# FDA-MIMO Beampattern Synthesis with an Analytical Method

Authors: Changlin Zhou; Chunyang Wang; Jian Gong; Ming Tan; Yingjian Zhao; Mingjie Liu
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

Since the beampattern has the characteristics of range-angle dependence, frequency diverse array multiple-input multiple-output (FDA-MIMO) radar has a good application prospect. There have been many studies to improve the performance of the beampattern by optimizing the frequency offset. However, on the basis of fully understanding the time parameters, the relationship between the array element frequency offset and the beampattern performance still needs to be clarified. Based on a new FDA-MIMO radar framework, this paper presents an analytical solution of the beampattern, which removes the influence of the time parameter. Taking the minimum main lobe as the objective function, an analytical method for solving a better frequency offset is given. Then, a method of using the window function was proposed to reduce the high side lobes of the range dimension. Comparing with the existing FDA radar beampattern design methods, it can achieve a more focusing beampattern. The simulation results verify the correctness of the theory.


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## Body

## 1. Introduction

Since FDA has been first proposed in [1], it has received extensive attention and research from scholars at home and abroad, because its beampattern is range-angle-depend and time-variant [2, 3]. FDA radar has great application value because of its ability of range-angle-dependent [4–6]. To improve its application value, the literature [7–11] studied the time-variant problem of FDA radar. The literature [11] considers the FDA-MIMO radar scheme proposed in [12, 13], which form time-invariant beampattern.The study of FDA beampattern is a vital issue. A logarithmic increase frequency offset scheme was proposed in [14], which can form spot beams in space. The design of frequency offset using window function is another important research direction [15–17]. These methods are all numerical solutions based on the comparison of effects, but there are countless possibilities for numerical solutions. And they did not consider the influence of time parameters. Considering the influence of time parameters [11], a method of increasing frequency combined with window function is proposed in [18]. It is still a numerical solution. In addition to numerical solutions, many scholars have explored the use of heuristic algorithms to obtain the optimal frequency offset [19–21]. However, when the target position changes, the frequency offset needs to be optimized again, which consumes a lot of computing resources. The problem is particularly prominent in large arrays. The most ideal solution is to obtain an analytical solution of the frequency offset, which was discussed in [22] using FDA radar. However, [22] did not consider the influence of time parameters, and the obtained beampattern is an instantaneous beampattern, which cannot be used to track targets.Therefore, it is vital to get rid of the disadvantages of the FDA beampattern such as range-angle coupling and time-variant. In this paper, we use the FDA-MIMO radar system to obtain the analytical solution of the beampattern. And the beampattern is range-dependent and time-invariant. The main contributions of this paper are as follows:(1)
Through phase modulation, the FDA-MIMO radar system we designed can form a spot beam at any position in space. Literature [11] activates the range-dependent property of FDA through the signal processing at the receiver, and the matched filtering at the receiver is related to the frequency offset of the transmitting array element. Opposed to it, this paper weights the transmitted signal so that all the parameters related to the frequency offset are designed at the transmitter. The modulation phase of each element is a function of frequency offset and target position. Moreover, the frequency offset is only related to the number of transmitting array elements(2)
In FDA-MIMO radar, the mathematical relationship between frequency offset and range-angle beampattern has not been clarified, such as [18]. Through the simplified approximation of the pattern, we get that the spot beam formed by the FDA-MIMO radar is a quadratic curve related to the frequency offset, which mathematically explains how the frequency offset affects the performance of the beampattern(3)
We minimize the main lobe area of the FDA-MIMO radar beampattern to obtain the optimal frequency offset that is only related to the number of array elements. Therefore, the frequency offset is optimal for the target at any position in the space. Finally, the side lobes of the range dimension are reduced by using a Kaiser window functionThe rest of the paper is organized as follows. In Section2, we rederived the beamforming theory of the FDA-MIMO radar. Then, a method is proposed to activate the range dependence of FDA-MIMO radar. In Section 3, through a series of approximate simplified calculations, we derive the analytical expression of the FDA-MIMO radar beampattern that depends on the frequency offset. By setting all frequency offsets to 0, the phased array radar beampattern expression is obtained, which further verifies the feasibility of this method. In Section 4, by minimizing the main lobe area of the beampattern, we derive a better offset to form the spot beampattern of the FDA-MIMO radar. In Section 5, we provide some simulations. Finally, concluding summaries are drawn in Section 6.Notation. We use boldface for vectors a and matrices A. Scalar a is denoted by italicized. The transpose and the conjugate are denoted by the symbols •T and •∗, respectively. IM andEM denote M×M-dimensional identity matrix and all-one matrix, respectively. The letter j represents the imaginary unit (i.e., j=−1).


