Colocated MIMO Radar Waveform Design for Transmit Beampattern Formation
In this paper, colocated MIMO radar waveform design is considered by minimizing the integrated sidelobe level (ISL) to obtain beampatterns with lower sidelobe levels than competing methods. First, a quadratic programming problem is formulated to design beampatterns by using the minimal-ISL criteria. A theorem is derived that provides a closed-form analytical optimal solution that appears to be an extension of the Rayleigh quotient minimization for a possibly singular matrix in the quadratic form. Such singularities are shown to occur in the problem of interest, but proofs for the optimum solution in these singular matrix cases could not be found in the literature. Next, an additional constraint is added to obtain beampatterns with desired 3dB beamwidths, resulting in a nonconvex quadratically constrained quadratic program which is NP-hard. A semi-definite program and a Gaussian randomized semi-definite relaxation are used to determine feasible solutions arbitrary close to the solution to the original problem. Theoretical and numerical analyses illustrate the impacts of changing the number of transmitters and orthogonal waveforms employed in the designs. Numerical comparisons are conducted to evaluate the proposed design approaches.
Multiple-input multi-output (MIMO) Radar, transmit beampattern, convex optimization, relaxation.
Multi-input multi-output (MIMO) radars which emit different waveforms from different transmit antennas can provide extra degrees of freedom in waveform design to allow significant coherent gains [1, 2, 3, 4] when the antennas are colocated. Thus, flexible waveform design for transmit beampattern formation is of interest for colocated multi-input multi-output (MIMO) radars. Beampattern formation is carried out by designing either a waveform covariance matrix [5, 6] or a waveform coefficient matrix . The goal is to focus the transmit power into a range of interesting angles while minimizing the transmit power for other angles . Various methods can be found in [1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 7] and in the references therein. Among the references on beampattern design, the most common strategy is called shape approximation, i.e., beampattern design according to a known shape under the criteria of minimum mean square error (MMSE) [1, 11, 13, 14, 16, 9] or minimum difference (MD) [13, 7]. To accomplish these goals, gradient search algorithms [1, 11], barrier methods , iterative algorithms [14, 16], and convex optimization [9, 7] have all been used. Since the sidelobe level is one of the most important performance indexes in antenna radiation theory [18, 19, 20], some minimum sidelobe level design strategies have received attention and this is also the topic of this paper. Published methods are the minimum peak sidelobe level (PSL) method [12, 9, 21, 22], the discrete prolate spheroidal sequences-based design (DPSSD) method  and the minimum integrated sidelobe level (ISL) method . The minimum PSL method constrains the definition of the sidelobe region as the region outside a band of angles twice as wide as the mainlobe, which is a severe restriction that limits flexibility to employ wider mainlobes. DPSSD used a maximization of the ratio of energy in mainlobe to the total energy to derive a closed-form solution using some approximations. Further, the sidelobe levels obtained by maximizing the ratio of energy in mainlobe to total energy can never be smaller than obtained by maximizing the ratio of energy in mainlobe to the energy in sidelobes . Alternatively, in this paper a closed-form analytical solution is given for the criterion of minimum ISL and no approximations are employed. To the best of our knowledge, this solution and the proof justifying it, have not appeared in any publications to date. Our results appear to be an extension of the Rayleigh quotient minimization [25, Sec. 8.2.3] for a possibly singular matrix in the quadratic form. Such singularities are shown to occur in the problem of interest, but proofs for the optimum solution in these singular matrix cases could not be found in the literature. To augment the closed form results, the minimum ISL criterion is next considered with an additional constraint to require a desired 3dB beamwidth, This results in a nonconvex quadratically constrained quadratic program which is NP-hard. A semi-definite program and a Gaussian randomized semi-definite relaxation is used to determine feasible solutions arbitrary close to the solution to the original problem. Extensive numerical comparisons are conducted to evaluate the proposed design approaches. Additionally, a novel theoretical analysis describes the impact of the number of transmitters and the number of transmit waveforms on the designed beampatterns.
