Antenna Beampattern With Range Null Control Using Weighted Frequency Diverse Array

The range-angle-dependent beampattern of frequency diverse array (FDA) makes it capable of dealing with range-dependent interference, which cannot be handled by conventional phased array. However, current publications pay more attention to the dot-shaped beam forming, which is not suitable for the range-dependent jamming signal suppression. To address this problem, this paper proposes a weight designing method to control the nulls distribution among range bins in the mainbeam. With the additional degrees of freedom provided by FDA, it is possible to flexibly adjust the positions and notch widths of the range nulls. The performance and effectiveness of the proposed weighted FDA are validated with numerical simulations.


I. INTRODUCTION
Frequency diverse array (FDA) is an array whose array elements have a variety of transmit frequencies, which is apparently different from the conventional phased array [1]- [3]. Phased array adjusts its phase shifts of array elements to steer its beam to the interested azimuthal angle. However, the beampattern of phased array depends only on angle, and phase shifts limit its flexibility in range. By contrast, the array elements of frequency diverse array have different frequencies, which brings in additional degrees of freedom, finally leading to a range-angle-dependent beampattern [4].
Due to its range-dependent advantage, FDA has attracted a growing attention and many research works have been published to explore its further application. As first proposed by Antonik et al., FDA generates an 'S'-shaped beampattern, which scans periodically in range and angle [5]. [6] designed a continuous FDA system and evaluated its performance by simulations. In [7], FDA technique was employed into MIMO The associate editor coordinating the review of this manuscript and approving it for publication was Irfan Ahmed . radar system. Reference [8] studied the periodicity of the beampattern of FDA in terms of range, angle and time. The coprime FDA with coprime frequency offsets was reported in [9] for multi-target localization. In [10], the multipath characteristics of the FDA over the ground plane was investigated. Reference [11] analyzed the range-angle coupled beamforming with the FDA. As stated in [12], a linear frequency modulated continuous waveform system for FDA is exploited and proved to be feasible. Additionally, FDA was applied into synthetic aperture radar (SAR) imaging, the range-dependent beampattern increases the synthetic aperture in spotlight SAR, resulting in a higher azimuth resolution [13]. In [14], FDA was adopted to increase the range resolution in SAR processing. Reference [15] exploited the FDA technique in space-time adaptive processing (STAP) radar to improve the range ambiguous clutter suppression. Besides, FDA also has been employed for forward-looking ground moving target indication (GMTI) radar [16].
The traditional FDA beampattern is range-angle-coupled. To address this issue, many researches have been carried out. Reference [17] reported an FDA with nonuniform elements VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ distribution to implement range-angle imaging of interested targets. In [18], an FDA with logarithmically increasing frequency offset was proposed, the range and angle of the beam were decoupled to some extent. In order to overcome the time-dependent effect, an FDA scheme with time-dependent frequency offset was studied and a time-independent beampattern for a fixed range and angle would be formed [19]. In [20], a time-modulated optimized frequency offset with optimal algorithm was introduced to FDA to obtain a timeinvariant spatial fine focusing beampattern. We have studied the symmetrical logarithmic FDA with multi-carrier frequency technique in [21] to accurately focus the mainbeam.
To the best of our knowledge, current researches on FDA are focused on concentrating beam energy to a given spot in terms of beampattern. In the practical field, sometimes it is necessary to suppress the beam energy to one or several positions, such as SAR anti-jamming technique [22]. Generally, we tend to use conventional phased array to build notch filters to obtain some desired null directions in its beampattern [23]. However, as mentioned before, phased array can only implement null placements in certain directions and cannot deal with the range-dependent interference due to its limitation on degrees of freedom [24], [25]. Thus, we investigate into the FDA and establish a range-dependent interference suppression filter based on the FDA. An easily implemented method is proposed to obtain some range nulls according to the given positions. The remainder of this paper is organized as follows. Section II presents the range-dependent problem existing in the current radar system. The basic concept of FDA is given in Section III. Then the range-dependent interference suppression approach is provided in Section IV. To validate the effectiveness of the proposed method, Section V shows some numerical simulations demonstrating the performance of the propose weighted FDA. The conclusion is drawn in Section VI.

