Joint antenna selection and waveform design for coexistence of MIMO radar and communications

As the problem of spectral congestion is becoming severe, the coexistence between two primary spectrum users, radar and communications, has spurred extensive research interest. To reduce the mutual interferences between the two functions, MIMO radar waveform design needs to consider the compatibility in both spectral and spatial domains, where the former is achieved by null forming in the frequency domain and the latter is achieved by shaped beampattern synthesis. Additionally, high power efficiency and low system overhead are two desirable characteristics for MIMO radar system design. To this end, we first introduce a new realistic waveform constraint, peak-to-valley-power-ratio (PVPR) constraint per antenna to improve the power efficiency. Then, combined with PVPR constraint, we propose a switchable individual antenna power control scheme to jointly optimize waveforms and antenna locations. We adopt a max–min beampattern matching criterion and impose the ℓ2,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{2,1}$$\end{document}-norm penalty on the waveform matrix to promote the sparsity of the array. To solve the resultant non-smooth and non-convex problem, we develop a modified alternating directions method of multipliers, where a surrogate subproblem over primal variables is solved instead of the original problem, and its local convergence is analyzed. Finally, numerical experiments demonstrate the effectiveness and superiority of the proposed method over counterparts, especially obtaining the lowest sidelobe level and deeper spectral nulls using much fewer antennas.


