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Beampattern analysis of frequency diverse array radar: a review
EURASIP Journal on Wireless Communications and Networking volume 2021, Article number: 189 (2021)
Abstract
Electronic beam steering is an essential feature of stateoftheart radar systems. Conventional phased array (PA) radars with fixed carrier frequencies are wellknown for electronically steering their beam with high directivity. However, the resulting beampattern is angledependent but rangeindependent. Recently, a new electronic beam steering concept, referred to as frequency diverse array (FDA) radar, has attracted increasing attention due to its unique rangeangle dependent beampattern. More importantly, the FDA radar employs a small frequency increment across the array elements to achieve beam steering as a function of angle, range, and time. In this paper, we review the development of the FDA radar since its inception in 2006. Since the frequency offset attaches great importance in FDAs to determine the beampattern shape, initially much of the research and development were focused on designing the optimal frequency offsets for improved beampattern synthesis. Specifically, we analyze characteristics of the FDA beampattern synthesis using various frequency offsets. In addition to analyzing the FDA beampattern characteristics, this study also focuses on the neglected propagation process of the transmitted signals in the early FDA literature, and discuss the timevariant perspective of FDA beampatterns. Furthermore, FDA can also play a significant role in wireless communications, owing to its potential advantages over the conventional PAs. Therefore, we highlight its potential applications in wireless communication systems. Numerical simulations are implemented to illustrate the FDA beampattern characteristics with various frequency offset functions.
1 Introduction
One of the most interesting and useful pursuits in radars today is the study of electronic scanned arrays. Electronic scanned arrays have long been of great research interest in numerous applications such as radar, sonar, navigation and wireless communications [1,2,3,4,5,6,7,8,9,10,11]. A considerable amount of effort has been expended in the investigation of methods for electronic scanning of antenna systems. During World War II, the United States and Britain were actively engaged in research on array antennas [12,13,14]. It was not until the 1950s that the first fully electronic scanning was realized [15,16,17]. In the interim, several mechanical scanners were invented and used [18]. The significant advances in the 1950s and early 1960s have made the electronic scanning using array antennas feasible with the development of ferrite phase shifters in 19541955. Consequently, the development of Lincoln Laboratory’s phased array (PA) radar began around 1958 [19]. Historically, electronically steerable phased arrays (ESA) are designed in two ways, the passive ESA and the active ESA [20]. The former utilizes a single transmitter and receiver, the latter utilizes multiple Transmit/Receive modules, typically one per element, to provide amplitude and phase control. Although scanning antenna arrays had been developed for various electronic communications applications, radar system was the first application in which PA was broadly used for the first time [20,21,22].
Conventional PA radars using fixed carrier frequencies are widely recognized for their capability to electronically steer a beam with high directivity [23, 24]. The effective beam of a PA radar can be electronically steered in the desired direction using phaseshifters across the array elements [25, 26]. The resulting beampattern is angledependent but rangeindependent [27]. Therefore, PA radars cannot be deployed in applications involving rangedependent sources. On the other hand, the cost and complexity are other major concerns in PA radars because phaseshifters deployed at each antenna element represent a considerable portion of the overall system cost [23,24,25,26,27]. In addition, since the elementspacing in a PA system is usually fixed at halfawavelength corresponding to the carrier frequency, it is necessary to expand the aperture size by employing more sensors to achieve a more directed beam [20,21,22,23,24,25,26,27]. However, this will further increase the complexity, and cost. Antenna spacing could be increased above halfawavelength, however, this will come at the expense of multiple large sidelobes or gratinglobes in the real space when scanning the main beam [28]. Therefore, the array size dictates the number of elements in a PA beamformer [29]. Although substantial advances have taken place in PA technology since the early 1970’s, affordability is still a major concern, and the design factors such as aperture sizing, pattern synthesis, and beamswitching speeds have changed little [27].
To overcome these disadvantages, and emphasizing aspects of simplicity, reliability, and versatility required by modern systems, it is necessary to find more innovative array beam steering techniques [27]. Consequently, more attention is paid to new concepts for electronic scanning of antenna systems. New concepts for electronic beam scanning proposed in this connection include waveform diversity techniques i.e., to vary the excitation across the array in either the time or frequency domain. These new configurations regarded as timemodulated arrays [30,31,32,33] and frequency scanning arrays [34, 35] can provide beam steering ability without using the expensive phaseshifters. In timemodulated arrays, high speed RF switches are used instead of expensive phase shifters to electronically steer and shape the radiation pattern [36]. The RF switches periodically turned on and off the excitations of the individual array elements according to prefixed time sequences [37]. In frequency scanned arrays, beam steering is achieved by the variation of frequency as a function of time, and the same signal is simultaneously applied to all the spatial channels. However, the resulted beampatterns of these techniques are still rangeindependent, which limit their applicability in applications involving rangedependent sources, e.g., rangedependent interference suppression, and localization of multiple targets with the same direction but distinct ranges, etc.
Recently, a new concept, namely, frequency diverse array (FDA) radar has emerged as a popular beam steering technique that generates a rangeangle dependent beampattern [38]. The essential difference between a PA radar and an FDA radar is the carrier frequency of each array element. In contrast to PA radars using fixed carrier frequencies, FDA radars employ a small frequency increment across each array element to achieve beam steering in both the angle and range domains [39, 40]. The concept of FDA radar was first proposed by Antonik et. al in [41], where a progressive incremental frequency shift is applied to the array signals. This progressive frequency shift generates a natural timedependent progressive phase difference across the array elements, which enables the FDA beampattern to scan in range and angle domains as a function of time [42]. Consequently, the expensive phaseshifters are dispensed in FDA radars to steer their beam in a particular direction. The FDA radar [43] has attracted noticeable attention because the rangeangle dependent beampattern is attractive to various realtime array signal processing applications including radar, sonar, wireless communications and acoustics [44,45,46].
