Beampattern Optimization for Frequency Diverse Array With Sparse Frequency Waveforms

Multiple-input multiple-output (MIMO) radar equipped with a frequency diverse array (FDA) can produce a range-dependent beampattern and increase the degrees-of-freedom of the antenna array. In this paper, a new method of designing the MIMO radar beampattern with sparse frequency waveforms is proposed for the FDA, which randomly samples multiple distance points such that the MIMO radar beampattern with the both sparse frequency spectrum and constant modulus constraints are realized by the proposed beampattern design framework. The main steps are as follows. We first obtain the covariance matrix of the transmitted signal by a given ideal beampattern, and formulate the problem of designing the realizable beampattern as a nonconvex optimization problem, which includes the constraints of the both constant modulus of transmitted signals and sparse frequency spectrum. Then, a cyclic optimization algorithm is proposed, which guarantees the monotonic decrease of the objective function as the algorithm proceeds. The simulation results illustrate that the proposed method can achieve smaller errors than the traditional method, which does not consider the frequency diversity.

SECTION I. Introduction

The concept of the frequency diverse array (FDA) was first introduced by Antonik in [1], where a frequency increment between the adjacent elements was considered such that a range-dependent transmit beampattern can be realized [2]–​[6]. Compared with the traditional phased array which omits the frequency diversity consideration, the FDA radar can achieve beam space scanning without phase shifters. The property can greatly simplify the array structure and reduce the system power consumption [2], [5].

Due to the presence of the frequency difference among the array elements, the beamforming direction of the FDA will be changed as a function of range, angle and time delay [4], [5]. Hence, the FDA beampattern design problem contains both time and angle periodicity [2], resulting in a more complicated optimization problem. From an optimization perspective, the space and time variables are coupled such that the nonconvex objective function of optimizing the beampattern is not separable over variables and further the variables in the constrains are coupled in a non-linear way. To the best of our knowledge, the current nonconvex optimization algorithms do not apply for this problem [7]–​[10].

In recent years, the characteristics of the FDA range-dependent beampattern have been widely studied in [3]–​[5], and the application and technical development of the FDA are also investigated from various aspects. In these applications, the area of the MIMO radar with the FDA ([4], [6] and references therein) attracts more attention. There are two main reasons: first, the FDA architecture can produce a range-dependent beampattern; second, from the perspective of the waveform design, the MIMO radar system can increase the degrees-of-freedom of the antenna array.

A. Related Works

In the design of the MIMO radar beampattern with a linear array, the relation between the transmit beampattern and radar waveforms has been studied in the literature [11]–​[18]. The problem of designing the beampattern can be converted to optimize the covariance matrix of the radar transmit waveforms [11], [12] and solved by the positive semi-definite programming [13], [14]. Besides knowing the covariance matrix, the corresponding radar waveforms are also necessarily to be with a constant modulus. In [15], the constant modulus constraint of the radar signals was relaxed to the peak-to-average power ratio (PAR) constraint [16], [17] such that the desired radar waveforms can be easily obtained when a required PAR condition was satisfied.

In the frequency domain, a sparse frequency waveform can have multiple passbands and stopbands. This property can avoid interference from/to other users operating in the same band in most practical applications. Also, the sparse frequency waveform can increase the spectrum efficiency of cognitive radio systems. Sparse frequency waveform design is based on the ambiguity function (AF) theory, which was first proposed by Woodward [19] to analyze the radar waveform. Then, the theory is extensively studied by many others (for example, [20], [21] and references therein). More related techniques on the sparse frequency radar waveform design are proposed consequently in [22]–​[32]. Specifically, the basic method of the sparse frequency waveform design focuses on imposing some constraints on the power spectral density (PSD) and the waveform auto-correlation function (ACF) [23], [27], [31], i.e., suppress the peak sidelobe level (PSL) and reduce the integral side lobe (ISL) level [24], [25]. For solving this type of problems, an alternating projection (AP) algorithm was proposed in the radar waveform design problem [26], and later adopted in the design of the sparse frequency waveform in [28], [29]. As a result, a good auto-correlation characteristic of the waveform can be obtained by using the PSD matching method to design the sparse frequency waveform [29]. The application of the sparse frequency waveform was further introduced in [18]. During the process of keeping the constant modulus of the radar waveforms by using the covariance matrix, the waveform frequency constraint was also taken into account in [18], where the sparse frequency radar waveforms with good beampattern characteristics are obtained by a cyclic algorithm. However, to the best of our knowledge, the beampattern design with the sparse waveforms for the frequency dependent arrays, e.g., FDA, is still an open problem. Different from the previous optimization problems, the variables in spatial domain and frequency domain are coupled in both objective function and constraints, which makes the problem much complicated than the above mentioned beampattern design problems.

B. Contributions

In this paper, we discuss the design problem of the MIMO radar beampattern with the sparse frequency waveforms for the FDA and formulate it as a nonconvex optimization problem. Given an ideal beampattern, we start to obtain the covariance matrix of the transmitted signal with the help of the semi-definite programming (SDP). Since the beampattern of the FDA radar is range-dependent, after taking some distance points, we can get a series of the covariance matrices. Then the corresponding waveform matrices can be obtained from each covariance matrix, where the constant modulus constraint and sparse frequency constraint are both imposed in the process of solving the waveform design problem. In this way, the resulting constant modulus waveforms can have the sparse frequency characteristics. From these waveform matrices, we select one that minimizes the objective function value and take it into the next iteration. Finally, after the termination condition is satisfied, the resulting radar waveforms will not only satisfy the sparse frequency constraint in the frequency domain, but also have a good auto-correlation characteristic. The closeness between the optimized beampattern and the ideal beampattern can be measured by the sum of the objective function at each distance point.

Compared with the results which do not consider the frequency diversity, the waveforms designed by the proposed method can make the FDA radar beampattern very close to the ideal beampattern at any distance point. The problem in [18] which only takes a certain single distance point for the beampattern design is a special case of the problem considered in this paper. In other words, the problem in [18] is only the case that the carrier frequency increment between the two array elements is zero. Also, the proposed algorithm can be implemented in a parallel way with convergence guarantees, where the complexity of the proposed method is still proportional to the classic ones [18], [28].

C. Paper Organization

The remaining sections are organized as follows. Section 2 introduces the basic structure of the FDA, and shows the derivations of the signal model based on the covariance matrix and radar waveforms obtained through the radar transmit beampattern. In Section 3, a beampattern design procedure of the FDA with the sparse frequency waveforms is proposed, and the design flow chart is given. Simulation results as well as the discussion are presented in Section 4. Finally, conclusions are drawn in Section 5.

SECTION II. The Signal Model

A. The Mimo Radar Model

Consider a MIMO radar system with $M$ elements. The distance between the two elements is $d$ . A uniform linear array is shown in Figure 1.

FIGURE 1.

Geometric relation between the target and the elements of a linear array.

Assume that there is a far-field target, with the distances $R_{1},\ldots ,R_{M}$ to each array element. In particular, $R_{1}$ denotes the distance between the target to the antenna array. The angle between the target direction and the array normal direction is $\theta$ , and the range difference of the signals received between the two adjacent array elements is $d\sin \theta$ .