Introduction
There are generally two ways of synthesizing a desired transmit beampattern, indirect way and direct way. The former adopts a two-step process which first optimizes a waveform covariance matrix (WCM) to form a desired beampattern and then performs the probing waveform optimization to approximate the obtained WCM. For example, the authors in [13] proposed to design a desired transmit beampattern by matching an ideal transmit beampattern in square error (SE) sense via using a semidefinite quadratic programming (SDQP) technique to optimize the WCM. Afterward, the literature [14] proposed a cyclic algorithm (CA) to design constant modulus (CM) or low peak-to-average-power ratio (PAR) waveforms to approximate the optimized WCM for obtaining a shaped-good beampattern. In addition, the work in [15] utilized the obtained WCM to synthesize binary-shift keying coded waveforms. Other optimal WCM design methods are also investigated in [16][17][18]. The latter emphasizes directly optimizing transmit probing waveforms without designing the WCM for a desired transmit beampattern. For instance, the authors in [22] applied the alternating direction method of multipliers (ADMM) [23] to directly synthesize CM waveforms based on the SE-minimization in the beampattern matching criterion. Employing the same matching criterion, [24] solved a quartic optimization problem under the majorization-minimization (MM) framework [25] to design waveforms subject to multiple constraints. Moreover, the work in [26] suppressed the peak sidelobe level (PSL) of the synthesized beampattern with restricted mainlobe ripples. Nevertheless, it is worth stressing that none of the aforementioned works pays attention to the radio frequency (RF) spectrum congestion in the waveform design.
Furthermore, high power efficiency and low system overhead are two desirable characteristics for MIMO radar system design from the viewpoint of practical applications, which are also the cores of this work. To achieve high power efficiency, CM or low PAR constraints are usually imposed on the directly-designed waveform since the radar transmitters only operate in saturation to deliver maximum power efficiency. However, the CM requirement is too strict to improve the performance and it usually is imposed on both temporal and spatial dimension, which may cause an unnecessary loss of degrees of freedom (DoFs). Besides, the PAR constraint has a key shortcoming as explained in [27] that it is insufficient to control the transmit power uniformity across the transmit antennas, which may result in a large power dynamic range and thus reduces the power efficiency. To overcome this issue of PAR constraint, we introduce a more realistic waveform power constraint, referred to as peak-to-valley-power-ratio (PVPR) constraint. The power dynamic range on the transmit waveforms of the lth antenna, {x l (n)} N n=1 in Fig. 1, is maintained within an allowable degree of power fluctuation by controlling the value of PVPR. In particular, when PVPR = 1 , it implies that the waveforms transmitted by the lth antenna satisfy the uniform elemental power requirement, i.e., |x l (n)| 2 = µ l , ∀n . Different from CM, various uniform power {µ l } L l=1 can be assigned to the L antennas, which apparently increases more DoFs than the common CM. To achieve low system overhead, a sparse array (SA) synthesized with MIMO transceiver is further designed via antenna selection, as full MIMO arrays are expensive attributed to the front-end channel associated with each antenna and complicated digital signal processing [28][29][30]. In addition, SA with an enlarged inter-element spacing can reduce the mutual coupling between antenna elements, specifically at millimeter wave frequencies with a very small wavelenth [31]. Also, a nonuniform SA with the same number of antennas owns more spatial DoFs brought by configuration flexibility to improve the system performance, such as enhancement of the aperture efficiencies and high resolution in the angular domain [32,33].
In the recent studies [34][35][36][37], the authors investigated the joint optimization on the WCM and antenna position, which is formulated a NP-hard quartic boolean combinational optimization. The convex relaxation methods adopted there usually cannot be guaranteed to be globally optimal. In this paper, different from the existing works, we directly design the probing waveforms circumventing the WCM and the binary selection vector to achieve the array sparsity. Specifically, a ℓ 2,1norm is imposed on the transmit waveform matrix to promote the group sparsity of the designed waveforms, whose sparse structure indicates the selected antenna positions. Moreover, in light of the fact that each antenna is fed by an individual RF power amplifier (PA), as shown in Fig. 1, it would be pragmatic to control the power uniformity separately on each antenna. Hence, we consider to add a switch to control the connection status of the front end of individual antenna, which is referred to as switchable individual antenna power control (SIAPC). Intuitively, our SIAPC scheme combined with unit-PVPR constraint possess more DoFs than the common CM. Thus, it is expected that the transmit beampattern synthesized by the proposed SIAPC strategy incorporated with PVPR constraint can obtain a lower sidelobe level (SLL) than the existing schemes.
The main contributions of this paper are highlighted as follows: 1 A switchable individual antenna power control (SIAPC) strategy combined with a waveform temporal constraint, peak-to-valley-power-ratio (PVPR), is proposed in this paper to flexibly control the power uniformity per antenna and power distribution across the array. The SIAPC scheme is able to obtain more designate DoFs while using the smallest number of antennas, which thus allows a high power efficiency and low system overhead. 2 To achieve the coexistence of the radar and communications, we propose a maxmin criteria to design spectrally-compatible waveforms for shaping desired transmit beampatterns, where the spectral energy nulls are formed in the specified spatialspectral domain. Meanwhile, a ℓ 2,1 -norm penalty term is directly imposed on waveform matrix to promote the group sparsity of the designed waveform. By doing so, we achieve a joint optimization on the antenna positions and transmit waveform. 3 To solve the proposed problem, we develop an efficient method based on a modified alternating directions method of multipliers since the conventional ADMM is hard to tackle such non-convex and non-smooth problem. Specifically, we approximately solve a surrogate subproblem over primal variables such that the non-convex PVPR constraint is solved by a projection operation. We prove that such an approximate solution keeps the original interior monotonicity of the augmented Lagrangian. Numerical experiments verify the effectiveness of the proposed method and improved performance over counterparts.

Notation
Vectors and matrices are denoted by boldface lowercase and uppercase letters, respectively. The � · � denotes the Frobenius norm while (·) T , (·) H , (·) * and ∠(·) are the transpose, conjugate transpose, complex conjugate and phase operators, respectively. ℜ{} and ℑ{} mean the real and imaginary parts of a complex number. I n denote the identity matrix of size n × n . The symbols R and C stand for the complex and real space, respectively. The expression max (A) means taking the maximum eigenvalue of the matrix A.