Notation: Throughout this paper, we use lowercase italic letters to denote scalars, lowercase and uppercase letters in bold to denote the vectors and the matrices, respectively. Superscripts of , and represent the conjugate, transpose and complex conjugate-transpose (or Hermitian) operators of a matrix, respectively, while represents the trace of a square matrix and represents the vectorization operator which creates a column vector from a matrix by stacking its column vectors. denotes the Euclidean (or ) norm of a vector and denotes Kronecker product. The notation denotes identity matrix and denotes all-zero vector or matrix. , and denote the real, complex and positive integer spaces, respectively.
2 Problem formulation
Consider a colocated uniform linear array (ULA) MIMO radar system equipped with transmit antennas, where the antennas are separated by a half wavelength. Assume each transmitter emits a weighted sum of independent orthonormal baseband waveforms which are expressed by . The weightings are described by a coefficient matrix , where and is used to control the amount of the th waveform added into a sum at each transmit antenna . The total transmit power is fixed at by setting . Thus the signals transmitted by the antennas can be described by
After applying a steering vector , the waveforms transmitted to a far-field target at with can be expressed as
where , and . Note that both and are Hermitian positive semidefinite matrices for .
Divide , the set of all possible , into two disjoint sets called , the mainlobe region and , the sidelobe region. To concentrate the transmit power into as much as possible, we can formulate the following minimum integrated sidelobe level (ISL) optimization
Problem (5) is a quadratic programming problem with a ratio objective (QP-R). When () is a positive definite matrix with full rank, then (5) just can be converted into a Rayleigh quotient [25, Sec. 8.2.3] minimization, and an analytical optimal solution can be obtained [27, 28, 29]. However, in our case may not be a full rank matrix which means will not be positive definite. Here we provide a numerical demonstration of this by considering a case when we vary the width of mainlobe from to in Fig. 1. From the figure, we can see that only when the width of is greater than is full rank. Thus, a more general method to obtain the analytical optimal solution of (5) needs to be considered. However, to the best of our knowledge, we have not found any literature giving an analytical solution to (5) for not full rank.
The problem in (5) seeks the minimal ISL for the defined and . However, in some applications a particular dB beamwidth is also desired, requiring the addition of some constraints to (5) to obtain the altered optimization by
where denotes the maximum power point in , which is usually chosen to be at the center point of the main lobe. The above problem is a nonconvex quadratically constrained quadratic programming (QCQP) problem  which is NP-hard.
In the following section, we will propose the best suitable methods to solve the above two problems, respectively.
3 Waveform desgin for Beampattern formation
3.1 Beampattern formation for minimal-ISL only
Although we have demonstrated that might not be a positive definite matrix, we can confirm that is positive definite as shown in the following Lemma 1.
Lemma 1: Given and , where , and , then is always positive definite.
Proof of Lemma 1:
For is positive semidefinite, thus given and , we have
Obviously, is always positive definite, hence, the proof of Lemma 1 is completed.
Based on Lemma 1, then we can use the following theorem (which can be viewed as an extension of the Rayleigh quotient minimization) to get the minimal solution for problem (5)(Note that can normalized without loss of any generality for the ratio objective function, thus the constraint in (5) is ignored).
Theorem 1: Define Hermitian matrices , and assume is positive definite while is non-negative definite with . Focusing on and a minimization problem formulated as
If , the minimal solution of (9) is given by
where () denotes the eigenvector corresponding to the minimum eigenvalue of ; such that ; is an invertible diagonal matrix obtained from the eigen-decomposition of by , and , and are the submatrices of given by
If , the minimal solution of (9) is given by
where () denotes the eigenvector corresponding to the minimum eigenvalue of .
Proof of Theorem 1:
is a Hermitian and non-negative matrix with rank by assumption. Using eigen-decomposition, we can factorize as
where is a unitary matrix whose columns are eigenvectors argumented with a set of vectors to form a basis and is an invertible diagonal matrix with positive eigenvalues of along the diagonal. Further, is defined as
Then (9) can be formulated as
where is invertible and for .
