Joint linear array structure and waveform design for MIMO radar under practical constraints

1.   Introduction

The MIMO radar was developed as a result of the multiple-input multiple-output (MIMO) technology in communication systems ^[1,2]. The main difference between MIMO radar and the phased array radar is that each antenna of MIMO radar can independently transmit arbitrary waveforms, and the receiving end can perform diversity processing on the signal ^[3,4]. Under the condition of the same number of antennas, the diversity of waveforms allows the MIMO radar to outperform the phased array radar in terms of target detection ^[5,6], such as higher spatial resolution ^[7,8] and DOA estimation ^[9]. Therefore, the waveform design problem for the MIMO radar is a hot spot. Based on different design goals, the MIMO radar waveform optimization is divided into the following categories. The first type is to design the emission waveform to achieve the desired pattern when the target or interference airspace direction is prior information ^[10,11,12]; the second type is the orthogonal emission waveform design for achieving signal diversity at the receiving end ^[13,14]; the third category is to design waveform based on the joint signal processing of the radar transceiver system for different goals, such as maximizing the output signal-to-interference-noise ratio (SINR) ^[15,16] or maximizing the mutual information (MI) of target response and the echo ^[17,18]. In addition, real-time agile cognitive waveform design through environmental perception and echo feedback to the transmitter can be achieved ^[19,20].

All of the above studies are based on the consideration of fixed arrays. Here, we consider the joint design of transmit waveform and array structure, which supplies additional degrees of freedom at the transmitter. In ^[21] and ^[22], an efficient method of waveform covariance matrix design and antenna selection was developed. However, few works have been devoted to the joint design of array structure and waveform based on the consideration of signal reliability or quality at the receiver under interference and noise conditions, which is more challenging and practical for MIMO radar systems. This study aims to obtain a high-quality signal at the receiver of the MIMO radar by joint optimization of array structure and emission waveform with the practical constraint of constant modulus and similarity constraint (SC) in a signal-dependent interference environment. The main task of this study is to successfully optimize the joint objective of maximizing the SINR and proper array structure under a specific number of transmit and receive array elements with the constant modulus constraint (CMC) and SC. Specifically, to solve the resulted non-convex joint optimization problem, an efficient alternate iterative optimization algorithm based on fractional programming and generalized power-like iteration (GPLIM) is developed such that the SINR is improved while the responding radar array structure is obtained. The numerical results demonstrate the effectiveness and efficiency of the proposed algorithm.

The remainder of this paper is organized as follows. The signal and array structure models are introduced in Section 2. In Section 3, the practical constraints of emission waveform and partial-array structure design problems are discussed, and a joint optimization model is established. Based on fractional programming and GPLIM, an iterative optimization algorithm for the joint design is developed in Section 4. Section 5 presents various numerical simulations and analyses of the experimental results. Finally, the conclusions are drawn in Section 6.

Notation: We use ${\mathit{\boldsymbol{a}}}$ (lowercase, boldface) or $A$ (uppercase, italics) for vector, and ${\bf{A}}$(uppercase, boldface) for matrix. ${\left(\cdot \right)^T}$, ${\left(\cdot \right)^*}$, and ${\left(\cdot \right)^H}$ denote the transpose, conjugate, and conjugate transpose, respectively. ${\rm{vec}}\left({\bf{A}} \right)$ denotes a column vector obtained by stacking the columns of ${\bf{A}}$ on top of one another. ${\bf{A}} \succ 0\ $ indicates that the matrix ${\bf{A}}$ is positive definite. ${\bf{I}}_N$ represents the $N \times N$ identity matrix. Finally, we use $\odot$ and $\otimes$ to denote the Hadamard product and Kronecker product, respectively. $E\left(\cdot \right)$ represents the expectation of a random variable; $\arg \left(x \right)$ represents the angle of $x$.

