Alternating Direction Method of Multipliers-Based Constant Modulus Waveform Design for Dual-Function Radar-Communication Systems

In this paper, we design constant modulus waveforms for dual-function radar-communication (DFRC) systems based on a multi-input multi-output (MIMO) configuration of sensors for a far-field scenario. At first, we formulate a non-convex optimization problem subject to waveform synthesis for minimizing the interference power while maintaining a constant modulus constraint. Next, we solve this non-convex problem, iteratively, using the alternating direction method of multipliers (ADMM) algorithm. Importantly, the designed waveforms approximate a desired beampattern in terms of a high-gain radar beam and a slightly high gain communication beam while maintaining a desired low sidelobe level. The designed waveforms ensure an improved detection probability and an improved bit error rate (BER) for radar and communications parts, respectively. Finally, we demonstrate the effectiveness of the proposed method through simulation results.


Introduction
The exponential growth of the wireless communication industry, producing billions of connected devices, has led to a severe problem of frequency spectrum congestion [1][2][3]. Unfortunately, new mobile network operators and emerging technologies are facing spectrum scarcity. Consequently, the auction prices of the wireless frequency spectra have risen sharply in recent years [4][5][6]. Therefore, different methods have been proposed to improve the coexistence between active sensing systems (i.e., spectrum sharing) [7][8][9][10][11]. On the other hand, the comprehensive studies have shown that the radar bands are mostly under-utilized and can be used for communication purposes. Therefore, the use of radar bands for communication purposes has increased in recent years. For example, the L-band (1-2 GHz) is shared by the long-range surveillance and air traffic control (ATC) radars with supported communication technologies, such as LTE and 5 G NR. Additionally, milli-meter wave (mmWave) bands  are shared by automotive and high-resolution imaging radars with supporting IEEE 802.11ad/ay and WLAN communication technologies. Similarly, S-band (2-4 GHz) and C-band (4)(5)(6)(7)(8) are also being shared for communication purposes. These spectrum sharing methods gave birth to a new technology named as joint radar-communication designs, which is also called communication-radar spectrum sharing (CRSS) [12], RadCom [13], or integrated sensing and communication (ISAC) [14,15] designs in a broader sense. The schemes, in which the radar and communication systems Different methods have been proposed in the literature to deal with the non-convex CMC problem [49,50], e.g., semidefinite relaxation is used in [49] to design code radar waveforms and sequential optimization procedures are proposed in [50] to design constant modulus waveforms for MIMO radar. Additionally, a manifold-based algorithm has been proposed in [51] to solve the constant envelope precoding problem. Moreover, a strategy based on antenna selection is proposed in [52] for MIMO DFRC systems where several sparse antenna selection and permutation methods are used for embedding communication bits in the radar emissions. However, as full array sensors have not been used, emissions have wide mainlobes and high sidelobes problems. Additionally, different optimization methods have been proposed in [53] to decrease the downlink multi-user interference in communication operation and generate the desired radar beampatterns. However, because the waveform formulation in the radar direction is unconstrained, the radar detection performance deteriorates drastically. In [46], different methods have been presented for integrated waveforms through a waveform synthesis (WS) constraint. These methods synthesize the desired radar and communication waveforms in different directions; however, they give a high peak-to-average power ratio (PAPR) or high sidelobe levels.
In this paper, we investigate a constant modulus waveforms design to approximate a desired beampattern for dual-function radar-communication based on a MIMO system. Note that the desired beampattern consists of a high-gain radar main beam with a slightly high gain communication beam while maintaining the desired low sidelobe levels. First, we formulate the waveform design problem as an optimization problem. Because the constant modulus constraint makes the problem non-convex and NP-hard, traditional methods cannot be applied to solve the problem. Therefore, we use a well-known alternating direction method of multipliers (ADMM) algorithm to obtain an iterative solution to the problem. The ADMM blends the idea of the augmented Lagrangian method (ALM) with the dual decomposition method [54,55] to achieve an improved performance. The designed waveforms ensure an improved detection probability and bit error rate (BER) for radar and communications parts, respectively. Finally, the simulation results validate the efficiency of the proposed design in terms of convergence, approximation of the desired waveform, and beampattern synthesis. This paper is organized as follows. Section 1 gave a brief introduction of the study field and a review of the relevant literature. The signal model is explained in Section 2. Section 3 explains how the waveform has been formulated mathematically as non-convex optimization. In Section 4, the design problem is manipulated such that ADMM can be applied to it. The simulation results are provided in Section 5. Finally, the discussion of this paper is summarized and concluded in Section 6.
Notations: The notations R and C, used in this paper, represent the real and complex sets, respectively, while (·) and (·) denote the real and imaginary parts of the argument, respectively. The notation ⊗ represents the Kronecker product, | · | represents the absolute value of the argument, · 2 represents the l 2 norm, and I N represents the N × N identity matrix. Table 1 provides a list of the abbreviations used in this paper.