Let . Further simplification of (15) leads to
where is defined as
The Lagrange function of (18) can be written as
To get the stationary points of , we use the theorems in , where for a real function of the complex variable vectors and , the stationary points are obtained by solving or . Here we use the gradients of the conjugates of the vectors to obtain the necessary condition for a stationary point as
On the other hand, let , and left and right multiply the equation with and , respectively, then we have
Since is positive definite, then the matrix on the right-hand side (RHS) of (23) is positive definite, thus should also be positive definite. Therefore based on (21) we have . Then combining (20) and (22), all the stationary points should satisfy
Based on the Rayleigh-Ritz theorem [32, Sec. 4.2.2], then the minimal solution of (26) becomes the eigenvector corresponding to the minimum eigenvalue of , where based on (17). If we assume is the eigenvector, then the minimum value of (9) is achieved by
where is the eigenvector corresponding to the minimal eigenvalue of .
Therefore the proof of Theorem 1 is completed.
3.2 Beampattern formation for desired beamwidths
In the following, we will propose different convex methods to solve the problem of (7) according to the relationship between the number of transmitters and transmit waveforms .
Since , we can use a Hermitian semi-definite matrix () to represent without loss of any generality. Then (4) can be reformulated as
Thus (7) can be reformulated as
where , , and . Also in (30) denotes that should be a semi-definite matrix.
Since for and , then (7) is equivalent to
Observing (32), we find that it is not a convex optimization due to the last nonconvex constraint (). Therefore, to solve (32), we consider using semidefinite relaxation (SDR) [36, 37]. Relaxing the constraint to a convex positive semidefinite constraint , we can obtain a lower bound on the optimal value of (32) by solving the following convex problem
Hence problem (33) is the SDR of (7) and can be solved with CVX. Suppose the optimal solution of (33) is . This may not be the solution of (7). To get an accurate approximate solution for (7) based on , Gaussian randomization  can be used. Generate a sufficient number of samples by assuming is a Gaussian variable with , then choose the samples satisfying the constraint of (7) to find the best feasible point among them to make the objective function the smallest.
4 Numerical Results
In this section, some theoretical and numerical analyses are performed to show how the variables (number of waveforms) and (number of transmitters) impact the designed beampatterns. Next, some comparison simulations are conducted to verify the performance our methods.
4.1 Beampattern analyses with respect to the number of waveforms and transmitters
Analysis on the impact of different on the designed beampatterns
First we give a Lemma which will be useful to our analyses.
Lemma 2: Given Hermitian non-negative definite matrices , if there exists such that , then , where is a positive semidefinite Hermitian matrix.
See the proof in Appendix 6. Given a non-zero vector , where and , we have
where and . Based on (34), we have
where and the inequality holds because the problem on the RHS is a SDR of the problem on the LHS. On the other hand, we have since setting the extra components in to zero gives the same answer as the problem on the RHS. However, no matter what and are, will always hold based on Lemma 2. Thus . Then we can conclude that the values of will not have impact on beampatterns obtained by the analytical method proposed in Section 3.1. Fig. 2 has verified our conclusion since all the cases with different have the same shaped beampattern. Now we consider the convex method proposed in Section 3.2. To begin our analyses, a lemma is provided.
Lemma 3: Given a real-valued function subject to , , where , and
are all Hermitian matrices
See the proof in Appendix 7. It’s not hard for us to prove that the SDRs of the minimization (7) for a case with and a case with can be expressed in the form of and , respectively, as defined in Lemma 3. Therefore, the SDRs for a case with and a case with have the same minimal value based on Lemma 3. Since we find the final solutions for different Q based on different SDRs but with the same minimal value, we can conclude that the beampatterns designed by the convex method for different Q will generally be the same except for minor differences that may exist because we find the best solution for each case among a Gaussian randomized sample. Fig. 3 shows the numerical results with different values of . We can see that there are only minor differences between the different beampatterns, which not only shows the correctness of our conclusion but also shows the good precision of Gaussian randomization.