For the traditional array radar, if the carrier frequency is $f_{0}$ , the wavelength of the transmitted signal is $\lambda _{0}=c/f_{0}$ , where $c$ denotes the speed of light. Then the phase difference $\Delta \Psi _{a}$ between the two adjacent elements can be calculated as $$\begin{align} \Delta \Psi _{a}(\theta )=&2\pi \left ({\frac {R_{m}}{\lambda _{0}}-\frac {R_{m+1}}{\lambda _{0}}}\right )=-2\pi \frac {d}{\lambda _{0}}\sin \theta , \notag \\&~\qquad \qquad \qquad m=1,\ldots ,M-1. \end{align}$$

For the FDA radar, we assume that the array frequency is linearly increased and the frequency increment is denoted as $\Delta f$ [1]. The carrier frequency at the $m$ th element is $$\begin{equation} f_{m}=f_{0}+(m-1)\Delta f, \quad m=1,\ldots ,M. \end{equation}$$

Then, we can get the phase difference $\Delta \Psi _{b}$ between the two adjacent elements, which is stated in a formal way as follows.

Lemma 1:

Let $R_{1}$ denote the distance between the target and antenna array, $\theta$ be the angle between the target direction and array normal direction, $\Delta f$ be the frequency increment between two adjacent elements. Then, the phase difference between the two adjacent elements is $$\begin{align}&\hspace {-1.7pc}\Delta \Psi _{b}(\theta ,\Delta f,{R_{1}},m) \notag \\[-2pt]=&-2\pi \frac {d}{\lambda _{0}}\sin \theta \underbrace {-4\pi dm\frac {\Delta f}{c}\sin \theta -2\pi \Delta f\frac {R_{1}-d\sin \theta }{c}}_{\Delta \widetilde {\Psi }_{b}(\theta ,\Delta f, R_{1},m)}, \notag \\[-2 pt]&\qquad \qquad \qquad \qquad \qquad m=1,\ldots ,M. \end{align}$$

Proof:

See Section V-A

Compared with the phase difference of the traditional array, there is a new phase $\Delta \widetilde {\Psi }_{b}(\theta ,\Delta f, R_{1},m)$ involved in (3) due to the introduction of the frequency diversity. It is clear that the phase differences among the array elements are associated with the target distance $R_{1}$ and the introduced frequency increment $\Delta f$ . So the direction of the beampattern caused by the phase difference is associated with $R_{1}$ and $\Delta f$ as well. Assume that the phase shift of (3) makes the beam focus on a particular angle $\widetilde {\theta }$ , which can be expressed similarly as (1) by $$\begin{equation} \Delta \Psi _{b}=-2\pi \frac {d}{\lambda _{0}}\sin \widetilde {\theta }. \end{equation}$$

From (3) and (4), we can get the closed-form of $\widetilde {\theta }$ after some manipulations, i.e., $$\begin{align}&\hspace {-1.2pc}\widetilde {\theta }\notag \\=&\sin ^{-1}\left ({\sin \theta \!+\!\frac {2m\Delta f\lambda _{0}}{c}\sin \theta \!+\!\frac {\Delta f\lambda _{0}R_{1}}{cd}\!-\!\frac {\Delta f\lambda _{0}}{c}\sin \theta }\right )\!,\notag \\ {}\end{align}$$ which characterizes the relationship between the beam direction $\widetilde {\theta }$ at the distance $R_{1}$ and the beam direction $\theta$ at the array plane. If the beam direction at the array plane is $\theta =0$ , then (5) can be written as $$\begin{equation} \widetilde {\theta }(R_{1},\Delta f)=\sin ^{-1}\left ({\frac {\Delta f\lambda _{0} R_{1}}{cd}}\right ), \end{equation}$$ which clearly indicates that the FDA radar can obtain a range-dependent transmit beampattern by changing the signal carrier frequency of the array, so the spatial beam scanning can be realized effectively without the phase shifter [1]. This characteristic is one of the most important applications of the frequency diversity.

For the traditional array MIMO radar, assume that the number of samples during a transmitted signal pulse repetition period is $N$ . Let $x_{m}(n)$ denote a discrete-time radar transmitted baseband signal from the $m$ th radar at time instant $n$ . Then, the transmitted signal of the $m$ th radar transmitter becomes $$\begin{equation} \mathbf x_{m}=[x_{m}(1), \ldots , x_{m}(N)]^{ \mathsf {T}},\quad m=1,\ldots ,M \end{equation}$$ where the superscript symbol $^{ \mathsf {T}}$ denotes the transpose of a vector.

The MIMO radar waveforms at the $n$ th moment can be expressed as $$\begin{equation} \mathbf x(n)=[x_{1}(n), \ldots , x_{M}(n)]^{ \mathsf {T}},\quad n=1,\ldots ,N. \end{equation}$$ Consequently, the matrix $\mathbf X$ with size $N\times M$ represents the MIMO radar waveform matrix, i.e., $$\begin{align} \notag \mathbf X=&\left [{ \mathbf x_{1}, \ldots , \mathbf x_{M}}\right ] \\=&\left [{\begin{array}{ccc}x_{1}(1) & \ldots & x_{M}(1)\\ \vdots & \ddots & \vdots \\ x_{1}(N) & \ldots & x_{M}(N)\end{array}}\right ]. \end{align}$$

Then, the power radiated at the target $\theta$ by the radar array during a pulse repetition period is $$\begin{equation} P_{a}(\theta )= \mathbf w^{ \mathsf {H}}_{a}(\theta ) \mathbf R \mathbf w _{a}(\theta ) \end{equation}$$ where the superscript symbol $^{ \mathsf {H}}$ denotes the conjugate transpose of a vector, $\mathbf R$ denotes the covariance matrix of the transmitted signal $\mathbf x(n)$ , i.e., $$\begin{equation} \mathbf R=\frac {1}{N}\sum ^{N}_{n=1} \mathbf x^{ \mathsf {H}}(n) \mathbf x(n), \end{equation}$$ and $\mathbf w_{a}(\theta )$ is the steering vector of the traditional linear array MIMO radar system, which can be expressed by $$\begin{equation} \mathbf w_{a}(\theta )=\left [{1, e^{j\Delta \Psi _{a}(\theta )}, \ldots , e^{j(M-1)\Delta \Psi _{a}(\theta )}}\right ]^{ \mathsf {T}}. \end{equation}$$

When the radar carrier frequency $f_{0}$ is determined, $\mathbf w_{a}(\theta )$ is only related to $\theta$ . $P_{a}(\theta )$ is also a function of $\theta$ , which actually reflects the radar transmitting energy distribution in space, i.e., so called “transmit beampattern”.