2.   MIMO radar signal model and array structure

This study mainly deals with the fast-time radar waveform design; therefore, the Doppler frequency of the target is ignored. In addition, information regarding interference and target, such as the direction and the radar cross section can be acquired from a professional database or obtained through the cognitive working mode of the radar. Based on these instructions, a centralized MIMO radar system equipped with $M$ antennas is considered, assuming the array is a uniform linear array with half-a-wavelength element separation. Suppose that the total emission energy of the radar system is $E = {N_t}*K$. As shown in Figure 1, the numbers of effective emission and receiving antennas of the optimized partial-array are ${N_t}$ and ${N_r}$, where ${N_t} \le {N_r} \le M$.

The effective vector of array elements can be expressed as

when ${p_m}$ is 1, it means that the $m$th antenna of the array element position is valid; when ${p_m}$ is 0, it means that the $m$th antenna of the array element position does not work. Further, ${P_t}$ and ${P_r}$ denote the effective vectors of transmit and receive array, respectively.

Suppose ${{\mathit{\boldsymbol{s}}}_m}$ is the sampling sequence of the transmitted signal of the $m$th array element, and $K$ represents the sequence length. The emission signal of MIMO radar can be expressed as

where ${{\mathit{\boldsymbol{s}}}_m} = {[{s_m}(1), {s_m}(2), ..., {s_m}(K)]^T}$, and ${\mathit{\boldsymbol{s}}}(k) = {[{s_1}(k), ..., {s_2}(k), ..., {s_M}(k)]^T}$ denotes the sequence at the $k$th sampling time of the full-array. Considering array structure optimization, the actual transmit waveform can be expressed as

To facilitate the calculation, an array element position optimization matrix $P_t^{*}{\text{ = }}{P_t} \cdot ones(1, K)$ is introduced, and Eq (4) can be recast as

Assuming that the target direction ${\theta _t}$ is a priori, the reflected signal of the target in this direction received by MIMO radar is

where ${\alpha _t}$ denotes the complex amplitudes of target; ${{\mathit{\boldsymbol{a}}}_{tp}}({\theta _t})$ and ${{\mathit{\boldsymbol{a}}}_{rp}}({\theta _t})$ are the steering vectors of the radar transmit and receive arrays, respectively. Further, ${{\mathit{\boldsymbol{a}}}_{tp}}({\theta _t})$ and ${{\mathit{\boldsymbol{a}}}_{rp}}({\theta _t})$ can be expressed as

When the transmit and receive array vectors are equal, ${{\mathit{\boldsymbol{a}}}_{rp}}({\theta _t}) = {{\mathit{\boldsymbol{a}}}_{tp}}({\theta _t})$.

By taking the target location as the reference point, the direction of the $n$th signal-dependent interference is ${\theta _n}$, $n = 1, ..., {N_c}$, ${\theta _n} \ne {\theta _t}$. The clutter signal at the receive array is given by

where ${\alpha _n}$ denotes the complex amplitude of the $n$th signal-dependent interference. Let ${\bf{N}} \in {C^{M \times K}}$ be the received white Gaussian noise, and the variance be ${\sigma ^2}$. Because both the steering vector ${{\mathit{\boldsymbol{a}}}_{tp}}({\theta _t})$ and emission signal ${{\bf{S}}^{*}}$ are multiplied by ${P_t}$, the baseband equivalent of signals at the receive array can be described as