System and signal model
Consider a colocated MIMO radar consisting of a uniform linear array (ULA) of L antennas, as shown in Fig. 1, where two adjacent transmit antennas are equally spaced by d .
There is one front-end for each antenna, and the connection status of the front-end is controlled by a switch to determine whether it is on or off. Note that the scenario in Fig. 1 enables the detection of targets in the focused main-lobe and simultaneously spectrally coexists with multiple communication users which share certain overlapped frequency bands. Assuming that the discrete-time waveform emitted by the lth antenna is denoted by x l (n), n = 1, . . . , N , where N denotes the sample number of each radar pulse. Then, we define the waveform vectors along the spatial dimension and the temporal dimension as, and the corresponding concatenated waveform vectors and paralleled matrices are given by, Note that there is a relationship between x and x , i.e, x = Ex , where (e T l ⊗ I N ⊗ e l ) with e l ∈ R L×1 being a unit-basic vector whose elements are 0 except the lth one is 1. Under the case that the propagation is nondispersive and the transmitted signal is narrow-band, the base-band signal received at the angular direction θ p is given by [13,26], with c being the speed of propagation, and a(θ p ) is the transmit array steering vector defined by, The transmit beampattern, which describes the cumulated spatial power of the transmitted signal at the direction θ p , is then given by [13], For ease of discussion, the spatial angular regions are divided into P uniformly spaced samples, which cover the mainlobe region � m = {θ m } M m=1 and the sidelobe region � s = {ϑ s } S s=1 . For the MIMO radar transmit beampattern synthesis, a main issue is to design waveform x for shaping a good transmit radiation power distribution on the mainlobe region m , and the sidelobe region s . In addition to that, in light of the limitation of the nonlinear PA in the RF circuitry in practical applications, some practical constraints are required to impose on the waveform x . For example, the common CM constraint with a fixed amplitude µ 0 , i.e., is usually imposed on x to maximize the transmit power efficiency. Usually, the relaxed PAR constraint [14,38], i.e., where δ ∈ [1, N ] , is employed instead of the rigorous CM constraint. Unfortunately, as revealed in [27,39], Eq (9) accompanied with the per-antenna energy constraint (EC), i.e., �x l � 2 = κ 2 e L is equivalent to, where κ e = δκ 2 e NL with κ 2 e being the total energy budget. From (10), we see that PAR constraint has a major drawback that there may exist a large dynamic power fluctuation for the lth transmit antenna in the N samples since no lower bound is imposed on (10). Note that this drawback is not beneficial to the maximization of the transmission power efficiency. To fill this gap, we introduce a realistic power constraint into waveform design, referred to as peak-to-valley-power ratio (PVPR) constraint, which can control the power uniformity of each transmit antenna for improving the power efficiency.

Per-antenna PVPR constraint
We define the waveform PVPR constraint on the lth antenna as, |x l (n)| 2 ≤κ e , l = 1, . . . , L, n = 1, . . . , N , which restricts the ratio of the maximum power to the minimum one over N samples emitted by the lth antenna less than a preset threshold σ l , so as to control the difference of transmit power among the N samples per antenna. Here, σ l plays a role of controlling the power uniformity among N samples for the lth transmit antenna and different σ l for l = 1, . . . , L can be assigned to control the varying power uniformity on L antennas. Regarding the PVPR constraint (11), we discuss some points as follows: 1 Remark 1 Since the RF amplifier for each front antenna feed is individual per Fig. 1

Remark 2
The PVPR constraint is actually a general power control constraint. As when we set σ l = 1 in (11), it degenerates to unit-PVPR constraint, i.e., Note that µ l is a scaling factor to be optimized. This is different from the conventional CM constraint (8), which constrains the uniformity in both temporal and spatial dimensions, while the unit-PVPR relaxes the spatial dimension and focuses on elemental uniformity, thus earning more designate DoFs. On the other hand, when the PVPR constraint is imposed on both spatial dimension and temporal dimension, i.e., the unit-PVPR constraint in (13) with σ 0 = 1 under the total energy constraint, i.e., �x� 2 = κ 2 e coincides with the CM constraint, i.e., This manifests the generality of the proposed PVPR constraint.