For the FDA MIMO radar, the signal power $P_{b}$ is both related to the carrier frequency increment $\Delta f$ and the range $R_{1}$ . According to (3), we have the steering vector $$\begin{align}&\hspace {-1.7pc} \mathbf w_{b}(\theta ,\Delta f,R_{1})\notag \\=&\left [{1, e^{j \Delta \Psi _{b}(\theta ,\Delta f,R_{1},m=1)},\ldots ,e^{j \sum ^{M-1}_{m=1} \Delta \Psi _{b}(\theta ,\Delta f,R_{1},m)}}\right ]^{ \mathsf {T}},\notag \\ {}\end{align}$$ which involves both $\Delta f$ and $R_{1}$ . Then, the signal power can be expressed by $$\begin{equation} P_{b}(\theta ,\Delta f,R_{1})= \mathbf w^{ \mathsf {H}}_{b}(\theta ,\Delta f,R_{1}) \mathbf R \mathbf w _{b}(\theta ,\Delta f,R_{1}), \end{equation}$$ which is a function of $\theta$ , $\Delta f$ and $R_{1}$ .

B. From Ideal Beampattern to Covariance Matrix $\mathbf R$

For a traditional array radar, we can first initialize an ideal beampattern $\Gamma _{a}(\theta )$ and then design a covariance matrix $\mathbf R$ such that the beampattern of $\mathbf R$ should be as close to $\Gamma _{a}(\theta )$ as possible. According to the previous work in [13], the optimization problem can be formulated as $$\begin{align}&\min _{\alpha , \mathbf R}\quad \left ({\alpha \Gamma _{a}(\theta )- \mathbf w_{a}^{ \mathsf {H}}(\theta ) \mathbf R \mathbf w _{a}(\theta )}\right )^{2} \notag \\&\textrm {s.t.}\quad r_{mm}=1, m=1,\ldots ,M, \notag \\&\qquad \mathbf R \succeq 0 \end{align}$$ where the optimization variable $\alpha$ is the ratio which controls the magnitude of the beampattern; $r_{mm},m=1,\ldots ,M$ denote the diagonal elements of $\mathbf R$ ; $r_{mm}=1$ imposes the equal power constraint for each transmitter channel, and $\mathbf R\succeq 0$ denotes that the covariance matrix is positive semi-definite. It can be seen that problem (15) is a positive semi-definite quadratic programming (SQP) problem [13], which can be solved by optimization tools efficiently, such as SEDUMI [14].

The ideal beampattern $\Gamma _{a}(\theta )$ can be designed as a rectangle. The magnitude of the calculated beampattern by $\mathbf R$ is shown in Figure 2, which is a smooth curve shown by red color.

FIGURE 2.

From the ideal beampattern to the beampattern of covariance matrix R.

For the FDA MIMO radar, the ideal beampattern is $\Gamma _{b}(\theta ,\Delta f,R_{1})$ . The effects of distance and frequency increment are both reflected in $\mathbf w_{b}(\theta ,\Delta f, R_{1})$ . Replacing $\mathbf w_{a}(\theta )$ by $\mathbf w_{b}(\theta ,\Delta f,R_{1})$ in (15), we have the following optimization problem: $$\begin{align}&\min _{\alpha , \mathbf R}~\left ({\alpha \Gamma _{b}(\theta ,\Delta f,R_{1})- \mathbf w_{b}^{ \mathsf {H}}(\theta ,\Delta f,R_{1}) \mathbf R \mathbf w _{b}(\theta ,\Delta f,R_{1})}\right )^{2} \notag \\&\textrm {s.t.}\quad r_{mm}=1, m=1,\ldots ,M, \notag \\&\qquad \mathbf R \succeq 0. \end{align}$$ When $\Delta f$ and $R_{1}$ are given, the optimization problem (16) can be also solved by the SEDUMI tool, which is the same as solving problem (15). In the following discussion, the calculation is under the condition that the covariance matrix $\mathbf R$ is known.

C. From Covariance Matrix $\mathbf R$ to Waveform Matrix $\mathbf X$

First we analyze the relationship between the covariance matrix $\mathbf R$ and the radar waveform matrix $\mathbf X$ . From (11) we have $$\begin{equation} \mathbf R=\frac {1}{N} \mathbf X^{ \mathsf {H}} \mathbf X. \end{equation}$$ When the covariance matrix $\mathbf R$ is known and there is no constraint in (17), $\mathbf R$ has one-to-many relationship with respective to $\mathbf X$ . In general, the number of samples of the radar waveform pulse is much larger than the number of the radar transmitters, i.e. $N>M$ , so $\mathbf X$ is a tall matrix. The key problem is that how to obtain the tall matrix $\mathbf X$ from the square matrix $\mathbf R$ . One possible way of solving this issue is to introduce a $N\times M$ orthogonal matrix $\mathbf V$ that satisfies $\mathbf V^{ \mathsf {H}} \mathbf V= \mathbf I$ [15], where the identity matrix has the same dimension as the covariance matrix $\mathbf R$ . Then according to $\mathbf R= \mathbf R \mathbf V ^{ \mathsf {H}} \mathbf V$ and (17), we have $$\begin{equation} \mathbf X=\sqrt {N} \mathbf V \mathbf R ^{1/2}. \end{equation}$$

From (18) we can observe that when the covariance matrix $\mathbf R$ is known we can calculate the radar waveform matrix through the transforming matrix $\mathbf V$ . At the same time, it is worth noting that the choice of matrix $\mathbf V$ will decide the properties of radar waveform matrix $\mathbf X$ directly. In a practical radar system, the main design concern is to guarantee the constant modulus of the radar waveform signals. Let $\mathcal {Q}$ represent the set of $\mathbf X$ that meets the constraint of the constant modulus property. Then the optimization problem of calculating the radar waveform matrix $\mathbf X$ through the covariance matrix $\mathbf R$ becomes $$\begin{align}&\min _{ \mathbf X, \mathbf V}\quad \| \mathbf X-\sqrt {N} \mathbf V \mathbf R ^{1/2}\|^{2} \notag \\&\textrm {s.t.}\quad \mathbf X \in \mathcal {Q}, \notag \\&\qquad \mathbf V ^{ \mathsf {H}} \mathbf V= \mathbf I \end{align}$$ where $\mathbf V^{ \mathsf {H}} \mathbf V= \mathbf I$ is the identity matrix constraint of the matrix $\mathbf V$ , $\mathbf X\in \mathcal {Q}$ denotes the constant modulus constraint of matrix $\mathbf X$ . According to [15], optimization problem (19) can be solved by an alternating algorithm and the basic idea is as follows

• Fix matrix $\mathbf X$ , minimize $\| \mathbf X-\sqrt {N} \mathbf V \mathbf R ^{1/2}\|^{2}$ with respective to $\mathbf V$ ,

• Fix matrix $\mathbf V$ , minimize $\| \mathbf X-\sqrt {N} \mathbf V \mathbf R ^{1/2}\|^{2}$ with respective to $\mathbf X$ .

These two steps are implemented alternately until some stopping criterion is satisfied. Let $\|\Delta \mathbf V \|$ be the difference of matrix $\mathbf V$ between the two adjacent iterations. When $\|\Delta \mathbf V \|$ is smaller than some threshold, we stop the iteration. With leveraging this alternating method, we can obtain the radar waveform matrix $\mathbf X$ which has the constant modulus.

