}}$ is the maximum non-ambiguous range, and ${N_{\mathrm{a}}}$ denotes the maximum number of ambiguous ranges. Subsequently, the transmit spatial frequency after compensating for the principal range is written as

where T denotes the pulse repetition period. Note that the transmit spatial frequency includes the index of range ambiguity p. Furthermore, the corresponding equivalent transmit beampattern is given by

where $Z\left( {p,\theta } \right) = p\Delta fT + {d}/{{{\lambda _0}}}\left( {\sin \left( \theta \right) - \sin \left( {{\theta _0}} \right)} \right)$. Compared with the beampattern of the conventional radar framework, for different numbers of delayed pulses p, the mainlobe direction of the original beampattern shifts due to the phase term related with $p\Delta fT$ in $P_{{\text{ET}}}^{}(p,\theta )$. As shown in Fig. 8, the equivalent beampatterns have different directions as p changes, meaning that an equivalent spatial “flag” is introduced between the transmit pulses of FDA-MIMO radar. Therefore, different transmit pulses can be distinguished according to different equivalent mainlobe directions, i.e., discriminating the echoes corresponding to different range ambiguity regions. Moreover, multiple pulses are transmitted to different spatial angles, i.e., achieving full spatial coverage by utilizing the slow-time resources.

Figure  8.  Principle of resolving the range ambiguity in FDA-MIMO radar

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In fact, extending the $m{\text{-th}}$ transmit signal of FDA-MIMO radar to the $k{\text{-th}}$ pulse yields

where ${\phi _m}(t)$ denotes the orthogonal waveform transmitted by $m{\text{-th}}$ element. A phase difference ${{\mathrm{e}}^{{\text{j}}2\pi \left( {m - 1} \right)\Delta f\left( {k - 1} \right)T}}$ is observed between different transmit pulses (i.e., slow time), and this term changes with transmit channel m and pulse k. Actually, if set $\gamma \triangleq \Delta fT$, the EPC-MIMO radar^[80] is realized. Open literature implies that the EPC-MIMO radar has an equivalent capability in range ambiguity resolution as the FDA-MIMO radar^[8]. However, both the FDA-MIMO and EPC-MIMO radars have limitations in suppressing false targets located in the same transmit pulse as the true target is emphasized.

4.2 Mainlobe jammer suppression based on resolving range ambiguity

In this subsection, self-defense jammers are taken into consideration, where the angles of the jammers are equal to that of the true target. The transmitted waveforms are intercepted by the jammers, generating several false targets after modulation (i.e., the entire pulse is transmitted). For range pull-off jamming, the tracking guidance radar utilizes the technique of tracking the pulse front to extract information in the range domain of the true target. Furthermore, the transmit spatial frequencies of the true and false targets after RDC are expressed as^[39]

As shown in Eq. (29), after compensation, the transmit spatial frequency of the true target is the same as the receive spatial frequency, i.e., the true target is located on the diagonal line of the 2D transmit-receive spatial domain. However, the false targets with p pulses delayed (i.e., the range ambiguity index is p) are randomly distributed in the transmit spatial domain with the offset $\Delta {f_{\text{T}}} = - ({{2\Delta fp{R_{\mathrm{u}}}}})/{{\mathrm{c}}}$. As depicted in Fig. 9, the actual equivalent range of the false target must be larger than the actual range of the true target^[40], although the ranges of the true and false targets may have the relation of lead and lag in a non-ambiguous region (i.e., the range ambiguous region 2 in Fig. 9). Note that the jammer suppression method fails if the false target is located in the same range ambiguity region as the true target by fast retransmission (as shown by false target 1 in Fig. 9). In other words, in a non-ambiguous region, any false target that is ahead of the true one can be suppressed, but those behind the true target within the same transmit pulse cannot be suppressed effectively. In summary, the FDA-MIMO radar only has the ability to effectively suppress deceptive jammers that are range ambiguous.

