Waveform Design for MIMO Radar Detection in a Heterogeneous Clutter Environment

In this study, the problem of transmitted waveform design for a multiple-input-multiple-output (MIMO) radar in a heterogeneous clutter environment is considered. Under a Bayesian framework, the waveform design problem is used to maximize the generalized likelihood ratio for a given false alarm probability. In addition, the proposed problem is nonconvex under the constant modulus constraint. Hence, the alternating direction method of multipliers-based and the Broyden–Fletcher–Goldfarb–Shanno method are adopted to solve this nonconvex optimization problem. Finally, the clutter suppression performance of the proposed algorithms is evaluated in real-world data by numerical simulations. The simulation results show that the proposed methods have better performance on clutter suppression and target detection.


I. INTRODUCTION
In comparison to a phase-array radar, a multiple-inputmultiple-output (MIMO) radar manifests significantly better performances in target detection and parameter identification [1]- [4]. Depending on different antenna configurations, MIMO radars can be divided into two classes-distributed MIMO radars that possess widely separated antennas [2] and colocated MIMO radars that possess closely spaced antennas [4].
Studies on the waveform design for colocated MIMO radars can be divided into two main classes: In the first category, MIMO radar waveforms are designed to control the transmitted power distribution in the spatial domain by setting the desired beam pattern. In order to address this problem, two main strategies are usually adopted. In the first strategy, the transmitted beam pattern is synthesized through the optimization of a waveform covariance matrix [4]- [8], and then transmitted waveforms are acquired from the obtained covariance matrix under practical constraints [5], [7], [9]. Lipor et al. [8] utilized discrete-Fourier-transform (DFT) coefficients and Toeplitz matrices to form a covariance matrix, which matched the desired beam pattern at low complexity. Finally, with The associate editor coordinating the review of this manuscript and approving it for publication was Filbert Juwono . the obtained covariance matrix, practical constraints, such as constant modulus (CM) [9] and the peak to average power ratio [7], were taken into consideration to synthesize these transmitted waveforms. In the second strategy, transmitted waveforms are directly designed to realize the desired beam pattern [10]- [14]. For example, Cheng et al. [13] tried to minimize the mean squared error between the synthesized and desired beam patterns based on the alternating direction method of multipliers (ADMM) algorithm. Fan et al. [14] also applied the ADMM algorithm; however, paid more attention to the restrictions of the sidelobe and the main lobe of the transmitted beam pattern.
In the second category, transmitted waveforms and receiver filters are designed jointly. MIMO waveforms are generally designed to maximize the output signal to interference plus noise ratio (SINR), which enhances the target detection performance of a MIMO radar and also suppresses signal-dependent interferences (clutter) [15]- [17]. In the literature [15], two sequential optimization procedures were adopted to design the desired waveform by maximizing the SINR value under a CM constraint and a similarity constraint. Sadjad et al. [16], with the given prior information of the target and interference angle locations, presented an iterative approach to jointly optimize transmitted waveforms and receiver filters. Tang and Tang [17] applied a cyclic algorithm to achieve an improved SINR under practical constraints. Some researchers [18]- [20] also presented the principle of MIMO waveform design based on the information theory. In the literature [18], [19], the minimum mean-square error (MMSE) was used to optimize MIMO waveforms in colored noise and clutter. In the study [20], instead of solving the intractable problem of maximizing the detection probability for a given false alarm, the relative entropy was employed to improve the detection performance of a MIMO radar.
Instead of maximizing the SINR value, in this study, the waveform design problem is performed by maximizing the likelihood ratio test or generalized likelihood ratio test (GLRT) for a given false alarm probability. In the literature [21]- [25], the GLRT expressions of different environments with prior information (clutter map, statistical information of clutters) were discussed. In some earlier studies [26]- [30], a Bayesian framework was adopted to develop the GLRT expression without secondary data. Therefore, we also consider the GLRT with the Bayesian framework.
In this study, considering the CM constraint, the waveform design problem is nonconvex, and two different approaches are proposed to accomplish the maximization of GLRT. In the first approach, the nonconvex problem is solved by the ADMM method [32]- [34]. In the second approach, the nonconvex problem is first relaxed as an unconstrained optimization problem and then solved by the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [38]. Finally, the detection probabilities of our designed waveforms the existing orthogonal linear frequency modulation (LFM) waveform are compared through numerical results. In addition, the CM waveform designed by [17] algorithm 2 is also provided to compare.
The remaining paper has been organized as follows: The data model for the proposed problem is formulated in Section II. The first and second algorithms are introduced in Sections III and IV, respectively. Numerical results are presented in Section V, and finally, conclusions are drawn in Section VI.
Notations: The transpose and conjugate transpose operators are denoted by (·) T and (·) H , respectively, whereas and ⊗ represent the Hadamard product and the Kronecker product, respectively. The letter j represents the imaginary unit (j = √ −1). · 2 denotes the Euclidean norm of a vector, and |·| signifies the absolute value. The lower case letter 'a' and the upper-case letter 'A' denote vectors and matrices, respectively. vec (A) represents the column vector obtained by stacking its columns, and I N stands for the N × N dimensional identity matrix. {·} and Im {·} respectively imply the real and imaginary parts of a complex value or a matrix. etr (·) stands for the exponential of the trace of the matrix argument, and det (·) denotes the determinant of a matrix.