Spectral coexistence with communications
In a spectrally crowed scenario, the radar system is required to coexist with communication devices while simultaneously detecting the targets of interest [39][40][41][42][43].
In this respect, we desire that the synthesized radar waveforms are not only able to focus energy in the spatial direction of targets but also to suppress mutual interferences on overlaid communication services, especially those in the mainlobe. To this end, we assume that there are K communication devices and each of them operates on a frequency band k = [f k,1 , f k,2 ] for k = 1, . . . , K , where f k,1 and f k,2 are the lower and upper normalized frequencies for the kth device, respectively. Besides, we suppose that the kth communication device is located on the spatial angle range . Then, the integrated energy on the kth communication in the specific space-frequency region is given by [42], and U k ∈ C L×L being the frequency matrix and angle matrix of the kth space-frequency region, respectively, which are defined by [40][41][42], and Accordingly, the total energy produced on the all K communication devices is given by, To facilitate the MIMO radar in spectral coexistence with communications, it is desirable to minimize the total energy E s in order to reduce mutual interferences to other communication devices. In the ensuing section, we will incorporate the PVPR constraint (11) into the waveform optimization which involves the transmit beampattern synthesis and spectral coexistence with communications.

Problem formulation
For the radar operation, it usually desires to concentrate the radiation power in the mainlobe region while lowering the power level in the sidelobes [13,17,44] for sake of target detection and radar clutter/jamming suppression. To this end, we propose a min-max design under PVPR constraint to fully explore the DoFs for suppressing the sidelobes and shaping the mainlobe while introducing a ℓ 2,1 -norm regularizer to promote the waveform group sparsity so as to achieve antenna selection. More precisely, our goal is to minimize the maximum matching error between the synthesized beampattern P(θ p ) and the desired one D p subject to PVPR constraint, which is formulated as the following optimization problem, where w p for p = 1, . . . , P are the weight coefficients that emphasize different certain angular region over others, τ > 0 is the regularization parameter that compromises between the beampattern shape and the sparsity on antennas, β represents the penalty parameter on the spectral energy suppression, and S denotes the support set containing the indices of nonzero waveform vectors in the set {x l } L l=1 .

Remark 3
In the proposed design, we do not consider the waveform total energy constraint, i.e., �x� 2 = κ 2 e . The reason is that once we obtain the solution, e.g., Notice that such a scaled x o does not affect the PVPR constraint, neither the beampattern shape, as well as the relative spectral energy distribution under the normalization processing.

Remark 4
Note that problem (17) is very challenging to solve. The difficulties are that the objective functions are non-smooth attributed to the discontinuous function max{·} and particularly contain the inseparable quadratic terms �Ȃ H θ p x� 2 2 in x l , which makes the per-antenna PVPR constraint on x l hard to tackle. To deal with this issue, we resort to a splitting method to solve the resultant problem as in the next section.

