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 range-angle decoupling of the FDA beampattern. The final subsection presents the time-modulated frequency offsets to achieve the time-invariant range-angle dependent FDA beampattern.

### FDA signal model

Assume a ULA composed of *M* omni-directional elements with half a wavelength inter-element spacing as shown in Fig. 4. The signal transmitted by the *m*-th element is [43,44,45]

$$\begin{aligned} s_m(t)= w_m \exp \lbrace -j2\pi f_mt\rbrace ,\quad 0\le m \le M-1, t\in [0,T], \end{aligned}$$

(8)

where *T* denotes the transmitted pulse duration, and \(f_m\) is the radiation frequency of the *m*-th element given by [38,39,40,41,42,43,44,45]

$$\begin{aligned} f_m= f_0+m\Delta f. \end{aligned}$$

(9)

Here, \(f_0\) is the carrier frequency radiated by the first element, and \(m\Delta f\) is the frequency offset of the *m*-th 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]

$$\begin{aligned} \begin{aligned} S(t,r,\theta )=\sum _{m=0}^{M-1}s_m(t-\frac{r_m}{c})=\sum _{m=0}^{M-1}w_m\exp \left\{ -j2\pi f_m (t-\frac{r_m}{c})\right\} , \end{aligned} \end{aligned}$$

(10)

where \(r_m\cong r-md\sin \theta\) is the target slant range with respect to the *m*-th element [45], and *c* is the wave speed. Substituting \(r_m\cong r-md\sin \theta\) and (9) into (10), we have [26, 45]

$$\begin{aligned} \begin{aligned} S(t,r,\theta )=&\sum _{m=0}^{M-1}w_m \exp \Big \{-j2\pi (f_0+m\Delta f)\Big [t-\frac{r-md\sin \theta }{c}\Big ]\Big \}\\ =&\exp \Big \{ -j2\pi f_0(t-\frac{r}{c})\Big \} \sum _{m=0}^{M-1}w_m \exp \Big \{ -j2\pi \Big [ m\Delta f \\&(t-\frac{r}{c})+{m}^2\Delta f(\frac{d\sin \theta }{c})+f_0(\frac{md\sin \theta }{c})\Big ]\Big \}. \end{aligned} \end{aligned}$$

(11)

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]

$$\begin{aligned} \begin{aligned} AF^{FDA}(t,r,\theta )=\sum _{m=0}^{M-1}w_m \exp \left\{ -j2\pi \left[ m\Delta f (t-\frac{r}{c}) + f_0(\frac{md\sin \theta }{c})\right] \right\} . \end{aligned} \end{aligned}$$

(12)

Consider the case that \(w_0=w_1=...w_{m-1}=1\), the FDA transmit beampattern is given by

$$\begin{aligned} \begin{aligned} B^{FDA}(t,r,\theta )=\bigg | \sum _{m=0}^{M-1} \exp\left\{ -j2\pi \left[ m\Delta f (t-\frac{r}{c}) + f_0(\frac{md\sin \theta }{c})\right] \right\} \bigg |^2 . \end{aligned} \end{aligned}$$

(13)

The beampattern can also be expressed as [44, 45]

$$\begin{aligned} B^{FDA}(t,r,\theta )=\bigg | \frac{\sin [M\pi ( \Delta f t+f_0d\sin \theta /c-\Delta fr/c)]}{\sin [\pi ( \Delta f t+f_0d\sin \theta /c-\Delta fr/c)]} \bigg |^2. \end{aligned}$$

(14)

Note that when \(\Delta f=0\), the beampattern becomes [45]

$$\begin{aligned} B^{PA}(\theta )=\bigg | \frac{\sin [M\pi ( f_0d\sin \theta /c)]}{\sin [\pi (f_0d\sin \theta /c)]} \bigg |^2, \end{aligned}$$

(15)

which is only angle-dependent 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]

$$\begin{aligned} w_m=\exp \left\{ -j2\pi \left[ \frac{m\Delta fr_d}{c} - f_0\frac{md\sin \theta _d}{c}\right] \right\} . \end{aligned}$$

(16)

The corresponding array factor becomes

$$\begin{aligned} \begin{aligned} AF^{FDA}(t,r_d,\theta _d)=\sum _{m=0}^{M-1} \exp&\left\{ -j2\pi \left[ m\Delta f (t-(\frac{r-r_d}{c})) \right. \right. \\&\left. \left. + f_0(\frac{md(\sin \theta -\sin \theta _d)}{c})\right] \right\} , \end{aligned} \end{aligned}$$