By stacking all the columns of ${\bf{Y}}$ into a vector, Eq (9) can be recast as

where ${\mathit{\boldsymbol{y}}} = {\rm{vec}}({\bf{Y}})$; ${\mathit{\boldsymbol{x}}} = {\rm{vec}}({\bf{S}}){\text{ = [}}{{\mathit{\boldsymbol{s}}}^T}(1), {{\mathit{\boldsymbol{s}}}^T}(2), ..., {{\mathit{\boldsymbol{s}}}^T}(K){{\text{]}}^T}$; ${\mathit{\boldsymbol{n}}} = {\rm{vec}}({\bf{N}}) \in {C^{K{N_r}*1}}$ is a circular complex Gaussian noise vector with zero mean and covariance matrix $\sigma _n^2{{\bf{I}}_{{N_r}K}}$; ${\bf{A}}({\theta _t}) = {{\bf{I}}_K} \otimes ({{\mathit{\boldsymbol{a}}}_{rp}}({\theta _t}){{\mathit{\boldsymbol{a}}}_{tp}}^T({\theta _t}))$ and ${\bf{A}}({\theta _n}) = {{\bf{I}}_K} \otimes ({{\mathit{\boldsymbol{a}}}_{rp}}({\theta _n}){{\mathit{\boldsymbol{a}}}_{tp}}^T({\theta _n}))$ denote the transceiver joint steering matrix of the MIMO radar in target and jammer directions, respectively.

3.   Practical constraints and problem formulation

Because the target detection performance is determined by the output SINR of the MIMO radar system, this section takes the SINR maximization as the optimization goal to design the array structure and emission waveform simultaneously for the MIMO radar in the presence of signal dependent interferences. From (10), the output SINR can be modeled as

Denote ${\rm{SNR}} = {{E\left({{{\left| {{\alpha _t}} \right|}^2}} \right)} \mathord{\left/ {\vphantom {{E\left({{{\left| {{\alpha _t}} \right|}^2}} \right)} {\sigma _n^2}}} \right. } {\sigma _n^2}}$; ${\rm{IN}}{{\rm{R}}_n} = {{E\left({{{\left| {{\alpha _n}} \right|}^2}} \right)} \mathord{\left/ {\vphantom {{E\left({{{\left| {{\alpha _n}} \right|}^2}} \right)} {\sigma _n^2}}} \right. } {\sigma _n^2}}$, ${\bf{\Sigma}} \left({{\theta _n}} \right) = \sum\limits_{n = 1}^{{N_c}} {{\rm{IN}}{{\rm{R}}_n}{{\bf{A}}^H}\left({{\theta _n}} \right)} {\bf{A}}\left({{\theta _n}} \right)$, $n = 1, ..., {N_c}$; $E({{\mathit{\boldsymbol{n}}}^H}{\mathit{\boldsymbol{n}}}) = K{N_r}\sigma _n^2$; $\forall \theta, \ {\bf{A}}(\theta) = {{\bf{I}}}_{K}\otimes ({\bf{P}}\odot ({{\mathit{\boldsymbol{a}}}}_{r}(\theta){{\mathit{\boldsymbol{a}}}}_{t}{}^{T}(\theta))) = {{\bf{I}}}_{K}\otimes ({\bf{P}}\odot {\bf{D}}(\theta))$, ${\bf{D}}(\theta) = {{\mathit{\boldsymbol{a}}}_r}(\theta){{\mathit{\boldsymbol{a}}}_t}^T(\theta)$; ${\bf{P}} = {P_r}{P_t}^T$. Simplifying the above formula, we obtain

The output SINR of the MIMO radar system is determined by the radar transmit waveform and the joint structure matrix of the transmit and receive array under the condition that the target and clutter direction, scattering power, and other information are known a priori. The structure matrix ${\bf{P}}$ of the array and the transmit waveform vector $x$ can effectively improve the quality of the radar output signal.

Considering the nonlinear limitation of the power amplifier in the radar system, it is reasonable to enforce a peak-to-average ratio (PAR) constraint on each antenna, that is,

When $\rho = 1$, constant modulus codes can be obtained, and a modulus value of 1 can be calculated according to the emission energy. The CMC is the strictest PAR constraint. In addition, to ensure some characteristics of emission waveform, a SC should also be imposed on the waveform, that is,

where ${{\mathit{\boldsymbol{x}}}_0}$ is a reference waveform with desired properties, and parameter $0 \le \varepsilon \le 2$ rules the extent of the similarity ^[23,24]. By considering the constant modulus constraint, the similarity constraint (13) can be further recast as

where ${\gamma _i} = \arg {{\mathit{\boldsymbol{x}}}_0}(i), i = 1, ..., MK$, and $\alpha = \arccos (1 - {\varepsilon ^2}/2)$.