## 2. Analysis of FDA-MIMO Radar System

We consider a collocated narrowband FDA-MIMO radar system withM transmitting antennas and N receiving antennas. Both the transmitting array and the receiving array are uniform linear arrays. The distance between transmitting array elements is dT, and the distance between receiving array elements is dR. The carrier frequency of the transmitted signal is f0. Its configuration is shown in Figure 1.Figure 1
The conventional configuration of FDA-MIMO.The frequency offset of them-th element is fm; then, the transmitting frequency is
(1)fm=f0+fm,m=1,2,⋯,M.Consider [11], the transmitted signal at time t of the m-th element is
(2)smt=ψmtej2πfmt,where ψmt is the baseband envelope of the m-th transmitting array element, which satisfies orthogonality:
(3)∫ψm1∗t⋅ψm2t−τdt=0,m1≠m2,∀τ.We add phase modulation to each transmitting array element, which is easy to implement for MIMO radar. Therefore, equation (2) can be rewritten as:
(4)smt=ψmtejΦmtej2πfmt.Observed from the desired target atr,θ. The received signal related to the m-th transmitting element and the n-th receiving element can be expressed as:
(5)sm,nt−τm,n=ψnt−τm,nejΦmt−−τm,n⋅ej2πf0+fmt−τm,n,n=1,2⋯,N,where τm,n=2r/c−mdTsinθ/c−ndRsinθ/c represents the transmit delay.In the case of far-field narrowband, (5) can be approximately expressed as [11]:
(6)sm,nt′=ψmt2r/cejΦmt′ejφnt′=ψmt′ejΦmt′ejφnt′,where t′ is the time index within pulse and φnt′ can be expressed as:
(7)φnt′=2π∫0t′+mdTsinθ/c+ndRsinθ/cf0+fmdx=2πf0t′+fmt′+f0mdTsinθc+f0ndRsinθc+fmmdTsinθc+fmndRsinθc.Becausefm≪f0, (7) can be reapproximated as:
(8)φnt′=2π∫0t′+mdTsinθ/c+ndRsinθ/cf0+fmdx=2πf0t′+fmt′+f0mdTsinθc+f0ndRsinθc.Bring (8) into (6) and note that t=t′+2r/c, (6) can be rewritten as:
(9)sm,nt′=ψmt′ejΦmt′,ej2πf0t′+fmt′+f0mdTsinθ/c+f0ndRsinθ/c=ψmt′ejΦmt′,ej2πf0t−f02r/c+fmt−fm2r/c+f0mdTsinθ/c+f0ndRsinθ/c.The received signal is first mixed withe−j2πf0t in the analog signal processor and then mixed with ψmt′ in the digital signal processor, as shown in Figure 2.Figure 2
The signal processing at receiver.Then, the relative output of them-th transmitting array element and the n-th receiving array element can be expressed as:
(10)sm,nOutputt′=ξsejΦmt′,ej2π−f02r/c+fmt−fm2r/c+f0mdTsinθ/c+f0ndRsinθ/c.ξs is the signal complex coefficient after matched filtering. The array factor can be expressed as:
(11)AF=∑m=1M∑n=1Nsm,nOutputt′=ξse−j2πf02r/c⋅∑m=1M∑n=1NejΦmt′ej2πfmt.ej2π−fm2r/c+f0mdTsinθ/c+f0ndRsinθ/c.In order to detect the target atrt,θt, the phase modulation of the m-th transmitting array element is given as:
(12)Φmt′=−2πfmt+2πfm2rtc−2πf0mdTsinθtc−2πf0ndRsinθtc.Bring (12) into (11), then (11) can be rewritten as:
(13)AF=ξse−j2πf02r/c∑m=1Mej4π/cfmrt−rej2πf0mdTsinθ−sinθt/c⋅∑n=1Nej2πf0ndRsinθ−sinθt/c.Therefore, the transmit-receive normalized beampattern is expressed as:(14)B=∑m=1Mej4π/cfmrt−rej2πf0mdTsinθ−sinθt/c2⋅∑n=1Nej2πf0ndRsinθ−sinθt/c2.The transmitting-receiving beampattern can be further equivalent to the transmitting beampattern and the receiving beampattern at the receiving end, and the transmitting beampattern can be expressed as:(15)BT=∑m=1Mej4π/cfmrt−rej2πf0mdTsinθ−sinθt/c2.The receiving beampattern can be expressed as:(16)BR=∑n=1Nej2πf0ndRsinθ−sinθt/c2.(14) indicates that the FDA-MIMO radar produces a range-dependent beam with time-invariant characteristics. And regardless of the value of the frequency offset, the FDA-MIMO radar beampattern can achieve the maximum M2N2 at the target point. (15) and (16) indicate that the equivalent receiving beampattern only affects the angle characteristics, while the equivalent transmitting beampattern can affect the angle and range characteristics. Therefore, when designing the FDA-MIMO radar, the variables related to the frequency offset should be implemented at the transmitter. The configuration is shown in Figure 3.Figure 3
Our proposed configuration of FDA-MIMO.Φmt′ is the variable about frequency offset and target space position. Therefore, we can activate the range dependence of FDA-MIMO radar by setting Φmt′.