(17)

and its magnitude squared, known as transmit beampattern, is given as

$$\begin{aligned} B^{FDA}(t,r_d,\theta _d)=\vert AF^{FDA}(t,r_d,\theta _d)\vert ^2. \end{aligned}$$

(18)

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 angle-dependent but range-independent 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 S-shaped 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]

$$\begin{aligned} \Delta ft+\frac{f_0d\sin \theta }{c}-\frac{\Delta fr}{c}=i, \quad i=0,\pm 1,\pm 2,..... \end{aligned}$$

(19)

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]

$$\begin{aligned} t = \frac{i}{\Delta f}-\frac{f_0d\sin \theta }{c\Delta f}+\frac{r}{c}, \end{aligned}$$

(20)

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]

$$\begin{aligned} r = ct-\frac{ci}{\Delta f}-\frac{f_0d\sin \theta }{\Delta f}, \end{aligned}$$

(21)

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]

$$\begin{aligned} \sin \theta = \frac{ci-\Delta f(ct-r)}{f_0d}, \end{aligned}$$

(22)

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 17-element FDA with 10 kHz inter-element frequency offset.

It is clear from (20)–(22) that the FDA beampattern is time-dependent, and its maxima drifts in space with the time, which facilitates the auto-scanning feature of FDA radars. That is to say, the entire space can be scanned without using the expensive phase-shifters. This unique auto-scanning 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 S-shaped 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 auto-scanning 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 forward-looking-radar 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 phased-MIMO radar for range-dependent 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 two-dimensional imaging of targets and suppressing the range-dependent clutters in FDAs are, respectively, given in [84, 85] and [86, 87].

Since the range-angle 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 double-pulse FDA radar was proposed in [88] for the range-angle 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 (SFP-FDA) radar is proposed, which can be regarded as an upgraded version of the double-pulse FDA in [88]. The first pulse of SFP-FDA 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 MIMO-FDA 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 range-dependent targets and interferences is proposed in [92], where the ranges and angles of targets can be solely estimated with MUSIC-based algorithm. Similarly, two FDA-MIMO hybrid radar transmitter design schemes, namely, FDA–MIMO radar, and transmit subarray FDA–MIMO radar are proposed in [93] for range-dependent target localization, where the targets are localized using the beamspace-based multiple signal classification algorithm. Although these methods have improved the localization performance, the range-angle 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 range-angle imaging of targets. Likewise, a new FDA framework is proposed in [95], where the logarithmically spaced array elements were distributed symmetrically to produce a well-shaped “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 range-angle 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.

### FDA with nonuniform frequency offset

Frequency offset design has attracted great interests in the range-angle 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 window-based nonuniform frequency offsets [99], Costas sequence modulated frequency offsets [100], piecewise trigonometric frequency offsets [101], and nonuniform logarithmic frequency offsets [102].

A multi-carrier 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 single-dot and multi-dot shaped range-angle dependent beampatterns are obtained in [104] by using multi-carrier frequency offsets and convex optimization. In [105], a MIMO-log-FDA radar is proposed, where the range bins concept along with logarithmic offset in each subarray of MIMO-FDA is used to produce single maxima for multiple targets present in different range bins. A uniformly spaced linear FDA with logarithmically increasing frequency offset (log-FDA), and nonuniform but symmetric frequency offsets calculated using well known mu-law 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 two-target localization. The unambiguous target localization is achieved in [111] by combining the subarray-based FDA and full-band FDA, as the transmitter and the receiver, respectively. In [112], a gridless compressed sensing-based range-angle estimation algorithm is proposed for FDA-MIMO radar via Atomic Norm Minimization and Accelerated Proximal Gradient. By optimizing frequency offsets with a genetic algorithm, a single-dot and multi-dot 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 inter-element frequency offset is not uniform, therefore the frequency input at *m*-th element is

$$\begin{aligned} f_m= f_0+\Delta f_m,\quad 0\le m \le M-1 \end{aligned}$$

(23)

where \(\Delta f_m\) is the frequency offset of the *m*-th element.

The Hamming window based frequency offsets can be expressed as [99]

$$\begin{aligned} \Delta f_m= \delta \left\{ 0.54-0.46\cos \left( \frac{2\pi m}{M}\right) \right\} . \end{aligned}$$

(24)

This is the general Hamming window equation where \(\delta\) is an adjustable parameter.