To maximize the SINR under the CMC and SC ^[23], the following problem should to be addressed:

Equation (15) expresses a multivariate joint optimization problem. Compared with the existing approaches, the feasible set of waveforms is broadened owing to the simultaneous optimization of the transmit and receive arrays.

4.   Optimization algorithm for the joint design of transmit waveform and array structure

The optimization problem in Eq (15) is non-convex and has no closed-form solution; thus, in the following section a sequential optimization algorithm is introduced to address this NP-hard problem. Specifically, this study iteratively optimizes one variable of $(x, {\bf{P}})$ with the other variable fixed by maximizing the objective of Eq (11). Meanwhile, the values of ${P_t}$ and ${P_r}$ are updated with the achieved ${\rm{SINR}}$ at each iteration.

4.1.   Optimizing ${\bf{P}}$ with fixed ${\mathit{\boldsymbol{x}}}$

According to Eq (11), the CMC and SC have no relationship with ${\bf{P}}$. Hence, with fixed ${\mathit{\boldsymbol{x}}}$, optimization problem in Eq (15) with respect to ${\bf{P}}$ can be recast as

As the array structure matrix ${\bf{P}}$ is a matrix of 0-1 elements, ${\bf{P}} = {P_r}{P_t}^T$. ${P_t}$ and ${P_r}$ can be generated through enumeration, and then ${\bf{P}}$ can be synthesized.

4.2.   Optimizing ${\mathit{\boldsymbol{x}}}$ with fixed ${\bf{P}}$

With a fixed ${\bf{P}}$, Eq (15) with respect to ${\mathit{\boldsymbol{x}}}$ can be equivalent to

As the power emitted by an antenna is $\sum\limits_{k = 1}^K {{{\mathit{\boldsymbol{s}}}^H}(k)} {\mathit{\boldsymbol{s}}}(k){\text{ = }}K$, the amplitude of the $k$th code is $1$. Using fractional programming as in ^[25], the above problem can be approximately equivalent to

where ${\bf{U}}({\theta _t}) = {\rm{SNR}} \cdot {({{\bf{I}}_K} \otimes ({\bf{P}} \odot {\bf{D}}({\theta _t})))^H}({{\bf{I}}_K} \otimes ({\bf{P}} \odot {\bf{D}}({\theta _t})))$, ${\bf{\Sigma}} \left({{\theta _n}} \right){\text{ = }}\sum\limits_{n = 1}^{{N_c}} {{\rm{{\bf{I}}N}}{{\rm{R}}_n} \cdot \left\{ {{{({{\bf{I}}_K} \otimes ({\bf{P}} \odot {\bf{D}}({\theta _n})))}^H}({{\bf{I}}_K} \otimes ({\bf{P}} \odot D({\theta _n})))} \right\}}$; and the superscript $r$ denotes the iteration number of fractional programming for addressing Eq (17). The positive parameter ${\lambda ^{(r)}}$ denotes the output SINR of the $r$th iteration. Let

where $\eta$ is a non-negative constant to ensure ${{\bf{W}}^{(r)}} \succ 0$.

