Beampattern Synthesis for Frequency Diverse Array Based on Time-Modulated Double Parameters Approach

Brazilian Microwave and Optoelectronics Society-SBMO received 12 Feb 2018; for review 08 Mar 2018; accepted 01 Aug 2018 Brazilian Society of Electromagnetism-SBMag © 2018 SBMO/SBMag ISSN 2179-1074 Abstract—The basic frequency diverse array (FDA) using linearly increasing frequency increments generates a range-angle dependent beampattern. However, it is coupled in range and angle dimensions and is also periodic in range and time, making its applications limited. In this paper, a novel FDA beampattern synthesis approach is proposed utilizing the time-modulated double parameters based on the chaos sequence. The chirp signal mechanism is used instead of the single-frequency signal mechanism. Meanwhile, the multi-carrier architecture is used for range-angle decoupling. Satisfactory time-invariant range-angle beampattern can be synthesized for both single and multiple targets locations. Simulation results show the effectiveness of the proposed FDA scheme. Furthermore, comparative study with the existing technology indicates that the proposed approach can provide better performance in spatial focusing and side-lobe suppressing.


I. INTRODUCTION
First proposed in [1] and [2], frequency diverse array (FDA) technology generates beampatterns that depend on the range, angle and time by introducing a tiny frequency offset among the array elements.In a conventional array, the beampattern can be focused to the desired angle utilizing the progressive phase shift (PPS) between the elements [3][4], but the beam steering is fixed in an angle for all range cells.The FDA transmits the beampattern over the desired range and angle thus it can suppress range-dependent interferences and clutters which is desirable in radar applications.Besides, the FDA provides new degrees of freedom in the range, angle and time for designing the array factor, which enables the beam to scan without the need of phase shifters or mechanical steering [5].
Therefore the FDA can offer much capabilities such as synthetic aperture radar imaging [6], auto scanning [7] and localization of targets [8], etc.
In the basic FDA, the linearly increasing frequency increments are applied and result in a beampattern coupled in range and angle domains, which has a 'S' shape [5].This characteristic limits FDA's applications in unambiguously localizing targets in range-angle dimension [9].Moreover, the basic FDA beampattern is periodic in range and time, deteriorating the resulting signal-to-Beampattern Synthesis for Frequency Diverse Array Based on Time-Modulated Double Parameters Approach Tong Mu, Yaoliang Song, Zhonghan Wang School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing, China.mutong@njust.edu.cn;ylsong@njust.edu.cn;zhwang@njust.edu.cninterference-plus-noise ratio (SINR) and target detectability.The range and time periodicity of FDA beampattern is analyzed in [10], and the range-angle coupled beamforming has been investigated using frequency-diverse chips in [11].A logarithmic frequency offset scheme is proposed in [12] which can decouple the beampattern and eliminate the range periodicity.However, its reduced rangeangle resolution and enhanced side-lobes may not be acceptable for most applications.A FDA system based on multi-carrier frequencies with logarithmic frequency offset is proposed in [13], which requires additional convex optimization algorithm for synthesizing the complex element weights to generate dot-shaped beampatterns.Particularly, the beampatterns generated by these approaches are time-variant, causing the loss in SINR and seriously affecting the target detectability because of extremely short duration of the beam illumination at the target position.Time-independent beampattern based on time-modulated frequency offset (TMFO) is proposed in [14], but the range periodicity is still not eliminated.Then, time-invariant beampattern with a single maximum at the target location is achieved in [15] using the time-modulated logarithmically increasing frequency offset (TMLFO).Also in [15], time-modulated optimized frequency offset (TMOFO) is developed for obtaining better spatial focusing performance.Recently, multi-carrier architecture combined with nonlinear distributed frequency offset is proposed in [16] and pencil-shaped time-invariant beampattern is achieved.
The existing research are based on the single-frequency signal mechanism and thus only single parameter modulation is considered, namely the frequency offset modulation.In this paper, a novel FDA beampattern synthesis approach is proposed based on the chirp signal mechanism where frequency offset as well as chirp constant modulation are both considered.The Logistic mapping chaos sequence is applied for the double parameter modulation, and the multi-carrier architecture is employed to decouple the beampattern in range and angle dimensions.Time-invariant beampattern at the desired location can be achieved by the proposed approach.Also, the time window version of the proposed FDA is used for multiple targets scenario.Simulation experiments are carried out to evaluate the performance of the proposed approach.The results are also compared with the Multi-Carrier-FDA (MC-FDA) recently proposed in [16] and it turned out that the proposed FDA can provide improved spatial focusing performance with more effective side-lobe suppression simultaneously.

