Robust design and evaluation of phase codes for radar performance optimization with a ﬁnite alphabet constraint

This letter considers the robust design of radar waveform with dis- crete phase codes under constant modulus constraint to enhance target detectability. To solve the resultant max-min non-convex optimization problem, a robust discrete phase coding algorithm via iterative technology (R-DPCA-IT) is developed which monotonically improves the worst-case SNR and has an upper bound ensuring to converge a ﬁnite value. At each iteration, the original multidimensional quadratic op- timization problem is turned into multiple one-dimensional optimization problems solved globally via line search. Finally, the computational time and objective value of the proposed algorithm are assessed in comparison with the existing methods.

✉ E-mail: xianxiangy@gmail.com This letter considers the robust design of radar waveform with discrete phase codes under constant modulus constraint to enhance target detectability. To solve the resultant max-min non-convex optimization problem, a robust discrete phase coding algorithm via iterative technology (R-DPCA-IT) is developed which monotonically improves the worst-case SNR and has an upper bound ensuring to converge a finite value. At each iteration, the original multidimensional quadratic optimization problem is turned into multiple one-dimensional optimization problems solved globally via line search. Finally, the computational time and objective value of the proposed algorithm are assessed in comparison with the existing methods.
Introduction: Radar waveform design has been received wide attention in the literatures during the last decades. Additionally, high performance computer, cognitive radio technology, and powerful digital arbitrary waveform generators have paved the way for waveform design exploiting the priori information with respect to the operating environment and target to significantly enhance the target detection capability and tracking performance [1][2][3][4][5][6][7][8]. However, the cognitive systems are hard to obtain the accurate prior information. Recently, robust waveform design has received considerable attention in the literatures [2, [9][10][11][12][13][14][15][16][17]. For example, in [2], focusing on the worst-case signal-to-interferenceplus-noise ratio (SINR) with colored interference, a robust design of a slow-time transmitter sequence under peak-to-average-power ratio (PAR) constraints is solved by semi-definite program (SDP) relaxation and randomization. Based on mutual information and minimum mean-squared error metric, a minimax robust waveform design has been introduced in [9]. For signal-dependent clutter suppression, [11] considered the joint design of the radar waveform and a Doppler filter bank using a related generalized Dinkelbach's procedure to improve the worst-case SINR at the output of the filter array. Also aiming at improving the worst-case SINR, the authors focused on uncertainties both in the received useful signal component and interference covariance matrix [12]. For extended targets, [13][14][15] have introduced several methods for robust waveform design in the presence of signal-dependent interference. A robust transmit code and receive filter is jointly designed under PAR constraint and an energy constraint for an extended target in [16]. Most of the aforementioned design methods are based on SDP-related techniques which share a huge computation complexity probably limiting its usage from a practical point of view.
In this letter, we focus on the robust slow-time waveform with discrete phase modulation design under constant modulus constraint assuming that coloured interference with known covariance matrix. According to the prior information of the rough target Doppler frequency, a max-min quadratic optimization problem is found to improve the worstcase signal-to-noise ratio (SNR). To tackle the resultant non-convex problem, we propose a coordinate descent based (CD-based) algorithm to iteratively and sequentially update the slow-time code. Precisely, we split the multidimensional problem into a set of trigonometric function problems which can be solved by 1D search. Finally, the performance of the proposed algorithm is assessed via several numerical simulations. Results show the proposed algorithm is a reasonable trade-off between performance gains and complexity in comparison with the SDP-related and majorization-minimization (MM)-related [20] techniques.
System model and problem formulation: Consider a monostatic radar system which emits a coherent burst of N slow-time coded pulses denoted by s = [s 1 , s 2 , . . . , s N ] T ∈ C N , where C N denotes the sets of Ndimensional vectors of complex numbers and (·) T is the transpose operation. Each pulse at the receiver end is down-converted to baseband and then is sampled after matched filtering. Considering a far-field target located at the range-azimuth cell under test, the N-dimensional column where α 0 is a complex parameter accounting for the target radar cross section (RCS), channel propagation effects, and other terms involved into the radar range equation. p 0 = [1, e j2πv0 , . . . , e j2π (N−1)v0 ] T is the Doppler frequency vector of the target where v 0 = T f 0 is the normalized target Doppler frequency, with T and f 0 denoting the pulse repetition interval (PRI) and the actual Doppler frequency of the target, respectively. n ∈ C N represents a zero-mean complex circular Gaussian vector with known positive definite covariance matrix, that is E[nn † ] = M, where (·) † and E[·] denote the conjugate transpose and statistical expectation, respectively. If the target Doppler frequency is a priori known, the SNR of the target is defined as [1] where R = M −1 (pp † ) * while (·) * denotes complex conjugate. Note that R is positive definite due to s † Rs > 0 for any s = 0. Besides, R is a Hermite matrix because of M and (pp † ) * are both a Hermite matrix.
While lack of a priori information, we consider a robust system (i.e. worst-case SNR) accounting for v k , k = 1, . . . , K with the number of all possible Doppler shifts K. Hence, Equation (2) can be transformed into with SNR k (s) = |α 0 | 2 s † R k s and R k = M −1 (p k p † k ) * . Therein, from practical considerations, a finite alphabet phase code is adopted due to the limited number of bits are available in digital waveform generators. Finally, we can formulate the waveform design problem with the constant modulus constraint as follows: where M denotes the number of discrete alphabet. Note that P 1 is a general NP-hard problem which can not be solved in polynomial time.
Herein, we proposed a CD based algorithm to monotonically increase the objective value of the original NP-hard problem.