Hence, Eq (18) can be recast as

In ^[26], a generalized power-like iteration method (GPLIM) is adopted to address the non-convex problem in Eq (20). $q$ is used the internal iteration number of the GPLIM. By ignoring the SC of Eq (20), an unimodular vector ${{\mathit{\boldsymbol{x}}}^{(q{\text{ + }}1)}}$ can obtained by maximizing the objective of the problem in Eq (20), i.e.,

Further, the phases of the obtained vector ${{\mathit{\boldsymbol{x}}}^{(q{\text{ + }}1)}}$ should be optimized according to the SC. In particular, denote ${\mathit{\boldsymbol{\phi}}} = \arg ({{\bf{W}}^{(r)}}{{\mathit{\boldsymbol{x}}}^{(q)}}) = {[{\phi _1}, ..., {\phi _k}, ...., {\phi _{MK}}]^T}$ and let the phase ${\varphi _{\mathit{\boldsymbol{k}}}}$ of ${\hat {\mathit{\boldsymbol{x}}}^{(q{\text{ + }}1)}}(k)$ satisfy the following equation.

Finally, the obtained transmit waveform that satisfies the CMC and SC of Eq (20) via the $(q + 1)$ th GPLIM is given by

Suppose a prescribed threshold is $\kappa$. Let ${\delta ^{(q + 1)}}$ denote the objective function value of the problem given in Eq (20) after the $(q + 1)$ th iteration, that is, ${\delta ^{(q + 1)}} = {{\mathit{\boldsymbol{x}}}^H}{{\bf{W}}^{(r)}}{\mathit{\boldsymbol{x}}}\left| {_{{\mathit{\boldsymbol{x}}} = {{\mathit{\boldsymbol{x}}}^{(q + 1\ )}}}} \right. \ \ $; thus, if $\left| {{\delta ^{(q + 1)}}{\text{ - }}{\delta ^{(q)}}} \right| < \kappa$, the GPLIM is complete, and the optimal solution of Eq (20) can be obtained. The GPLIM algorithm is summarized as follows.

4.3.   Joint Optimization of ${\mathit{\boldsymbol{x}}}$ and ${\bf{P}}$

In this section, the overall joint design of the transmit waveform and array structure for the MIMO radar under interference and noise conditions is summarized in Algorithm 2.

The inspection of Algorithm 2 reveals that Algorithm 2 is combined with the sequential optimization, fractional programming, and the GPLIM. Because the sequential optimization and fractional programming algorithms are all convergent ^[26,20], only the convergence of the GPLIM is considered in this case. Note that at the beginning of Algorithm 2, the initial emission waveform is ${\mathit{\boldsymbol{x}}}_0$, so that the input ${{\mathit{\boldsymbol{x}}}^{(r)}}$ of Algorithm 1 is always within the feasible sets of Eq (20). Denote ${\beta ^{(q)}} = {{\bf{W}}^{(r)}}{{\mathit{\boldsymbol{x}}}^{(q)}}$, according to the phase optimization rule in Eq (22), then

As ${{\mathit{\boldsymbol{x}}}^{(q{\text{ + }}1)}}$ and ${{\mathit{\boldsymbol{x}}}^{(q)}}$ are both unimodular, there is

With the definition of Eq (19), ${{\bf{W}}^{(r)}}$ is a positive Hermitian matrix, that is, ${{\bf{W}}^{(r)}} \succ 0$, therefore,

If ${{\mathit{\boldsymbol{x}}}^{(q{\text{ + }}1)}} \ne {{\mathit{\boldsymbol{x}}}^{(q)}}$, we have

which implies that

The inequality chain in Eq (28) proves that Algorithm 1 is monotonously non-decreasing. Further, objective ${{\mathit{\boldsymbol{x}}}^H}{{\bf{W}}^{(r)}}{\mathit{\boldsymbol{x}}}$ is evidently upper bounded by the maximum eigenvalue of ${{\bf{W}}^{(r)}}$. Thus, Algorithm 1 is convergent, and as a result, Algorithm 2 is also convergent.