II. MOTIVATION
The conventional radar systems suffer from the threat of range-dependent interference. Let us take deceptive jamming in SAR for example [26]. As denoted in Fig. 1, the radar platform transmits beam to the interested spot and receives the echo for the following processing. Suppose that P is a target in the scene and J represents a hostile jammer. When the jammer intercepts the radar signal, it generates a series of false replicas of the intercepted waveform and confuse the radar by sending back the false simulated waveform repeatedly. The false targets dwells in the radar main beam, and thus it is difficult to distinguish the true target from the false ones [27].
Although phased array can be utilized to generate several null directions to avoid certain azimuth-dependent incoming waves, it is obvious that the interference source may appear in the mainbeam and dwell at certain range bin, as shown in Fig. 1, and the phased array is incapable of solving this problem. Instead, the FDA provides a new way of avoiding the range-dependent deception. By introducing additional degrees of freedom, the FDA can generate a joint range-angle-dependent beampattern. However, most current investigations are focused on the beam energy concentration on certain range bins, researches seldom consider the notch formulation among range bins. Thus, it is necessary to design a practical method to suppress the range-dependent interference.

III. SIGNAL MODEL OF FDA
The signal model of FDA is given in this section. Suppose the target is located at (r, θ), and the far field assumption is adopted. There are M array elements of the FDA. The frequency offset of each element is denoted as f m , and it can be expressed as where f 0 represents the carrier frequency, f is a tiny frequency shift. Usually, f is far less than the carrier frequency. The incremental frequency offset is f m = m f . In order to implement the FDA scheme, the architecture of FDA is denoted by Fig. 2. The signal of each array element is weighted before being transmitted. The complex weights are composed of amplitudes and phases. The amplifier a m can be regarded as the amplitude weight and the phase shifter φ m the phase weight.
Thus, according to the above array structure, the transmitted signal by the mth element of FDA can be expressed as where ψ m means the phase difference between the mth element and the reference element. For a target located at (r, θ), ψ m at t = 0 can be written as where c is the light speed. d is the element spacing. Note that the far field assumption is adopted in (3), and thus the range r m between the mth element and the target can be given by If the weights are all set to one, the transmit beampattern of FDA can be expressed as and thus the array factor of FDA can be solved as given by The beampattern of FDA can be then depicted as Fig. 3. In the simulation, the carrier frequency is 10GHz, the incremental frequency 1kHz, and the number of elements is 9. Suppose that there is a jammer in the scene. It can be found from Fig. 3(b) that it is impossible for phased array to keep the beam away from the jammer, if it happens to dwell in the path of mainbeam pointing to the interested scene. By contrast, the FDA can manage to avoid the jammer with its rangedependent bent beampattern.

IV. RANGE-DEPENDENT INTERFERENCE SUPPRESSION BASED ON FDA
In order to decouple the range and angle for FDA, some schemes such as Log-FDA are proposed, and their key aim is to focus the energy to a certain spot. Instead of steering the beam to a particular range bin, we need to allocate a minimum of energy into particular range cells in the range-dependent interference suppression, like the case in deceptive jamming suppression. To achieve this goal, we propose a weighting design method by selecting a series of proper weights of the FDA so that nulls (transmitted energy is zero) go to particular range cells in the beampattern. In this manner, the range-dependent undesirable interference, noise, or jamming signals can be reduced or completely removed in range dimension.
In order to make nulls going to particular range bins, it is necessary to construct proper weights for the FDA. According to (6), the weighted array factor of FDA can be written as where w m denotes the weight of the mth element. For simplicity, let It should be noted that the interference or jamming signals are in the mainbeam. Thus, the range pattern along the VOLUME 8, 2020 mainbeam is taken into account and herein, let us consider the case of side-looking SAR, in which its squint angle is zero, that is, θ = 0. Then, can be rewritten as Observe (8) and (9), we can find that, with help of the range-dependent FDA, there are M − 1 degrees of freedom in the weighted array factor. Note that the maximum number of the nulls that can be built in range is dependent on the number of the number of the array element. If the number of element is M , the number of the nulls in range should not be larger than M − 1. Herein, we define the degrees of freedom as the maximum number of nulls in range that the FDA is capable to generate. Thus, the degrees of freedom can be regarded to be M − 1. This in turn tells us that the number of nulls is dependent on the number of array elements. Notice that the polynomial in (8) is numbered starting from zero, and we will have M − 1 nulls starting from 0 to M − 2. Then the array factor can be rewritten as where ψ m is the mth null appearing at the wanted range bin. Let (8) equals to (10), and the weights can be figured out.
Note that for simplicity, suppose w M −1 = 1, and we have . . .
Thus, due to the fact that the FDA has additional degrees of freedom in range, it is possible to make M − 1 nulls along range, resulting in range-dependent interference suppression.

