Analysis of beampattern dwell time for planar frequency diverse array

Hui Chen, School of Information and Communication Engineering, University of Electronic Science and Technology of China, No.2006, Xiyuan Ave, West Hi‐Tech Zone, 611731, Chengdu, Sichuan, P.R.China. Email: huichen0929@uestc.edu.cn Abstract Frequency diverse array (FDA) has received much attention due to its both range– angle‐dependent beampattern. However, most of the existing works focus on the linear FDA, and time‐varying characteristics of its beampattern only give a rough qualitative analysis. To further study the spatial beampattern dwell time issues, the pulsed‐planar FDA measurement model is first given, and then, the quantitative analysis of beam dwell time which can be affected by observation point location (azimuth, elevation, and range) and array antenna frequency offset coding method for a given pulse transmitting signal is presented. Finally, the theoretical analysis is proved to be consistent with the simulations.


| INTRODUCTION
Linear frequency diverse array (FDA) radar was first proposed and analyzed by Antonik and Wicks et al. [1] for range-angle-dependent beamforming. The Ref. [2] has analysed the beampattern periodicity in range, angle and time for continuous wave. Recently, most existing works focus on its angle-range-dependent beampattern characteristics, and FDA has been extensively suggested for applications due to its angle-range-dependent beampattern, such as target range-angle localization [3][4][5], secure wireless communication [6][7][8], and synthetic aperture radar imaging [9,10]. While angle-range coupling of the FDA beampattern has also been shown in Ref. [2], which means that the FDA is not conductive to controlling beam energy focusing. So many frequency offset coding methods [11][12][13][14] are designed to decouple the angle-range beampattern. Note that they only consider the static beampattern without the impact of time. To address this problem, a quasi-static beampattern is obtained by solving a convex optimization problem [15] or adding a constraint to alleviate the time-variant beampattern [16]. Besides, several time-invariant beampattern design schemes have been proposed [17,18] by using timemodulated frequency offset, which has been proved impossible by the studies [19][20][21][22]. What is more, only qualitative analysis is studied for the time-varying FDA beampattern in the latest relevant research.
Another research topic of FDA is to exploit the array geometry characteristics [23][24][25][26]. As we know, linear FDA has the azimuth-range-dependent beampattern, whose radiating directionality of its beampattern cannot satisfy practical usage. So, the planar FDA is explored for some special applications, such as secure communication and identification of friend or foe fields. Unlike conventional planar phased arrays that allow the synthesis of the far-field radiation patterns as function of the azimuth angle and elevation angle, the beampatterns generated by planar FDAs also depend on the range and the time t. In Ref. [23], the authors studied the beamforming theory for planar FDA and gave spatial pattern snapshots from the transmit and receive aspects. Ref. [26] mainly focuses on the beampattern characteristic analysis of planar FDA and auto-scanning feature of its beampattern. To further study the time-variant characteristic of the planar FDA beampattern, the quantitative analysis of beam dwell time for pulsed planar FDA at a given observation point in theory, which is also the biggest contribution of our work, is first provided.
A FDA with more general geometry is proposed to analyse spatial beampattern dwell time, which benefits studying the FDA beampattern characteristics quantitatively and practical application. The remainder of the paper is organized as follows. Section 2 introduces the formulation of planar pulsed-FDA signal model. Section 3 derives theoretical analysis of beampattern dwell time. Next, numerical results and discussions are provided in Section 4, and conclusions are finally drawn in Section 5.

| FDA SIGNAL MODEL
Without loss of generality, consider a N � M-element planar FDA, as shown in Figure 1. Take the antenna located at (0, 0, 0) as a reference, the frequency of the (n, m)th antenna is and Δf nm are the reference frequency and frequency offset, respectively, of the (n, m)th element with Δf nm ≪ f 0 .
The transmit signal of the (n, m)th antenna is where a nm (t) is the propagation fading factor, and then the delayed signal observed at the observation point is Then, the eclectic field of a continuous-wave FDA at an arbitrary far-field observation point (r, θ, ϕ) is approximately as Without loss of generality, the propagation fading factor a nm (t) can be ignored, that is a nm (t) = 1, for any t, and the phase of the (n, m)th element is φ nm . The time delay τ nm ¼ r nm − r → nm ⋅b r c with the light speed c, the position vector r → nm of the (n, m)th element, and the uniform propagation vector b r. More specifically, r where x nm , y nm , and z nm represent the coordinates of the (n, m)th element in the global Cartesian coordinate system; b u x , b u y , and b u z are the unit vectors along the x−, y−, and z−axis, respectively. The uniform propagation vector is b r ¼ ub u x þ vb u y þ cosθb u z with u = sin θ cos ϕ and v = sin θ sin ϕ.
Consider the planar pulsed-FDA, whose receiving model can be written as where U(t) is the rectangular pulse with a pulse duration T d and pulse repetition interval T, satisfying T d ≤ T. Note that, without special stated, one pulse planar FDA is employed to derive the transmit beampattern and analyse its corresponding dwell time.