Figure  9.  Generation of false targets

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By utilizing the advantage of FDA-MIMO radar in resolving the range ambiguity, existing jammer suppression methods mainly focus on spatial beamforming, including data-independent and data-dependent beamforming^[40]. Fig. 10(a) and Fig. 10(b) demonstrate the profiles of the equivalent transmit beampattern based on data-independent and data-dependent beamforming in the 2D transmit-receive spatial domain, respectively. As shown in Fig. 10, the transmit spatial frequency of the false target is relative to the true target, and the offset $\Delta {f_{\text{T}}}$ is related to the number of delayed pulses. By designing the frequency increment, the false targets can be located at the nulls of the equivalent beampattern and further suppressed by nulling at the equivalent beampattern^[45,81]. Additionally, the frequency increment is designed according to the relationship between the nulls of the beampattern and the directions of the false targets.

Figure  10.  Mainlobe deceptive jammer suppression with beamforming in FDA-MIMO radar

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On the contrary, a data-dependent beamformer based on the MVDR criterion is constructed, as shown in Fig. 10(b), which forms nulls adaptively in the range-angle 2D spatial domain, and the false targets that mismatch in the range domain are suppressed^[33,39,40]. The data-dependent beamformer is expressed as

where, R_j+n denotes the jammer-plus-noise covariance matix, ${\boldsymbol{w}} = \mu {\boldsymbol{R}}_{{\text{j + n}}}^{ - 1}{\boldsymbol{u}}({R_0},{\theta _0})$ is the weight vector.

In practice, there is a performance loss in jammer suppression by utilizing beamforming methods. Therefore, research on jammer suppression based on robust beamforming methods is necessary. For the jammer suppression method based on data-independent beamforming, the error in the array or system leads to the deviation of the target parameters from the theoretical values, and the false targets cannot be positioned at the nulls. To address this, the jammer can be fully suppressed by broadening the nulls and widening the mainlobe to improve the robustness of jammer suppression. A Preset Broadened Nulling Beamformer (PBN-BF) algorithm based on artificial interference is proposed in Ref. [13]. A Preset BeamPattern Synthesis (PBPS) method to broaden the mainlobe and nulls of the beampattern is proposed by orthogonally decomposing the weight vector of the transmit-receive 2D beampattern^[46]. As demonstrated in Fig. 11(a) and Fig. 11(b), the output SINR is significantly enhanced by employing the aforementioned beampattern designing methods, and the results of matched filtering have the maximum output response only at the position of the true target. Regarding the data-dependent beamforming method for jammer suppression, the method to reconstruct the jammer-plus-noise covariance matrix is necessary because the distribution of the false targets does not satisfy the I.I.D. condition. Therefore, a robust nonhomogeneous sample detection method is proposed by selecting the nonhomogeneous samples and rejecting the samples, including the true target signal. In Ref. [43], a subspace of the receive mainlobe is constructed in the transmit-receive 2D spatial domain to select the mainlobe deceptive jammer using the orthogonal projection method. In Ref. [44], a two-step GoDec algorithm is proposed based on the alternating minimization method to suppress barrage jamming and burst jamming. As shown in Fig. 11(c), the aforementioned methods can suppress the jammer that is mismatched in the range domain and produce high output SINR.

Figure  11.  Output SINR with robust beamforming for jammer suppression in FDA-MIMO radar

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Our team has established the frequency diverse system, including antenna arrays, RF components, signal processor, data memory, radar system display, and control parts, and conducted the field experiment of retransmission mainlobe deceptive jammer suppression. The principal prototype of the frequency diverse system and the results of jammer suppression are given in Fig. 12. The results indicate that phased array and MIMO radars cannot suppress mainlobe deceptive jammers due to the lack of sufficient DOFs in the spatial domain. In contrast, the FDA-MIMO radar can effectively suppress mainlobe deceptive jammers by introducing DOFs in the range domain, obtaining only the maximum power at the position of the true target. The measured results are in agreement with the theoretical results after processing^[8,40,82].

Figure  12.  Verification of jammer suppression with FDA system

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The FDA-MIMO radar indeed excels in achieving range-angle-Doppler 3D processing, enabling concurrent suppression of clutter and jammers by resolving range ambiguity^[6,30,31]. In Ref. [29], an airborne MIMO STAP radar introduces a signal model, and Fig. 13(a) illustrates the distribution of true targets, clutter, suppression jamming, and deceptive jamming in the transmit-receive-time 3D domain. Notably, true targets and clutter share the same plane, whereas suppression jamming is solely angle-dependent in the receive domain. False targets, however, exhibit distinguishable characteristics in the 3D domain. Leveraging angle, Doppler, and range information in the FDA-MIMO STAP radar allows for the effective suppression of false targets while detecting true ones. Additionally, Ref. [83] proposes a Space-Time-Range 3D Adaptive Processing (STRAP) method employing subspace projection. This technique suppresses deceptive jamming through pre-filtering subspace projection, estimates the covariance matrix of clutter and suppression jamming using compensated data, and achieves target detection in the transmit-receive-time 3D domain. Fig. 13(b) displays the loss of output SCJNR, demonstrating the successful integrated suppression of clutter and jammer through these methods.