FDA radar is different from orthogonal frequency division multiplexing (OFDM) radar [47,48,49] and multipleinput multipleoutput (MIMO) radar [50,51,52]. OFDM radar uses orthogonal subcarriers, but nonorthogonal carriers are employed in FDA radar. MIMO radars either emit unique and independent waveforms using multiple antennas to achieve waveform diversity (i.e., collocated MIMO) or employs orthogonal signals from widely separated antennas (i.e., widely spaced MIMO) over multiple independent paths to provide spatial diversity, whereas FDA radar transmits overlapping signals with closely spaced frequencies to provide additional functionalities, such as rangedependent beamforming, increased range resolution and degreeoffreedoms (DOFs), effective mitigation of rangeangle dependent interference sources and clutter. Likewise, FDA radar is also different from conventional frequency scanning radar [53, 54]. Conventional frequency scanning radar using frequency offsets as a function of time has the same frequency at each element in a given time, but FDA frequency offsets are characterized by the element index. The timemodulated array antenna [55, 56] proposed by H. E. Shanks in 1962 is a new technique for electronic scanning, which weights each element using on/off switching operation. It is noteworthy that the synthetic antenna and impulse radar (RIAS) pioneered by French researchers in early 1970’s [57,58,59], and sparsearray synthetic antenna and impulse radar (SIAR) [60, 61] developed by Chinese researchers in 1980’s are typical kinds of MIMO radar even before the concept of MIMO radar was put forward. They exploit both spatial and spectral diversity to achieve more degrees of freedom, higher angular resolution and target detection probability. MIMO radar is mainly adopted from the idea of SIAR. RIAS and SIAR simultaneously transmits a unique pulse – orthogonally phased, frequency modulated and coded and receive echoes via multiple receive antennas [62]. The concept of RIAS and SIAR also paved a way for the development of the FDA radar.
Conventional PA radar is well developed in the literature for electronically steering the beam direction with high directivity. However, the rangedependent sources cannot be unambiguously estimated from its inherent rangeindependent beampattern. Although spacefrequency or spacetime adaptive processing (STAP) based techniques can estimate the range of sources, they usually have a high computational cost and complexity [63,64,65,66]. FDA radar looks particularly promising technique for the electronic beam steering. Reference [44] overviews the basic FDA system architecture in radar and navigation applications, and its potential applications in rangedependent energy control and technical challenges in system implementation are discussed in [45]. Since then, there have been a number of recent advances in FDA beampattern synthesis techniques. With the increasing importance of rangeangle dependent beampattern, and recent advances in FDAs, a review incorporating more recent findings has become necessary. This paper aims to briefly analyze the rangeangle dependent beampattern of the FDA radar from its origin to the current stateoftheart FDA beampatterns. While the rangeangle dependent beampatterns achieved in FDAs are encouraging, the realism of the simulations is limited. More attention has been paid to achieve rangedependent beampatterns, which neglects the practical constraint of wave propagation. Therefore, we have dedicated a section to discuss the wave propagation in FDAs. In addition, we also discuss the potential applications of FDAs in wireless communications.
The outline of this paper is organized as follows. In Sect. 2, we present the PA signal model and beampattern synthesis. In Sect. 3, we describe the preliminaries of the FDA, including signal model and the timeinvariant FDA beampattern synthesis. In Sect. 4, we explain the neglected propagation process of the transmitted signals in FDAs. Recent trends in FDAs are discussed in Sect. 5, and the areas that are suitable for continuing research are provided in Sect. 6. Finally, Sect. 7 concludes this study.
2 Phased arrays
Consider the simple case of an Melement uniform linear array (ULA) of isotropic radiating antenna elements with halfawavelength interelement spacing as shown in Fig. 1. The farfield array radiation pattern, or array factor (AF) in this case, in terms of standard spherical coordinates, can be expressed by superposition of the M radiating currents as [20,21,22,23]
where \(w_m\) is the complex weight related to the mth element, \(\kappa = 2\pi /\lambda\) is referred to as the wavenumber with \(\lambda\) denotes the wavelength of the signal, d is the spacing between adjacent antenna elements, and \(\theta\) is the direction of the target.
In what follows, the operations of transposition, complex conjugation, and Hermitian transposition are denoted by superscripts T, \(*\), and H, respectively. Lowercase and uppercase boldface characters denote the vectors and matrices, respectively.
In vector form, Eq. (1) can be rewritten as
where \(\varvec{w}\) and \(\varvec{a}(\theta )\), respectively, denote the weight vector and the array steering vector, which are given as [19]
and
The beam can be steered by controlling the phase shift in the weight associated with individual element. With uniform weights, i.e., \(w_{m}=1\), the array factor can be expressed in closedform as [27]
The beampattern for the PA antenna is given as [27]
For the linear PA antenna to steer its main beam to angle \(\theta _d\), \(w_m\) must be equal to \(\exp (jm\kappa d\sin \theta _d)\), so that \(B^{PA}(\theta )\) gives a maximum magnitude at that value [20,21,22,23,24].
In Fig. 2, we plot the normalized beampattern of a 20elements PA. Figure 2a, b shows the PA beampattern in rangeangle dimension, and angledimension with the main lobe steered to \(30^{\circ }\), respectively. It is observed that the PA has an angledependent but rangeindependent beampattern.
To further illustrate the characteristics of the PA beampattern, several array patterns with different number of elements M and interelement spacing d as a function of wavelength \(\lambda\) have been plotted in Fig. 3. Note that phaseshift has not been applied, and the main lobe is in the broadside direction. From Fig. 3a, it is observed that as the number of elements increases, the array directivity increases. Extending the interelement spacing toward \(\lambda\) leads to narrow main lobe but suffers high grating lobe effect as shown in Fig. 3b. The maximum grating lobe amplitude equal to the main lobe magnitude at an element spacing \(\lambda\). Clearly, increasing the interelement spacing results in a narrower main lobe but it comes at the expense of multiple large sidelobes or gratinglobes, whereas more antenna elements not only further decrease the main lobe width but also increase the maintosidelobe ratio.
3 Frequency diverse arrays
In this section, we briefly investigate the FDA beampattern characteristics. The first subsection formulates the signal model of FDA radars. The next subsection discusses the nonlinear frequency offsets for rangeangle decoupling of the FDA beampattern. The final subsection presents the timemodulated frequency offsets to achieve the timeinvariant rangeangle dependent FDA beampattern.
