Time ‐ variant focusing range ‐ angle dependent beampattern synthesis by uniform circular frequency diverse array radar

The beam steering of phased array (PA) radar is fixed in one direction for all the ranges, thus the resulting beampattern is angle ‐ dependent but range ‐ independent. Compared with the PA radar, the frequency diverse array (FDA) radar can achieve range ‐ angle dependent beampattern by employing a small frequency increment across the array elements. Numerous studies have been conducted using linear FDAs, while other array configurations are seldom studied. On the other hand, the range ‐ angle dependent beampattern generated by the FDA is also time ‐ variant. However, previous studies on time ‐ invariant spatial ‐ focusing FDA beampatterns neglected the propagation process of the transmitted signals, leading the focused beampattern difficult to achieve when the transmit delay caused by wave propagation is considered. Taking the time ‐ variant property of the FDA beampatterns into consideration, in this study, we propose a novel uniform circular FDA (UC ‐ FDA) radar for short ‐ range beampattern synthesis. The UC ‐ FDA radar


| INTRODUCTION
Conventional phased array (PA) radars using fixed carrier frequencies are well-known for their capability to electronically steer a beam for target detection, and to generate an angledependent but range-independent beampattern [1][2][3]. The PA radar can steer the beam electronically by varying the phase of a signal at each array element [4]. However, it is costly to implement expensive phase-shifters at the element level [5].
Recently, a novel electronic scanning array, namely frequency diverse array (FDA), has emerged as an active area of research due to its ability to generate a range-angle dependent beampattern [6]. Different from conventional PA radars, the FDA radar employs a small frequency increment across each element to achieve beam steering as a function of angle, range and time [7]. Moreover, the adoption of frequency shift between the adjacent array elements enables the FDA to provide the beam scanning ability without using the expensive phase-shifters [6,7]. Motivated by these attractive features, the FDA radar has recently become a hot research topic owing to its potential advantages over the conventional PA radar. The range-angle dependent beampattern makes FDA attractive to numerous applications related to localization [8], such as wireless communications, sonar, radar and others.
The concept of FDA with progressive incremental frequency offsets across the array elements was originally proposed by Antonik et al. [6] in 2006. The standard FDA using progressive incremental frequency offsets generates a periodic range and angle coupled S-shaped beampattern. The range-angle coupled beampattern and beamforming using progressive incremental FDA have been investigated in [9,10], whereas the time and angle periodicity of an FDA beampattern were analysed in [11]. However, it is difficult to estimate the range and angle of a target unambiguously from such a range-angle coupled beampattern [12]. To overcome this disadvantage, a number of improved methods have been proposed. One way to mitigate the range-angle coupling in FDA beampattern is to alter the array configuration, for example, non-uniform linear array [13]. Likewise, a multi-carrier FDA framework with symmetrically and logarithmically spaced array elements is proposed in [14] to produce a dot-shaped beampattern for target indication. However, the physical relocation of array elements with accurate placement of transmitter and receiver is impractical at each scanning, and using this technique for beampattern decoupling is therefore not feasible. Another intuitive method to decouple the range-angle beampattern is through frequency increment design. By employing random frequency offsets [15] in the uniform linear array (ULA), the FDA beampattern can be decoupled into the range and angle dimensions, which is much more effective than the array configuration design. Therefore, early studies on FDA were mainly focused on the design of frequency offsets, particularly for decoupled range-angle dependent beampattern.
In past years, remarkable functions have been introduced to generate non-linear frequency offsets, such as square increasing and cubic increasing frequency offsets [16], Hamming windowbased non-uniform frequency offsets [17], Costas sequence modulated frequency offsets [18], piecewise trigonometric frequency offsets [19] and non-uniform logarithmic frequency offsets [20]. Although the range-angle periodicity in FDA beampatterns has been eliminated by the methods proposed in [9][10][11][12][13][14][15][16][17][18][19][20], the time periodicity still prevails. Since the beampatterns generated by the above-mentioned methods are actually timevariant, it will cause difficulties in controlling the main beam direction and improving the target detectability. To deal with this time-periodic beampattern, an FDA with time-dependent frequency offsets (TDFOs) [21] and a pulsed-FDA [22] with constraints on both pulse duration and frequency shift, were proposed to achieve a time-independent beampattern at the target location. Though the beampatterns achieved in [21,22] are time-independent for a particular range-angle pair, 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 [23] to obtain a timeinvariant range-angle-decoupled FDA beampattern. Recently, various improved methods based on a unified configuration of non-linear and time-modulated frequency offsets have been proposed in [24][25][26] to design time-invariant spatial focusing beampatterns. However, the propagation process of the transmitted signals had not been taken into account. The constant range for the focusing point of FDAs would be unattainable when the time-variant property is considered.
As highlighted in [27,28], the FDA beampattern is actually time-variant, but the propagation process of the transmitted signal is usually ignored in the early FDA literatures. Since the FDA beampattern cannot be controlled independently in time and range dimensions [29], the aforementioned FDA techniques are no longer applicable for time-variant beampattern synthesis. In addition, the time-invariant beampattern inevitably leads to a range-invariant beampattern as achieved in conventional arrays [30], and constructing such a time-range invariant beampattern is feasible by using conventional arrays instead of FDAs. It must be remembered that achieving a dotshaped beampattern with FDAs in an instance of time is not in contradiction with the physical phenomenon of electromagnetic (EM) wave propagation. However, it is impossible to form a range-dependent time-invariant beampattern by FDA [29]. More recently, some studies [31,32] also address the propagation process of EM waves in FDA beampatterns. A time-variant short-range focusing beampattern is obtained in [31] by incorporating an arc-tangent function-modulated frequency scheme. Similarly, a multi-carrier short-range symmetrical FDA radar with frequency diverse chirp signal is proposed in [32] to produce a time-variant focused range-angle dependent transmit beampattern by compensating the propagation delays of transmitting signals. Although much attention has been paid to the linear array configurations [33], other array configurations remain very scarce in the FDA literatures. Some interesting investigations on exploring different array configurations for improved performance in far-field beampattern synthesis have been made in [34][35][36][37]. The beampattern analysis on the planar FDA in terms of its auto-scanning ability has been presented in [34]. The three-dimensional (3D) beam steering capabilities and basic analysis of the uniform circular FDA (UC-FDA) and elliptical FDA were discussed in [35,36], respectively. A hemi-spherical FDA is exploited in [37] to demonstrate the unique advantage of conformal FDA. Nevertheless, all these aforementioned FDA techniques only dealt with the long-range focusing beampattern synthesis, but seldom studied the short-range focusing beampattern synthesis. Hence, more investigations are required in FDAs with new application potentials.
In view of these issues, we propose a novel short-range UC-FDA radar in this study to provide time-variant focusing beampattern by adopting pulse-dependent non-linear frequency offsets. The proposed UC-FDA radar can scan the beam azimuthally through 360°, providing a potential solution of 3D beam steering. In addition, the multi-carrier technique is adopted in the proposed UC-FDA radar to get a higher resolution. The performance of the proposed UC-FDA radar is demonstrated with numerical simulations.
The rest of this study is organized as follows. Section 2 introduces the required background on the time-invariant FDA radar. Section 3 proposes the UC-FDA radar for time-variant short-range beampattern synthesis. Section 4 provides the numerical simulation results and related discussions. Finally, Section 5 concludes this study.

