Closed-Form Waveform Design for MIMO Radar Transmit Beampattern Synthesis via Integral Equality

Recently, due to the adequate use of waveform diversity, MIMO technology has been widely adopted, resulting in the waveform design for MIMO radar beampattern synthesis becoming a hot issue. However, most previous works regarding this problem optimize the waveform by introducing the global mean squared error (MSE) as the cost function, this requires large computation in applications. Different from these works, we adopt the idea from the shaped beampattern synthesis problem of 2-D array in this paper, that is, to form a flat-top beampattern in the desired area with small ripple. By considering the physical meaning of the flat-top beampattern from scratch, the MIMO waveform design problem is turned to designing the waveforms that make its beampattern integral (i.e. the energy transmitted by array) equal in the desired area. Subsequently, we put forth a closed-form method and give its mathematical proof. Numerical simulations and comparisons to known MIMO radar waveform design methods are provided to verify its effectiveness and outperformance.


I. INTRODUCTION
In the past decade, multiple-input multiple-output (MIMO) technology has attracted widespread attention and been widely used in the fields of communication [1], [2], beamforming technique [3], [4], source localization [5]- [7] and target tracking [8]. MIMO has higher degree of freedom (DoF) than phased array due to its adequate use of waveform diversity [9], [10]. According to the antenna distribution, MIMO array can be classified into distributed MIMO and colocated MIMO. In MIMO system, each antenna transmits independent waveform to obtain higher resolution and larger array aperture, and this waveform can be designed to meet different requirements.
In MIMO communication, much attention is focused on the orthogonality of waveform to improve the communication quality [11], [12]. However, in MIMO radar, people are more concerned about how to design its waveform to take full advantage of the increased DoF of MIMO.
The associate editor coordinating the review of this manuscript and approving it for publication was Kai Lu . Current research regarding the MIMO radar can be divided into two main categories: designing both transmit waveform and receiving filter; designing the transmit waveform only. The former one jointly designs the transmit waveform and the receiving filter to maximize the output signal-to-interferenceplus-noise ratio (SINR) [13]- [22]. Specifically, for colocated MIMO radar, the transmit waveforms are designed to improve the SINR performance in the presence of signal clutter in signal [13]- [20]. For distributed MIMO radar, the transmit waveforms are designed by maximizing the mutual information (MI) between the receiving echoes and the target response [21], [22]. The latter category only designs the transmit waveform to flexibly control the distribution of the array transmit energy in space: that is the MIMO radar transmit beampattern synthesis problem [23], [24]. Recently, MIMO waveform design for its transmit beampattern matching is a hot issue in MIMO research and also the focus of this paper [25], [26].
In the latest works about MIMO beampattern matching problem, there are two types of methods: two-steps and one-step methods. The two-steps methods firstly design the waveform covariance matrix R that matches the desired beampattern [27], [28]. However, it is difficult to directly obtain the waveform matrix S from its covariance matrix R. Calculating S from R requires special relaxation processing such as Semidefinite Quadratic Programming (SQP) and Cyclic Algorithm (CA) [9]. To avoid the relaxation processing, the one-step approaches such as AMDD [23] and DFTbased method [24] design the waveform S directly. However, all these methods, including the two-steps and the one-step methods, introduce certain optimizing strategies to design the waveform, requiring complicated calculations. On the contrary, the DFT-based method is not the case: it is a closedform method. The DFT-based method can design the waveform that produces the beampattern matching the desired beampattern with low computational complexity. Nonetheless, the resolution of the DFT method is poor when the number of array element is small. In practice, increasing the number of array elements is usually inconvenient, indicating the limitation of the DFT-based method. In addition, in order to evaluate the performance of the designed waveforms, all these works use the global Mean-Squared Error (MSE) in the whole spatial domain as the optimized cost function and the performance evaluation criteria, leading to the fact that the designed waveforms may not have the best performance in the desired region. In engineering, for the problem of beampattern synthesis, more attention is focused on how to transmit energy to the desired area accurately.
Fundamentally, the increased DoF in MIMO radar comes from its time-varying waveform. In other words, MIMO radar adopts the strategy that takes time for space, and the MIMO array can be seen as a 2-D array with time-invariant parameters [29]. Thus, the shaped beampattern synthesis with 2-D time-invariant array (also a hot topic) is a similar problem to the MIMO radar beampattern synthesis. In the problem of shaped beampattern synthesis with time-invariant parameters, accurately transmitting the array energy to certain desired areas means forming flat-top beampatterns with smallest local MSE in the desired areas [30], [31]. However, this problem usually requires lots of optimizations as well [32]- [35]. Inspired by the shaped beampattern synthesis, unlike the previous work considering global MSE, we adopt the evaluation criterion of shaped beampattern synthesis [30], i.e., the local MSE in the desired region in this paper. Specifically, by designing the waveform directly, the MIMO beampattern is determined to match our desired beampattern as close as possible in the desired region [36]. By doing so, the problem turns to synthesizing a flat-top beampattern with small ripple in the desired region.
From the physical essence, a flat-top beampattern in the desired region means the energy transmitted by MIMO array is equal everywhere within the desired region [37], [38]. In fact, the transmit energy is the integral of transmit beampattern. However, because the beampattern expression does not have an analytic form, this integral is difficult to calculate directly. In this paper, we put forth a closed-form method that directly designs the MIMO waveform without optimizing. Our method first designs the phase excitation of waveform to make the beampattern expression analytical. Subsequently, we introduce a Taylor approximation to simplify the beampattern integral and deduce the magnitude excitations of waveform that can ensure the beampattern integral (i.e. the energy transmitted by array) equal in the desired region. The mathematical proof and the simulations are provided to verify its effectiveness and show the outperformance of our method.
The remainder of this paper are organized as follows. Section II formulates the beampattern synthesis problem for MIMO radar. Our closed-form method and its mathematical proof are provided in section III. Numerical simulations and comparison to known MIMO radar waveform design are given in Section IV. Section V presents the conclusion. Some necessary mathematical proofs are given in Appendices A and B.