II. DATA MODELING AND PROBLEM FORMULATION
The system under consideration is a colocated MIMO radar placed on an airborne platform. The radar is equipped with N T transmitting antennas and N R receiving antennas. Now, assume the radar transmits a burst of M pulses at a constant pulse repetition frequency (PRF) f r and s n (l) , l = 1, 2, · · · , L; n = 1, 2, · · · , N T denote the discrete time radar waveform radiated by the n-th antenna (L is the number of time samples of each radar waveform). Furthermore, s l = [s 1 (l) , . . . , s N T (l) T indicates the N T dimensional vector that collects transmitted waveforms at l th time; thus S = [s 1 , s 2 , . . . , s L ] T can be defined as the transmitted waveform matrix. Let x be the received signal in the cell under test (CUT). The problem of detecting targets in the presence of a heterogeneous clutter can be formulated through the following binary hypothesis testing: where x t and x c are the vectors that contain target signals and the clutter, respectively.

A. TARGETS MODEL
We assume the propagation is nondispersive, and the transmitted probing signals are narrowband. Considering there are I targets in the space, the i-th target is moving with a speed v t i relative to the radar platform, then received target signals at the direction of arrival (DOA) θ t i of the m-th pulse can be expressed as [6], where α i is the target amplitude, b θ t i is the received array steering vector at θ t i , a θ t i is the transmit array steering vector, f d i = 2v t i λ is the target Doppler frequency (λ denotes the wavelength), and T r = 1 f r is the pulse repetition interval (PRI). Let x t i = vec T X t,1 , . . . , vec T X t,M T be the i-th target response, and then we have T , then the targets responses can be given as B. CLUTTER MODEL Clutter can be described as unwanted echoes from scatters; hence, it can be regarded as a signal-dependent interference.
In the present study, the measured clutter is modelled as the VOLUME 8, 2020 superposition of signals associated with a large number of patches distributed in the azimuthal direction in the same range cell of the target. The measured signal-dependent clutter associated with the m-th pulse and the κ-th patch in the azimuth can be modeled as where δ c,m,κ is the amplitude associated with the m-th pulse and the κ-th patch, and the parameters f c,κ and θ c,κ , respectively, denote the Doppler frequency and the DOA of the κ-th patch in the range cell. Let x c,κ = vec T X c,1,κ , . . . , vec T X c,M ,κ T ; hence, the measured clutter at the κ-th patch in the azimuth can be expressed as with h c,κ =ũ f c,κ ⊗ a θ c,κ ⊗ b θ c,κ andũ f c,κ = diag δ c,1,κ , . . . , δ c,M ,κ u f c,κ . Now, by taking the clutter from N c clutter patches, the measured clutter model can be formulated as Furthermore, the clutter covariance matrix R SCM can be expressed as where = N c κ=1 h c,κ h H c,κ .