Analysis on the impacts of different on the designed beampatterns
Considering a vector , then holds. Thus given , we have
where is an extension of by padding a zero at the end of each elements. Therefore, the beampattern we obtain at by (5) can be viewed as a special case of with zero-components. Thus when we increase the number of transmitters, we obtain a beampattern with a lower sidelobe level because the case of will provide a lower optimal value than the case of which means a lower sidelobe level. Fig. 2 shows the simulation results of the beampatterns for different values of . From the figure, it can be seen that the sidelobe level of the beampattern decreases when increases, as the analysis indicated. When is large enough, the mainlobe width of the beampattern is steady at the desired width of . However, when is not, though increasing can reduce sidelobe levels, it will result in a reduction in the 3dB beamwidth. Thus when using the analytical method proposed in Section 3.1, one should choose carefully in beampattern formation if it is desirable to obtain a good trade-off between the 3dB beamwidth and the sidelobe level.
Similarly, based on (36) we also can convert the case of for (7) into a special case of though (7) has an addtional constraint comparing with (5), which again means that increasing will lower the sidelobe level. Fig. 3 has verified our conclusion.
4.2 Comparisons among the different design methods
Comparisons between the two proposed methods
For and , we consider three types of desired beampatterns: the one-mainlobe case, the two-mainlobe case and the three-mainlobe
Comparisons with conventional methods
The minimal-ISL only design method is compared with the DPSSD method proposed in , since both of them can obtain closed-form solutions. Fig. 5 shows the numerical results of the two methods (where , , and the width of the defined mainlobe region is ). We can see that our method obtains a lower sidelobe level, which means the closed-formed minimal solution of our method has better performance in terms of lowering sidelobe level. The minimal-ISL design method for the desired 3dB beamwidth is compared with the shape approximation methods (SAMs) which also provide the desired 3dB beamwidth. Now consider two specific SAM methods. The first one is the SAM proposed in [9, Sec. III-C] (denoted as ’’). Though is a covariance matrix () design method, it can obtain the globally optimal solution by using the criteria of minimum MSE. The second method is the SAM proposed in  (denoted as ’’), which used shape approximation by minimizing the maximum difference. Fig. 5 shows the simulation results of a desired beampattern with a beamwidth, where and . From the figure, we can see that the beampattern designed by our method has the lowest sidelobe level.
Narrow-beam comparisons with conventional phased-array radar
In some cases it is desired to focus energy on a single angle . This can be accomplished by setting in (5) and then applying Theorem 1. In Fig. 6, we present some comparisons of results obtained from this slight modification of the minimal-ISL only criterion with those obtained from the conventional phased-array (CPA) radar , which can obtain narrowly focused beampatterns . The three subfigures in Fig. 6 show that the beampatterns designed using our method have lower sidelobe levels when compared to those obtained by the CPA radar. When we carefully inspect the 3dB beamwidths of the beampatterns, we can observe that our method can achieve a mainlobe width which seems to approach to that of the conventional one as we increase . As Fig. 6 illustrates, the differences of the 3dB beamwidths for , and are less than , and , respectively.
Colocated MIMO radar waveform design for transmit beampattern formation by minimizing ISL has been considered in this paper. Both analytical and convex design methods using the criteria of minimum ISL are proposed, respectively, to obtain beampatterns with lower sidelobe levels with different goals. Under the minimum ISL design criterion, both theoretical and numerical analyses have shown that the number of waveforms doesn’t have impact on the quality of beampatterns while the number of transmitters does. Further, the larger the value of , the lower the value of the sidelobe level. Finally, numerical comparisons have shown our methods can obtain beampatterns with lower sidelobe levels than conventional methods. \appendices
6 Proof of Lemma 2
Define and such that . Use eigen-decomposition to factorize as . Without loss of generality, we assume for and is the minimal solution of , then
Note that can be an arbitrary feasible point of , thus . On the other hand, we have for is an SDR of . Therefore the proof of Lemma 2 is completed.
7 Proof of Lemma 3
Write as , since if we let then holds, hence, we have
Define , where , and , then we can rewrite as