| BASIC FDA RADAR AND PROBLEM PRELIMINARIES
In this section, we will review the time-invariant transmit beampatterns respectively synthesized by linear FDA radar and circular FDA radar, which are relevant to the proposed UC-FDA radar in Section 3. AHMAD ET AL. -63
where w m represents the complex weight related to the m-th element, T is the transmitted pulse duration, and f m is the radiation frequency of the m-th element given by [21][22][23][24][25][26][27].
Here, f 0 and Δf m respectively denote the reference carrier frequency and the frequency offset of the m-th element. The overall signal arriving at an arbitrary point Pðr; θÞ (r and θ are the range and the elevation angle with respect to the first array element) in the space can be expressed as [21][22][23][24][25][26][27].
where r m ≅ rÀ md sinθ is the target slant range with respect to the m-th element [38,39], and c is the wave speed. Substituting r m ≅ rÀ md sinθ and Equation (2) into Equation (3), we have [40,41].
where Δf m ð md sin θ c Þ < � π 4 because the maximum frequency offset is far less than the carrier frequency, that is, max fΔf m g ≪ f 0 [23]. Hence, the quadratic phase term Δf m ð md sin θ c Þ can be ignored in the further analysis. The array factor (AF) of uniform linear FDA radar is then derived as [23,24].
and the beampattern towards the desired target located at Pðr d ; θ d Þ is given as [17].
where r d and θ d denote the range and angle of the desired target, respectively.