Proposed method
In this section, we derive an efficient algorithm via modifying the standard ADMM framework to solve the formulated problem in (17). Specifically, since solving (17) directly is very difficult due to the existence of the discontinuous function max{·} , we first introduce an auxiliary variable η to rephrase (17) as, where A p =Ȃ θ p for notional brevity.  (20) is still difficult to handle as η is coupled in the lower and upper bounds, which makes (20) intractable. To solve (20), we again introduce an auxiliary variable v p and impose the equality constraints, i.e., Â H p x = v p for p = 1, . . . , P on (20), yielding the following equivalent problem, where d p = D p w p . From (21), we can see that the constraints d p − η ≤ �v p � 2 ≤ d p + η and PVPR(x l ) ≤ σ l will act their roles in respective subproblems with respect to (w.r.t) variables v p and x l , respectively. However, v p and x are coupled together in the additional constraint (21a). This variable splitting characteristic enables us to utilize the ADMM strategy for solving (21). Specifically, we first construct the augmented Lagrangian function of (21) in terms of (21a) as, where ρ > 0 is the penalty parameter and {ũ p } P p=1 are dual variables. By introducing the scaled dual variable u p =ũ p ρ , we can recast (22) as, It follows the ADMM alternates minimizing the augmented Lagrangian function L ρ w.r.t x and v p along with the dual ascent update on u p to produce the following iteration steps: where t denotes the count of iterations.
In what follows, we discuss the solutions to the subproblems (24) and (25), respectively.
1) The solution to (24): The x-subproblem in (24) by omitting the irrelative items, is given by, To clearly solve (27), we further simplify (27)  However, we find that the quadratic term x H Bx in (28) is inseparable for x l , l = 1, . . . , L , which makes the update of x t+1 l subject to PVPR constraint complicated. To this end, we propose to update x t+1 l in an approximated manner. More specifically, we first introduce the following Lemma 1 to help us seek a simpler surrogate problem for (28).
Lemma 1 [45]: Let L be N × N Hermitian matrix and M be another N × N Hermitian matrix such that M L . Then, for any point x t ∈ C N ×1 , the quadratic function x H Lx is majorized by: According to Lemma 1, we define L = B and M = δI with δ = max (B) and linearize the quadratic term x H Bx as its a local upper bound function g(x, x t ) at x t , i.e., It is worth stressing that differing from the original ADMM strategy that directly solves the problem (28), our scheme is to update x t+1 by replacing x H Bx with g(x, x t ) in (28) and ignoring the constant terms, which results in the following surrogate problem, where y l is the lth subvector of y = [y T 1 , . . . , y T L ] T defined by, It is obvious that such a transformation in (31) makes the objective function become separable in x l for l = 1, . . . , L and much easier to solve x l constrained in the PVPR constraint. Specifically, the lth subproblem of (31) is given by, Note that the PVPR constraint in (33) depends on the support set S , which is explicitly related to the sparsity-inducing term τ L l=1 �x l � 2 . Hence, we have to first determine the support set S . To this end, regardless of the PVPR constraint, we find that the unconstrained optimization (33) admits a closed-form solution as given by the block soft thresholding formula: where q l = (1 −τ �y l � 2 )y l with τ = τ ρδ . From (34), we can naturally obtain the support set S as, (30) and under its complementary set N S = {l|�y l � 2 <τ } , we accordingly set x l , l ∈ N S as zero vector, i.e., 0 N . For x l , l ∈ S , we turn to solve the following projection problem as, It can be seen that (36) is intractable due to the non-convex PVPR constraint. To facilitate solving (36), we tactfully introduce a scalar µ to equivalently rewrite (36) as, By defining ζ = √ µ and ν l = √ σ l , we find that (37) is equivalent to, Fortunately, the solution to (38) admits a closed-form solution ζ o , which can be determined by the technique in [46]. After obtaining ζ o , we then have the optimal µ ⋆ = ζ 2 o and each element of x t+1 l , l ∈ S can be updated by the following projection: for n = 1, . . . , N . Once obtaining the solution x t+1 l for both cases of l ∈ S or not, then the concatenated vector x t+1 is constructed by Note that when the common CM constraint (8) is considered in x-subproblem with τ = 0 , the optimal x t+1 is readily updated by an element-wise manner as x t+1 = [µ 0 e ∠y(1) , . . . , µ 0 e ∠y(NL) ] T .
2) The solution to (25): The v p -subproblem in (25) by omitting the irrelative items, is given by, Clearly, once given η , v t+1 p can be updated by solving the following problem, which admits a closed-form solution as, (37)  where η = max p {d p } P p=1 and G(·)| C denote the condition function that equals 1 if the argument satisfies C and equals 0 otherwise. It can be seen that the objective function in (43) is a piecewise nonlinear one that relies on the value of the condition function G(·)| C . To this end, we select J reasonable and non-overlapped turning points that are less than η from the set �ṽ p � 2 − d p P p=1 and sort them as {ȓ j } J j=1 in ascending order, which naturally divides the feasible domain of η into J + 1 subintervals. Evidently, the optimal solution η ⋆ to (43) corresponds to the smallest one among the objective values in the J + 1 subintervals. Therefore, we first study the local minimum in each subinterval. Specifically, at the jth subinterval, i.e., η ∈ [ȓ j−1 ,ȓ j ] , the subproblem of (43) is given by, where Denote the cost function in (44) as f j (η) and its first-order derivative and the secondorder derivative are given by, respectively, (43) Clearly, f ′ (η) is increasing over η and f ′′ (η) > 0 , which implies that the subfunction f j (η) is convex. Thus, the local optimal solution to the jth subproblem in η ∈ [ȓ j−1 ,ȓ j ] can be determined from one of the following three cases: (1) When f ′ (ȓ j−1 ) > 0 , the subfunction f j (η) is increasing in the interval [ȓ j−1 ,ȓ j ] and the local minimizer η j =ȓ j−1 .
(2) When f ′ (ȓ j ) < 0 , the subfunction f j (η) is decreasing in the interval [ȓ j−1 ,ȓ j ] and the local minimizer η j =ȓ j . After obtaining η j and its objective value f j (η j ) for j = 1, . . . , J , we then choose the smallest one from {f j (η j )} J j=1 and set the corresponding η j as the optimal η ⋆ , that is the update of η t+1 . Once getting η t+1 , v t+1 p can be updated by inserting η t+1 into (42). Finally, we summarize the above steps in Algorithm 1 and repeat it until the maximum iteration number T m is reached or the maximum residual, i.e., max p �Â H p x t − v t p � 2 is less than the stop tolerance value ε.