The logarithmically increasing frequency offsets are computed as [106]

$$\begin{aligned} \Delta f_m= \log \left( m+1\right) \delta . \end{aligned}$$

(25)

The piecewise frequency offsets based on a simple trigonometric are given as [101]

$$\begin{aligned} \Delta f_m= h(m)\delta , \end{aligned}$$

(26)

where

$$\begin{aligned} h(m)={\left\{ \begin{array}{ll} \cos (2M m+\phi _1), \quad -M_s\le m \le 0\\ \cos (M m+\phi _2), \qquad 1\le m \le M_s. \end{array}\right. } \end{aligned}$$

(27)

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]

$$\begin{aligned} \Delta f_m={\left\{ \begin{array}{ll} m^{2}\Delta f, \\ m^{3}\Delta f. \end{array}\right. } \end{aligned}$$

(28)

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 range-angle 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 time-variant. They are range-angle dependent only for a fixed time, neglecting the influence of time. Nevertheless, the performance degradation is inevitable because the time-periodicity still exists.

### FDA with time-modulated frequency offsets

In order to overcome the time-periodicity in FDA beampatterns, a number of improved methods have been proposed [114,115,116,117,118,119,120,121]. Among those are the time-dependent frequency offsets (TDFO) proposed by Khan and Qureshi [114] and a pulsed-FDA with constraints on both pulse duration and frequency shift proposed by Xu et.al. [115]. Instead of using fixed frequency offsets, the TDFO-FDA considers time-modulated frequency offsets to generate a range-angle dependent as well as time-independent beampattern. The pulsed-FDA is proposed to form a quasi-static range-angle-dependent beampattern by properly choosing the pulse duration and the frequency shift. However, the beampatterns achieved in [114, 115] are time-independent only for a particular range-angle pair but they remain time-dependent for other ranges and angles. To address this issue, a time-modulated optimized frequency offset FDA (TMOFO-FDA) has been proposed in [116], which exploits the combination of time-modulated and nonlinear distributed frequency offset across the array elements to obtain a time-invariant range-angle-decoupled beampattern. By now, various improved methods based on the unified configuration of nonlinear and time-modulated frequency offsets have been proposed in [117,118,119,120,121] to design time-invariant spatial focusing beampatterns.

In general, the time-varying frequency offsets are given as

$$\begin{aligned} \Delta f_m(t) = \frac{\Delta f(m)}{t-\frac{r_d}{c}}, \quad t\in [0,T], \end{aligned}$$

(29)

where \(\Delta f(m)\) is a nonlinear function with respect to the element index *m*.

As an example, we review here the TDFO-FDA [114], time-modulated logarithmically increasing frequency offset FDA (TMLFO-FDA) [116], and the time-modulated double-parameter FDA (TMDP-FDA) [120].

The TDFO applied to the \(m-th\) element is [114]

$$\begin{aligned} \Delta f_m^{TDFO}(t) = m\Delta f (t) = m\frac{1-\frac{f_0\sin \theta _d}{c}}{t-\frac{r_d}{c}}, \end{aligned}$$

(30)

and the array factor at the target point is expressed as

$$\begin{aligned} \begin{aligned} AF^{TDFO}(t,r_d,\theta _d) = \sum _{m=0}^{M-1} \exp &\left\{ j2\pi m \left[ \frac{(t-\frac{r}{c})}{(t-\frac{r_d}{c})}(1- \right.\right.\\&\left. \left. \frac{f_0d\sin \theta _d}{c}) + \frac{f_0d\sin \theta }{c}\right] \right\} . \end{aligned} \end{aligned}$$

(31)

The TMLFO applied to the \(m-th\) element is given as [116]

$$\begin{aligned} \Delta f_m^{TMLFO}(t) = \frac{g(m)-\frac{f_0\sin \theta _d}{c}}{t-\frac{r_d}{c}}, \end{aligned}$$

(32)

where *g*(*m*) is a logarithmic function defined as

$$\begin{aligned} g(m) = \ln {(m+1)}^k, \end{aligned}$$

(33)

where *k* is a control parameter for the frequency offset. The array factor at the target point is then expressed as

$$\begin{aligned} AF^{TMLFO}(t,r_d,\theta _d) =\sum _{m=0}^{M-1} \exp&\left\{ j2\pi \left[ \ln {(m+1)}^k(\frac{t-\frac{r}{c}}{t-\frac{r_d}{c}}-1)\right. \right. \\&\left. \left. +(m\frac{f_0d}{c}) (\sin \theta -\sin \theta _d\frac{t-\frac{r}{c}}{t-\frac{r_d}{c}})\right] \right\} . \end{aligned}$$

(34)

The time-modulated logistic map based frequency offsets and chirpiness constant considered in the TMDP-FDA are defined as [120]

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta f_{m, n}^{TMDP}(t) = \Delta f \frac{p_{m, n}}{t-\frac{r_d}{c}}, \\ u_{m,n}^{TMDP} = \Delta u \frac{2q_{m, n}}{(t-\frac{r_d}{c})^2}, \end{array}\right. } \end{aligned}$$