5.   Numerical results

In this section, we evaluate the performance of the developed algorithm for joint design emission waveform and array structure of the MIMO radar system with SINR maximization criteria, considering interferences and noise. Unless otherwise stated, we assume that the number of antennas in full-array is $M = 10$, and the numbers of transmit and receive antennas is ${N_t} = 4$ and ${N_r} = 8$, respectively. The code length is $K = 36$, and the total transmit energy is $E = M \cdot K$. Without loss of generality, we choose the orthogonal linear frequency modulation (LFM) as the reference waveform, which can be expressed as

where $m = 1, ..., M$ and $k = 1, ..., K$. The reference waveform vector ${\mathit{\boldsymbol{x}}}_0$ can be obtained by stacking the column of ${\bf{S}}_0$. There is only one spatial target, and reflecting energy of target is ${\left| {{\alpha _t}} \right|^2} = 20dB$ from direction ${\theta _t} = 20^\circ$. The number of the spatial interferences is ${N_c} = 3$, the locating azimuth set is $\left\{ { - 50^\circ, 0^\circ, 60^\circ } \right\}$, the corresponding reflecting energy is ${\left| {{\alpha _1}} \right|^2} = {\left| {{\alpha _2}} \right|^2} = {\left| {{\alpha _3}} \right|^2} = 30dB$, and the noise energy is $\delta _n^2 = 30dB$.

By keeping the above simulation parameters constant, the number of optimization combinations for ${\bf{P}}$ is $Q = 9450$. The number of the external iteration is $l = 5$, and the number of internal iterations for GPLIM is $r = 10$. The tolerance of the SINR is $\kappa = {10^{ - 2}}$. The simulation is performed in MATLAB 2018b version, running on a standard PC (with a 2.5GHz Core i7 CPU and 32GB RAM).

5.1.   Optimization of waveform ${\mathit{\boldsymbol{x}}}$ with fixed transmit array vector ${P_t}$ and receive array vector ${P_r}$

In this subsection, we assess the performance of Algorithm 1 in a fixed receive and transmit array structure. Assume that the vectors of transmit and receive array are ${P_t}{\text{ = }}{[1\ \ 0\ \ 1\ \ 0\ \ 1\ \ 0\ \ 0\ \ 0\ \ 0\ \ 1]^T}$ and ${P_r}{\text{ = }}{[1\ \ 1\ \ 1\ \ 0\ \ 1\ \ 1\ \ 0\ \ 1\ \ 1\ \ 1]^T}$, which is just one of 9450 combinations. During alternating optimization, the array vectors ${P_t}$ and ${P_r}$ that match with the fixed waveform are selected from all the combinations, thus maximizing the output SINR. Here, the similarity parameter takes $\varepsilon = 1$. The pulse compression curves of the optimized waveform and reference waveform are shown in Figure 2, which indicates that the pulse compression performance of the optimized waveform is close to the reference waveform under similar constraints; however, the sidelobe is slightly higher.

Figure 3 illustrates the output SINR versus the inner and outer iteration numbers. The first five curves represent the results of the five executions of Algorithm 1, which is the inner iteration, and each curve consists of 10 internal iteration SINR data. The 6th curve represents the outer iteration, and the SINR converges after five iterations. It can be noted that the objective function of Algorithm 1 based on the GPLIM gradually increases, and the SINR converges to 55.0289 dB after three times for both inner and outer iterations.

After the waveform phase array is transformed into a vector, the phase comparison of optimized sequence and reference sequence is shown in Figure 4. The left two figures are the phase curve of the reference sequence and optimized sequence, and the right is the phase bias between the two phases. The calculations indicate 294 symbols in the optimized waveform sequence have the same phase as that of the reference sequence. This implies that the similarity constraint works well. As expected, most symbols of the optimized waveform have equal or very similar phase values with the reference waveform.

To demonstrate the convergence of the algorithm, we save the output data of Algorithm 1 as shown in Table 1. As expected, the result indicates that both the inner and outer iteration converge after a relative limited number of executions. The total running time is approximately 2 s.

