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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 17, NO. 1, JANUARY 2018
Transmit Beampattern Synthesis for the FDA Radar Baoxin Chen
, Xiaolong Chen, Yong Huang, and Jian Guan
Abstract—The frequency diverse array (FDA) radar has drawn great attention due to the periodicity of the beampattern in range, angle, and time. In this letter, we restudy the recent work that designed a time-invariant beampattern of the FDA radar, which can focus the transmit energy in a desired position. We reanalyze the derivation of the FDA beampattern synthesis and point out the neglected constraint condition, which leads the research of the FDA beampattern synthesis to an impractical direction. By comparing the replication of one of the prior works with our result, we draw a conclusion that it is impossible to obtain a beampattern merely focusing on some specific spatial positions and lasting for some specific time. Index Terms—Beampattern synthesis, frequency diverse array (FDR), range–angle-dependent beampattern, time-invariant beampattern, transmit beamforming.
I. INTRODUCTION HE frequency diverse array (FDA) radar [1] has recently gained great interest due to its range–angledependenteampattern. In [2], the periodicity in range, angle, and time is analyzed based on the continuous wave (CW). It is because of the periodicity of the FDA beampattern that some researchers strive to seek a beampattern that can focus the transmit energy in a desired position and lasting for some time, i.e., the time-invariant and dot-shaped beampattern. The beampattern with a single maximum at the target location is designed in [3]–[8] and in the derivation of which the time variable t is set to be a fixed value, such as 0. In [9], a time-dependent frequency offset is proposed to obtain a time-independent beampattern for a given range–angle position, which is claimed to remain illuminated constantly for the whole pulse duration. A quasi-static beampattern is derived in [10] to alleviate the problem of the time-variant beampattern for pulsed FDA by adding a constraint to neglect the time variable t. In [11], the time-invariant beampattern is synthesized by three convex optimization-based algorithms. Based on [9], two schemes of the frequency offset are modified [12], and the timemodulated optimized frequency offset FDA is then applied in
T
Manuscript received August 29, 2017; revised October 21, 2017; accepted November 9, 2017. Date of publication November 23, 2017; date of current version January 10, 2018. This work was supported in part by the National Natural Science Foundation of China under Grant 61501487, Grant 61401495, Grant U1633122, Grant 61471382, and Grant 61531020, in part by the National Defense Technology Fund under Grant 2102024, in part by the Aeronautical Science Foundation of China under Grant 20162084005, Grant 20162084006, and Grant 20150184003, and in part by the Special Funds of Taishan Scholars of Shandong and Young Talents Program of the China Association for Science and Technology. (Corresponding author: Baoxin Chen.) The authors are with Naval Aviation University, Yantai 264001, China (e-mail:
[email protected];
[email protected]; huangyong_
[email protected];
[email protected]). Digital Object Identifier 10.1109/LAWP.2017.2776957
Fig. 1.
ULA FDA.
the short-range scenario [13] and the multitarget scenario [14]. Wang et al. [15] combine the frequency offset in [12] with the multicarrier FDA. However, the above-mentioned letters [3]–[15] neglect a practical constraint, i.e., leave the propagation process of the transmitted signals out of consideration. Thus, the time variable t cannot be assumed to be a fixed value or merely starting from 0 to T (T is the pulse duration for pulsed FDA). In this letter, by taking the practical constraint into consideration, we find that it is impossible to obtain a beampattern merely focusing on some specific spatial positions and lasting for some specific time. II. ANALYSIS OF FDA BEAMPATTERN SYNTHESIS Consider a uniform linear array (ULA) with M transmit antennas, spaced at d, as shown in Fig. 1. The pulse signal transmitted by the mth antenna is ∗ j 2π f m t sm (t) = wm e ,
m = 0, . . . , M − 1, t ∈ [0, T ]
(1)
where T is the pulse duration, fm is the radiated signal frequency given by fm = f0 + Δfm , and wm is the complex weight used to steer the beam to a specific direction. Then, the overall signal arriving at an arbitrary point (r, θ) (r and θ are the range and the azimuth angle with respect to the first array element) in the far field is S (t, r, θ) =
M −1
∗ wm sm (t − rm /c)
m =0
=
M −1
∗ −j 2π (f 0 +Δ f m ) (t− wm e
r −m d s i n θ c
)
m =0
= e−j 2π f 0 (t−r /c)
M −1
∗ wm
m =0 −j 2π Δ f m (t−r /c)−j 2π f 0 m d sin θ /c−j 2π Δ f m m d sin θ /c
e
1536-1225 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
(2)
CHEN et al.: TRANSMIT BEAMPATTERN SYNTHESIS FOR THE FDA RADAR
99
Fig. 2. Replication results of [12]. (a) Angle–range dimensions at t = 0 s. (b) Range–time dimensions at θ = −15°. (c) Angle–time dimensions at r = 500 000 m.