Figure  13.  Integrated clutter and jammer suppression in FDA-MIMO STAP radar

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5. Integrated SAR Imaging and Moving Target Detection

5.1 Principle of integrated SAR imaging and moving target detection

SAR imaging and MTI constitute two primary functions of airborne radar systems, with crucial considerations regarding antenna beam scanning and system bandwidth. SAR imaging necessitates comprehensive target area illumination, requiring the entire space to be scanned to detect moving targets. Additionally, for enhanced range resolution in SAR imaging, broad spectrum signals are advantageous, and detecting moving targets benefits from transmitting narrowband signals to address target migration issues across range gates. This integration poses two fundamental challenges: signal bandwidth incompatibility and beam scanning incompatibility.

To facilitate integrated SAR imaging and MTI, orthogonal LFM signals with distinct carrier frequencies are transmitted. In FDA radar, spectrum overlapping arises due to slight frequency increments between adjacent transmit elements. By increasing the frequency increment to the bandwidth, i.e., $\Delta f = B$, spectrum overlapping can be averted, yielding orthogonal frequency-division signals^[75,76]. The signal transmitted by the $m{\text{-th}}$ element can be expressed as

Where ${f_m} = {f_0} + (m - 1)B$. The key difference from FDA-MIMO radar lies in the frequency increments: FDA-MIMO employs small increments, achieving waveform orthogonality through code division. In contrast, frequency-division orthogonal radar uses bandwidth for increments, as depicted in Ref. [2]. Fig. 14(a) illustrates the configuration, where each channel has a 500 MHz bandwidth, occupying the 11 GHz to 15 GHz frequency band for the entire transmit signal. Fig. 14(b) displays the transmit pattern of nine array elements.

Figure  14.  Signal model of orthogonal frequency diverse LFM signal

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To achieve MTI, narrowband signals corresponding to different channels are separated, and spectrum overlapping and alignment in different spatial channels are achieved through spectrum shifting. At the receiver, processing methods such as bandpass filtering, spectrum alignment, and matched filtering extract the separated narrowband transmit signals for MTI. Additionally, splicing and synthesizing narrowband signals from different spatial channels in the frequency domain yield broadband signals. This further facilitates the separation of ambiguous echo signals and unambiguous SAR imaging based on differences in the transmit frequency domain.

5.2 Moving target detection and HRWS SAR imaging

To realize the detection of moving targets under the background of ground/sea clutter, the joint processing method of FDA-MIMO radar in the space-time-frequency domain can be used to mitigate the range-Doppler ambiguity clutter^[29]. After separating the transmit waveform at the receiver, an equivalent transmit pattern can be formed, and the moving target detection in different angles, range ambiguity regions, and Dopplers can be achieved through 3D search. Considering that in the FDA-MIMO radar, the clutters corresponding to the range ambiguity regions are separated in the joint transmit-receive domain, the processing method of FDA-MIMO for reduced dimension in the multidimensional beam domain is designed in Ref. [29]. For any angle, multiple beams can be formed in the transmit and receive spatial domain around the main range ambiguity region. Fig. 15 shows the localized multi-beam for moving target detection in the first range ambiguity region and the normal direction of the antenna^[4].

Figure  15.  Diagram of space-time-frequency localized processing with FDA-MIMO

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The adaptive range-angle-Doppler processing method based on 3D Localization (3DL) can be expressed as^[29]

where ${\boldsymbol{\tilde R}}$ represents the clutter plus noise covariance matrix with reduced dimension, ${\boldsymbol{\tilde t}}\left( {{p_0},{\psi _0},{v_0}} \right)$ is the transformed presumed steering vector of the target, and the index of range region, conic angle, and velocity of the target are respectively determined by^[29]

where ${{\boldsymbol{\tilde x}}_l}$ represents the transformed training samples of the ${{\boldsymbol{\tilde x}}_l}$ range bin. Fig. 16(a) and Fig. 16(b) present the responses in three dimensions and two dimensions, respectively. As shown in Fig. 16(c), the SCNR loss closely approximates the ideal curve, whereas the full-dimensional processor incurs more SCNR loss^[29].