| Transmit beampattern of UC-FDA radar
Assume a uniform circular array (UCA) composed of M omni-directional elements with radius a as shown in Figure 2. The slant range with respect to the m-th element is given as [35].
where φ is the azimuth angle, and φ m ¼ 2πm M . Substituting Equation (2) and Equation (7) into Equation (3), the overall signal arriving at the point Pðr; θ; φÞ in the space can be written as Similarly, the quadratic phase term Δf m ð a sin θ cosðφÀ φ m Þ c Þ can be ignored in the further analysis because the maximum frequency offset is far less than the carrier frequency.
Finally, the AF of UC-FDA radar is given as [35].
and the beampattern towards the desired target located at Pðr d ; θ d ; φ d Þ is given as

| Problem preliminaries
The problem of central interest herein is that the propagation process of the transmitted signals is ignored in the existing FDA literature. The AF is a function of time t, which indicates that the EM waves propagate at the velocity of light. The essential point is that the term "t À r/c" indicates that the range r and time t are correlated, and their relationship cannot be ignored [27]. As a matter of fact, the time consumed by EM waves propagating at the velocity of light to the desired target location from the array point is neglected in the FDA literature [29]. The time-invariant FDA beampatterns are obtained by assuming the time variable t to be a fixed value (e.g., t ¼ 0) or defining the variable t in the range from 0 to T [32]. Consequently, most signal models in time-invariant theories are actually inappropriate because the design of time-invariant beampattern of FDA will lead to a range-invariant beampattern. It is important to note that these FDA schemes cease to focus the transmit energy at the desired target location when the propagation process of the transmitted signals is considered [30]. In the transmit beampattern derived in (5), the maximum field can be obtained when the phase term satisfies the condition Δf m ðt À r=cÞ þ f 0 ðmd sinθ=cÞ ¼ i, where i ¼ 0, � 1, � 2,… It can be observed that the maximum field depends not only on the angle and range but also on the time. However, the beampatterns of time-invariant FDAs are obtained by neglecting the influence of time on the range that is, t ¼ 0, and a focused range-angle dependent beampattern is generated. On the other hand, when the propagation process of the transmitted signals is taken into account, it leads to the movement of focusing point with time. Therefore, these FDA schemes are unable to focus the transmit energy at the desired target.
On the other hand, ULA configuration is the most common form employed in the FDAs. Nevertheless, so far, little attention has been paid to other array configurations. In general, ULAs cannot provide an appropriate solution to the scenarios that require 360°fields of view. Although 360°scanning of the radiation beam is possible to be achieved by combining multiple linear arrays, the complexity will be significantly increased [42,43]. In contrast to the ULAs, the UCA has many advantages and is suitable for scanning 360°without any variation in gain and radiation pattern.
To demonstrate the distinctive features of UC-FDA, we compare the transmit beampatterns of the PA, standard linear FDA, Hamming FDA and UC-FDA with progressive frequency increment in Figure 3. In this experiment, the simulation parameters are as follows: M ¼ 10, d ¼ λ 0 2 , f 0 ¼ 10 GHz, Δf ¼ 1 KHz and the target location ðr d ; θ d ; φ d Þ¼ (500 km, 30°, 90°). For the Hamming FDA, symmetrical array configuration with 2M þ 1 ¼ 15 elements is used, and for the UC-FDA, the radius is set to a ¼ 0.9λ 0 . Note that Δf ¼ 0 is employed for the PA. The results indicate that the PA has angle-dependent but range-independent To clarify the significance of the propagation process of the transmitted signals in FDAs, in Figure 4, we compare the original time-invariant beampatterns generated by a shortrange FDA [26], a time-modulated logarithmically increasing frequency offset (TMLFO) FDA [23], and a TDFO FDA [21], with their respectively corrected beampatterns. Among them,

| PROPOSED SHORT-RANGE UC-FDA RADAR
In this section, we propose a novel short-range UC-FDA radar for realizing time-variant focusing range-angle dependent beampattern synthesis. The first subsection formulates the mathematical model of short-range UC-FDA radar. The next subsection designs a novel pulse-dependent non-linear frequency offset function for the short-range UC-FDA radar. The third subsection presents the short-range transmit beampattern synthesis of the proposed UC-FDA radar. The final subsection provides performance analysis.