Robust discrete phase coding algorithm via iterative technology (R-DPCA-IT):
In this section, we focus on solving the problem P 1 in a polynomial time. An inspection of the P 1 reveals that the constraints function concerning (s 1 , s 2 , . . . , s N ) are separate. Consequently, a CD-based [18,19] R-DPCA-IT is proposed to sequentially optimize (s 1 , s 2 , . . . , s N ) to maximize the worst-case output SNR. Specifically, we sequentially optimize s i while other codes in s are fixed until all codes in s have been updated. Let s (n) and SNR (n) T = SNR T (s (n) ), respectively, denote the solution of problem P 1 and the worst-case SNR at the nth iteration. Hence, s (n) i can be obtained from with a ik = |α k | 2 R k s −i and R k,i,i the (i, i)th entry of R k , where (x) represents the real part of x. Imposing the constraint |s i | = 1, Equation (5) can be transformed into where After ignoring the constant terms in Equation (7), P s (n) i can be equivalently recast as where a ik,i is the ith element of a ik , and ϕ ik = arg(a ik,i ). The above subproblem can be solved via the 1D search. Let ν(P (n) si ) denotes the optimal value of the Equation (8). At each iteration, we can find an optimal solution of P ϕi via CD framework, and we have Further, the following inequality holds true with λ min (R k ) the minimum eigenvalue of R k . The above expression reveals that the worst-case SNR monotonously increases and has an upper bound of SNR U T to ensure convergence. Herein, we can continue the sequential optimization procedure until |SNR (n) where is a user selected parameter to control convergence. The formal description of the alternating optimization procedure is reported in Algorithm 1. It is worth observing that the total computational complexity is associated with the size of s and K. More in detail, each iteration involves the solution of P ϕi which requires to compute the a ik with the computational complexity of O((N ) 2 K ).
Simulation: In this section, we assess the performance of the proposed algorithm also compared with the SDP-related and MM-related technique [20]. Specifically, the interference covariance matrix M is modelled as M i, j = ρ |i− j| [1], where one-lag correlation coefficient ρ = 0.8. We set |α 0 | 2 =0 dB and v k = 0.1 +k v with v =0.025 for k = 1, . . . , K, and pulses number N = 64. Besides, we consider the exit condition = 10 −7 for R-DPCA-IT, resort to the CVX toolbox [21] to solve the SDP and set the number of random trial to 500 involved in randomization approach. The running computation time is analyzed using Mat- In Figure 1, the curves of SNR T and the corresponding global computational times of RDPC-IT, SDP and MM algorithm versus the alphabet size M (M = 2 i , i = 1, . . . , 7) are plotted for K = 4, where we also show the continuous phase case" curve obtained by SDP to provide the upper bound for the discrete phase case. Note that the proposed algorithm outperforms MM algorithm in terms of worst-case SNR. In particular, for M =2, the proposed algorithm has a performance gain of 0.38 and 0.53 dB with respect to MM and SDP, respectively. Besides, SNR T obtained via R-DPCA-IT, SDP and MM algorithms show an improved trend as M increases, while R-DPCA-IT and SDP show performance close to the upper bound SNR U T with M is greater than 32. Figure 1b highlights that the larger the M, the more the computational times for all algorithms. Specifically, R-DPCA-IT possesses a fast convergence time than SDP, and the computational times of R-DPCA-IT and MM are comparable. Figure 2 plots the curves of SNR T and the corresponding global computational times of R-DPCA-IT, SDP and MM algorithm versus K(K = 1, . . . , 10) for M = 2 and M = 64. As expected, all SNR T values achieved using R-DPCA-IT, SDP and MM algorithm decrease as K increases. Interestingly, R-DPCA-IT achieves the best SNR T than SDP and MM while K is no greater than 7. Besides, the worst-case SNR tends to a constant of R-DPCA-IT and MM while K is greater than 7, which is reasonable because the more K equivalently means more constraints need to meet and smaller feasible set to match. In Figure 2b, it highlights that the proposed algorithm shows much faster convergence than SDP and analogous to MM for K = 1, . . . , 10. Besides, we assess the worst-case detection performance for the designed waveform considering different M. According to the Neyman-Pearson criterion, an analytical expression of the detection probability P d  , where Q(·, ·) denotes the Marcum Q function of order 1 and P f a stands for a desired value of the false alarm probability. In Figure 3, we fix K = 4, and plot the detection probability P d assuming P f a = 10 −6 against |α 0 | 2 (in dB) for M = 2, 64. As expected, the larger the M and |α 0 | 2 the bigger P d for all considered algorithms. In particular, R-DPCA-IT gain superior detection performance (increase detection probability by 10%) than SDP and MM under the same values of M = 2.  11,15,19,23 We redefine the v k = 0.1 + 0.025k for k = −L + 1, . . . , L + 1, and let K = 2L + 1. Finally, the robustness of R-DPCA-IT with respect to target Doppler shifts is studied in Figure 4. Specifically, we plot the SNR and the worst-case P d versus v 0 (v 0 ∈ [−0.5, 0.5]) for M = 64 and K=1, 11,15,19,23. Inspection of the curves in Figure 4a shows that SNR tends to flat as K increases, which benefits to Doppler frequency unknown target detection. Besides, it's worth noting that, in the presence of significant Doppler mismatches (v 0 ∈ [−0.4, −0.2]), R-DPCA-IT with K = 23 shows a quite flat shape about Doppler shifts, while the values of R-DPCA-IT with K = 1 approach very close to zero.

Conclusion:
In this letter, we have addressed the design problem of radar waveform with discrete phase under constant modulus and proposed the R-DPCA-IT to sequentially improve the worst-case SNR. Results have shown that the proposed algorithm achieves a reasonable trade-off between performance gains and complexity.