II. TIME-MODULATED DOUBLE-PARAMETER FDA
A. Signal model Consider an antenna array of M identical transmit elements with uniform spacing d .As shown in Fig. 1, P is an observation point in the far field, and m r is the distance from P to the th m element.Multi-carrier framework is considered in the system thus each transmit element emits a weighted summation of a series of basic signals.Unlike the previous research, the chirp signal instead of the single-frequency signal is used here as the carrier.Suppose the number of carriers at each element is N, then the signal transmitted from the th m element is where , nm a , , nm f and , nm u are the complex weight used for steering the maximum of the beampattern at the target location, radiation frequency and chirp constant for the th n carrier at the th m element, respectively.T is the duration of the transmitted pulse signal.The frequency component , nm f can be expressed as , 0 , , 0 1, 0 1 where 0 f is the reference frequency and , nm f  is the frequency offset.
Therefore, the overall signal observed at an arbitrary far field point ( , ) where c is the speed of light in free space, r and  are the range and azimuth angle from the point to the first element respectively, and m r is the distance between the point and the th m element.
Taking the far field approximation into consideration yields Substituting ( 2) and ( 4) into (3) yields In practical FDA system, the frequency offset satisfies ,, Therefore, ( 5) can be further approximated as ) Then the corresponding array factor (AF) can be obtained q are generated by the chaos sequence based on the Logistic mapping and they are given by 0, , ),1 1 ),1 1 where 0,m p and 0,m q are the initial values for the th m element and satisfy 0, respectively.The transmit beampattern can be decoupled as range and angle parts by applying the multi-carrier architecture [16], therefore the resulting array factor can be expressed as ,,

( )
respectively.Without loss of generality, the time variable is set as 0 t  , and by substituting ( 16) and ( 17) into (15) the array factor can be further deduced as Therefore, the beampattern maximum is generated at the desired location based on the designed complex weights.

C. Beampattern synthesis for multiple targets
For multiple targets scenario, the time window version of the proposed FDA can be used to generate multiple maximums at the positions of different targets.Suppose there are I targets in the space located at ( , )   ii r  respectively, 1,2,..., iI  .Then the duration of single transmit pulse can be uniformly divided into I time windows, as shown in Fig. 2.During the th i time window, the frequency offset and chirp constant for the th n carrier at the th m element are defined as , , 0 = () respectively where 0 () rt is given by and 1, 0 () 0, 0 As a result, by combining the time window subdivision with the chirp signal mechanism, the solution for multi-targets beampattern synthesis is given by  16) and ( 17), the complex weights for steering the maximum at the th i desired target position ( , )   ii r  are designed as ,,

( )
respectively.It can be seen that the angle complex weight needs to be changed for different subpulses, while the range complex weight remains unchanged during the total pulse duration.and reference frequency is 0 5 GHz f  .

III. NUMERICAL SIMULATIONS
In order to avoid the grating lobes in angle dimension, the interelement spacing is set as  respectively.The logarithmic function used in MC-FDA for controlling the frequency offset is also shown in Fig. 3 (c) for comparison.It can be seen that for P and Q , the sequence with respect to each element is nonlinear, non-monotonic, irregular and different from each other.As for Fig. 3 (c), the sequence with respect to each element is nonlinear, monotonic increasing and the same with each other.