| Mathematical formulation of the shortrange UC-FDA radar
where w m,n is the complex weight associated with the n-th pulse of signal radiated from the m-th element, and is the corresponding frequency with the frequency offset Δf m;n . The overall signal arriving at the point Pðr; θ; φÞ in the space can be expressed as Substituting Equations (7) and (12) into Equation (13), we have Sðt; r; θ; φÞ ¼ ∑ It is noted that unlike the long-range scenarios, the quadratic phase term Δf m;n ð a sin θ cosðφÀ φ m Þ c Þ must be preserved in the analysis of short-range beampattern synthesis considered in this study. Consequently, the phase contribution of the quadratic phase term must be compensated to ensure shortrange focusing performance.
Finally, the AF of the short-range UC-FDA radar is derived as

| Frequency offsets design
Frequency offsets are of great importance to determine the shape of the FDA's beampattern. Uniform frequency offsets yield an S-shaped range-angle coupled and periodic beampattern. The non-linear frequency offsets can produce a dot-shaped uncoupled beampattern. Here, we consider multi-carrier non-linear frequency offsets to achieve an uncoupled FDA beampattern with a single maximum at the target location. The non-linear frequency offset of the n-th pulse from the m-th element is defined as Δf m;n ðtÞ ¼ 2 6 4 αgðm; nÞ À f 0 where, is a non-linear function generated by a logistic map, and α is a configurable parameter to control the non-linear function g (m,n).
Since the transmitted signals can propagate to the target the focused beampattern can only be achieved by assuming the value of t within this range. Hence, the final form of pulsedependent frequency offsets becomes where, the introduced coefficient β satisfies the condition 0 ≤ β ≤ 1. The distribution of various frequency offsets across the array elements are compared in Figure 5. Obviously, the distribution of the proposed frequency offsets is highly random and non-monotonic. Therefore, it will facilitate to achieve an improved uncoupled range-angle dot-shaped transmit beampattern.

| Transmit beampattern synthesis
By substituting Equation (18)   -69 To achieve maximum of the beampattern at the desired target location ðr d ; θ d ; φ d Þ, the complex weights are designed as Consequently, the AF at the desired target position Pðr d ; θ d ; φ d Þ turns out to be ð21Þ and the transmit beampattern of the short-range UC-FDA radar can be written as From (21)-(22), we can see that the AF is a function of time, range and angle. Therefore, the transmit beampattern achieves the maximum at certain instant of time depending on the desired target position.

| Performance analysis
According to the derived transmit beampattern in (21), it is worth highlighting that the maximum of the beampattern can be achieved at the target location Pðr d ; There is only one maximum in the space at a given time instant, and the single-maximum beampattern is time-dependent. Therefore, a time-variant spatial focusing beampattern at the target location is obtained.
In order to achieve a range-angle dependent beampattern, the frequency offsets should satisfy | Δ f m;n | >0. If | Δ f m;n | ¼0, the FDA beampattern degenerates to angle dependent only as that of PA. To ensure a narrowband system, the frequency offsets must satisfy the condition Since the maximum value of frequency offsets with a reference carrier frequency of 10 GHz should be less than 1 GHz, to ensure the frequency offsets condition in (24), we first derive the maximum value of the non-linear function g (m,n) in (18). For μ ¼ 4 and 0<g m,nÀ 1 <1 in (17), it is easy to see that all subsequent values of g (m,n) also lie in this range. In fact, the largest value of g (m,n) (which occurs for g m,nÀ 1 ¼ 1/2) is equal to 1. Besides μ ¼ 4, for almost all initial values, g (m,n) will eventually leave the interval [0,1]. If the initial value g m,nÀ 1 is 0 or 1, all the subsequent values of g (m,n) would be zero. The essential of logistic map is the non-linearity of the proposed frequency offsets. Therefore, the non-linearity of the proposed frequency offsets are assured provided the initial value g m,nÀ 1 is not a 0 or 1. Figure 6 shows the distribution of logistic map versus the initial values g m,nÀ 1 and parameter μ.
As such, if we assume a pulse duration of 20 ns, then the lower and upper bound of α ensuring the frequency offsets condition in (24) is given by Here the bound is obtained with an intuitive understanding that the lower limit ensures the randomness of the frequency offsets, whereas the upper limit guarantees a narrowband system.