II. BEAMPATTERN SYNTHESIS PROBLEM FOR MIMO RADAR
In this paper, we consider a colocated Uniform Linear Array (ULA) MIMO. It is composed of M transmitting elements with uniform half-wavelength spacing d, as shown in for l = 1, 2, · · · , L, m = 1, 2, · · · , M ; (1) where α l (m) and ϕ l (m) are the magnitude and phase excitations, respectively. The entry in the l-th row and m-th column of the waveform matrix S ∈ C L×M is s l (m), as shown in FIGURE 1.
All the waveforms in the waveform matrix S arrive at the far-field direction θ and coherently summate. As a result, the total energy radiated by the array seen at the direction θ is is the steering vector, R SS = S H S L the waveform covariance matrix, and f l (θ) the radiating power of sample l. For the MIMO radar beampattern synthesis problem, the core issue is to control the spatial distribution of transmit power to match our desired beampattern shape by designing the waveform matrix S. Different from the previous work designing S by optimizing global MSE, in this paper, as shown in the FIGURE 2, we introduce the 'distance' ε between the desired beampattern shaped d (θ ) and the MIMO radar transmit beampattern F (θ ) from the shaped beampattern synthesis problem [30]. In other words, we want to synthesize a flat-top beampattern with small ripple ε that leads to minimum local MSE in desired region. Then the synthesis problem can be written as where DB and SL are the Desired Beam region and the SideLobe region respectively. The sidelobes are kept below ρ, where the ρ indicates the overall sidelobe level. FIGURE 2 provides the graphic illustration. This non-convex problem with multiple parameters is difficult to solve and may have more than one optimal solution. As mentioned in Section I, optimizing strategies such as CVX [30], CA [9] and ADMM [23] are used to solve this problem in many previous works. In this paper, for MIMO radar beampattern synthesis problem (3), we put forth a closed-form method with high accuracy.