## 3. Analytical Expression of the FDA-MIMO Radar Beampattern

We hope to form a spot beampattern that can only rely on (15), because (16) shows that the equivalent receiving beampattern can only affect the angular resolution. We use half-power beamwidth to analyze the FDA-MIMO radar equivalent transmitting beampattern. The FDA-MIMO radar equivalent transmitting beampattern can achieve the maximum M2 at the target point. rh,θh are those points where the power decays by half.
(17)BT=∑m=1Mej4π/cfmrt−rhej2πf0mdTsinθh−sinθt/c2=∑m=1Mejym2=M22,where ym=Afm+Bm, A=−4π/crh−rt, B=2πf0dT/csinθh−sinθt. By Euler’s formula and Taylor approximation, it can be expressed as [22]:
(18)∑m=1Mejym2=∑m=1Mcosym2+∑i=1Mcosyi2=∑m=1M∑i=1Mcosym−yi≈∑m=1M∑i=1M1−12ym−yi2=M22.Simplifying further, we can get:(19)∑m=1M∑i=1Mym−yi2=M2.Bringym into (19), and expand (19) to:
(20)∑m=1M∑i=1Mym−yi2=A2af+2ABbf+B2gf=M2,(21)af=∑m=1M∑i=1Mfm−fi2,(22)bf=∑m=1M∑i=1Mfm−fim−i,(23)gf=∑m=1M∑i=1Mm−i2=M2M2−16=G.Whenθh and θt are close, we introduce an approximation:
(24)sinθh−sinθt≈θh−θtcosθt.This is true when forming a spot beam. Bringing (24) into (20), we can further get:
(25)M2=af4πc2rh−rt2−bf4πc2f0dTcosθtθh−θt+G2πf0dTcosθtc2θh−θt2.(25) represents the analytical expression of the equivalent transmit beampattern produced by the frequency offset f. It is a quadratic curve with rh,θh as the boundary and the target point rt,θt as the center in the angle-range plane. Using the Cauchy-Schwarz inequality, it is easy to prove Δ=afG−bf2≥0. Therefore, its beampattern can only be ellipses (Δ>0) and parallel lines (Δ=0).When all frequency offsets are set to zero, it is equivalent to phased array radar. At this timeaf=0, bf=0, and Δ=0.At this time, (25) can be expressed as:
(26)θh−θt2=6M2−12πf0dTcosθt/c2.Obviously, the pattern has no range dependence, and its main lobe is a set of parallel lines centered onθt and bounded by θh. The results are consistent with the phased array radar, further verifying the correctness of the derivation.