V. NUMERICAL SIMULATIONS
In order to validate the performance of the proposed method, numerical experiments are carried out in this section. In these experiments, unless stated otherwise, the carrier frequency is set to be 10GHz at X-band, then wavelength λ = 0.03m, the element spacing d is half wavelength 0.015m, the frequency offset f = 1kHz, the range we observe is fixed from 300km to 600km for comparison.
First let us consider the case that the number of element M = 3. Suppose that we want to construct two nulls of the mainbeam in range dimension, whose range bins are located at r 0 = 350km and r 1 = 525km, respectively (that is, 1/6 and 3/4 of the range region). Then, two nulls of the array factor are deduced accordingly, as given by Let w M −1 = w 2 = 1 for simplicity. The array factor then becomes Let (13) where w denotes the weight vector and the superscript T is the transpose symbol. The weights can be figured out based on the given positions of nulls. The beampattern in range is depicted by Fig. 4. It can be seen from the figure that the conventional FDA (green line) without proper weights will generates two fixed nulls at 350km and 550km, respectively, whose positions are fixed and cannot be adjusted. The Log-FDA can only concentrate the energy to a maximal point to some extent, and no nulls would be made. By contrast, the proposed weighted method (blue line) has the capability to generate whatever nulls we want within the given range region by properly adjusting the weights. Two range nulls of the mainbeam appear exactly as we specified. Note that Fig. 4 is presented to compare the capability of different FDAs in constructing range nulls at specified range location. It is necessary to emphasize that the so-called mainbeam interference is an important problem existing in the phased array. Our final goal is not to divide the main beam, but to make several range nulls for eliminating jamming and keep the real interested area illuminated, no matter the jammer appears in the redundant mainlobe or in sidelobes. Actually, the main beam is comparatively large and redundant, particularly in the phased array case. Only part of the 50110 VOLUME 8, 2020 main beam is used for the interested scene. Thus, unless the jammer appears in the scene, which is definitely impossible to eliminate by designing beampattern, the jamming signal from the mainbeam of phased array can be suppressed by controlling the nulls in range with weighted FDA.
In order to observe the performance clearly, the beampatterns of different FDAs are presented in Fig. 5. The performance of the standard FDA can generate an S-shaped pattern, but the nulls are fixed and cannot be adjusted. It is impossible to control the null to the jammer. The Log-FDA can focus the energy to a certain spot. However, no null can be made in this case. By contrast, the proposed weighted FDA can control the positions of the nulls in range and maintain the patten at the same time to facilitate the SAR imaging.

2) CASE 2
In most FDA cases, the aim of the beampattern synthesis is to focus its energy to a certain spot for target detection. However, in the application of SAR deceptive jamming suppression, the range-dependent beampattern should be maintained for the imaging. The detailed discussion about the azimuth resolution of FDA-SAR and its imaging process will be given in Section V-C. Current optimization problems for FDA pattern are mainly focused on the concentration of energy to a specified spot as well. The cost function can be then built through minimizing the energy of other regions and maintaining the energy at the interested location as a constrained condition. In this paper, it is important to maintain the S-shaped beampattern to increase the SAR resolution and make several wanted nulls in range, which can not be realized by conventional phased array and the standard FDA. Thus, the common optimization technique is not the most proper solution to the problem encountered, although it is verified to be effective in many other cases [28], [29]. Fig. 6 indicates the beampattern synthesis result of the FDA using genetic optimization algorithm [30]. In the simulation, same parameters are employed and the interested target is set at (450km, 0 • ). It is obvious from the figure that a dot-shaped pattern is well formed and the energy is focused on the interested spot. However, it cannot be perfectly employed into the SAR system. The azimuthal observation angle is too small to achieve a high azimuth resolution of the final image. Thus, the linear FDA structure should be maintained and the S-shaped beampattern is needed to keep the advantage of FDA in the SAR processing.

3) CASE 3
Note that the limitation of the proposed method is that only M − 1 nulls at most can be designed, which means that the number of nulls are dependent on the number of elements. If more nulls are wanted, it is necessary to increase the number of elements. Fig. 7 shows the case when M = 5. There are four nulls in the scene, locating at 350km, 450km, 487km, and 524km. It is obvious that four notches are perfectly constructed.