3.1 FDA signal model
Assume a ULA composed of M omnidirectional elements with half a wavelength interelement spacing as shown in Fig. 4. The signal transmitted by the mth element is [43,44,45]
where T denotes the transmitted pulse duration, and \(f_m\) is the radiation frequency of the mth element given by [38,39,40,41,42,43,44,45]
Here, \(f_0\) is the carrier frequency radiated by the first element, and \(m\Delta f\) is the frequency offset of the mth element. The overall signal arriving at an arbitrary point \(P(r,\theta )\) (r and \(\theta\) are the range and the azimuth angle with respect to the first array element) in the space can be expressed as [44]
where \(r_m\cong rmd\sin \theta\) is the target slant range with respect to the mth element [45], and c is the wave speed. Substituting \(r_m\cong rmd\sin \theta\) and (9) into (10), we have [26, 45]
According to the fundamental condition of the FDA radar, the quadratic phase term \({m}^2 \Delta f \frac{d\sin \theta }{c}\) satisfies the condition \({m}^2 \vert \Delta f \vert \frac{d\sin \theta }{c}<\frac{\pi }{4}\) because the maximum frequency offset is far less than the carrier frequency i.e., max\(\lbrace \Delta f\rbrace \ll f_0\) [26, 45]. Hence, it can be ignored in the further analysis. In (11), the terms inside the summation sign are determined by the array geometry and frequency offsets of the FDA, therefore, the array factor can be expressed as [38,39,40,41,42,43,44,45]
Consider the case that \(w_0=w_1=...w_{m1}=1\), the FDA transmit beampattern is given by
The beampattern can also be expressed as [44, 45]
Note that when \(\Delta f=0\), the beampattern becomes [45]
which is only angledependent just like conventional PA radar.
For steering a target positioned at \(P(r_{d}, \theta _{d})\), the weights \(w_m\) for the transmitted beampattern can be computed as [26]
The corresponding array factor becomes
and its magnitude squared, known as transmit beampattern, is given as
Assume the following parameters: \(M=10\), \(f_0=10\) GHz, \(\Delta f=3\) kHz, and \(d=\frac{\lambda _0}{2}\). In Fig. 5, we plot the normalized transmit beampatterns of the PA, the standard FDA, and the symmetrical FDA. Note that \(\Delta f=0\) is employed for the PA. From Fig. 5a, it can be observed that the PA has an angledependent but rangeindependent beampattern. In contrast, from Fig. 5b we can see that the standard FDA with a progressive frequency increment yields a periodic range and angle coupled Sshaped beampattern. The symmetrical FDA has range–angle decoupled beampattern as shown in Fig. 5c.
In the transmit beampattern derived in Eq. (14), the maximum field can be obtained when the phase term satisfies the condition [44, 45]
It can be observed from Eq. (19) that the maximum field depends not only on the angle but also on the range and time. Moreover, when only one parameter is fixed, there are multiple solutions for the unfixed parameters. On the other hand, when two parameters are fixed, the pattern periodicity depends on the unfixed variable.
For instance, if Eq. (19) is solved for time t, we have [44, 45]
This implies the periodic nature of the FDA beampattern in time. When the range r and angle \(\theta\) are fixed, the fundamental period is \(\frac{1}{\Delta f}\).
Similarly, the periodicity in range dimension is derived by solving Eq. (19) for range r as [44]
whose fundamental period is \(\frac{c}{\Delta f}\) for a fixed \(\theta\) and t.
In the same manner, if Eq. (19) is solved for \(\theta\), we can obtain [45]
Obviously, \(\sin \theta\) depends on both the time and range variables. The periodicity in time, range, and angle is also shown graphically in Fig. 6 for a 17element FDA with 10 kHz interelement frequency offset.
It is clear from (20)–(22) that the FDA beampattern is timedependent, and its maxima drifts in space with the time, which facilitates the autoscanning feature of FDA radars. That is to say, the entire space can be scanned without using the expensive phaseshifters. This unique autoscanning feature of FDAs can also be useful in wireless communication systems such as at the base station and communication devices [29].
The standard FDA using progressive incremental frequency offsets generates a periodic range and angle coupled Sshaped beampattern. The range dependency of FDA beampatterns is investigated in [67,68,69,70,71], whereas its ability in increasing DOFs is studied in [38, 72]. The autoscanning property of the FDA radar is analyzed in [42, 69]. Higgins and Blunt [73] explored the range–angle coupled beamforming in FDAs, and Secmen et al. [68] described the time and angle periodicity of FDA beampatterns. In [74], Eker et al. introduced a practical FDA system using linear frequency modulated continuous waveform, wherein both the transmit and receiver architectures together with the waveform processing is being analyzed in detail. The application of the FDA for forwardlookingradar ground moving target indication, and in bistatic radar is studied in [75] and [76, 77], respectively. In [78,79,80], the application of FDA is extended to synthetic aperture radar for improved performance, and the frequency diversity applied to the phasedMIMO radar for rangedependent beamforming is proposed in [81]. The multipath characteristics of FDAs over a ground plane are studied in [82]. In [83], the localization performance of FDA is analyzed. Additional investigations on the twodimensional imaging of targets and suppressing the rangedependent clutters in FDAs are, respectively, given in [84, 85] and [86, 87].
Since the rangeangle coupled beampattern introduces ambiguity into target indication, the range and angle of a target cannot be unambiguously estimated using FDAs [84, 85]. There has been a long line of research in FDAs, where a series of efforts have been made to decouple its beampattern in range and angle dimensions. Among the various methods, utilizing nonlinearly increasing frequency offsets in a ULA and utilizing linearly increasing frequency offsets in nonuniform linear array are the two most representative methods to decouple the FDA beampattern in range and angle dimensions [44, 45].
Numerous other techniques were also proposed to get improved localization performance. To name a few, a doublepulse FDA radar was proposed in [88] for the rangeangle localization of targets. A pulse with zero frequency increment degenerates the system into a PA radar detecting the targets in the angle dimension, and then localize them in the range dimension using a nonzero frequency increment pulse. In [89], a stepped frequency pulse FDA (SFPFDA) radar is proposed, which can be regarded as an upgraded version of the doublepulse FDA in [88]. The first pulse of SFPFDA radar is same as conventional FDA with a bit of frequency increment. There is an additional small frequency increment from pulse to pulse. Another doubled pulsed MIMOFDA has been proposed in [90], where the FDA transmit array is partitioned into subarrays and then transmits a unique waveform from each subarray with zero and nonzero frequency increment pulse, respectively. A vertical FDA, which applies frequency diversity in the vertical of a planar array, is explored in [91] to circumvent the range ambiguity problem in STAP radar. An FDA MIMO adaptive beamforming and localization scheme for rangedependent targets and interferences is proposed in [92], where the ranges and angles of targets can be solely estimated with MUSICbased algorithm. Similarly, two FDAMIMO hybrid radar transmitter design schemes, namely, FDA–MIMO radar, and transmit subarray FDA–MIMO radar are proposed in [93] for rangedependent target localization, where the targets are localized using the beamspacebased multiple signal classification algorithm. Although these methods have improved the localization performance, the rangeangle decoupling is not achieved completely.