5.2.   Joint Optimization of transmit array vector${P_t}$, receive array vector${P_r}$ and waveform ${\mathit{\boldsymbol{x}}}$

In this subsection, the performance of Algorithm 2 for the joint optimization of the waveform and array structure is evaluated. Without loss of generality, the numbers of transmit and receive antennas remain ${N_t} = 4$ and ${N_r} = 8$, and the code length is still $K = 36$. The similarity parameter is $\varepsilon = 1$, and other simulation parameters remain constant as in part A.

Because the number of ${P_t}$ and ${P_r}$ combinations is 9450, it implies that there are many subsequent iterations. To improve the efficiency of optimization, ${\bf{U}}({\theta _t})$ and ${{\bf{\Sigma }}\left({{\theta _n}} \right)}$ under each different group of ${P_t}$ and ${P_r}$ are renewed first in Algorithm 2. According to the results in part A, Algorithm 1 converges quickly. The number of outer and inner iterations is set to 5 and 20, respectively. Figure 5 illustrates the optimized array structure, as well as the output SINR versus the inner and outer iteration numbers. Compared with the performance under a fixed pair of ${P_t}$ and ${P_r}$ in part A, it can be concluded that with the same prior information of the target, interference, and noise, the joint optimization of array structure and waveforms can obtain higher SINR, which indicates that the joint optimization improves the design freedom and provides better performance. Table 2 shows the detailed output of Algorithm 2 under the optimized array structure; whose convergence is consistent with Algorithm 1.

Similarly, the optimized waveforms are compared with the reference waveform. Figure 6 shows the pulse compression of three different waveforms corresponding to the optimized waveform under the fixed array structure, the optimized waveform under the combined optimization of the transmit waveform and array structure, and the reference waveform, respectively.

By analyzing the pulse compression and SINR figures, Figures 5 and 6, and the corresponding SINR tables, Tables 1 and 2, it is evident that the higher the achieved SINR, the poorer the pulse compression performance. Consequently, it can be concluded that the improved performance of the SINR under the optimized waveform alone or the jointly optimized array structure and waveform is at the price of the waveform pulse compression. Further, it can be observed from Tables 1 and 2 that the former several iterations cause a significant improvement in SINR.

Finally, the similarity constraint of the optimized waveform is examined. The phase and difference between the optimized and reference waveforms are similar to those in Figure 4. In addition, the total number of optimized sequences owning the same phase with the reference vector is 287. Here, Figure 7 adds a subplot to demonstrate the phase error of the optimized waveform relative to the reference waveform.

6.   Conclusions

In this study, the problem of MIMO radar emission waveform and array structure joint design was addressed under the practical constraints of constant modulus and similarity in the presence of signal-dependent interferences and noise. In contrast to the existing methods, which consider waveform optimization or array structure optimization alone, this study takes SINR as the optimization objective and optimizes the design of the transmit waveform and array structure simultaneously, which significantly increases the design freedom. To solve the non-convex problem, a fractional planning and alternating iteration optimization algorithm was deduced. Furthermore, the proposed algorithm was assessed in terms of the performance of both the waveform design under special array structure and waveform design combined with array structure, respectively. In the former scenario, the GPLIM generated waveforms with good pulse compression performance and SINR and converged quickly. In the latter scenario, joint optimization of array structure improved the SINR while keeping the pulse compression performance almost constant. The simulation results demonstrate the effectiveness of the proposed algorithm. Furthermore, some interesting topics, such as finding a balance between waveform similarity and maximizing SINR, addressing non-independent interference of the signal, and reducing the complexity of joint optimization, will be discussed further in future work.

Acknowledgments

The work was supported by the National Nature Science Foundation of China under Grant 62131020 and 61971434, and in part supported by Hainan Provincial Natural Science Foundation of China under Grant 621QN0872. The authors sincerely thank all reviewers and editors for their careful reading, valuable comments and suggestions on this paper.

Conflict of interest

The authors declare there is no conflict of interest.