Signal Model
Consider a multi-input, multi-output joint radar-communication system, which is equipped with a uniform linear array (ULA), consisting of M transmit antenna sensors, as shown in Figure 1. Moreover, the radar and communication receivers consist of M transmit antenna elements. The antenna array transmits an integrated waveform for radar target detection that is also decoded at the communication end for detecting encoded information.
Let s m (n) ∈ C denote the nth sample of a discrete waveform, consisting of N samples, emitted by the mth antenna, where m = 1, . . . , M and n = 1, . . . , N. Let s(n) denote a vector that collects the nth samples of the waveforms transmitted by all antennas, i.e., s(n) = [s 1 (n), . . . , s M (n)] T . Then, the far-field waveform in the direction θ is given by is the transmit steering vector, with λ being the wavelength and d the inter-element spacing between the individual antenna elements. Let S = [s 1 , . . . , s N ] be the M × N space-time transmit waveform matrix.

Equations (3) and (4) can be combined as
where

Problem Formulation
The problem under consideration is to design a transmit waveform matrix S so that the power radiation in the sidelobe region can be minimized. Moreover, there are two constraints. The first constraint is the waveform synthesis (WS) constraint, i.e., the transmit waveform matrix S synthesizes the desired radar waveform x R and the desired communication waveform x C in the direction of the radar target and communication users, respectively, as given by Equation (5). The second constraint is the constant modulus constraint which prevents the nonlinear signal distortion in the amplifiers to increase the efficiency of the transmitter. The constant modulus constraint is expressed as The problem can be formulated as an optimization problem given by where Θ = [θ R , θ C ] is the collection of radar and communication direction angles while Θ = [θ 1 , θ 2 , · · · , θ K ] is the collection of angles of K sidelobes. The CM constraint Equation (6) renders the optimization problem (7) as non-convex. Being NP-hard, this problem is difficult to solve using any convex optimization methods. The problem (7) can be re-formulated for ease of analysis in two steps: the vectorization step and the realization step.

Vectorization
In the vectorization step, the matrices S and X are vectorized by stacking all of their respective column vectors into single column vectors. Correspondingly, matrices A(Θ) and A Θ are also updated. This is given by The CM constraint is given in terms of s as which is, equivalently, given by where where 0 ≤ m, n, i ≤ MN. At the end of the vectorization step, the problem Equation (7) can be expressed as

Realization
In the realization step, the complex-valued variables are converted to the real-valued version. For example, the realization of s takes the real part of s in one column vector and the complex part in another vector and then stacks the column vectors together to give s r .
The realization of s r , x r , A r (Θ), and A r Θ is given as In terms of vectorized real-valued variables, the CM constraint is given by where and 0 ≤ m, n, i ≤ 2MN. At the end of the realization step, the problem Equation (12) can be expressed as The optimization problem in Equation (16) can be solved to obtain s r−opt , which is the vectorized and real-valued version of S opt . So, the reverse operation, i.e., can be performed to obtain S opt , where s r1−opt contains the first MN elements, the real part, and s r2−opt contains the other MN elements, the imaginary part.

ADMM Formulation and Solution
The optimization problem Equation (16), like Equation (7), is non-convex and NP-hard. Analytical solutions to problems such as this are challenging to obtain and alternatives such as numerical or heuristic techniques are employed instead to obtain approximate solutions. Even using heuristic techniques, such as a genetic algorithm (GA), it may be difficult to formulate the CM constraint. Therefore, we use the ADMM-based iterative technique to approximate a solution to this problem.
An auxiliary variable r r is introduced in Equation (16) and the following equivalent version is obtained: minimize It can be observed that for the WS constraint in Equation (16), s r is expressed as two times s r and one of them is replaced by r r in Equation (18). Moreover, the CM constraint, consisting of MN equations in Equation (16), is expressed in its compact form in (18) as G(r r , s r ) = G(s r , r r ) = 0 (19) where G(r r , s r ) ∈ R 2MN×2MN is a vector given by where 1 and 0 are 2MN × 1 vectors, all 1s and 0s, respectively, and The augmented Lagrangian of Equation (18) is given as where u ∈ R 4N×1 , v ∈ R MN×1 , and w ∈ R 2MN×1 are the dual variables and ρ 1 , ρ 2 , ρ 3 > 0 are the adjustable penalty parameters.
The (m + 1)th iteration of the algorithm, in terms of the different variables, is given as follows: r m+1 r := arg min As can be seen from Equation (23), the updates Equations (23c)-(23e) are straightforward. The subequations Equations (23a) and (23b) are convex and give closed-form solutions. The details of the updates of variables r r and s r are presented next.