| THE DWELL TIME ANALYSIS
In this section, taking uniform rectangular planar array as an example, assuming all elements of the planar FDA are ideal omnidirectional and noiseless, the spacing along the xÀ axis and yÀ axis are d x and d y , respectively, the location of the reference element is (0, 0, 0), and the linear frequency offset is adopted in the array, that is Δf nm ¼ nΔf x þ mΔf y , n ¼ 0, …, N À 1; m ¼ 0, …, M À 1, and Δf x ≪ f 0 , Δf y ≪ f 0 ; The geometry of the planar frequency diverse array Eðt; r; θ; ϕÞ ¼ ∑ then, the electric field at the observation point (r, θ, ϕ) is simplified as Equation (6) with the delay τ from the reference antenna to observation point.
To simplify function, we let Then, the beampattern can be obtained by Equation (6) as Fðt; r; θ; ϕÞ ¼ Eðt; r; θ; ϕÞ j j ¼ sin From Equation (7), we can see that the beampattern peak appears when φ Exactly when the beampattern peak occurs at spatial position (r, θ, ϕ) with given Δf x and Δf y . Now take 3dB bandwidth as the boundary to discuss the dwell time of the beampattern, that is 0:707 ≤ sin which is equivalent to following constraints ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 0:707 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 0:707 We know both According to we can obtain the Equation (13)  As shown in Equation (13), the dwell time of the beampattern depends on the frequency offset, observation position, and array geometry. If t 1 and t 2 do not intersect, which means there is no big enough radiation energy in the observation point (r, θ, ϕ).

| SIMULATIONS AND DISCUSSIONS
In order to prove our theoretical derivation, we have carried out the following beampattern dwell time verifications for planar pulsed FDA with Td ¼ 1.2 ms.
In the first simulation, we give the whole pulse transmit beampattern of 4 � 4-element uniform planar FDA using linearly increasing frequency offsets with Δf x ¼ Δf y ¼ 200 Hz at t ¼ 0.24 ms, as shown in Figure 2, which is different from that of the phased planar array, as shown in Figure 3.  Table.1, we can see that the start/end times of t 1 , t 2 for observation position 'P1' are the same, as well as observation position 'P2', while the start/end times of t 1 and t 2 are not equal for observation positions 'P3' and 'P4', which means that the beam energy focusing effect is better, but the duration of beam energy is shorter. In contrast, the beam energy focusing effect is slightly worse, but it may lasts longer at the observation positions, which is also consistent with the simulation curves shown in Figure 4. From Figure 4, we can see that the beam energy focusing effect is better for observation positions 'P1' and 'P2' due to their same start/end times of t 1 and t 2 , while the dwell times of the beam energy at observation positions 'P3' and 'P4' are slightly longer. In addition, 'P1' and 'P2' are closer to the FDA emission source than 'P3' and 'P4', so the radiating energy first arrives 'P1' and 'P2', which is also congruent with electromagnetic wave propagation theory. Note that Figure 4 plots curves of the received signal amplitude over time, while Table 1 shows the start and end times for the 3 dB bandwidth.
Then, we plot the beampattern amplitude with time t under different frequency-offset coding schemes (such as linear frequency offset coding, logarithmic frequency offset coding and square frequency offset coding) for a given observation 'P1', as shown in the Figure 5, which demonstrates that the frequencyoffset coding method affect FDA radiation energy distribution, and here, the linear frequency offset coding method is the best choice for energy focusing and long residence time requirements.
From Figures 4 and 5, it can be seen that the observation point location (azimuth, elevation, and range) and array antenna frequency offset coding method affect the radiation energy distribution of FDA, which gives us a new scheme to design specific energy distribution of the array.

| CONCLUSION
The quantitative analysis of beam dwell time for pulsed planar FDA is first performed, and then, a number of experimental results and theoretical deduction mutually corroborated each other, which will guide our further application research of FDA time-varying characteristics.

ACKNOWLEDGMENTS
This work was supported in part by the National Natural Science Foundation of China (61,871,092) and Fundamental Research Funds for the Central Universities (ZYGX2018J005).