C. PROBLEM FORMULATION
Generally, a heterogeneous clutter can be characterized by the well-known complex compound-Gaussian model: where R is the covariance matrix. In the field of the multivariate statistical analysis, the inverse Wishart distribution is used as the prior distribution of R [31], and in practical, the real radar data of SAR and polarimetric SAR (Pol-SAR) show that the covariance of the radar clutter fits the inverse Wishart distribution well [26]- [30]. Therefore, it is assumed that R is a random matrix drawn from a complex inverse Wishart distribution here; namely, where CW −1 ( , v) signifies the complex inverse Wishart distribution with the concentration matrix ( > 0) and the freedom v (v > MN R L). Moreover, the probability density function (p.d.f) of the covariance matrix R can be expressed as where (·) is the Eulerian Gamma function. Usually, the target amplitude α is unknown in detection problem. Thus, based on the Neyman − Pearson Theorem [35], the Bayesian GLRT expression for a given false alarm probability can be written as In addition, Letα denote the maximum likelihood estimate (MLE) of α under H 1 , and substitute Eqs. (5) and (14) into Eq. (13), yields where ∝ implies ''proportional to'', and according to [30], the MLE expression of α is given bŷ Furthermore, let us consider the concentration matrix is unknown, then an alternative model of the concentration matrix can be modelled as where σ is the positive diagonal loading coefficient. According to the Neyman − Pearson theorem, for a given false alarm probability, GLRT expresses the maximized detection probability. Therefore, by substituting (17) into Eq. (15), it can be noticed that the GLRT value is dependent on the transmitted waveform S. Hence, the waveform design problem that maximizes GLRT can be formulated under the CM constraint as follows: The optimization model in Eq. (19) involves the minimization of a nonconvex cost function and a nonconvex CM constraint, so this is an NP-hard problem. In general, iteration system is used to solve the NP-hard problem. However, the target amplitude vector is unknown and needs to be estimated first at each iteration. Therefore, the specific iteration system is shown in Fig.1, whereα k is the MLE of the target amplitude vector with k denoting the iteration time. Then two approaches in Sections III and IV, are proposed to optimize the waveforms respectively.
Hence, the CM constraint in Eq. (19) can be reformulated as Subsequently, Eq. (19) can be transformed into the following equivalent problem: where 1 is a LN T dimensional vector in which all elements are equal to 1. The augmented Lagrangian (formulated using scaled dual variables) [32] of Eq. (22) can be defined as where ρ > 0 is the step size and u ∈ R LN T ×1 is the scaled dual variable. Finally, ADMM is applied to Eq. (23) to determine {s R , s I , u}: where k is the time mentioned before in Eq. (19). Clearly, the constrained optimization problem in Eq. (19) can be divided into several subproblems (Eqs.24(a −(e)). The first subproblem of updating s R in Eq. (24a) and the third subproblem of updating s I in Eq. (24c) are both unconstrained. The final step in Eq. (24e) defines common update rules of scaled dual variables [32].
Let 2 2 and both g A (s R , s I ) and g B (s R , s I ) are implicit functions of s R and s I . The solutions of the subproblems in Eqs. (24a) and (24c) can be derived as follows: Update of The optimization problem in Eq. (24a) can be written as Therefore, Eq. (25) is unconstrained. However, it is hard to obtain s k+1 R by directly setting ∂L ρ s R , s k I , u k ∂s R = 0. Fortunately, the Quasi-Newton method is one of the most effective methods to solve unconstrained nonlinear optimization problems. Therefore, the BFGS method [38], a type of quasi-Newton method, is adopted to solve Eq. (25). Furthermore, the key is transformed to calculate the gradient of L ρ s R , s k I , u k at s R . Based on the derivative of a real function with respect to a complex variable [36], the derivative of g A s R , s t I with respect to s R can be given as which can be computed from Proposition 1.
Proposition 1: The derivative of g A with respect to vec (S * ) can be calculated as where the matrix [37].
Proof: See Appendix A. In addition, with the given s k I , u k , the derivative of g B s R , s k I , u k with respect to s R can be written as where s 3 R is defined by taking a cube of each element in s R . Proof: See Appendix B.
By combing Eqs. (26) and (28), the gradient of L ρ (s R , s k I , u k ) with respect to s R can be expressed as Herein, let τ be the iteration number and A τ be the approximation to the Hessian matrix of L ρ s R , s k I , u k with respect to s R . Then the BFGS method [38] to update s R with a fixed step length ur is summarized in algorithm 1.

Algorithm 1 Update of s R
Inputs: s k R , s k I , u k , the MLEα k of k-th time, step size ur, ρ, iteration stop tolerance εr, and the maximum iteration number τ max .
R , s k I , u k using Eq. (29), and determine: Update of s I : Due to the symmetry of s R and s I , the update method of s I (Eq. 24c) is akin to that of s R (Eq. 24a). Let . Hence, the derivative of g B s k+1 R , s I , u k with respect to s I can be expressed as where s 3 I is defined by taking a cube of each element in s I . Furthermore, using the expression of Eqs. (27) and the results of algorithm 1, the derivative of g A s k+1 R , s I , u k with respect to s I can be written as Therefore, the gradient of L ρ s k+1 R , s I , u k with respect to s I can be expressed as In summary, the design process of the ADMM-based waveform (ADMM-W) is summarized in algorithm 2.