SECTION III. Sparse Frequency Waveform Design for FDA

In the above section, the FDA radar signal model and the relationship between the covariance matrix $\mathbf R$ and the waveform matrix $\mathbf X$ are introduced. In this section, we mainly discuss how to design the sparse frequency radar waveforms for the FDA, which includes three parts: i) waveform design with the sparse frequency constraint, ii) beampattern design for the FDA, and iii) the flow diagram of the design steps.

A. Sparse Frequency Constraint

First, in the case where is only a single array element, we can use the PSD matching method to obtain a sequence that has sparse frequency characteristics from an ordinary constant modulus sequence.

For a continuous signal, we assume that a radar waveform $y(t)$ is with an impulse duration $T$ . Given an expected PSD with sparse characteristics, represented by $g_{\mathrm{exp}}(f)$ , then the PSD matching problem can be formulated as $$\begin{align}&\min _{y(t)}\quad \int ^{\infty }_{-\infty }||\mathscr {F}(y(t))|^{2}-g_{\mathrm{exp}}(f)|^{2}df \notag \\& {s.t.}\quad |y(t)|=1,\quad t\in [0,T] \end{align}$$ where $\mathscr {F}(y(t))$ denotes the Fourier transform of $y(t)$ ; $|y(t)|=1$ is the constant modulus constraint for the signal.

Formulation (20) gives the representation of a continuous signal waveform. For the corresponding discrete representation, the sequence $\mathbf y=[y(1),\ldots ,y(N)]^{ \mathsf {T}}$ can be used to represent the sparse frequency radar waveform. Each element of $\mathbf y$ is $$\begin{equation} y(n)=\frac {1}{\sqrt {N}}\exp (j\phi _{n}),\quad n=1,\ldots ,N, \end{equation}$$ where the factor $1/\sqrt {N}$ normalizes the waveform energy, $\phi _{n}$ represents the phase of the radar waveform at the $n$ th snapshot. Let $\Theta$ denote the phase vector which consists $N$ phase values, i.e., $$\begin{equation} \Theta =[\phi _{1},\ldots ,\phi _{N}]^{ \mathsf {T}}. \end{equation}$$ Since every element $y(n)$ in $\mathbf y$ is only related to the phase $\phi _{n}$ in the current moment, the radar waveform $\mathbf y$ can be written as a function of $\Theta$ , denoted as $\mathbf y(\Theta )$ . Let $\mathbf u=[u_{1}, {\dots },u_{N}]^{ \mathsf {T}}$ denote the discrete representation form of the ideal PSD $g_{\mathrm{exp}}(f)$ which is sparse. Then, the objective function of solving the sparse frequency waveform problem with known $\mathbf u$ can be expressed as $$\begin{equation} \min _{\Theta }\|( \mathbf A \mathbf y (\Theta ))\odot ( \mathbf A \mathbf y (\Theta ))^{*}- \mathbf u\|^{2} \end{equation}$$ where the superscript symbol * denotes the conjugate of a vector; $\odot$ means the Hadamad multiplication between two matrices; $\mathbf A$ represents the discrete Fourier transform matrix and every element in $\mathbf A$ is defined as $\mathbf A_{mn}=\exp (-j2\pi mn/N)$ where $m,n=1,\ldots ,N$ .

The optimization problem (23) turns out to be a quartic unconstrained non-convex optimization problem. Unfortunately, there is no general method to give the global optimal solution of this kind of problems. We might only get a local minimum with adopting the Quasi-Newton method. Compared with the Newton method, the Quasi-Newton method has a superlinear convergence rate with less computational complexity in most cases [33]. Similar to the steepest descent method, the Quasi-Newton method is also required to calculate the gradient of the objective function in each iteration. In this paper, we use the Quasi-Newton method and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) correction to solve problem (23).

Let $f(\Theta )$ represent the objective function, and define $\mathbf a_{n}=[ \mathbf A_{n1},\ldots , \mathbf A_{nN}]^{ \mathsf {T}}$ where $n=1,\ldots ,N$ . Then problem (23) can be written as $$\begin{equation} \min _{\Theta }f(\Theta )=\sum ^{N}_{n=1}| \mathbf y^{ \mathsf {H}}(\Theta ) \mathbf A_{n} \mathbf y(\Theta )- \mathbf u_{n}|^{2} \end{equation}$$ where $\mathbf A_{n}= \mathbf a_{n} \mathbf a^{ \mathsf {H}}_{n}$ . Then the gradient $\mathbf g$ of the objective function $f(\Theta )$ on $\Theta$ can be calculated by $$\begin{align} \notag \mathbf g=&\frac {\partial f(\Theta )}{\partial \Theta } \\=&2\sum ^{N}_{n=1}( \mathbf y^{ \mathsf {H}}(\Theta ) \mathbf A_{n} \mathbf y(\Theta )-u_{n})2\textrm {Im}(\textrm {diag}( \mathbf y^{ \mathsf {H}}(\Theta )) \mathbf A^{ \mathsf {H}}_{n} \mathbf y(\Theta ))\notag \\ {}\end{align}$$ where $\textrm {diag}( \mathbf y^{ \mathsf {H}}(\Theta ))$ denotes a diagonal matrix with the elements of the vector $\mathbf y^{ \mathsf {H}}(\Theta )$ on the main diagonal; the symbol “Im”means the imaginary part of a complex number.

Let the subscript symbol $l$ represent the index of each iteration, and use the subscript “out” to represent the final result. Then the steps for solving problem (24) are summarized in Algorithm 1, where $\varepsilon$ denotes the threshold of the iteration termination, $\|\boldsymbol {\delta }_{l}\|$ measures the difference of $\Theta$ between the two adjacent iterations, $\mathbf S_{l}$ is the approximation of the inverse of the Hessian matrix, the calculation of $\mathbf S_{l}$ is consistent with the BFGS correction, and more corresponding issues in details on the derivation and implementation can be referred to [33], [34].

Algorithm 1 Algorithm for Solving Problem (24)

1:

Input: $\Theta _{0}$ , $\varepsilon$

2:

Set $l=1$ , $\mathbf S_{l}= \mathbf I$ . Initialize $\Theta _{0}$ . Calculate $\mathbf g_{l}$ by (25).

3:

Calculate search direction $\mathbf d_{l}=- \mathbf S_{l} \mathbf g_{l}$ .

4:

Line search: $\alpha _{l}=\arg \min f(\Theta _{l}+\alpha _{l} \mathbf d_{l})$ .

5:

Let $\boldsymbol {\delta }_{l}=\alpha _{l} \mathbf d_{l}$ and implement $\Theta _{l+1}=\Theta _{l}+\boldsymbol {\delta }_{l}$ .

6:

if $\|\boldsymbol {\delta }_{l}\|\le \varepsilon$ then

7:

Output $\Theta _{{out}}=\Theta _{l+1}$

8:

else

9:

Calculate $\mathbf g_{l+1}$ by (25) and let $\gamma _{l}= \mathbf g_{l+1}- \mathbf g_{l}$ .