Considering the array configuration design, several researchers exploit the nonuniform array in FDA radars for improved localization performance. For example, a nonuniform FDA radar is proposed in [94] for rangeangle imaging of targets. Likewise, a new FDA framework is proposed in [95], where the logarithmically spaced array elements were distributed symmetrically to produce a wellshaped “dot” main beam, facilitating the application in target indication. An FDA scheme utilizing nonuniform element spacing and nonlinear frequency offset is proposed in [96] to provide a rangeangle decoupled beampattern with narrower main lobe and no periodicity. Moreover, a nonuniformly distributed FDA is proposed in [97], where both the frequency offsets and element positions are optimally determined via particle swarm optimization algorithm. However, it is difficult to alter the carrier frequency or frequency offsets in real time, and the relocation requirement for the accurate physical placement of transmitter and receiver at each scanning is impractical [26]. Therefore, this approach is not feasible in practice.
Another viable solution to decouple the FDA beampattern is through frequency offset design [26, 44]. By employing nonlinear and random frequency offsets, the FDA beampattern can be efficiently decoupled into the range and angle dimensions. Therefore, the nonlinear frequency offsets are intensively researched and extensively applied in FDA’s mainly for a range and angle decoupled beampattern.
3.2 FDA with nonuniform frequency offset
Frequency offset design has attracted great interests in the rangeangle decoupling of FDA beampatterns. Until recently, several attractive functions have been proposed to design the nonlinear frequency offsets such as square increasing and cubic increasing frequency offsets [98], Hamming windowbased nonuniform frequency offsets [99], Costas sequence modulated frequency offsets [100], piecewise trigonometric frequency offsets [101], and nonuniform logarithmic frequency offsets [102].
A multicarrier nonlinear frequency modulation FDA scheme based on logistic map is presented in [103], which is capable to reduce the main lobe width and the sidelobe peaks simultaneously. Both singledot and multidot shaped rangeangle dependent beampatterns are obtained in [104] by using multicarrier frequency offsets and convex optimization. In [105], a MIMOlogFDA radar is proposed, where the range bins concept along with logarithmic offset in each subarray of MIMOFDA is used to produce single maxima for multiple targets present in different range bins. A uniformly spaced linear FDA with logarithmically increasing frequency offset (logFDA), and nonuniform but symmetric frequency offsets calculated using well known mulaw formula are, respectively, proposed in [106] and [107] to achieve a beampattern with a single maximum at the target location. Moreover, an adaptive frequency offset selection scheme for FDA radar is proposed in [108], where the frequency offset is determined at each step with an iterative algorithm. Likewise, an FDA framework with Taylor windowed frequency offsets is proposed in [109] where the adjustable tapering windows determine the optimized parameters of the window function. The relationship among ambiguity, frequency pattern and target relative location is derived in [110] to identify the unambiguous frequency patterns for twotarget localization. The unambiguous target localization is achieved in [111] by combining the subarraybased FDA and fullband FDA, as the transmitter and the receiver, respectively. In [112], a gridless compressed sensingbased rangeangle estimation algorithm is proposed for FDAMIMO radar via Atomic Norm Minimization and Accelerated Proximal Gradient. By optimizing frequency offsets with a genetic algorithm, a singledot and multidot shape transmit beampatterns is synthesized in [113].
For illustration purpose, few typical nonlinear frequency offset functions are briefly reviewed here, and their comparative beampatterns are also analyzed.
Since the interelement frequency offset is not uniform, therefore the frequency input at mth element is
where \(\Delta f_m\) is the frequency offset of the mth element.
The Hamming window based frequency offsets can be expressed as [99]
This is the general Hamming window equation where \(\delta\) is an adjustable parameter.
The logarithmically increasing frequency offsets are computed as [106]
The piecewise frequency offsets based on a simple trigonometric are given as [101]
where
Here, \(\phi _{1}=0\), and \(\phi _{2}=\frac{\pi }{2}\). Since a symmetrical array configuration is utilized here, therefore \(M=2M_s+1\).
The square increasing and cubic increasing frequency offsets are derived as [98]
Graphical representations of the aforementioned frequency offset functions are shown in Fig. 7.
In Fig. 8, we compare the FDA transmit beampatterns generated using the Hamming window based frequency offsets, logarithmically increasing frequency offsets, and piecewise frequency offsets based on a simple trigonometric. In this experiment, the simulation parameters are as follows: M=17, \(f_0=10\) GHz, \(\delta =2\) kHz, \(d=\frac{\lambda _0}{2}\), and target location, \((r_d, \theta _d)=\) (500 km, \(30^{\circ }\)). The results indicate that all the three FDAs achieve a focused rangeangle decoupled beampattern with a single maximum pointing at the target location.
These nonlinear frequency offsets can decouple the FDA beampattern into range and angle dimensions. However, the actual beampatterns of aforementioned FDAs are timevariant. They are rangeangle dependent only for a fixed time, neglecting the influence of time. Nevertheless, the performance degradation is inevitable because the timeperiodicity still exists.
3.3 FDA with timemodulated frequency offsets
In order to overcome the timeperiodicity in FDA beampatterns, a number of improved methods have been proposed [114,115,116,117,118,119,120,121]. Among those are the timedependent frequency offsets (TDFO) proposed by Khan and Qureshi [114] and a pulsedFDA with constraints on both pulse duration and frequency shift proposed by Xu et.al. [115]. Instead of using fixed frequency offsets, the TDFOFDA considers timemodulated frequency offsets to generate a rangeangle dependent as well as timeindependent beampattern. The pulsedFDA is proposed to form a quasistatic rangeangledependent beampattern by properly choosing the pulse duration and the frequency shift. However, the beampatterns achieved in [114, 115] are timeindependent only for a particular rangeangle pair but they remain timedependent for other ranges and angles. To address this issue, a timemodulated optimized frequency offset FDA (TMOFOFDA) has been proposed in [116], which exploits the combination of timemodulated and nonlinear distributed frequency offset across the array elements to obtain a timeinvariant rangeangledecoupled beampattern. By now, various improved methods based on the unified configuration of nonlinear and timemodulated frequency offsets have been proposed in [117,118,119,120,121] to design timeinvariant spatial focusing beampatterns.