Algorithm 2 ADMM-Based Waveform (ADMM-W) Design
Inputs: step size ρ, iteration stop tolerance ς , and the maximum iteration time K max Initialize: s R , s I , s 0 = s 0 R + js 0 I and k = 0; 1: while max L ρ s k+1 R , s k+1 First, the convergence performance of the ADMM-based algorithm is discussed. ADMM can guarantee the global convergence for a convex optimization problem [32]. The theoretical proof of convergence for a nonconvex problem is proposed in Theorem 1 [14], [33], [34]. The following theorem reveals that sequences generated by the ADMM-W algorithm are convergent under some mild conditions. Theorem 1: Let s k R , s k I , u k be a sequence generated by ADMM-W with ρ > 0 (Eqs.24(a)−(e)). Now, if lim t→∞ u k+1 − u k = 0, then the limit point s R , s I , u is an optimal solution of Eq. (22).
Hence, the sequences s k R and s k I are bounded, and a stationary point s R , s I , u exists between them.
lim t→∞ s k R , s k I , u k = s R , s I , u .
Next, we analyze the computational complexity of ADMM-W algorithm.
As to the computational complexity of ADMM-W, it mainly depends on the number of (outer) iterations and the number of operations at each iteration. Herein, we focus on the analysis of the complexity involved in each (outer) iteration. For Algorithm ADMM-W, the update ofα(with (16)) requires O (LMN R ) 3 operations, and the optimization of s R requires O τ R (LN T ) 3 operations with τ R denoting the number of (inner) iterations of Algorithm 1. Similarly, the optimization of s I requires O τ I (LN T ) 3 operations with τ I denoting the number of (inner) iterations. Last, the update of s requires O (LN T ) operations.

IV. PHASE-ONLY WAVEFORM DESIGN
Since the ADMM-W algorithm is sensitive to the parameter ρ and the algorithm is quite complicated, it is necessary to find a simpler method to solve the optimization problem (19). In the present section, the phase-only waveform design algorithm, also based on the BFGS method [38], is employed to solve Eq. (19).

A. PHASE-ONLY ALGORITHM
In order to solve this nonconvex problem, CM constraints can be simplified by introducing auxiliary phase variables.
Hence, Eq. (19) is converted into the following optimization problem without performing any modulus operation: Now, by substituting S = e j into the cost function of Eq. (37), the optimization problem can be relaxed as an unconstrained nonlinear optimization problem g A (ϕ) associated with : which is an unconstrained problem. Similar to algorithm 1, the BFGS method is also adopted to solve Eq. (38). According to the derivative theorem of a scalar real-valued function with respect to the complex matrix variables Z and VOLUME 8, 2020 [36], the gradient of g A ( ) with respect to can be given as And ∂g A ∂S * can be calculated from Proposition 1. Let ϕ = vec T ( ); thus the gradient of g A ( ) with respect to can be expressed as Based on the BFGS method, the phase vector ϕ k+1 at iteration k + 1 can be updated as where u k is the selected step size in the current search direction and B k is the approximation to the Hessian matrix of g A (ϕ k ) with respect to ϕ k . Algorithm 3 summarizes the phase-only waveform design (POW) process using the BFGS method.

B. CONVERGENCE PERFORMANCE AND COMPUTATIONAL COMPLEXITY ANALYSIS
POW is based on the BFGS method, and the convergence process of the BFGS can be found in literature [38], thus the convergence analysis of POW is omitted here.
The computational complexity of POW mainly depends on the number of (outer) iterations and the number of operations at each iteration. In precise, at each iteration, the update of α(with (16)), same with ADMM-W, requires O (LMN R ) 3 operations. The optimization of ϕ requires O (LN T ) 3 . Compared with ADMM-W, the computational complexity of POW per iteration is lower.

V. SIMULATION
In the current section, a MIMO radar system consisting of 4 transmit antennas (N T ) and 4 receiving antennas (N R ) is considered. The proposed waveforms are tested using MCARM data (obtained from the MCARM file re050575) [39]. In this file, datacubes are comprised of 22 channels, 128 pulses, and 630 unambiguous range cells. The element 1:4, the channel 1:4:13, and the pulse 1:16 are, respectively, assumed as the receiver array, the transmit array, and the number of pulses in a CPI. The diagonal loading coefficient in Eq. (18) is set as Some numerical examples are also provided to demonstrate the performances of the waveforms designed by algorithms 1, 2 and 3. In both algorithms 1 and 3, the phase vector ϕ, s R , and s I are initialized to be random.