10:

Calculate $\mathbf S_{l+1}$ by $$\begin{equation} \mathbf S_{l+1}= \mathbf S_{l}+\left ({1+\frac { \boldsymbol \gamma ^{ \mathsf {H}}_{l} \mathbf S_{l} \boldsymbol \gamma _{l}}{ \boldsymbol \gamma ^{ \mathsf {H}}_{l}\boldsymbol {\delta }_{l}}}\right )-\frac {\boldsymbol {\delta }_{l} \boldsymbol \gamma ^{ \mathsf {H}}_{l} \mathbf S_{l}+ \mathbf S_{l} \boldsymbol \gamma _{l}\boldsymbol {\delta }^{ \mathsf {H}}_{l}}{ \boldsymbol \gamma ^{ \mathsf {H}}_{l}\boldsymbol {\delta }_{l}}. \end{equation}$$

11:

Let $l\leftarrow l+1$ and go back to step 3.

12:

end if

In Algorithm 1, the main computational load of the Quasi-Newton method and the BFGS correction lies in the computation of the objective function as well as the $\mathbf S_{l}$ . After the algorithm is terminated, we can get the radar waveform $\mathbf y(\Theta _{{out}})$ as well as the output phase vector $\Theta _{{out}}$ such that $\mathbf y(\Theta _{{out}})$ has the sparse frequency spectrum. Because the iteration process only changes the value of the phase vector $\Theta$ , there is no effect on the modulus value of the signal. Therefore, if the initialized sequence $\mathbf y(\Theta _{0})$ has the constant modulus, then the resulting waveform $\mathbf y(\Theta _{{out}})$ still keeps the constant modulus property as well.

For the traditional array MIMO radar system, we assume that the $M$ radar transmitters have the same constraint of the sparse frequency spectrum, meaning that the corresponding ideal PSD is $\mathbf u^{(1)}=\ldots = \mathbf u^{(M)}= \mathbf u$ . Then, the optimization problem of the sparse frequency waveforms design is $$\begin{equation} \min _{\Theta ^{(1)},\ldots ,\Theta ^{(M)}}\sum ^{M}_{m=1}\|( \mathbf A \mathbf y _{m}(\Theta ^{(m)}))\odot ( \mathbf A \mathbf y _{m}(\Theta ^{(m)}))^{*}- \mathbf u\|^{2}\qquad \end{equation}$$ where $\Theta ^{(1)},\ldots ,\Theta ^{(M)}$ denote the phase vectors of the $M$ radar transmit waveforms.

Let the $m$ th transmit waveform of the MIMO radar transmitter be $\mathbf x_{m}= \mathbf y_{m}(\Theta ^{(m)})$ . The optimization problem of designing the beampattern with the both constant modulus waveforms and sparse frequency spectrum can be written as $$\begin{align}&\notag \min _{ \mathbf X, \mathbf V,\Theta ^{(1)},\ldots ,\Theta ^{(M)}}~\sum ^{M}_{m=1}\|( \mathbf A \mathbf y _{m}(\Theta ^{(m)}))\odot ( \mathbf A \mathbf y _{m}(\Theta ^{(m)}))^{*}- \mathbf u\|^{2} \\&\qquad +\,\| \mathbf X-\sqrt {N} \mathbf V \mathbf R ^{1/2}\|^{2} \notag \\&\textrm {s.t.}\quad \mathbf x _{m}= \mathbf y_{m}(\Theta ^{(m)}),\;m=1,\ldots ,M, \notag \\&\qquad \mathbf X \in \mathcal {Q}, \notag \\&\qquad \mathbf V ^{ \mathsf {H}} \mathbf V= \mathbf I. \end{align}$$

We deal with optimization problem (28) with a cyclic algorithm by introducing a threshold $\tau$ for the termination of the iterations. The steps are shown in Algorithm 2, where $\|\Delta \mathbf V \|$ denotes the difference of matrix $\mathbf V$ between the two successive iterations.

Algorithm 2 Algorithm for Solving Problem (28)

1:

Input: $\mathbf X=[ \mathbf y_{1}(\Theta ^{(1)}),\ldots , \mathbf y_{M}(\Theta ^{(M)})]$ , $\tau$

2:

Update $\mathbf V$ by solving

$\min _{ \mathbf V}\| \mathbf X-\sqrt {N} \mathbf V \mathbf R ^{1/2}\|^{2}\;\textrm {s.t.} \mathbf V^{ \mathsf {H}} \mathbf V= \mathbf I$

3:

Update $\mathbf X$ by solving

$\min _{ \mathbf X}\| \mathbf X-\sqrt {N} \mathbf V \mathbf R ^{1/2}\|^{2}\;\textrm {s.t.}\quad \mathbf X \in \mathcal {Q}$

4:

Update $\Theta ^{(1)}_{{out}},\ldots ,\Theta ^{(M)}_{{out}}$ by solving (24)

5:

Update $\mathbf X$ by $\mathbf x_{m}= \mathbf y_{m}(\Theta ^{(m)})$

6:

if $\|\Delta \mathbf V \|\le \tau$ then

7:

Output $\Theta _{{out}}$

8:

else

9:

Go back to step 2.

10:

end if

In Algorithm 2, the main computation load concentrates on Step 2, Step 3 and Step 4. The specific efficient implementations of solving matrix $\mathbf V$ from $\mathbf X$ in Step 2 and solving matrix $\mathbf X$ from $\mathbf V$ in Step 3 can refer to [15]. The purpose of Step 4 is to obtain the waveforms with the sparse frequency characteristics and constant modulus. The specific operation is roughly the same as that of solving optimization problem (24), where the only difference between them is just that for the MIMO radar system it is required to obtain the $M$ waveforms.

After the implementation of Algorithm 2, an optimized waveform matrix $\mathbf X$ with the sparse frequency can be obtained such that the beampattern calculated through $\mathbf X$ is very close to the beampattern of the covariance matrix $\mathbf R$ , which is the case of traditional array MIMO radar system with $M$ elements. Next we will consider the beampattern design with sparse frequency waveforms for the FDA MIMO radar.

B. Waveform Design for FDA

When $\Delta f$ is given, we can use (6) to design an ideal FDA MIMO radar beampattern $\Gamma _{b}(\theta ,R_{1})$ . For example, $\Gamma _{b}(\theta ,0)$ refers that the FDA MIMO radar beam direction at the distance $R_{1}=0$ is set as $\widetilde {\theta }=0^{\circ }$ . Assume that the beampattern $\Gamma _{b}(\theta ,0)$ is an ideal rectangle. Then the beam direction is equal to the central position of the rectangle. Set the width of the rectangle as $B(0)$ . We have $$\begin{align} \Gamma _{b}(\theta ,0)=\begin{cases}0\textrm {dB}, & \theta \in \left [{-\frac {B(0)}{2}, \frac {B(0)}{2}}\right ] \\[0.6pc] -20\textrm {dB}, & \theta \in \left ({{-90^{\circ },-\frac {B(0)}{2}}}\right ) \\[0.3pc] -20\textrm {dB}, & \theta \in \left ({\frac {B(0)}{2},90^{\circ }}\right ) \end{cases} \end{align}$$