III. OUR CLOSED-FORM METHOD AND ITS MATHEMATICAL PROOF
The focus of designing the waveform matrix S is the excitations. It is natural to think of the case where the conventional phased array is used at each sample, and the DB region is covered by their pointing angles, while the amplitude excitations keep the same. However, this setting cannot achieve optimal performance because the energy transmitting into certain region by phased arrays with different pointing angle is different.
Nonetheless, based on the above discussion, it is possible to simplify the problem. In our method, as shown in the FIGURE 3, the magnitude and phase excitations of m-th element in sample l are directly set as follows where θ l is a given angle and they cover the DB region. We still choose the phase excitations from the phased array as the phase excitations of our method, resulting in the array radiating power f l (θ ) having a closed-form expression. In this way, the problem of designing all excitations can be simplified into only designing the amplitude excitations. It is noted that the waveforms are costant-modulus across the array only in sample l. It can be achieved by the power divider in engineering. Furthermore, since all the excitations in (4) are only related to θ l and they are given directly, our method has another advantage that each waveform is chosen from a finite alphabet, which decreases the physical implementation complexity.
Taking (4) into (2) yields which makes f l (θ ) having a closed-form expression as mentioned above. However, due to that the relationship between θ and f l (θ ) is still not a combination of basic functions, equation (5) requires further simplification. As shown in FIGURE 3, in the mainlobe region l of θ l , θ is close to θ l , hence the following two approximations can be introduced to simplify the equation (5) sin π (sin θ − sin θ l ) 2 ≈ π (sin θ − sin θ l ) 2, (6a) The approximation (6a) comes from the equivalent infinitesimal of sine function and the derivation of (6b) can be seen in the APPENDIX A. Taking the approximations (6) to the equation (5), the equation (5) can be simplified as So far, equation (7) is approximated to the simplest and analytical form. The physical meaning of equation (7) is the array radiating power at direction θ in sample l, which is mainly focused near θ l . Back to the MIMO radar beampattern synthesis problem, the goal of problem (3) is synthesizing a flat-top beampattern with small ripple ε. Put differently, the array radiating energy on its mainbeam region l ∈ [θ l − l , θ l + l ] is an integral which should keep constant. This means that no matter how θ l changes, the array radiating energy on its mainlobe region l remains unchanged by controlling the magnitude excitations α l . By doing so, the flat-top beampattern synthesis problem can be transformed into finding the α l that keeps the integral in (8) equal. FIGURE 3 provides the detailed graphic illustration.
Unfortunately, the integral in (8) is a transcendental integral that does not have a solution with mathematically-closed form. In this paper, in order to analytically calculate the integral (8), we introduce the Taylor approximation as The detailed Taylor expansion for equation (9) is provided in APPENDIX B. The integral interval l of the integral (8) is the mainlobe region of θ l and its width l can be obtained from equation (9) 1 Taking (9) and the integral interval (10) into the integral (8) yields Thus, equation (11) provides the mathematical proof to confirm that the magnitude excitations α l = √ cos θ l of our method (4) can ensure that the array radiating energy on spatial region l remains equal. So far, we solve the problem (3) analytically.

IV. SIMULATIONS
In the simulation section, we assume a ULA composed of M = 10 identical transmitting elements with uniform halfwavelength spacing d = c (2f 0 ), where the carrier frequency f 0 = 1GHz. For comparison, the simulation setting is exactly the same as [23], in which the sample number is L = 32 and the simulation range of angle dimension is −90 • , 90 • with spacing 1 • .
With this setup, the simulations and comparisons to known MIMO radar waveform design methods such as CA [9], ADMM [23] and DFT-based [24] are provided to show the performances. It is worth to point out that we compare our method with the optimizing methods [9], [23] and the analytical method [24].
As what discussed in previous sections, we adopt the local MSEs in the desired region from the shaped beampattern synthesis problem as the criterion to evaluate the performances. The local MSE in DB region is defined as , θ k ∈ DB; (12) whereF (θ) is the normalized transmit beampattern. In this section, as same as [23], we also consider two classical cases: one mainlobe beam and multiple mainlobe beams in DB region. Meanwhile, the overall sidelobe performance in SL region and the computational complexities of each method are also provided.

A. ONE MAINLOBE BEAM
In this example, the one mainlobe beampattern is considered. The DB region is −30 • , 30 • with the desired beampattern The L pointing angles θ l = θ 1 , θ 2 , . . . , θ L related to phase excitations are uniformly distributed on the DB. The L magnitude excitations α l = α 1 , α 2 , . . . , α L are set as our method (4), that is α l = √ cos θ l . The simulation results are presented in FIGURE 4.
Whether compared with the optimization method [9], [23] or the analytical method [24], the beampattern of our proposed method with ripple ε = 0.008 is the closest one to the desired flat-top beampattern in the DB region. Moreover, the overall sidelobe of the proposed method in SL region are the smallest one.