Update of r r
To obtain the (m + 1)th update of r r , we take the gradient of Equation (23a) with respect to r r and equate the result to 0, i.e., The solution to Equation (24) is given by where and

Update of s r
Similar to the r r update, in the s r update we take the gradient of Equation (23b) with respect to s r and equate the result to 0, i.e., The solution to Equation (28) is given by where and

Termination Criteria of the Algorithm
Let the primal residuals at iteration m + 1 be defined as and the dual residuals be defined as Then, as suggested by [54], reasonable termination criteria are where pri 1 , pri 2 , pri 3 are the tolerances for the primal residual and dual 1 is the tolerance for dual residuals. These tolerances, in accordance with [54], are defined as Algorithm 1 summarizes the steps of the algorithm.

Penalty Parameter Selection
Choosing the penalty parameters properly is very important in ADMM. The values of penalty parameters are decreased or increased depending on the values of some predefined tolerances. Different methods can be used choose the penalty parameters, such as hit-andtrial, etc. Another method is to relate the values of the penalty parameters to iteration numbers so that the values of penalty parameters increase or decrease (from the initially defined value) in steps. One standard method is to relate the values of the residual norms with the tolerances by using the concept of 'residual balancing' as given by Equation (36).
where ρ k is the penalty parameter, and µ > 1 and η > 1 are constants, d m+1 pr1 is the primary residual, and pri 1 is the tolerance.

Simulation Results and Analysis
In this section, the performance analysis of the algorithm is discussed and the results of some numerical examples are presented to evaluate the performance of the proposed waveform design method. A ULA consisting of M = 32 antenna elements having halfwavelength enter-element spacing has been considered at the transmitter and receiver sides. The radar target is located at θ R = 0 • and the communication user at θ C = 45 • . The desired radar waveform is based on linear frequency modulation (LFM). Similarly, the desired communication waveform uses the QPSK modulation scheme. We have considered N s = 1 symbols and N b = 2 bits per symbol. Thus, each waveform carries 2 bits of information per pulse repetition interval (PRI).
Different experiments are performed to evaluate the performance in different scenarios. Because both radar and communication receivers expect some desired waveforms, coherent detection can be used to match the received signal waveform with the desired waveform. Monte Carlo simulations are conducted to evaluate the performance of communication for different values of the signal-to-noise ratio (SNR).
The proposed method is compared with the far-field radiated emission design (FFRED) [56], the iterative optimization technique (using directly normalized waveforms) [26], and the theoretical values. In the FFRED method, 0%, 10%, and 40% of the total power is allocated to the orthogonal complement waveform, of which the FFRED-40% has the best performance. The authors of [26] proposed several waveform design methods. One method designed non-constant modulus waveforms and had a closed-form solution to the waveform design problem. They also proposed an iterative method for constant modulus waveforms. However, being computationally complex, they used the results of the first method using non-constant modulus waveforms and used iterative optimization for further refining those waveforms. They defined this method as 'directly normalized' in their simulations.

Computational Complexity Analysis
The computational complexity analysis is used to estimate the amount of computational resources (such as time and memory) required to run an algorithm. It provides an understanding of how the algorithm scales with the input size and helps in optimizing the algorithm or selecting alternative approaches if the complexity is too high.
To calculate the computational complexity of the proposed algorithm, each part of the code is analyzed to determine the number of operations or iterations performed in terms of the input size. In the code, the main loop iterates 'iter' number of times. Within each iteration, there are multiple calculations and operations performed, such as matrix multiplications, norm calculations, and updates of variables. The complexity of each of these operations is analyzed and sums them up to obtain an overall complexity estimate for the code. In addition, the input size-dependent variables, such as M and L, and their impact on the complexity are also considered.
The proposed algorithm has high computational complexity as it is cubic in nature. This is because it involves a matrix inversion operation. After the matrix inversion operation, the other main time-consuming operations are matrix multiplication operations.
For r updates, the calculation of Ξ 1 takes O(KM 2 L 2 ) and the calculation of γ takes O(M 2 L 2 ); therefore, the complexity of the update of r using Equation (25) is O(KM 2 L 2 + M 2 L 2 + M 3 L 3 ). Overall, the computational complexity of the algorithm is O(2(KM 2 L 2 + M 2 L 2 + M 3 L 3 )) at each iteration.