When $\Gamma _{b}(\theta ,0)$ is known, the ideal rectangular beampatterns at different distances’ central positions can be obtained by (6), and the width is $$\begin{equation} B(R_{1})=\frac {B(0)}{\cos (\widetilde {\theta }(R_{1}))}. \end{equation}$$ Similarly, we can get several ideal rectangular beampatterns, corresponding to different distances. In the latter analysis, we uniformly take $q$ distance points in a certain distance range, which are $R_{1}(k),k=1,\ldots ,q$ . Consequently, we can get the $q$ ideal rectangular beampatterns $\Gamma _{b}(\theta ,R_{1}(k))$ where $k=1,\ldots ,q$ . By substituting them into (16), we have $$\begin{align}&\min _{\alpha , \mathbf R_{k}}~\left ({\alpha \Gamma _{b}(\theta ,R_{1}(k))- \mathbf w_{b}^{ \mathsf {H}}(\theta ,R_{1}(k)) \mathbf R_{k} \mathbf w_{b}(\theta ,R_{1}(k)}\right )^{2} \notag \\&\textrm {s.t.}\quad r_{kmm}=1,\quad m=1,\ldots ,M, \notag \\&\qquad \mathbf R _{k}\succeq 0, \notag \\&\qquad k=1,\ldots ,q \end{align}$$ where $r_{kmm}, m=1,\ldots ,M$ are the diagonal elements of $\mathbf R_{k}$ ; $\mathbf R_{k}, k=1,\ldots ,q$ are positive semi-definite matrices and $r_{kmm}=1, k=1,\ldots ,q$ denote the equal power constraint of each transmitter channel. Similar as problem (16), the SEDUMI tool can be adopted to solve problem (31) to get the $q$ optimized covariance matrices $\mathbf R_{1},\ldots , \mathbf R_{q}$ .

Further, these covariance matrices $\mathbf R_{1},\ldots , \mathbf R_{q}$ can be used through (19) to obtain the corresponding waveform matrices $\mathbf X_{1},\ldots , \mathbf X_{q}$ . According to (17), we have $$\begin{equation} \mathbf R_{k}=\frac {1}{\sqrt {N}} \mathbf X^{ \mathsf {H}}_{k} \mathbf X_{k},\quad k=1,\ldots ,q. \end{equation}$$

Similar as (28), we can see that the optimization problem of getting the radar waveform matrices $\mathbf X_{k},k=1,\ldots ,q$ with the sparse frequency constraint can be expressed as $$\begin{align}&\notag \min _{ \mathbf X_{k}, \mathbf V_{k},\Theta ^{(1)},\ldots ,\Theta ^{(M)}}~\sum ^{M}_{m=1}\|( \mathbf A \mathbf y _{m}(\Theta ^{(m)}))\odot ( \mathbf A \mathbf y _{m}(\Theta ^{(m)}))^{*}- \mathbf u\|^{2} \\&\qquad +\,\| \mathbf X_{k}-\sqrt {N} \mathbf V_{k} \mathbf R^{1/2}_{k}\|^{2} \notag \\&\textrm {s.t.}\quad \mathbf x _{m}= \mathbf y_{m}(\Theta ^{(m)}),\;m=1,\ldots ,M, \notag \\&\qquad \mathbf X _{k}\in \mathcal {Q}, \notag \\&\qquad \mathbf V ^{ \mathsf {H}}_{k} \mathbf V_{k}= \mathbf I \end{align}$$ where $\mathbf X_{k}\in \mathcal {Q}$ denotes the constant modulus constraint of $\mathbf X_{k}$ . The procedure of solving problem (33) is the same as that of problem (28).

However, for the FDA MIMO radar, we need to solve the unique optimal waveform matrix $\mathbf X$ through $\mathbf R_{k}$ , $k=1,\ldots ,q$ . If the sparse frequency constraint of the waveform spectrum is added, then the optimization problem of designing the beampattern with the sparse frequency waveforms for the FDA becomes $$\begin{align}&\notag \min _{ \mathbf X, \mathbf V_{k},\Theta ^{(1)},\ldots ,\Theta ^{(M)}}~f_{\mathrm{p}}=\sum ^{q}_{k=1}\| \mathbf X-\sqrt {N} \mathbf V_{k} \mathbf R^{1/2}_{k}\|^{2} \\&\quad +\sum ^{M}_{m=1}\|( \mathbf A \mathbf y _{m}(\Theta ^{(m)}))\odot ( \mathbf A \mathbf y _{m}(\Theta ^{(m)}))^{*}- \mathbf u\|^{2} \notag \\&\textrm {s.t.}\quad \mathbf x _{m}= \mathbf y_{m}(\Theta ^{(m)}),\;m=1,\ldots ,M, \notag \\&\quad \mathbf X \in \mathcal {Q}, \notag \\&\quad \mathbf V ^{ \mathsf {H}}_{k} \mathbf V_{k}= \mathbf I. \end{align}$$ In (34), the covariance matrices $\mathbf R_{1},\ldots , \mathbf R_{q}$ are calculated by SDP through ideal beampatterns $P_{b}(\theta ,R_{1}(1)),\ldots , P_{b}(\theta ,R_{1}(q))$ ; the known quantity $\mathbf u$ is the expected PSD of the waveforms with the sparse frequency constraint. At first glance, solving this problem is very hard. The reasons are as follows i) the first term of the objective function is non-separable over blocks because of variable $\mathbf X$ , ii) the blocks $\mathbf X$ and $\Theta$ are non-linearly coupled in the constraints, iii) the objective function over $\Theta$ , orthogonality constraint on $\mathbf V_{k}$ and constant modules constraint on $\mathbf X$ are all nonconvex.

In this paper, a cyclic algorithm is proposed for solving optimization problem (34). The detailed steps are shown in Algorithm 3, where the subscript $(l)$ represents the number of iterations, and $\Delta f_{\mathrm{p}}(k^{*})$ denotes the difference of the minimum objective function values between the two iterations.

Algorithm 3 Algorithm for Solving Problem (34)

1:

Input: $\mathbf X_{0(l)}$ , $l$ , $q$ , $\varsigma$

2:

for $k=1,\ldots ,q$ do

3:

Update $\mathbf X_{k(l)}$ using $\mathbf X_{0(l)}$ and $\mathbf R_{k}$ by solving problem (28).

4:

Calculate the objective value of problem (34) with substituting $\mathbf X_{k(l)}$ , denoted by $f_{\mathrm{p}}(k)$ .

5:

end for

6:

Find $k^{*}=\arg \min _{k}\{f_{\mathrm{p}}(k)|k=1,\ldots ,q\}$ and $\mathbf X_{k^{*}(l)}$ .

7:

if $|\Delta f_{\mathrm{p}}(k^{*})|\le \varsigma$ then

8:

Output $\mathbf X_{{out}}= \mathbf X_{k^{*}(l)}$ .

9:

else

10:

Let $\mathbf X_{0(l)}= \mathbf X_{k^{*}(l)}$ , $l\leftarrow l+1$ and go back to step 2.