(35)

where \(0\le n \le N-1\); Here \(\Delta f\) and \(\Delta u\) are the control coefficients for the frequency offsets and chirpiness constant, respectively. It must be remembered that TMDP-FDA utilizes multi-carrier architecture, and frequency diverse chirp signal. The parameters \(p_{m, n}\) and \(q_{m, n}\) are generated by logistic map as [120]

$$\begin{aligned} {\left\{ \begin{array}{ll} p_{m, n} = 4p_{m, n-1}(1-p_{m, n-1}), \quad 0< p_{m, 0}< 1\\ q_{m,n} = 4q_{m, n-1}(1-q_{m, n-1}, \qquad 0< q_{m, 0} < 1 \end{array}\right. } \end{aligned}$$

(36)

and the corresponding array factor at the target point is given as

$$\begin{aligned} AF^{TMDP}(t,r_d,\theta _d) =\sum _{m=0}^{M-1} \sum _{n=0}^{N-1}\exp& \left\{ -j2\pi \left[ \frac{f_0 md(\sin \theta -\sin \theta _d)}{c}\right. \right. \\&\qquad \left. \left. +\Delta f \frac{p_{m, n}(r-r_d)}{r_d}+\Delta u \frac{q_{m, n}(r^2-r_d^2)}{r_d^2}\right] \right\} . \end{aligned}$$

(37)

In this example, we assume a 10-elements ULA operating at a reference carrier frequency of \(f_0=10\) GHz. The inter-element spacing is equal to half-a-wavelength. The pulse duration is \(T = 1\) ms, and the target is located at \((r_d, \theta _d)=\) (500 km, \(0^{\circ }\)). Since the multi-carrier technique is adopted in the TMDP-FDA, the number of carriers considered is \(N = 10\). Fig. 9 provides comparisons of beampattern generated by TDFO-FDA, TMLFO-FDA and the TMDP-FDA. It is shown in Fig. 9a that the time-independent beampattern of TDFO-FDA is periodic in range, and coupled in the range and angle dimensions. The periodicity, and range-angle coupling is eliminated in the time-invariant beampattern of TMLFO-FDA as depicted in Fig. 9b. Although the TMLFO-FDA generates a single maximum beampattern, its spatial focusing performance is degraded due to the high spatial peak sidelobe levels (P-SLL) and a broader spatial half-power beamwidth (HPBW). Furthermore, as Fig. 9c shows, the TMDP-FDA generate a well-shaped “dot” main beam and outperforms the other two schemes with a more focused beampattern due to multi-carrier architecture.

To further demonstrate the resolution and side-lobe 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 TDFO-FDA beampattern exhibit periodicity. However, from Fig. 10b, it is observed that the TMDP-FDA outperforms the TDFO-FDA and TMLFO-FDA with narrow HPBW in range dimension. The TDFO-beampattern exhibit periodicity in range dimension, and the performance of TMLFO-FDA is not satisfactory due to the high P-SLL and the broad spatial HPBW. Instead of a single carrier signal, each array element of the multi-carrier TMDP-FDA transmit a weighted summation of signals with a small frequency offset. Therefore, the effect of multi-carrier architecture on the performance of the TMDP-FDA is also demonstrated in Fig. 10c. It is observed that as the number of carrier frequencies increases, the P-SLL 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 well-developed techniques, other array configurations are also very appealing for non-communication 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 sparse-FDA 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 range-angle estimation with a new type of array antenna, named random frequency diverse array. In [125], a joint optimization design scheme for FDA-MIMO radar is proposed, where virtual coprime planar array with ‘unfolded’ coprime frequency offsets framework is utilized to achieve three-dimensional (3D) localization without ambiguity. Similarly, a new waveform synthesis model of time modulation and range compensation FDA-MIMO is proposed in [126] to achieve a joint range-angle estimation based on the optimized time-invariant and dot-shaped beampattern. The beampattern analysis in terms 3D beam steering and auto scanning capabilities of the uniform circular FDA (UC-FDA) and the planar FDA were discussed in [127, 128] and [129], respectively. A flat-top range-angle dependent beampattern synthesis method has been proposed in [130] for FDA based on second-order cone programming. Similarly, a subarray-based FDA framework is devised in [131] to achieve a range-angle 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 time-invariant range-angle decoupled beampattern, however, some recent studies indicate that FDA beampatterns are always time-variant in free space [133]. The existing FDA techniques designed for the range-angle dependent beampattern synthesis have not considered the propagation process of the transmitted signals, and may suffer performance degradation caused by the wave-propagation [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 time-variant focusing capability.