Fig. 3. Simulation results based on (5) with the same transmitted signals as [12]. (a) Angle–range dimensions at t = 0.0017 s. (b) Range–time dimensions at θ = −15°. (c) Angle–time dimensions at r = 500 000 m.
where c is the speed of light and rm ≈ r − md sin(θ) is the distance between the point and the first array element. Hence, the corresponding array factor can be written as AF (t, r, θ) =
M −1
∗ wm
m =0 −j 2π Δ f m (t−r /c) −j 2π f 0 m d sin θ /c −j 2π Δ f m m d sin θ /c
e
=
e
M −1
e
∗ jψm wm e
(3)
n =0
where ψm = −2π[Δfm (t − r/c) + f0 md sin θ/c + Δfm md sin θ/c], and the transmit beampattern is B (t, r, θ) = |AF (t, r, θ)|2 .
(4)
Note that there is a constraint between t and r, i.e., t ∈ [r/c, r/c + T ] (assuming the transmitted signals are narrow band). This is because that only within the time of t ∈ [r/c, r/c + T ], the signal can reach the point (r, θ), and
beyond that time, there is no signal. Hence, the array factor should be written as ⎧ −1 ⎨ M w∗ ej ψ m , t ∈ [r/c, r/c + T ] AF (t, r, θ) = n =0 m (5) ⎩ 0, otherwise. However, some studies simply calculate the beampattern by assuming t = 0 [3]–[8] or t starting from 0 to T [9]–[15]. It is unreasonable and impractical because the pulse signals have not arrived at the desired position (r0 , θ0 ) at time 0 or T (usually r0 T · c). Take [12] for example. In order to get a timeinvariant beampattern in the position (r0 , θ0 ), Δfm is designed as a time-variant function of t, i.e., Δfm (t) =
ln (m + 1)3 − f0 md sin (θ0 ) /c , t − r0 /c
t ∈ [0, T ]
(6) which is directly substituted in (3), and then the beampattern of all space is calculated with t ∈ [0, T ]. This does not agree with practice situation, and the proper way is to substitute
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Fig. 4.
IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 17, NO. 1, JANUARY 2018
Simulation results of the basic FDA at (a) t = 0.001 s, (b) t = 0.0017 s, and (c) t = 0.0027 s.
Δfm (t − r/c) into (3). Hence, in this letter, we take the practical constraint into consideration and it shows a different result. III. SIMULATION RESULTS AND DISCUSSION In this section, we compare the beampattern with the constraint t ∈ [r/c, r/c + T ] and that with t ∈ [0, T ] in [12]. The simulation results are carried out with M = 10, f0 = 5 GHz, d = 0.5λm ax , Δfm ax = 1 kHz, and the pulse duration T = 0.001 s. The results of one of the methods in [12] are replicated in Fig. 2 with r0 = 500 000 m and θ0 = −15◦ . From Fig. 2(a), we can see that there is one single maximum at the target location, which focuses the transmit energy in a desired spatial region (r0 , θ0 ), and Fig. 2(b) and (c) shows the so-called timeinvariant attribute of the designed beampattern according to [12]. However, it is conformed with the real case, and we argue that with a simple calculation of division and multiplication. As the desired point is 500 000 m away from the array and it costs approximately 0.0017 s for the electromagnetic wave propagating to there, at time 0.001 s, the transmitted electromagnetic wave has not arrived the desired position yet. The propagation of the same transmitted electromagnetic wave is simulated based on (5), which takes the practical constraint into consideration and is shown in Fig. 3. Fig. 3(a) shows the beampattern of angle– range dimensions of the whole space at time 0.0017 s when the wavefront has just arrived at the desired range and indicates that it fails to focus the energy in a desired position. The beam in Fig. 3(c) is angle–time dimensions at r = 500 000 m forming from time 0.0017 to 0.0027 s and is mirror symmetrical to Fig. 3(a). It is worth noting that Fig. 3(b) shows that the beam keeps invariant with the propagation of the transmitted electromagnetic wave. It means that once the pulse signals are transmitted in the space by the FDA, the synthesized beampattern keeps invariant and cannot vary with the spatial position, i.e., that it is impossible to obtain a beampattern merely focusing on some specific spatial positions and lasting for some specific time. This can be explained by studying (3). Assume any two points in the same direction and different range. By the time the electromagnetic wave has just reached the two
points successively, the first term in ψm is zero for both of the two points; thus, the synthesized beampattern keeps invariant with the propagation of the transmitted electromagnetic wave. In (3), the maximum field is achieved when ψm equals an integer multiple of 2π, and for the basic FDA radar with a progressive incremental frequency offset, the following equation is satisfied: Δf (t − r/c) + f0 d sin θ/c = 2πk,
m = 0, ±1, ±2, . . . (7) in which the term Δf md sin θ/c is ignored. Thus, the angle spread Δθ in a pulse duration can be calculated as Δθ = |arcsin (−Δf T c/ (f0 d))| .