4) CASE 4
Let us take the squint SAR mode into consideration, namely, the squint angle does not equal to zero. The corresponding results in the squint mode with the proposed FDA is denoted by Fig. 8. In the simulation, the azimuthal angle is set to be π/4. It can be seen from the figure that, the S-shaped beampattern is well maintained. Due to the fact that the receive direction of the squint SAR system is fixed at π/4, we need to observe the performance in this particular angle. The range profile is shown in 8(b). It is obvious that the result VOLUME 8, 2020  appears to be satisfactory as it is in the side-looking mode. It further turns out to be that the proposed method can be utilized successfully in the squint SAR mode.

B. TWO NULLS IN RANGE WITH ADDITIONAL DEGREES OF FREEDOM 1) EXAMPLE 1
Let us consider the case when the number of nulls is less than M − 1, which means that additional degrees of freedom are in hand. The equations in (11) become invalid. Some modifications should be taken to meet the requirement. Thus, let us take two nulls for example. Suppose that the two notches are still located at 350km and 525km, the factor array can be rewritten as where c 0 and c 1 are two coefficients, which can be obtained by c 0 = ψ 0 ψ 1 and c 1 = −(ψ 0 + ψ 1 ), respectively. In order to measure the performance of (15), simulations are implemented as indicated in Fig. 9. In the simulation, M = 5, M = 9 and M = 13 are adopted, respectively. It can be seen from the figure that two nulls are formed for different M . The difference is that, with the increase of M , more shallow notches appear. This is due to the inherent character of the FDA. If it is necessary to avoid this phenomenon, high order nulls method can be employed.

2) EXAMPLE 2
When the number of nulls is less than M − 1, there are additional degrees of freedom left, which means that we can utilize these degrees of freedom to construct more notches. One plausible idea is to let several additional nulls positioned at the required place, that is, high order null can be formed. Let us still take the two nulls problem for example. Suppose that M = 5, and we have two more degrees of freedom. Herein, r 0 = 350km is set to be a first-order null and    (18) Note that in (18), w M −1 is still set to be 1. Thus, the normalized weights can be figured out according to (18), as given by where C n 3 , n = 1, 2, 3 denotes n-combinations of a set of 3. With the weights provided by (19), it is easy to draw the range profile of the weighted FDA, as shown in Fig. 10. By comparison with Fig. 9, it can be found that the third-order null leads to be a wide notch at r 1 , and no other unwanted notch appears. Meanwhile, the third-order null at r 1 has a larger notch width than the first-order null at r 0 . Thus, it demonstrates that a higher order null can be introduced, if it is necessary to build a wider notch in range. In addition, the red line indicates the the case that M = 9, r 0 is the first-order null and r 1 is the 7th null. By contrast, the 7th notch at r 1 is obviously wider than that of the third-order one.
In contrast, Fig. 11 shows the comparison between the case of first-order r 0 , third-order r 1 and that of both second-order  r 0 and r 1 . The weights can be solved in a similar way as indicated in (18). It can be concluded from the figure that, with the additional degrees of freedom, it is possible to design the size of range notches, further impacting the performance of interference suppression.
The case of M = 9 is depicted in Fig. 12, which demonstrates that the bigger the number of array elements is, the more flexible it is for the FDA to design nulls in range dimension.