## 4. Optimal Frequency Offset Design

It is a vital problem we must solve that how to evaluate the impact of different frequency offset on the performance of the main lobe when optimizing the frequency offset. Because it is an ellipse on the range-angle plane, we naturally think of minimizing the area of the ellipse (the beam energy is more concentrated) to optimize the frequency offset.In analytic geometry, the cross-term coefficientbf represents the degree of inclination of the ellipse, and its value has nothing to do with the area. Without loss of generality, we set bf to zero, (25) can be expressed as:
(27)M2=af4πc2rh−rt2+G2πf0dTcosθtc2θh−θt2.Simplify (27) into an elliptical standard form.
(28)rh−rt2h12+θh−θt2h22=1,h1=Mcaf4π,h2=McG2πf0dTcosθt.The area of the ellipse can be expressed as:(29)Sf=πh1h2=M2c28πf0dTcosθt1afG.To facilitate the analysis, we rewrite the polynomial into a matrix form.(30)afG=fTQf,(31)Q=M3M2−13IM−M2M2−13EM.The detailed derivation of (30) is provided by [22]. Bring (30) into (29), and rewrite (29) as:
(32)Sf=M2c28πf0dTcosθt1fTQf≤M2c28πf0dTcosθt1λQmaxfTf,(33)λQmaxfTf≥fTQf.λQmax is the maximum eigenvalue of matrix Q, and only when f=fmax is the eigenvector corresponding to λQmax, the two sides of (33) are equal. Therefore, the smallest ellipse area is
(34)Sf=M2c28πf0dTcosθt1λQmaxfmaxTfmax.When the frequency offset of the transmitting array element isfmax, the area of the ellipse is the smallest. In other words, at this time, the main lobe area is the smallest, and the energy is more concentrated. In addition, fmax is only related to matrix Q, and matrix Qis a real matrix that is only related to the number of elements in the transmitting array. Therefore, fmax is only related to the number of transmitting array elements. When the number of transmitting array elements is determined, we can determine the frequency offset required by each transmitting array element.When the FDA-MIMO radar scans the space, we only need to change the parameters related to the target pointrt,θt in (12), and the frequency offset does not need to be reoptimized. Therefore, the method in this paper has lower time complexity than the method of using the optimization algorithm to solve the frequency offset.