In general, the timevarying frequency offsets are given as
where \(\Delta f(m)\) is a nonlinear function with respect to the element index m.
As an example, we review here the TDFOFDA [114], timemodulated logarithmically increasing frequency offset FDA (TMLFOFDA) [116], and the timemodulated doubleparameter FDA (TMDPFDA) [120].
The TDFO applied to the \(mth\) element is [114]
and the array factor at the target point is expressed as
The TMLFO applied to the \(mth\) element is given as [116]
where g(m) is a logarithmic function defined as
where k is a control parameter for the frequency offset. The array factor at the target point is then expressed as
The timemodulated logistic map based frequency offsets and chirpiness constant considered in the TMDPFDA are defined as [120]
where \(0\le n \le N1\); Here \(\Delta f\) and \(\Delta u\) are the control coefficients for the frequency offsets and chirpiness constant, respectively. It must be remembered that TMDPFDA utilizes multicarrier architecture, and frequency diverse chirp signal. The parameters \(p_{m, n}\) and \(q_{m, n}\) are generated by logistic map as [120]
and the corresponding array factor at the target point is given as
In this example, we assume a 10elements ULA operating at a reference carrier frequency of \(f_0=10\) GHz. The interelement spacing is equal to halfawavelength. The pulse duration is \(T = 1\) ms, and the target is located at \((r_d, \theta _d)=\) (500 km, \(0^{\circ }\)). Since the multicarrier technique is adopted in the TMDPFDA, the number of carriers considered is \(N = 10\). Fig. 9 provides comparisons of beampattern generated by TDFOFDA, TMLFOFDA and the TMDPFDA. It is shown in Fig. 9a that the timeindependent beampattern of TDFOFDA is periodic in range, and coupled in the range and angle dimensions. The periodicity, and rangeangle coupling is eliminated in the timeinvariant beampattern of TMLFOFDA as depicted in Fig. 9b. Although the TMLFOFDA generates a single maximum beampattern, its spatial focusing performance is degraded due to the high spatial peak sidelobe levels (PSLL) and a broader spatial halfpower beamwidth (HPBW). Furthermore, as Fig. 9c shows, the TMDPFDA generate a wellshaped “dot” main beam and outperforms the other two schemes with a more focused beampattern due to multicarrier architecture.
To further demonstrate the resolution and sidelobe suppression performance, the corresponding range, and angle dimension beampatterns are plotted in Fig. 10. From Fig. 10a, it can be found that all the three FDAs have equal response in angle dimension except that the TDFOFDA beampattern exhibit periodicity. However, from Fig. 10b, it is observed that the TMDPFDA outperforms the TDFOFDA and TMLFOFDA with narrow HPBW in range dimension. The TDFObeampattern exhibit periodicity in range dimension, and the performance of TMLFOFDA is not satisfactory due to the high PSLL and the broad spatial HPBW. Instead of a single carrier signal, each array element of the multicarrier TMDPFDA transmit a weighted summation of signals with a small frequency offset. Therefore, the effect of multicarrier architecture on the performance of the TMDPFDA is also demonstrated in Fig. 10c. It is observed that as the number of carrier frequencies increases, the PSLL suppression and the HPBW gets improved.
Although much attention has been paid to the uniform linear array configurations in existing FDAs due to its simple structure and welldeveloped techniques, other array configurations are also very appealing for noncommunication applications such as radar, localization and positioning [122]. Numerous related works exploring FDAs with different array configurations for improved performance can be found in [123,124,125,126,127,128,129]. For instance, a multitarget localization algorithm for the sparseFDA radar is proposed in [123], which incorporates both coprime arrays and coprime frequency offsets. Likewise, the compressive sensing technique is applied to FDA in [124] for rangeangle estimation with a new type of array antenna, named random frequency diverse array. In [125], a joint optimization design scheme for FDAMIMO radar is proposed, where virtual coprime planar array with ‘unfolded’ coprime frequency offsets framework is utilized to achieve threedimensional (3D) localization without ambiguity. Similarly, a new waveform synthesis model of time modulation and range compensation FDAMIMO is proposed in [126] to achieve a joint rangeangle estimation based on the optimized timeinvariant and dotshaped beampattern. The beampattern analysis in terms 3D beam steering and auto scanning capabilities of the uniform circular FDA (UCFDA) and the planar FDA were discussed in [127, 128] and [129], respectively. A flattop rangeangle dependent beampattern synthesis method has been proposed in [130] for FDA based on secondorder cone programming. Similarly, a subarraybased FDA framework is devised in [131] to achieve a rangeangle decoupled beampattern. A circular FDA utilizing tangent hyperbolic function for frequency offsets selection scheme is proposed in [132], where three different configurations of an FDA can be generated by adjusting a single function parameter of the tangent hyperbolic function.
The aforementioned FDA techniques can obtain a timeinvariant rangeangle decoupled beampattern, however, some recent studies indicate that FDA beampatterns are always timevariant in free space [133]. The existing FDA techniques designed for the rangeangle dependent beampattern synthesis have not considered the propagation process of the transmitted signals, and may suffer performance degradation caused by the wavepropagation [133]. Consequently, it is necessary to revisit the FDA signal model and considering the wave propagation phenomenon into account, the new signal model is devised with timevariant focusing capability.
4 Wave propagation effect in FDAs
4.1 Correction of the time parameters
The original array factor of the FDA is a function of time t, which indicates that the EM waves propagate at the speed of light. More importantly, the term “\(t\frac{r}{c}\)” in (12) indicates that the range r and time t are correlated, and their relationship cannot be ignored [133, 134]. As highlighted in [134], the propagation process of the transmitted signals is neglected in the early FDA literature, and the timeinvariant beampatterns for FDAs were obtained by assuming the time variable t to be a fixed value (e.g., \(t = 0\)) or defining the variable t in the range from 0 to T [135]. This assumption that the signals propagated to the target location without time consumption is not valid in practice [133, 134].
Recall the array factor of TMLFOFDA given by Eq. (34) as follows
In (38), when \((r, \theta )=(r_d, \theta _d)\), we have
In (38)–(39), the maximum of the rangeangledependent beampattern stayed at the target location is independent of time as long as the range and angle are satisfied with \(r=r_d\) and \(\theta =\theta _d\). Furthermore, the beampattern is calculated during \(t\in [0, T]\). However, considering the propagation process of the transmitted signals, it is impractical that the EM waves propagate to a certain point \(P(r, \theta )\) without consuming time, i.e., at \(t = 0\). Similarly, it is also unconvincing to achieve the desired beampattern within the pulse duration that is at \(t = T\), because the desired range is usually taken as \(r_d\gg T.c\) [135].