C. DISCUSSION
The above analysis investigates the range performance of the weighted FDA. The azimuthal angle of the FDA seems to be still large compared with phased array. In some signal processing fields, such as direction of arrival (DOA) estimation, the aim is to narrow the angle to have a higher angle resolution for the incoming wave [31]. However, in the SAR deceptive jamming suppression, due to the character of SAR data collection mode, it is better to enlarge the angle of the transmit beam.
Take the stripmap SAR for example, the geometry is denoted as Fig. 13. The antenna moves with the platform along a straight line, and its beam is fixed steering to the broadside. With the radar moving forward, a strip observing area is formed. v is the velocity of the platform, λ the wavelength. Assume that P is a point target in the scene, the time duration that the scanning beam covers the point target is called synthetic aperture time T s . The synthetic VOLUME 8, 2020 aperture length can be expressed as When the observation angle of the target is θ, the corresponding Doppler frequency becomes [32] The observation angle varies during the time that the beam scans the target, and the related Doppler bandwidth can be written as where −θ 1 is the squint angle when the beam first illuminates to the target, θ 2 is the squint angle when the beam ends scanning the target. Due to the fact that both θ 1 and θ 2 are small, sin θ can be approximated as θ . Thus, we have where θ = θ 2 + θ 1 represents the beam width. The time width after azimuth pulse compression (azimuth focusing in SAR imaging) can be obtained based on the Doppler bandwidth of the target, as given by The azimuth resolution ρ a of the target can be achieved by multiplying the time width with the platform velocity, It should be noted that if a single horn antenna is employed, we have θ = λ/D, where D is the length of the cross-range antenna aperture. ρ a also can be rewritten as ρ a = λ/(2 θ ) = D/2 [33]. Therefore, for weighted FDA, it is obvious from (25) that a better resolution can be obtained if a bigger azimuth angle is given. This in turn proves that it is not necessary to narrow the azimuth angle of the transmit beam in SAR imaging. Fig. 14 shows the imaging result with the proposed weighted FDA. It is obvious that the target is well focused. Besides, [13] illustrates that the S-shaped beampattern of FDA can increase the virtual synthetic aperture and further giving rise to a higher azimuth resolution, as indicated in Fig. 15, in which the azimuth profiles of the imaging results of the proposed weighted FDA based SAR and the conventional SAR are compared. It can be found from the figure that the mainlobe width of imaged point target after interpolation with the weighted FDA-SAR (red solid line) is narrower than that of the conventional SAR (blue dashed line), showing that a higher azimuth resolution is obtained.
Note that most researches on the FDA beampattern synthesis relies on the frequency diverse transmitting to achieve beam narrowing effect in azimuth. However, this paper aims to exploit the advantage of FDA bent beampattern to facilitate SAR imaging and control the nulls in range to suppress the range-dependent jamming at the same time. Additionally, the range ambiguity problem often appears in the spaceborne SAR imaging, especially for the high-resolution wideswath case. Usually the PRF needs to be high enough to prevent the SAR system from azimuth ambiguity. However, when the task is to obtain a wide swath image, range ambiguity would occur due to the high PRF. Reference [34] provides an effective solution by introducing the frequency diverse array technique and decouple the range and angle to make each sub-swath unambiguous. This paper proposes a weighted FDA to control nulls in range for the rangedependent interference suppression, especially in the case of SAR anti-deceptive-jamming. The platform may be an aircraft, and the observation scene is not that large. The range ambiguity is not the major concern of this paper. The rangedependent interference suppression in the high-resolution wide-swath spaceborne SAR needs to be studied in the future.
To sum up, the adoption of weighted FDA can provide more degrees of freedom so that several nulls in range can be flexibly made, contributing to the range-dependent interference suppression in SAR anti-deceptive-jamming which cannot be handled by conventional phased array. Meanwhile, the azimuthal angle expansion of FDA would not degrade the performance of SAR imaging in azimuth. Instead, the beampattern of FDA helps increase the system's azimuth resolution.

VI. CONCLUSION
In this work, the transmit beampattern of weighted FDA is studied. To overcome the range-dependent interference or jamming signal in mainbeam, a weights designing method is proposed. Several nulls at arbitrary range bins can be accordingly formed. Besides, with additional degrees of freedom, the weighted FDA has the capability of building high order null, leading to broadening one/multiple target notch width. In particular, related analysis proves that FDA would not reduce the quality of azimuthal performance in SAR imaging system. He has published more than 80 academic articles and two books. He holds 36 national invention patents. His current research interests include radar signal processing, especially for marine target detection, moving target detection, micro-Doppler, and clutter suppression. He has given more than 20 speeches of radar signal processing, especially marine target. Since 1999, he has been with Duke University, Durham, NC, USA, where he is currently a Professor of electrical and computer engineering. He has authored over 400 articles in refereed journals and 500 papers in conference proceedings. His research interests include computational electromagnetics and acoustics, inverse problems, and their application in nanophotonics, geophysics, biomedical imaging, and electronic packaging. He is a Fellow of the Acoustical Society of America, the Electromagnetics Academy, and the Optical Society of America. He was a recipient of the 1996 Presidential Early Career Award for Scientists and Engineers from the White House, the 1996 Early Career Research Award from the Environmental Protection Agency, the 1997 CAREER Award from the National Science Foundation, and the ACES Technical Achievement Award, in 2017. He has served as an IEEE Antennas and Propagation Society Distinguished Lecturer, from 2014 to 2016. He is currently serving as the founding Editor-in-Chief for the new IEEE JOURNAL ON MULTISCALE AND MULTIPHYSICS COMPUTATIONAL TECHNIQUES, the Deputy Editor-in-Chief for Progress in Electromagnetics Research, an Associate Editor for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, and an Editor for the Journal of Computational Acoustics. VOLUME 8, 2020