Computational complexity analysis
As for the computational complexity of the proposed Algorithm 1, we mainly consider the count of multiplications in each iteration. In Step1, the maximum eigenvalue of B (47)  can be pre-computed before the iteration to save the computational burden. Hence, the main computation in Step1 is involved in the matrix-vector multiplication Bx t which takes O(N 2 L 2 ) complexity, since the PVPR projection only needs scalar multiplications with the complexity of O(NL) . Step2 is dominated by the matrix-vector multiplications similar to the Step3 with a complexity of O(N 2 PL) , as updating η t+1 also only needs scalar multiplications. In summary, the overall complexity of Algorithm 1 is O(2N 2 PL + N 2 L 2 ).

Convergence analysis
In the proposed ADMM-based Algorithm 1, we approximately update x t+1 by solving surrogate primal variable subproblems, which actually follows the majorization-minimization algorithm at one step [25]. Note that such a solution by this way can be guaranteed to make the original objective function value decent at each successive update and the proof can be found in [25,45]. Thus, the interior monotonicity of the original augmented Lagrangian function L ρ (x, η t+1 , v t+1 p , u t p ) is also guaranteed. Then, we apply some convergence results of the standard ADMM in [23,47,48] to study the local convergence of Algorithm 1 and show that �x t+1 − x t � 2 → 0 as the iteration number t → ∞ . The following theorem is provided to explain the convergence behavior of Algorithm 1 under some mild conditions. 1 Proof see Appendix A.