B. Results and discussions
If the time is fixed, the beampattern in a 3-D plot in range and angle domains can be illustrated which contains the spatial information of the beampattern.Fig. 4 shows the normalized beampattern of the proposed TMDP-FDA when 0 t  .From Fig. 4 (a) it can be seen that a single maximum of the beampattern appears right at the desired target location.The range periodicity is eliminated and good range-angle decoupling is realized, indicating the satisfactory spatial focusing performance of the proposed approach.For analyzing the side-lobe level (SLL), the generated beampattern is projected into range and angle dimensions, as shown in Fig. 4 (b) and (c), respectively.It shows that the peak SLL in both range and angle dimensions are below 0.1, indicating the effective side-lobe suppressing performance of the proposed method.The results at 0 t  obtained by the MC-FDA are depicted in Fig. 5.It shows that the MC-FDA can also produce fine spatial focusing beampattern with low SLL.
However, the main-lobe beam width in range dimension of the TMDP-FDA is narrower, which means more energy can be transmitted at the target location and target detecting with higher accuracy can be provided by the proposed FDA.The time window version of the proposed TMDP-FDA is used for multiple targets beampattern synthesis.For simplicity, it is supposed that there are two targets located at (450km, 20 )  and (700km, -30 )  respectively.The pulse duration is divided into two time windows, the length of each is 0.5ms .By fixing the time 0 t  and 0.5ms t  respectively, the resulting normalized beampattern for the two targets is shown in Fig. 10   At last, the initial values 0,m p and 0,m q are changed by using different pairs of pseudo random sequences, and it turned out that the resulting beampattern is steady in both spatial focusing and SLL.
It indicates that the TMDP-FDA is robust to the initial values thus initial value optimization is not necessary, which reduces the system design complexity.

IV. CONCLUSION
In this paper, a novel TMDP-FDA beampattern synthesis approach has been proposed.Timeinvariant range-angle beampattern is achieved by using the time-modulated double parameters based on the chirp signal mechanism.It is able to generate fine spatial focusing beampattern at the desired location for both single and multiple targets, and it also provides good performance in side-lobe suppressing.Numerical simulations are carried out to evaluate the performance of the proposed TMDP-FDA, and the main results are compared with the MC-FDA.It turned out that at a fixed time the TMDP-FDA has narrower main-lobe beam width in range dimension, and also, it has satisfactorily lower side-lobes with the time changing in a pulse duration.In addition, the beampattern focusing scope of the TMDP-FDA can be flexibly controlled by adjusting the corresponding coefficients and it also reduces the system design complexity.
In future research, the optimal FDA geometric configuration such as the application of planar or distributed geometry and the receive beamforming for the FDA radar need to be investigated.
Furthermore, for the practical implementation of the proposed FDA scheme, many issues need to be considered such as generating the waveform, eliminating the intermodulation products of the frequency mixers and reducing the structure complexity, etc [10,[17][18].

A
. Simulation setup In this section, simulation experiments are carried out to examine the effectiveness of the proposed TMDP-FDA.The numerical data are obtained by MATLAB R2013b on the PC with CPU frequency being 3.3GHz and RAM being 8GHz, and the processing time for the beampattern synthesis code of the TMDP-FDA is 5.27 seconds.The simulation parameters are set as follows: element number is15 M  , carrier number is 10 N , pulse duration is 1ms T   respectively, and the target location is (500km, 30 )  .The frequency offset and chirp constant controlling coefficients are set as 7 f  and 5 u  respectively.Two pseudo random sequences each containing M values between 0 and 1 are generated and stored to serve as the initial values of , nm p and , nm q .
(a).The beampattern projection on time-range and time-angle dimensions are depicted in Fig.10 (b) and (c) respectively.It is shown that time-invariant multiple targets beampattern with good spatial focusing can be achieved by the TMDP-FDA as well.

Fig. 10 .
Fig. 10.(a) Normalized range-angle beampattern for two targets, (b) projection on time-range dimension and (c) projection on time-angle dimension.Furthermore, the effect of changing the value of frequency offset and chirp constant controlling coefficients to the performance of the TMDP-FDA is investigated.Firstly, the frequency offset is kept as 7 f  and u  changes from 1 to 10 and the values of the -3dB main-lobe width in range dimension and the peak SLL for each u  are obtained.Then, the chirp constant controlling coefficient is kept as 5 u  and f  changes from 1 to 10 and the corresponding values for each f  are obtained.The results are depicted in Fig. 11.On the one hand, when one of the controlling coefficients is fixed and the other one increases, the -3dB main-lobe width in range dimension decreases, which means narrower beampattern can be obtained.On the other hand, the peak SLL maintains its small value around 0.1 when the coefficient changes.Therefore, based on the TMDP-FDA, one can flexibly control the beampattern focusing scope as needed by choosing the appropriate frequency offset and chirp constant controlling coefficients.