Similarly, in the FDA schemes proposed in [114,115,116,117,118,119,120,121], the timemodulated frequency offsets were directly substituted in (12), and the variation of \(\Delta f_m(t)\) is supposed to propagate to any position without time consumption [134,135,136,137,138]. Hence, the beampatterns at the target point remain static all the time for \(r = r_d\) and \(\theta = \theta _d\). It is worth noticing that the range factor is introduced by the time factor in FDAs, however, the direct substitution of \(\Delta f_m(t)\) also indicates that the range influence is ignored. In other words, the range r is bounded up with time t, i.e., where there is t, there is r, and where there is r, there is t.
Hence, the array factor of these FDAs should be corrected by incorporating \(t\frac{r}{c}\) as a single quantity variable of function f for the signals propagating in free space. Therefore, Eq. (29) can be revised as [134,135,136]
The corrected array factor of TMLFOFDA [116] is then derived as
It can be observed from (41), that the maximum of the correct beampattern of TMLFOFDA no longer stays at the target point \((r_d, \theta _d)\). The original and correct beampattern of the TMLFOFDA is compared in Fig. 11. From the beampatterns projected on the rangetime dimension shown in Fig. 11a, b, it is observed that the original beampattern of TMLFOFDA at the target point is timeinvariant, however, when the propagation process of the transmitted signals is taken into account, the focus line disappears. The correct TMLFOFDA beampattern is unable to focus the transmit energy at the desired target, when the practical signal propagation process is taken into account. The comparison of original and corresponding corrected models of some prominent FDAs is shown in Table 1.
The unnoticed misconceptions about the timerange relationship, and frequencyphase relationship in FDAs are briefly analyzed in [134]. Additionally, the difference between the timeindex within the pulse duration T, and the actual time t is also explained in [136]. It is shown that the variable t represents the actual time of the signal traveling in the space, whereas T is the time index within the pulse duration. More importantly, it is shown that the range parameter is related to the wave propagation i.e., time variable t, while phase of the wave signal is related to the pulse duration T, but not vice versa [136]. Based on the frequencyphase relationship, and the timerange relationship, these studies reestablish the correct signal models. It is proved that the beampattern of FDA is always timevariant as long as the electromagnetic waves travel at the velocity of light. It is worthy of note that existing FDA techniques mainly designed for timeinvariant beampattern synthesis will essentially lead to a rangeinvariant beampattern as in conventional antenna arrays [137]. In other words, the design of timeinvariant rangedependent beampattern is impractical. Consequently, these FDA schemes cease to focus the transmit energy at the desired target location when the wave propagation property is considered [138].
4.2 Scope of the time variable
Practically, \(c\Delta t\) movement in range is obvious for an elapsed time \(\Delta t\). For instance, the minimum time required for EM waves propagating with the speed of light to reach the target located at range \(r_d\) is \(t = \frac{r_d}{c}\). For a signal with pulse duration T, the target can be illuminated for T seconds and beyond that time, there is no signal. So, the scope of the time related to wave propagation for a desired target located at \(P(r_d, \theta _d)\) can be revised as \(t\in [\frac{r_d}{c}, \frac{r_d}{c}+T]\). The corrected array factor can then be expressed as [133]
where \(\psi _m=2\pi [\Delta f_m(t\frac{r}{c})+\frac{f_0md\sin \theta }{c}]\).
In summary, most signal models in timeinvariant theories are actually inappropriate because the timerange relationship is ignored. The neglected propagation process of the transmitted signals causes essential misconception and results in erroneous conclusions. The term “\(t\frac{r}{c}\)” indicates that maximum of the FDA beampattern cannot be independently controlled in time and range dimension, and the focusing point always moves with the velocity of light. Nevertheless, the performance degradation in timeinvariant FDAs is inevitable when the wave propagation is considered, leading to the movement of focusing point with time. However, it is important to note that constant range for the focus point of FDAs appears to be unattainable but achieving a dotshaped beampattern with FDAs in an instant of time is not in contradiction with the physical phenomenon of EM wave propagation [134, 138].
5 Recent trends in FDAs
5.1 Timevariant focusing FDA beampattern synthesis
Considering the propagation process of transmitted signals in FDAs, several researchers strive to focus the transmit energy on the desired target in an instant of time. For example, a shortrange symmetrical FDA radar is proposed in [135] by incorporating an arctangent function modulated frequency scheme. Likewise, a transmit beampattern synthesis method based on arctangent function is proposed in [139] for waveform diverse array radar to focus the signal energy on the desired target for a period of time. A multicarrier symmetrical FDA radar with frequency diverse chirp signal is proposed in [140] for shortrange applications. They achieve a timevariant focused rangeangle dependent beampattern by compensating the propagation delays of transmitting signals. In [141], a focused anglerange beamforming synthesis method with timevarying characteristics is proposed, where the frequency offsets, amplitude weighting and phase weighting are optimized by bat metaheuristic algorithm to achieve singledot and multidot shape transmit beamforming. A logarithmbased optimized static nonlinear frequency offset for FDA is proposed in [142] to alleviate its intrinsic timevariant problem, which depends on a frequency offset control function and the duration of a shorttime interval rather than introducing the time variable. More recently, investigations on exploring different array configurations for improved performance have been made in [143,144,145,146]. A dual function rangeangle dependent sidelobe control using FDAMIMO is proposed in [143] for joint radarcommunications. The proposed system considers the timevariance of an FDA beampatterm. A highspeed usercentric beampattern synthesis method for FDA is devised in [144], where the timevariant beampattern peak accompany the quicklymoving user without altering the frequency offsets. A UCFDA radar is proposed in [145] to realize timevariant spatial focusing in shortrange at the target location. The UCFDA radar is capable to provide \(360^{\circ }\) of coverage in the azimuth plane, which cannot be achieved by the linear arraybased FDA radar. Similarly, a hemispherical FDA is exploited in [146] to demonstrate the unique advantage of conformal FDA in terms of 3D localization ability. It alleviates the inherent timevariant problem, and generate a quasitimeinvariant 3D focusing beampattern. The FDA schemes presented in [145, 146] incorporates pulsedependent frequency offsets to achieve timevariant spatial focusing beampattern.