A. MCARM FILE re050575 DATA PREPROCESSING
In the re050575 file of range bin 290, only the clutter data of 22 channels are available; however, it is indispensable to obtain the clutter patch in the azimuth dimension. Therefore, an iterative adaptive approach (IAA) algorithm [40] is employed to estimate the clutter patch amplitude δ c,m,k described in Eq. (6). Figure 2a displays the origin clutter distribution at range bin 290, and the estimated clutter patch amplitude δ c,m,k is presented in Fig. 1b. Furthermore, the clutter sample covariance matrix R SCM in Eq. (9) is obtained to measure the concentration matrix in Eq. (17).
In this paper, we consider I point targets model with fixed target amplitudes from trial to trial. In this case, the signalto-clutter ratio (SCR) is defined as Besides, the space-time cross-ambiguity can evaluate the clutter suppression performance of the waveform. And according to the space-time cross-ambiguity expression for a target in [17], we can extend that the space-time cross-ambiguity for multiple targets as where w i = (R SCM + σ I) −1S h t i is the optimal receiver filter for the i-th target.

B. PERFORMANCES OF DIFFERENT ALGORITHMS FOR A SINGLE TARGET
In the present subsection, the performances of the phase-only algorithm and the ADMM-based algorithm are evaluated with a low SCR = 5dB of a target, i.e. I = 1. The target is artificially injected at Doppler bin 3, range bin 290, and angle bin 65. Parameters for MCARM data are presented in Table 1. The number of time samples and the freedom are considered as L = 16 and v = MN R L + 2, respectively.

1) CONVERGENCE PERFORMANCE AND CM PROPERTY OF ADMM-W
In the current experiment, the convergence performance and the CM property of the ADMM waveform (ADMM-W)based design algorithm are explored. The step sizes of algorithm 2 for each parameter ρ are set ur = 0.3, respectively. The convergence performance of the ADMM-W-based design algorithm under different step sizes (ρ) is investigated.  Figure 3 presents a comparison between the obtained objective values L ρ (s R , s I , u) at different step sizes. It is noticeable that the ADMM-based algorithm with a smaller step size (for example, ρ = 0.5) converged faster than that with a larger step size (for example, ρ = 4); however, a too small step size (for example, ρ = 0.5) causes a big oscillation in the obtained objective function values. In addition, the objective function gradually increases with the increasing step size. It should be noted that the choice of parameter ρ has a great influence on the performance of the ADMM-based algorithm. As to the computational complexity of ADMM-W, the run-time in each (outer) iteration is about 278s for each parameter ρ. Furthermore, the CM property of the obtained waveforms is analyzed by considering ρ = 1 and the iteration stop tolerance ς = 0.001. It is evident from Fig.4 that the values start to converge after 45 iterations. Let the symbol 'Tx' denote the transmit antenna. Figure 5 presents the modules, and the phase values of ADMM-W. We can see that both the real and imaginary parts exist between −1 and 1. Therefore, from Fig.5a, it can be inferred that the ADMM-W fulfilled the constant modulus constraint of the MIMO radar.

2) CONVERGENCE PERFORMANCE AND CM PROPERTY OF POW
In the current experiment, the convergence performance and the CM property of the POW algorithm are explored. VOLUME 8, 2020  The influences of step size on the convergence of the POW design algorithm are first examined. The curves of the objective value g A ( ) with different step sizes (µ, β) and the same initial phase value are plotted in Fig.6. It is observable that the POW design algorithm has the fastest convergence rate at µ = 0.4, whereas a large step size µ results in a large objective function value. In addition, the step size β has little effect on the objective function. When µ = 0.06, β = 0.01, g A ( ) obtained the local minimum value. As to the computational complexity of POW, the run-time in each (outer) iteration is about 31s, which is significantly lower than ADMM-W.
It is evident from Fig.7 that both real (Fig.7a) and imaginary parts (Fig.7b) of each antenna existed between −1 and 1. Moreover, the phase values of the POW algorithm are presented in Fig.7c.