Data Rate Performance
The communication data rate is where N b is the number of bits per symbol, N s is the number of symbols in one pulse, and f PRF is the pulse repetition frequency.

ADMM Convergence Analysis
Plots of the norms of the primal and dual residuals, d pr1 , d pr2 , d pr3 , d dr2 , d dr2 , and the stopping criteria limits    Figure 5 shows the transmit beampattern formed by the waveform matrix S designed through the proposed ADMM-based approach for a DFRC system with 32 antenna elements.  Figure 6 shows the transmit beampatterns as synthesized by the waveform matrix S designed through the proposed ADMM-based approach and that of iterative optimization with amplitude weighting (IO-AW) as reported in [46]. In both cases, the systems have 16 antenna elements, and the radar target is located at θ R = 0 • and the communication user at θ C = 45 • . Moreover, in both cases, the power of the desired radar waveform is designed to be 10 dB more than that of the communication waveform. As can be seen in the figure, the IO-AW method leaks power at −45 • or, in other words, makes a mirror lobe toward a direction where there is no communication user. Otherwise, the sidelobe levels of the two beampatterns are almost the same. Thus, the beampattern formed through the proposed method outperforms the beampattern formed through IO-AW.

Waveform Error Analysis
Normalized waveform error is a performance metric that can be used to describe how closely the desired waveforms match the synthesized waveforms. Moreover, it can also give convergence performance, as the faster it decreases with the increasing number of iterations, the better the performance of the algorithm. Mathematically, this metric, denoted here as η i , is defined as where

Radar Performance Analysis
The radar performance evaluation is provided in two figures: the first figure compares the desired LFM waveform for radar and the far-field synthesized waveform, whereas the second gives the detection probability (pD) versus the SNR.
The waveform synthesized in the radar direction is shown in Figure 9. As shown in Figure 9 (upper), the synthesized radar waveform and the desired radar waveform seem almost identical, which validates the efficiency of the proposed scheme. However, there are small differences between the waveforms, shown in Figure 9 (lower). These differences or sample errors are defined as (40) Figure 9. Synthesized radar waveform: (Upper) desired waveform vs. far-field synthesized waveform, (Lower) difference between the desired waveform vs. far-field synthesized waveform. The asterisk sign (*) represents vector-matrix multiplication. Figure 10 shows the graph of detection probability plotted against the SNR. The probability of a false alarm is set as 10 −4 . For comparison, the pD versus SNR graphs of other methods are also provided. As the figure shows, FFRED-40% [56] has the best pD, which is almost the same as that of the desired LFM. However, the graph of the proposed method is so close that the difference becomes visible upon zooming in on the plots. At the same time, the proposed method provides better pD than that of the directly normalized method [26].

Communication Performance Analysis
Like radar performance, communication performance, too, is evaluated by two figures: the first figure gives a comparison of the desired communication waveform and the far-field synthesized waveform, whereas the second figure gives the SER versus the SNR.
The waveform synthesized in the communication direction is shown in Figure 11. As with radar waveforms, the synthesized and the desired communication waveforms seem almost exact. The sample errors in this case, Figure 11 (lower), are defined as   Figure 12 shows the SER plotted against the SNR. Again, for comparison, the SER versus SNR graphs of other methods are provided. The '2 bits per symbol' graph represents the theoretical values. Again, the graphs of FFRED-40% [56] and the proposed ADMM method are very close, although FFRED-40% has a relatively better performance. Both methods outperform the directly normalized waveform method [26].

Conclusions
A method for designing the constant modulus waveforms for MIMO dual-function radar-communication systems was proposed in this paper. The design problem was mathematically formulated as an optimization problem subject to the constraints of waveform synthesis and constant modulus. The optimization problem thus formulated, being nonconvex and NP-hard, was solved iteratively using an ADMM framework. Importantly, the designed waveforms approximated a desired beampattern in terms of a high-gain radar beam and a slightly high gain communication beam while maintaining a desired low sidelobe level. The designed waveforms ensured an improved detection probability and an improved bit error rate (BER) for the radar and communications parts, respectively. Based on the simulation results, the effectiveness of the proposed scheme has been validated.

Conflicts of Interest:
The authors declare no conflict of interest.