11:

end if

From Algorithm 3 we know that each iteration of solving problem (34) is equivalent to solving every sub-problem (28). In the first iteration, we can obtain the waveform matrices $\mathbf X_{1(1)},\ldots , \mathbf X_{q(1)}$ through $\mathbf R_{1},\ldots , \mathbf R_{q}$ , where $\mathbf X_{k^{*}(1)}$ can be found by looking for the minimum objective value in (34). In the second iteration, after obtaining the resulting waveform matrices $\mathbf X_{1(2)},\ldots , \mathbf X_{q(2)}$ , we will put $\mathbf X_{k^{*}(2)}$ into the third iteration. During the update of $k^{*}$ and $\mathbf X_{k^{*}(l)}$ , the objective function (34) will at least not be increased. Since each update of the algorithm does not increase the objective function that is always nonnegative, the value of the objective function will eventually converge.

C. The Chart of the Design Steps

Here we use the chart to show the basic steps of designing the MIMO radar transmit beampattern with the sparse frequency radar waveforms for the FDA proposed in this paper.

As shown in Figure 3, the steps of solving (34) can be implemented by using Algorithm 3. The input is a random initialized constant modulus waveform matrix $\mathbf X_{0(l)}$ , and the output is the waveform matrix $\mathbf X_{\textrm {out}}$ with the required properties of the FDA MIMO beampatterns. The $q$ matrices $\mathbf X_{1(l)},\ldots , \mathbf X_{q(l)}$ in each iteration can be calculated independently in parallel. The dashed part of Figure 3 is the calculation of obtaining the waveforms with the sparse frequency characteristics from the constant modulus waveforms using the iterative method. The concrete expansion of this step in more details is shown in Figure 4.

FIGURE 3.

The flow of MIMO beampattern with the sparse frequency waveforms optimization steps for FDA.

FIGURE 4.

The expansion of the steps of obtaining the sparse frequency waveforms in Figure 3.

Figure 4 shows the steps of solving waveform matrix with the sparse frequency constraint, which are listed in Algorithm 1. In each iteration, a complete PSD matching method is performed to provide the MIMO radar waveforms with the sparse frequency spectrum. When the input is a constant modulus waveform matrix $\mathbf X$ , the output is the waveform matrix $\mathbf X_{{out}}$ with the sparse frequency property. The $M$ array elements of the MIMO radar can be calculated independently.

SECTION IV. Numerical Results

In this section, simulation experiments are implemented to verify the performance of the proposed method of designing the beampattern with the sparse frequency waveforms for the FDA.

First, we assume that the FDA MIMO system has $M=10$ radar transmitters, and the length of each transmit waveform sequence is $N=256$ . The carrier frequency of each element of the array is $f_{0}+(m-1)\Delta f,m=1,\ldots ,M$ where $f_{0}=10$ GHz and the frequency increment $\Delta f=5$ kHz. Set the element spacing be $d=\lambda _{0}/2$ .

In this experiment, we take $q=25$ points from 1–25km equally, i.e., $R_{1}(1)=1\textrm {km},\ldots ,R_{1}(25)=25\textrm {km}$ . Then, the ideal beampattern $\Gamma _{b}(\theta ,0)$ of the distance $R_{1}=0$ is set to be $$\begin{equation} \Gamma _{b}(\theta ,0)=\begin{cases}0\textrm {dB}, & \theta \in \left [{-5^{\circ }, 5^{\circ }}\right ] \\ -20\textrm {dB}, & \theta \in \left ({{-90^{\circ },-5^{\circ }}}\right ) \\ -20\textrm {dB}, & \theta \in \left ({5^{\circ },90^{\circ }}\right ) \end{cases} \end{equation}$$ which shows that the central position of the ideal beampattern $\Gamma _{b}(\theta ,0)$ is $\widetilde {\theta }(0)=0^{\circ }$ and the width of the rectangle is $B(0)=10^{\circ }$ . From (6) and (30) we can get the $q=25$ ideal beampatterns which are $\Gamma _{b}(\theta ,1\textrm {km}),\ldots ,\Gamma _{b}(\theta ,25\textrm {km})$ .

Take two distance points $R_{1}(1)=1$ km and $R_{1}(25)=25$ km as examples. The central position and the width of the rectangle can be calculated as $\widetilde {\theta }(1\textrm {km})=1.91^{\circ }$ , $B(1\textrm {km})=10.01^{\circ }$ , $\widetilde {\theta }(25\textrm {km})=56.44^{\circ }$ , and $B(25\textrm {km})=18.09^{\circ }$ .

We plot the ideal beampattern of the FDA at the distance $R_{1}(1)=1$ km and $R_{1}(25)=25$ km respectively in Figure 5. It can be seen that the ideal beampattern of the FDA at different distances are set to be the rectangular form. Plugging these $q=25$ ideal beampatterns $\Gamma _{b}(\theta ,1\textrm {km}),\ldots ,\Gamma _{b}(\theta ,25\textrm {km})$ into (31), we can get the new 25 covariance matrices through solving $$\begin{align}&\hspace {-2pc}\min _{ \mathbf R_{k}}(\alpha \Gamma _{b}(\theta ,R_{1}(k))- \mathbf w^{ \mathsf {H}}_{b}(\theta ,R_{1}(k)) \mathbf R_{k} \mathbf w_{b}(\theta ,R_{1}(k)))^{2},\notag \\&\qquad \qquad \qquad \qquad k=1,\ldots ,25. \end{align}$$ The SEDUMI tool is used to solve (36). Then, the constant modulus MIMO radar waveforms with the sparse frequency characteristics can be obtained through the proposed iterative algorithm.

FIGURE 5.

The ideal beampatterns of the FDA. (a) At the distance 1km. (b) At the distance 25km.

The expected PSD of the sparse frequency waveforms is assumed to have 3 passbands and 2 stopbands. The gain in the passband is 5dB and in the stopband is −40dB, i.e., $$\begin{equation} u_{k}=\begin{cases}5\textrm {dB}, & k\in \mathcal {S},\\ -40\textrm {dB}, & k\notin \mathcal {S}, \end{cases} \end{equation}$$ where $\mathcal {S}$ denotes the set of the passbands. We set $\mathcal {S}=[1,55)\cup [110,163)\cup [{218,256}]$ in MHz. Then the expected PSD of the narrowband radar waveform can be shown in Figure 6, which is the goal of designing the sparse frequency waveforms.

FIGURE 6.

Expected power spectral density.

Set tolerance thresholds $\tau =10^{-4}$ , $\varepsilon =10^{-3}$ , $\varsigma =10^{-6}$ . The initialization of the phase constant modulus waveforms $\mathbf X_{0(0)}$ is realized randomly. Through implementing Algorithm 3, we can get $\mathbf X_{\textrm {out}}$ which represents the waveforms from the 10 radar transmitters.