## 5. Simulation Results

Many scholars have explored this to form a better beam. Logarithmically increasing frequency offset (Log_FDA) [14], Hamming window-based frequency offset (Ham_FDA) [16], and Hamming window weighted linear frequency increments (HL_FDA) [18] have been proposed. The proposed frequency offset scheme in our work is analytical method (AM_FDA). The frequency offset of the log-FDA can be expressed as:
(35)Δfmlog=δ1logm,where δ1 is a constant. Δfm represents the frequency offset of the m-th transmitting array element. The frequency offset of the Ham_FDA is
(36)ΔfmHam=Bw0.54−0.46cos2πm2M1,where Bw is the bandwidth. M should satisfy:
(37)M=2M1+1.The frequency offset of the HL_FDA can be expressed as:(38)ΔfmHL=m−1δ20.54−0.46cos2πm2M1,where δ2 is a constant. To obtain the frequency offset of the AM_FDA, we could use the function “eig” in the MATLAB. After many numerical experiments, the rank of matrix Q is always M−1, and there are M−1 eigenvectors corresponding to the largest eigenvalue λQmax. Hence, the frequency offset of the AM_FDA is
(39)f=δ3k1fmax1+k2fmax2+⋯+kM−1fmaxM−1,∈spanfmax1,fmax2,⋯,fmaxM−1,where δ3 is a constant and f=f1,f2,⋯fm⋯fM. spanfmax represents the linear eigen subspace formed by fmax.Without loss of generality, we setk =1. (39) rewrite as:
(40)f=δ3fmax1+fmax2+⋯+fmaxM−1.The equivalent receiving beampattern is not affected by the frequency offset, and the frequency offset only affects the equivalent transmit beampattern. The equivalent receiving beampattern of each scheme is the same. Therefore, only the equivalent transmit beampattern is compared, in the simulation. We set the maximum frequency offset of the four methods to be similar to increase the credibility of the comparison, and the date below -20 dB is approximated to -20 dB in the beampattern measured in decibel to improve the comparing effects.The carrier frequency isf0=10Ghz, the number of elements is M=N=15, transmitting element spacing and receiving element spacing are both dT=dR=0.015m, and the target is located at (100 km, 20°). The parameters related to frequency offset are set as follows: δ1=10kHz, Bw=27.5kHz, δ2=3.5kHz, and δ3=18kHz. The frequency offsets of log-FDA, Ham-FDA, HL_FDA, and our proposed AM_FDA are compared as shown in Figure 4. To eliminate unfairness, the maximum frequency offsets of these methods are approximately equal.Figure 4
Frequency offset comparison.Based on using the time-invariant characteristics and range dependence of the FDA-MIMO radar system beampattern, we analyze the four schemes log_FDA, Ham_FDA, HL_FDA, and our proposed AM_FDA. The equivalent transmit beampattern of the four schemes at the receiver is shown in Figure5.Figure 5
The equivalent transmit beampatterns at the receiver: (a) the beampattern of log_FDA; (b) the beampattern of Ham_FDA; (c) the beampattern of HL_FDA; (d) the beampattern of AM_FDA.
(a)(b)(c)(d)It can be seen from Figure5 that the shape of the main beam of each scheme is elliptical, which further verifies the correctness of formula (25). The beamwidth of log-FDA is the largest, and the beamwidth of AM_FDA is the smallest, which has better beam focusing and spatial resolution performance. When only range is considered, Ham-FDA, HL_FDA, and AM_FDA all form side lobes. The pattern performance is shown in Figure 6.Figure 6
The equivalent transmit beampatterns in the profile of range.Figure6 further shows that AM_FDA has a narrow beamwidth, forming a better beam. But AM _FDA also has a higher side lobe. As we all know, the reduction of the main lobe width will inevitably lead to an increase in the sidelobe level, because the total energy is constant. We use Kaiser window function weighting to reduce the sidelobe level (WAM_FDA), which will inevitably lead to an increase in the width of the main lobe. The frequency offset of the WAM_FDA is
(41)fW=wMf,we could use the function “kaiser” in the MATLAB to obtain wM, where wM=kaiserM,β. We control the relationship between the main lobe and side lobes by adjusting β. The value of β can be determined according to the electromagnetic environment of the battlefield. Without loss of generality, we set β=2.Figures7 and 8 show that when the side lobes decrease, the main lobe width increases, verifying the correctness of the theory. However, the HAM_FDA effectively reduces the side lobes, while the width of the main lobe increases less. Hence, the performance of HAM_FDA is better than other schemes.Figure 7
The equivalent transmit beampattern at the receiver: (a) the beampatterns of WAM_FDA (range-angle plane); (b) the beampatterns of WAM_FDA (3D).
(a)(b)Figure 8
Comparative results of (a) the equivalent transmit beampatterns in the profile of range and (b) ellipses generated by different schemes.
(a)(b)In summary, using the FDA-MIMO radar system framework mentioned in this article can make full use of the range-related characteristics without being affected by time parameters. AM_FDA has good main lobe performance but has the disadvantage of the higher side lobe. WAM_FDA can better balance the relationship between the width of the main lobe and the level of the side lobe.


## 6. Conclusions

The design of frequency offset is a hot issue in FDA radar. In this paper, considering the influence of time parameters, taking the minimum main lobe as the optimized objective function, the optimal analytical solution of the frequency offset is obtained. Three aspects need to be emphasized. (1) The FDA-MIMO radar framework mentioned in this paper sets all frequency-related variables at the transmitter, and the frequency offset is only related to the number of transmit array elements, and finally, a range-dependent beampattern is obtained. (2) AM_FDA is significantly better than other solutions in range resolution. (3) Aiming at the problem of high sidelobe level, a window function method (WAM_FDA) is adopted to appropriately increase the width of the main lobe to reduce the sidelobe level.



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