| SIMULATION RESULTS AND ANALYSIS
In this section, simulation experiments are carried out to demonstrate the effectiveness of the proposed short-range UC-FDA radar. In our simulations, unless specified otherwise, we assume a UC-FDA composed of 21-element operating at a reference carrier frequency of f 0 ¼ 10 GHz. The radius of the UC-FDA is set to a ¼ 50 mm, and the number of carriers is N ¼ 10. The desired target is located at ðr d ; θ d ; φ d Þ ¼ (15 m, 30°, 150°), and the pulse duration is T ¼ 20 ns. The control parameters of frequency offsets are set to α ¼ 7 and β ¼ 0.5.
The resulted normalized beampatterns at t ¼ 60 ns are plotted in Figure 7. It shows that the proposed UC-FDA radar is able to provide 360°of coverage in the azimuth plane, which cannot be achieved by the linear array-based FDA radar. The desired target is localized in three-dimensions, that is, range, elevation and azimuth, thus, the proposed UC-FDA radar has the potential to be employed when 360°coverage is required in the entire plane of the array. Figure 7 Another group of simulations is performed to investigate the effect of different system parameters on the performance of the proposed UC-FDA radar. Figure 8 shows the comparison of the range profile of the beampattern with different array elements and carrier frequency number. The peak-to-sidelobe ratio (PSR) and half-power beamwidths (HPBWs) for the proposed UC-FDA radar in different parameter settings are tabulated in Tables 1 and 2, respectively. From Figure 8 and Tables 1 and 2, it is observed that as the number of elements increases, the PSR performance of the proposed UC-FDA radar gets improved. Moreover, with an increase in the number of carrier frequencies, the PSR increases and the HPBW gets narrow. Therefore, we can conclude that the HPBW and PSR cannot be arbitrarily small due to the trade-off between the main lobe beamwidth and the peak sidelobe level (P-SLL).
In addition, the effect of the frequency offsets control parameter α on the performance of the proposed UC-FDA radar is also considered, and the HPBW and P-SLL curves with respect to different values of α are plotted in Figure 9. From Figure 9(a), it can be found that when α increases within the bounds provided in Equation (25), the HPBWs for all array settings get narrow modestly. However, it is evident from Figure 9(b) that the P-SLL gets higher as α increases. By observing these results, we can conclude that the performance of the proposed UC-FDA radar has been improved evidently with the help of multi-carrier technique.
Although the proposed UC-FDA radar yields a better spatial focusing performance with the help of multi-carrier technique, a drawback is that it may increase the complexity. However, the multi-carrier technique is optional and can be adopted in the practical applications according to requirement. Moreover, the frequency offsets can be optimally computed by using optimization technique for improved radar performance.

| CONCLUSION
Previous research results have focused largely on evaluating FDA performance under the assumption of ULA geometry. Despite the implementation advantages of other array configurations, they have not been extensively investigated. Moreover, the propagation process of signals is usually ignored in FDA beampattern synthesis. In this study, a novel range and angle dependent beampattern synthesis approach for the UC-FDA radar is proposed. The UC-FDA radar possesses many advantages over linear array-based FDA radar, and the most outstanding of which is the ability to scan azimuthally through 360°fields. Since the UC-FDA radar offers controllable degrees-of-freedom in range, elevation and azimuth domains, 3D localization of the target is achievable. Abbreviations: HPBW, half-power beamwidth; PSR, peak-to-sidelobe ratio.

F I G U R E 8
Comparison of beampatterns in the range dimension with; (a) varying numbers of elements (b) varying numbers of carrier frequencies

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A pulse-dependent non-linear frequency offset function has been utilized in the proposed UC-FDA radar to realize the time-variant spatial focusing in short-range at the target location. The frequency offsets utilize a non-linear function generated by logistic map together with multi-carrier architecture to generate a well-shaped "dot" main lobe. Simulation results demonstrate the effectiveness and applicability of the proposed UC-FDA radar.