The array factor for the shortrange FDA radar proposed in [135] is given as
where \(\psi _m (t) =\Delta f_m \frac{2}{\pi } (t\frac{T}{2}) \arctan [\frac{\alpha }{T}(t\frac{T}{2})]  \Delta f_m \frac{T}{\alpha \pi } \ln [\frac{\alpha ^2}{T^2} (t\frac{T}{2})^2 +1]\), and \(\Delta \tau _m = \frac{r_m^\prime  r_d}{c}\) is the corresponding transmit delay, where \(r_m^\prime = r_d md\sin \theta _d\).
The pulsedependent nonlinear frequency offset proposed in [145] is defined as
where \(\varphi\) denotes the elevation angle, and g(m, n) is a nonlinear function generated by a logistic map with a control parameter \(\alpha\). Since the transmitted signals can propagate to the target location \(P(r_d, \theta _d, \varphi _d)\) within the time interval \(t\in [\frac{r_d}{c}, \frac{r_d}{c}+T]\), therefore, assuming the value of t within this range, the final form of pulsedependent frequency offsets becomes
where the introduced coefficient \(\beta\) satisfies the condition \(0\le \beta \le 1\).
Similarly, the optimized static nonlinear frequency offset (OSNFO) for general conformal FDA in [146] is given by
where \(g_{OSNFO}(m)\) is a nonlinear function to be optimized by the conventional evolutionary methodology.
In Fig. 12, we have demonstrated the significance of the wave propagation effect in FDAs. Figure 12a shows the original timeinvariant beampattern generated by the compensated TMOFO (CTMOFO) FDA [119], where a focused beampattern with a single maximum at the target position is achieved. Figure 12b shows its corresponding corrected beampattern projected on the timerange dimension when wave propagation is considered. Obviously, the focused point disappears, and the beampattern is unable to be steered to the target position. The beampattern of the shortrange FDA radar based on arctangent function [135] is shown in Fig. 12c. It is observed that the transmit energy is focused when the pulse signals reach the desired spatial region \(r_d = 6\) m at \(t = 25\) ns only. The analysis of timeinvariant FDAs, and their corrected models with simulated examples are briefly discussed in [134,135,136,137,138,139,140, 145].
5.2 Applications
While most current research activities mainly focus on radar applications, such as localization, FDA is also very appealing for communication applications. The security issue in wireless communication systems is of great concern due to its broadcasting nature of transmission [29]. In this scenario, the communications between the transmitter and the legitimate destination receiver are more likely to be intercepted easily by eavesdroppers. Therefore, an active research topic has been focused on how to prevent eavesdroppers access to the desired information in wireless communication systems.
Directional modulation (DM) is recognized as one of the most promising techniques in the pointtopoint physical layer secure communications [147]. DM projects digitally modulated signals into a predefined secure communication spatial direction, while simultaneously distorting the constellation of these signals in all other directions [148]. This can significantly decrease the risk of these signals being eavesdropped. As such, DM techniques using PA antenna have been widely applied in wireless communications, which only achieves onedimension security, i.e., angledependent directional modulation [149]. Therefore, the PA based DM techniques are less successful when an eavesdropper and the legitimate user are located in the same direction.
On the other hand, frequency diversity has been shown to be a highly effective technique in wireless communications, which provides a number of advantages, including increased system capacity and SNR, security and robustness. Motivated by the rangeangle dependent beampattern of FDAs, researches have been carried out on FDA based DM techniques to enhance physical layer security of wireless communications [150, 151]. Both the PA and FDA based DM techniques can be typically applied to secure communication in the lineofpropagation channels such as millimeter wave, satellite communications, unmanned aerial vehicles, and smart transportation. Compared with the PAbased DM techniques, the FDA based DM technique can achieve the physical layer security in both range and angle dimensions. Since the FDA apparent scan angle is different than its nominal scan angle, precise beam steering depends on both the range and the angle. Consequently, the actual beam steering direction in FDAs cannot be effectively predicted as conventional phased arrays. Equation (22) demonstrates how the scan angle changed with frequency increment and generated an apparent scan angle.The FDA based DM techniques have provable performance guarantee when the legitimate user and eavesdropper locate in the same direction but different ranges [152].
Until recently, several attractive FDA based DM techniques have been proposed to ensure secure communication with low probability of interception and low probability of detection. To name a few, a decoupled rangeangle dependent DM scheme using FDA with symmetrically and nonlinearly increasing frequency offsets is proposed in [150] for secure wireless communications. In [151], frequency offsets optimized with genetic algorithm are used to achieve the physical layer security in both angle and range dimensions. Two timeinvariant rangeangle dependent DM schemes based on timemodulated frequency offsets FDA were proposed in [152] to achieve timeinvariant spatial fine focusing pointtopoint physical layer secure communications. In the DM transmission schemes, artificial noise plays a pivotal role. In artificial noiseaided secure transmission, the transmitter also sends artificial noise as an interference signal of eavesdropper, which only interferes the eavesdropper without affecting the legitimate user. The DM with artificial noise based on random FDA, referred to as the RFDADMAN scheme, is presented in [153] to enhance physical layer security of wireless communications. Similarly, DM with an artificialnoiseaided approach, and FDA beamformingbased approach are, respectively, proposed for the security of proximal legitimate user and eavesdropper in [154] and [155]. In contrast to the singleuser FDADM schemes, multibeam DM synthesis based on FDA is investigated in [156,157,158,159].
As such, DM utilizing FDA is shown to be more robust than using PA antenna, and becomes a preferred choice to achieve enhanced physical layer secure communications. Therefore, FDA based DM has been an intensive research topic, owing to their superior performance to the PA based DM techniques [160,161,162,163,164]. Similar to FDA, a new transmit diversity technique, namely, hybrid code, is presented in [165], where the operation of the PA radar and the functionality of the coherent MIMO radar are unified for use in electronically steered phased array radars. The hybrid code demonstrates low range sidelobes and better angular selectivity. Instead of scanning, the hybrid code obtains the transmit beampatterns for any/all chosen angular directions by signal processing means. The hybrid code is a promising concept that can play an important role in future radar, and communication applications.
6 Future work
This work has looked at broad concepts that provide a basis for continuing research in FDAs. In particular, areas that are suitable for continuing research includes:

1
Optimization of array geometry: Further research may explore optimal geometry for FDA radars. Various array configurations merit further consideration, including parabolic geometries and multiring UCA.

2
Optimization of frequency offsets: More investigations should be carried out to optimally design computationally efficient FDA frequency offsets.

3
Waveform optimization: Most of the investigations to date have focused on FDAs with narrowband signal sources, further efforts are required to its application in wideband and ultrawideband systems.

4
Optimal signal processing: FDAs should be researched in more complex electromagnetic environment such as mainlobe and sidelobe clutters, jamming, and intense interference environment.

5
Advanced applications: Much of the recent work in FDAs has been focused on localization related applications such as navigation and radar. Although some studies discuss their application in various other fields including communication, more research is required to explore its advantages in other fields.

6
Wave Propagation: To mitigate the performance degradation caused by wave propagation, and ensure the applicability of FDA radars in practical applications, improved timevariant focusing techniques with longer dwell time is an emerging research direction in FDAs.

7
Multitarget localization: Although various FDAs with timevariant consideration is proposed for a single target scenario, extension to multitarget detection may also be included in future studies.

8
Hardware Implementation: A future effort may also focus on implementation of timevariant FDAs with hardware implementation.
7 Conclusion
This paper provides an overview of the FDA radar, and briefly discuss its performance and implementation issues with an emphasis on recent research. The FDA radar is a newly emerging concept, which has a rangeangle dependent beampattern. Unlike PA radars, which work with fixed carrier frequencies, the FDA radar employ a small frequency increment across the array elements to achieve DOFs in both the angle and range dimensions. We have briefly analyzed the characteristics of the FDA beampattern from its origin to the current stateoftheart. Many different approaches to FDA radar for improved performance have been discussed briefly. The conventional FDA radar using progressive incremental frequency offsets yields a rangeangle coupled beampattern. To decouple the FDA beampattern in rangeangle dimensions, nonlinear arrays or nonlinear frequency offsets were utilized. However, the resulting beampatterns exhibit timeperiodicity. To deal with these timeperiodic beampatterns, timemodulated frequency offsets were reported, which achieve rangeangle decoupled, and timeinvariant beampatterns at the target position. However, some recent studies indicate that the propagation process of transmitted signals was ignored in the FDA literature. The FDA beampattern is always timevariant in free space. Therefore, the focused beampattern for FDAs using timeinvariant techniques cannot be realized practically. Although some researchers were successful in focusing the transmit energy over the desired target location, the dwelltime is very short. Future work will likely address the timevariant focusing problem with longer dwell time, and extension to multitarget scenarios is also a subject for future work. FDAs were also shown to have provable performance in wireless communication applications such as physical layer secure pointtopoint communication. However, unlike the PA which is a mature technology, FDA has yet to achieve super resolution capabilities, and many practical problems still need to be investigated.
Availability of data and materials
Not applicable.
Abbreviations
 AF:

Array factor
 DOFs:

Degreesoffreedom
 DM:

Directional modulation
 EM:

Electromagnetic
 ESA:

Electronically steerable arrays
 FDA:

Frequency diverse array
 HPBW:

Halfpower beamwidth
 MIMO:

Multiinput multioutput
 OFDM:

Orthogonal frequency division multiplexing
 OSNFO:

Optimized static nonlinear frequency offset
 PA:

Phased array
 PSLL:

Peak sidelobe level
 SFPFDA:

Stepped frequency pulse FDA
 STAP:

Spacetime adaptive processing
 TDFO:

Timedependent frequency offsets
 TMDPFDA:

Timemodulated double parameter FDA
 TMLFO:

Timemodulated logarithmically increasing frequency offset
 TMOFOFDA::

Timemodulated optimized frequency offset FDA
 UCFDA:

Uniform circular frequency diverse array
 ULA:

Uniform linear array
 3D:

Threedimensional
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Ahmad, Z., Chen, M. & Bao, SD. Beampattern analysis of frequency diverse array radar: a review. J Wireless Com Network 2021, 189 (2021). https://doi.org/10.1186/s13638021020636
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DOI: https://doi.org/10.1186/s13638021020636