C. CLUTTER SUPPRESSION PERFORMANCES OF THE WAVEFORMS DESIGNED FOR SINGLE TARGET
In order to assess the space-time cross-ambiguity of different waveforms at SCR = 5dB of a target, their clutter suppression performances are evaluated.
In [17], the authors proposed some algorithms to jointly optimize the transmit CM waveform and receive filter for single target MIMO radar systems. Especially, the Algorithm 2 is one of the method to design CM waveform. Thus, the waveform designed by Algorithm 2 is provided here. In addition, the orthogonal LFM waveform [15] is also offered to compare, i.e., the corresponding waveform matrix is given by where n t = 1, 2, . . . , N T and l = 1, 2, . . . , L. Fig.8 shows the space-time cross-ambiguity (Eq.43) associated with the different waveforms. It is clear from Fig. 8 that the target is located in the main lobe (Doppler bin 3 and angle bin 65) for all the waveforms. Compared to Fig.8b, Fig.8c and Fig.8d, Fig.8a has the biggest region in yellow. Moreover, as shown in Fig.8b and Fig.8d, the Algorithms 2 of [17] has similar performance with POW. As a consequence, our designed waveforms can suppress the power of clutter to a desired low level of Doppler bins and angle bins.

D. DETECTION PERFORMANCE TEST FOR SINGLE TARGET
In the current subsection, the detection performances of our designed waveforms, the waveform designed by the VOLUME 8, 2020 Algorithm 2 of [17] and the orthogonal LFM waveform (Eq. 44) are compared. In order to save simulation time, the value of M is set to 1 and the other parameters are kept unchanged. According to Eq. (15), under the proposed Bayesian framework, the detector associated with a transmitted waveform for a given probability of false alarm P fa (referred as the Bayesian-Optimal-detector T Bayesian ) can be expressed as where γ is the threshold (estimated from the results of 100 P fa Monte Carlo independent trials). The values of detection probability (P d ) are also obtained from these independent trials. Figure 9 displays the detection probability P d versus SCR curves of different waveforms for given false alarm probability P fa = 10 −3 . It is evident that the performance of radar detection is improved obviously by waveforms design. And the T Bayesian detector had the best performance with POW, better performance with ADMM-W, worst with LFM. In addition, the waveform designed by Algorithm 2 of [17] has a close detection performance with POW.

E. CLUTTER SUPPRESSION PERFORMANCES OF THE WAVEFORMS DESIGNED FOR TWO TARGETS
In the present subsection, the space-time cross-ambiguity associated with LFM of (44), ADMM-W and POW are evaluated in the scenario for two targets, i.e., I = 2. The target 1 is artificially injected at Doppler bin 10, and angle bin 45. The target 2 is artificially injected at Doppler bin 3, and angle bin 65. And the rest parameters are not changed.
Since the method of [17] is only applicable to a single target scenarios, it is not appropriate to compare Algorithm 2 of [17] with our algorithm here. Behaviors similar to Fig.8, Fig. 10 assessed the space-time cross-ambiguity associated with the designed waveforms by ADMM-W with ρ = 2 and POW with µ = 0.06, β = 0.2. As shown in Fig.10, the space-time cross-ambiguity function has two mainlobes, such as the first target is located in Doppler bin 10, angle bin 45, and the second target is located in Doppler bin 3, and angle bin 65. We also can see that in Fig.10b, the blue area is the largest and the yellow area is the smallest, that is, the clutter suppression effect of ADMM-W is the best. As a result, the waveforms designed by us manifest significantly better performance in suppressing the clutter of Doppler bins and angle bins of two targets.

VI. CONCLUSIONS
In this study, the transmit waveform design for target detection problems in heterogeneous clutter was addressed for an airborne colocated MIMO radar. For the signal-dependent clutter model, a Bayesian framework was considered, and the corresponding GLRT expression for waveform design problem was established. The POW and the ADMM-W methods were proposed to solve this problem. Simulation results revealed that the optimal waveform significantly improved the detection performance and suppressed the strong clutter of MCARM dataset; therefore, it can be inferred that the Bayesian detector manifested a better detection performance. In addition, for a single target, the ADMM-W method is not recommended because of its high complexity, long time consuming and sometimes not easy to converge. However, for multiple targets, it should be considered a compromise between the two methods.

APPENDIX A PROOF OF PROPOSITION 1
Let we define that Based on Table 3.2 in literature [36], it can be obtained that According to the definition of the derivative df (Page no. 76 in literature [36]), the gradient of g A with respect toS * can be given as Hence, proposition 1 is proved.

APPENDIX B
With the given function g B s R , s k I , u k , it can be expressed that where C is unrelated to s R . Let (s R ) i denote the i-th element in s R , then Hence, the proof is completed.