(8)
For further and more indicative explanation, the beampatterns of angle–range dimensions for the basic FDA radar with the progressive incremental frequency offset at different times are given in Fig. 4(a)–(c) by using the aforementioned simulation parameters, which show that the shapes of the beampatterns are identical to each other. We also find that the angle spread is about 13° for T = 0.001 s and agrees with the theoretical value calculated by using (8). When the pulse duration T = 0.00001 s, the angle spread is only 0.1°. In this case, the beampattern of the pulsed FDA radar is similar to that of the phased array radar. Note that the replication in this section is based on the method in [12], and the other beampattern synthesis methods for the FDA radar have the same problem. The main differences are Δfm (t) or the structure of the FDA; nonetheless, they do not interfere with the conclusion. IV. CONCLUSION The periodicity of the FDA beampattern in range, angle, and time offers a new idea on electronic beam scanning without phase shifters. In fact, it is an extension of the concept of a conventional electronically steered array. It is important to remark that, whereas a carrier wave is transmitted, the pattern propagates throughout all the ranges with time. Thus, by reanalyzing the derivation of the FDA beampattern synthesis and comparing the replication results with ours, we draw the conclusion that it is impossible to obtain a beampattern merely focusing
CHEN et al.: TRANSMIT BEAMPATTERN SYNTHESIS FOR THE FDA RADAR
on some specific spatial positions and lasting for some specific time. Compared with the phased array radar, the FDA radar is a new scan system, and further research should be conducted on accurate estimation of parameters for CW FDA radar or basic theory such as a matched filter for the pulsed FDA radar. REFERENCES [1] P. Antonik, M. C. Wicks, H. D. Griffiths, and C. J. Baker, “Frequency diverse array radars,” in Proc. IEEE Radar Conf., Verona, NY, USA, Apr. 2006, pp. 215–217. [2] M. Secmen, S. Demir, A. Hizal, and T. Eker, “Frequency diverse array antenna with periodic time modulated pattern in range and angle,” in Proc. IEEE Radar Conf., Boston, MA, USA, Apr. 2007, pp. 427–430. [3] W. Khan, I. M. Qureshi, and S. Saeed, “Frequency Diverse array radar with logarithmically increasing frequency offset,” IEEE Antennas Wireless Propag. Lett., vol. 14, pp. 499–502, 2015. [4] W. Khan, I. M. Qureshi, A. Basit, and W. Khan, “Range-bins-based MIMO frequency diverse array radar with logarithmic frequency offset,” IEEE Antennas Wireless Propag. Lett., vol. 15, pp. 885–888, 2016. [5] H. Shao, J. Dai, J. Xiong, H. Chen, and W. Q. Wang, “Dot-shaped rangeangle beampattern synthesis for frequency diverse array,” IEEE Antennas Wireless Propag. Lett., vol. 15, pp. 1703–1706, 2016. [6] Y. Wang, G. Huang, and W. Li, “Transmit beampattern design in range and angle domains for MIMO frequency diverse array radar,” IEEE Antennas Wireless Propag. Lett., vol. 16, pp. 1003–1006, 2017.
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