B. THREE MAINLOBES BEAM
In this example, we provide a case in which DB has multibeams. The desired beampattern has three mainlobes: The center points of DB in this example are −40 • , 0 • and 40 • respectively, each mainlobe has a width 20 • . As same as Example A, the L = 32 pointing angles θ l = θ 1 , θ 2 , . . . , θ L are uniformly distributed on the DB either. Because 32 is not a multiple of 3, for the three mainlobes in this case, 11, 10 and 11 samples are allocated for each one. The magnitude excitations are also α l = √ cos θ l , as equation (4). The simulation result is shown in FIGURE 5.
FIGURE 5 demonstrates that in the case of DB has multiple mainlobes, the proposed method is also the closest one to the desired beampattern. Its overall sidelobes performance is better than the DFT-based method, but slightly inferior to the optimization method. That is consistent with the law of conservation of energy, because the proposed method improve the local MSE performance greatly in this case. Moreover, the two sides are slightly higher than the middle one, because there is one more sample on each side as allocated above. 5 show that the methods introducing optimizing strategies such as CA [9] and ADMM [23] VOLUME 8, 2020  have better performances than the closed-form one (i.e. the DFT-based method [24]) with the cost of higher computational complexity. However, high computational complexity has a limitation on the real-time implementation. Although the DFT-based method is also a closed-form approach, it cannot achieve good performance because the array element number is too small, and the DFT transform itself still requires calculation. Furthermore, the TABLE 1 demonstrates that the better the performance in DB region, the higher overall sidelobe in SL region. This is due to the law of energy conservation.

FIGURE 4 and FIGURE
For the MIMO waveform design problem, regardless of whether the DB has one mainlobe or multiple mainlobes, our proposed method is the closest one to the desired flattop beam in the DB. The local MSE in TABLE 1 also confirm the performance of our method. We achieve the flat-top beampattern by equalizing the array radiating energy on the desired spatial area.
Moreover, as can be seen from TABLE 1, our method not only has an improvement in the local MSE performance, but also minimizes the computational complexity. Meanwhile, its overall sidelobe performance remains below a low value. In fact, because our method is completely closed-form and all the excitations of the waveforms are designed directly as equation (4), the computational complexity of the proposed method is O (1).
In sum, in the MIMO waveform design for beampattern synthesis problem, our method can achieve the best performance with smallest computational complexity. As deduced above, this benefit comes from the fact that we consider the physical meaning of the flat-top beampattern from scratch and find the waveforms that equalize the beampattern integrals.
Despite this, as this paper considers the uniform array and instantaneously constant modulus for MIMO waveform design, we aim to design a more general waveform in the future publication.

V. CONCLUSION
In this paper, we put forth a closed-form method for MIMO radar transmit beampattern synthesis problem by reconsidering its physical meaning from scratch. Unlike previous work optimizing the waveform by introducing the global MSE as the cost function, we use the idea from shaped beampattern synthesis problem of 2-D array, which synthesis the beampattern with the smallest local MSE in the desired area. Therefore, designing the MIMO waveform for beampattern matching is turned to form a flat-top beampattern with small ripple in the desired area. Subsequently, by discussing the physical essence of the flat-top beampattern, we deduce a method that can keep its beampattern integral (i.e. the energy transmitted by array) equal everywhere in the desired area, indicating a flat-top beampattern. The necessary mathematical proof is provided to demonstrate the validity of our method. Moreover, simulation results and comparisons to known MIMO radar waveform design methods show that our method outperforms in MIMO radar beampattern synthesis problem.

APPENDIX A DERIVATION OF THE APPROXIMATION (6b)
In this section, we give a brief derivation of the approximation in equation (6b). As mentioned above, in the mainlobe region l of θ l , θ is close to θ l . The geometrical relationship can be written as where θ is a tiny angle difference. There is sin θ = sin (θ l − θ ) = sin θ l cos θ − cos θ l sin θ The first approximation in equation (16) comes from the Taylor approximation cos θ ≈ 1 − θ 2 2 and sin θ ≈ θ. Moreover, the tiny θ leads to θ 2 ≈ 0, resulting the second approximation.