Figure  16.  Results on dimension reduction processor in FDA-MIMO

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For SAR imaging, an HRWS SAR imaging method based on a sub-band frequency-diverse array is proposed in Ref. [84]. The transmit array comprises multiple subarrays, and the waveform transmitted by each array element in each subarray is orthogonal with a frequency increment slightly less than or equal to the signal bandwidth. This solves the range ambiguity problem using the 2D dependence characteristics of the range and angle in the subarray, and high-bandwidth signals are obtained by using spectral stitching to achieve high range resolution.

Fig. 17 illustrates the signal processing flow. Initially, the echoes undergo matched filtering to recover the DOFs of the transmitter. This step is succeeded by receive beamforming to determine the gain of full apertures. Subsequently, range-dependent compensation and transmit beamforming are employed to segregate the range-ambiguous echoes into multiple unambiguous segments within the subarray. Following this, phase compensation is applied to the unambiguous signals corresponding to each subarray, aiming to eliminate excess phase. The signals from all subarrays are then synthesized in the range-frequency domain utilizing spectral stitching technology to achieve wide swath signal synthesis. Finally, the stitched signals undergo imaging processing to derive the HRWS-SAR imaging result, as depicted in Fig. 18^[84].

Figure  17.  Procedure of HRWS SAR imaging based on multiple sub-band FDA

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Figure  18.  Imaging results of sub-band FDA-HRWS

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6. Conclusions and Potential Developments

6.1 Conclusions

In addressing the imperative need for accurate target detection within a complex electromagnetic environment, the inevitable developmental trend is observed in multifunctional integrated radar technology. FDA stands out for its transmit beampattern with broad spatial coverage, and flexible beamformers can be configured through equivalent transmit beamforming in the receiver. Through meticulous system design and signal processing, high-dimensional DOFs can be attained with FDA. This study has thoroughly examined and synthesized the theory and methodologies of multifunctional integrated FDA, emphasizing integrated detection and estimation technologies, integrated ambiguity resolution and jammer suppression, as well as integrated SAR imaging and moving target detection. However, there are areas that necessitate further exploration, such as the suppression of fast-repeated mainlobe jammers, orthogonal waveform design, and system design. Presently, multifunctional integrated FDA studies are predominantly theoretical, urging the development of practical technologies to enhance the capability of the system in processing multiple functions and tasks, thereby elevating situational awareness, integration degree, and overall performance.

6.2 Potential developments

Considering current trends in FDA system applications and practical requirements, potential avenues for studies on multifunctional integrated FDA are outlined:

(1) Waveform design based on joint spatial-temporal-frequential modulations

Given that FDA has a flexible beamforming nature, methods for designing transmit waveforms utilizing spatial-temporal-frequential modulations should be explored. Spatially, the technology of wide transmit beams and equivalent transmit-receive multiple beams should be investigated to address the spatial beam scanning contradiction of the multifunctional system. Temporal-frequential domain considerations involve broadening the instant beam width by combining with ultrawideband, fully leveraging controllable system resources in multiple dimensions. Establishing a waveform library tailored to specific tasks and environments becomes feasible through these means.

(2) Intelligent processing combined with cognitive radar

The rising demand for situational awareness necessitates acquiring environmental information through interactive learning within the multifunctional system. Parameters of transmitters and receivers are adaptively adjusted based on a priori knowledge and ratiocination, enabling closed-loop processing of information involving cognition, reaction, decision, evaluation, and optimization^[85,86]. To accommodate diversity and time-variety, the multifunctional integrated system evolves toward intelligence, digitization, modularity, and networking, achieving system optimization with reduced calculations.

(3) Implementation of the multifunctional integrated FDA system

The success of the multifunctional integrated FDA system hinges not only on theoretical innovation but also on practical implementation, evaluated through experimental tests. Additionally, digitalization trends prompt the exploration of monolithic integration, reducing the volume and cost of radar systems by integrating multiple functions and expanding the application areas of radar systems.