Numerical results
In this section, some numerical examples are provided to demonstrate the superiority of the proposed SIAPC scheme over the counterparts. In the first subsection, we examine Algorithm 1 to jointly design waveforms and SA for approaching desired transmit beampatterns {D p } P p=1 without considering compatibility, that is, β = 0 . In the second subsection, we take the spectral energy regularization into account and investigate the coexistence performance of radar and communications. Without loss of generality, we set σ l = σ , ∀l to control the power uniformity per antenna with the total energy budget κ 2 e = NL . Besides, in all simulations, we set T m = 10 4 , ε = 10 −5 in the proposed algorithm and the range of spatial angle is divided into 181 grid points with a uniform interval of 1 • .  Fig. 3b. The recent WCM-based SA design methods, such as dynamic programming (DP) in [35] and the preset array reconfigured method in [36] (simplified as DP-WCM and RM-WCM, respectively) are performed with 10 antennas selected for fair comparison without cross-correlation term. Meanwhile, a 10-antenna ULA based on optimal WCM design via semi-definite quadratic programming (SQP) method [13] is included for comparison. We can see that the synthesized beampatterns from the SA optimized by our scheme with unit-PVPR has a lower SLL compared to the other methods and outperforms the uniform ULA, which demonstrates the superiority of nonuniform SA over uniform array with the same number of antennas. Moreover, the maximal square error (MSE), defined by MSE = max p |B(θ p ) − D p | 2 , and the PSL are listed in Table 1, which shows that SIAPC scheme exhibits the smallest MSE and the lowest PSL.

Waveform design based on
To clearly visualize the optimized waveforms by our SIAPC scheme, the normalized amplitude distributions of the designed waveforms are depicted in Fig. 4a-b. It can be observed that the proposed PVPR constraint enables to control the amplitude dynamic range on each antenna within an acceptable fluctuation, which is beneficial to the individual PA working in a nonlinear region for improving power efficiency. Furthermore, in Fig. 5, we compare the obtained beampattern by SA with unit-PVPR to that of the 15-antenna ULA based on the optimal WCM by SQP and CM waveform. Interestingly, the optimized SA by SIAPC scheme with unit-PVPR exhibits slightly lower SLL than that of full ULA under CM waveform and their MSEs in order are − 20.0 dB, − 26.96 dB Finally, to verify the rationality and credibility of the selected antenna locations, we exhaustively search all antenna selection combinations of choosing 10 elements from the full 15-element array with the optimized waveform by the proposed SA method under unit-PVPR constraint. Figure 6 shows that the PSL of the synthesized beampatterns corresponding to all 3003 combinations. We can see that the proposed antenna location shown in Fig. 4a is globally optimal with the lowest PSL.  In the second example, we consider synthesizing multiple wide beams to track multiple potential targets, whereas the directions of the true targets are imprecisely known [49]. To  Table 2 shows that the proposed SIAPC utilizes more DoFs to synthesize the beampattern with the smallest MSE and the lowest PSL. Moreover, we also compare the beampattern synthesized by the optimized SA with unit-PVPR to that of the full ULA in Fig. 8. This figure shows that the proposed SIAPC scheme is able to achieve almost the same performance with a full ULA but employing a smaller number of antennas in the beampattern synthesis. . Evidently, the radar detection beam produces large energy leakage in the spatial-spectral directions of communications, which interferes the coexistence between radar and communications. To achieve the spectral coexistence between radar and communications, we set ρ = 0.7, τ = 0.1 and β = 100 in Algorithm 1 to synthesize spectrally-compliant beampatterns under different array configurations as shown in Fig. 9a, b,  The energy distributions of the designed waveforms in the spatial-spectral domain are depicted in Fig. 10, where the yellow rectangular box denotes the spatial-spectral stop-bands to be formed. As expected, all the optimized waveforms form deep spectral nulls in the stop-bands, which locally approaches − 50 dB. In particular, it can be observed from Fig. 10c that the optimized waveform by our SIAPC scheme under PVPR=1 forms a more pronounced spectral stop-band stripe than those ULAs in the CM case in Fig. 10a, b, which is attributed to the more DoFs provided by the PVPR constraint than the CM constraint. We can see that the beampattern from the proposed SA with unit-PVPR has the smallest PSL of − 17.23 dB, especially lower than that of the full array. Figure 11b shows that the