The FDA beampatterns of the waveforms in the range 1–25km are shown in Figure 7, where the results are represented in a two-dimensional projection diagram. It can be observed that when the distance changes from 1km to 25km the center position of the beampattern varies from 1° to 56° where the width of the beampattern is also increased. These characteristics are the specialties of the FDA radar. Now we analyze the difference between the beampattern of the obtained waveforms and the beampattern of the covariance matrices. We use $\gamma _{k}$ to measure this difference as the degree of the closeness, where $\gamma _{k}$ is a part of the objective function in problem (34) and defined as $$\begin{equation} \gamma _{k}=\| \mathbf X-\sqrt {N} \mathbf V_{k} \mathbf R^{1/2}_{k}\|^{2}. \end{equation}$$ Since there are more than one distance point, we can compare the degree of the closeness at each point. Also, we have the sum of $\gamma _{k}$ . Next we take 2 distance points 1km and 25km as examples to discuss, which are shown in Figure 8 and Figure 9.

FIGURE 7.

The FDA beampattern of the resulting waveforms in the range 1–25km.

FIGURE 8.

Contrast diagram between the beampattern of the resulting waveforms in the distance range 1km and the beampattern of ${\text {R}}_{1}$ . (a) $x$ -$y$ form. (b) Polar diagram form.

FIGURE 9.

Contrast diagram between the beampattern of the resulting waveforms in the distance range 25km and the beampattern of ${\text {R}}_{25}$ . (a) $x$ -$y$ form. (b) Polar diagram form.

In Figure 8, it shows the beampattern of the resulting waveforms at the distance 1km and the beampattern of the covariance matrix $\mathbf R_{1}$ , with the $x$ -$y$ axis form and the polar diagram form respectively. From the curves we can see that the beampattern of the resulting waveforms at the distance 1km is close to the beampattern of the covariance matrix $\mathbf R_{1}$ . We plug $k=1$ into (38) to calculate $\gamma _{1}$ , which measures the closeness of these two beampatterns at the distance 1km, and we can get 4.429. Similarly, in Figure 9, the beampattern of the resulting waveforms at the distance 25km is close to the beampattern of the covariance matrix $\mathbf R_{25}$ , where we have $\gamma _{25}=4.401$ .

At the other distances, the closeness $\gamma _{k}$ between the FDA beampattern of the waveforms obtained from the simulation and the ones from other covariance matrices $\mathbf R_{k}$ are shown in Table 1 in details. Also, we can calculate that $\sum _{k}\gamma _{k}=134.2$ .

TABLE 1 The Closeness $\gamma _{k}$ at Different Distances

When the frequency increment between the array elements is zero, i.e., $\Delta f=0$ , the optimization problem in [18] is a special case of problem (34) in this paper. Because there is no carrier frequency increment between the elements, the waveform matrix $\mathbf X$ is not range-dependent, i.e., the beampatterns at different distances are the same. If we use one distance point in the previous method [18] for the FDA, we can still get the optimized beampattern. Take the distance points of 1km, 25km as examples. The difference with respective to the closeness between the new method of this paper and the method in [18] is shown in Figure 10.

FIGURE 10.

The closeness value $\gamma _{k}$ at each distance point. (a) Use $R_{1}(1)=1$ km. (b) Use $R_{1}(25)=25$ km.

It can be observed that although at the distance point of 1km (or 25km), $\gamma _{1}$ (or $\gamma _{25}$ ) of the method in [18] is 1.434 (or 2.104), which is smaller than 4.429 (or 4.401) obtained from the new method, but $\sum _{k}\gamma _{k}$ of the method in [18] is 148.1 (or 152.4), which is larger than 134.2 resulted by the new method. The $\sum _{k}\gamma _{k}$ results using other distance points are shown in Table 2 in details.

TABLE 2 The Closeness $\sum _{k}\gamma _{k}$ at Different Distances

Table 2 lists $\sum _{k}\gamma _{k}$ at the distance points 1–25km by using the method in [18]. Combined with Figure 10, it can be seen that by using the method in [18], the lowest closeness point generally corresponds to the fact that which distance is used: the lowest closeness point in Figure 10(a) is 1km when $R_{1}=1$ km is adopted; the lowest closeness point in Figure 10(b) is 25km when $R_{1}(25)=25$ km is adopted. But we know that $\sum _{k}\gamma _{k}$ calculated by the new method is 134.2, which is less than the result of any distance point used by the method in [18]. That illustrates that using this new method can make the objective function (34) achieve a smaller value.

SECTION V. Conclusions

In this paper, we mainly study the design of the MIMO radar beampattern with the sparse frequency waveforms for the FDA. Under the structure of the FDA, the MIMO radar waveform matrix is obtained through the covariance matrix, where the sparse frequency constraint and the constant modulus waveforms are both considered. According to the characteristics of the range-dependent beampattern, several distance points are taken into consideration to solve the beampattern design problem. Also, an efficient cyclic algorithm is proposed for solving the nonconvex optimization problem with convergence guarantees. From the numerical simulations, the resulting beampattern by our proposed method, compared with the optimized beampattern using a single distance point, is much closer to the ideal beampattern.

Appendix

SECTION A. Proof of Lemma 1

The phase difference between the two adjacent elements can be obtained by $$\begin{align*} \Delta \Psi _{b}=&2\pi \left ({\frac {R_{m}}{\lambda _{m}}-\frac {R_{m+1}}{\lambda _{m+1}}}\right ) \\ \left [-2 \textit{pt} \right ]=&2\pi \left ({\frac {R_{m}f_{m}}{c}-\frac {R_{m+1}f_{m+1}}{c}}\right ). \end{align*}$$

According to (2), we have $$\begin{equation*} \Delta \Psi _{b}=2\pi \left ({\frac {R_{m}(f_{0}+(m-1)\Delta f)}{c}-\frac {R_{m+1}(f_{0}+m\Delta f)}{c}}\right ). \end{equation*}$$

Based on the geometric relations of $R_{1}$ and $R_{m}$ where $m=2,\ldots ,M$ , we arrive at $$\begin{align*} \Delta \Psi _{b}=&\frac {2\pi }{c}\big ((R_{1}+md\sin \theta -d\sin \theta )f_{0} \\[-2pt]&+\,(R_{1}+md\sin \theta -d\sin \theta )m\Delta f \\[-2pt]&-\,(R_{1}+md\sin \theta -d\sin \theta )\Delta f \\[-2pt]&-\,(R_{1}+md\sin \theta )f_{0}-(R_{1}+md\sin \theta )m\Delta f\big ) . \end{align*}$$

After some manipulations, we can obtain $$\begin{align} \notag \Delta \Psi _{b}=&\frac {2\pi }{c}\big (-df_{0}\sin \theta -dm\Delta f\sin \theta -R_{1}\Delta f \\[-2pt]&\notag \qquad -md\Delta f\sin \theta +d\Delta f\sin \theta \big ) \\[-2pt]=&\frac {2\pi }{c}\left ({-df_{0}\sin \theta \!-\!2dm\Delta f\sin \theta \!-\!R_{1}\Delta f\!+\!d\Delta f\sin \theta }\right )\notag \\[-2pt]=&-\,2\pi \frac {d}{\lambda _{0}}\sin \theta -4\pi dm\frac {\Delta f}{c}\sin \theta -2\pi R_{1}\frac {\Delta f}{c}\notag \\&+\,2\pi d\frac {\Delta f}{c}\sin \theta . \end{align}$$