Sequential optimisation of orthogonal waveforms for MIMO radar

The existing methods to design orthogonal waveforms for multiple-input multiple-output radar mainly focus on the optimisation of autocorrelation and cross-correlation properties. Their performance will degrade severely in the presence of Doppler shifts. To overcome this limitation, the authors take an unknown Doppler shifts range into consideration and formulate a new waveform optimisation problem. Since the optimisation problem is highly non-linear, the authors propose an algorithm, called sequential cone programming, to tackle it. The key idea is to use the first-order Taylor expansion to approximate the constraints at each iteration. The authors show that the approximation can be solved via second-order cone programming. In addition, the autocorrelation peak sidelobe level and cross-correlation peak level could be further reduced by setting an appropriate threshold function. Simulation results demonstrate the efficiency of the proposed method compared with state-of-art methods.


Introduction
Multiple-input multiple-output (MIMO) radar allows transmitter to transmit independent waveforms [1][2][3][4]. It can provide more degrees of freedom and offers significant performance enhancement compared with its phased-array radar counterpart. In order to reduce interferences and to gain independent information from various target returns, orthogonal waveforms are highly desired in MIMO radar. However, ideal orthogonal waveforms do not exist. To design quasi-orthogonal waveforms, heuristic algorithms have been proposed. In [5], Deng proposed to design polyphase sequences with low autocorrelation peak sidelobe level (APSL) and cross-correlation peak level (CCPL) for orthogonal netted radar systems based on the simulated annealing algorithm. To further improve the performance of polyphase code, a number of optimisation methods including genetic algorithm [6], and complementary code [7], have been proposed to design orthogonal waveforms. Moreover, the authors in [8] proposed space-time coding to mitigate the waveform cross-correlation in MIMO radar. Besides the polyphase code sets, other types of orthogonal waveforms are also used for MIMO radar (see, e.g. in [9,10]). However, the above methods mainly focus on reducing the APSL and CCPL and do not take the target Doppler shifts into consideration. Thus, when the target is moving, the system transmitting such waveforms will degrade significantly even if the Doppler shift is small.
The optimisation of waveforms with tolerance of Doppler shifts have been considered in [11,12] via the construction of Hadamard matrix and introducing the cross-entropy technique. However, the length of the polyphase sequences that can be designed therein is limited. To obtain low level of APSL and CCPL as well as good Doppler shifts tolerance with arbitrary length sequences, [13] considered the Doppler shifts in the optimisation model. The optimisation problem therein was tackled by the adaptive clonal selection optimisation method.
In this paper, we propose an algorithm to design the orthogonal waveforms with low APSL and CCPL as well as good Doppler tolerance. We construct a mathematical model which considers different Doppler shifts. We adopt Taylor approximation to transform the original non-convex problem into a sequence of sequential cone programming problems. Moreover, we constrain the threshold of phase increment as a linear function in order to further improve the performance. Notation: Throughout this paper, boldfaced and italic letters denote matrices and vectors, respectively. (·) T and (·)* represent transpose and conjugate, respectively. ∥ ⋅ ∥ denotes the Euclidean norm. ⊗ and ⊙ represent Kronecker product and Hadamard product, respectively. diag(·) denotes a diagonal matrix. Ê P represents a P × 1 vector with elements of all ones.

Problem formulation
Consider a MIMO radar system with L antenna elements. All the antennas transmit phased-code waveforms consisting of N subpulses. We denote the transmit waveform of the lth antenna as follows: s l (n) = exp(jϕ l (n)), n = 1, . . . , N, l = 1, . . . , L where ϕ l (n) denotes the phase of the nth sub-pulse. Without loss of generality, we assume that ϕ l (n) ∈ [0, 2π]. Denote the Doppler shift frequency of the target by f d and the sub-pulse width by T. Due to the presence of Doppler shift, the additional phase increments from one sub-pulse to the next can be written as 2π f d T /N. Similar to reference [13], we define the aperiodic autocorrelation function A(s l , k) and cross-correlation function C(s p , s q , k), l = 1, . . . , L, k = − N + 1, . . . , N − 1 ， p ≠ q = 1, . . . , L, as follows: J. Eng where s l, d ′*(n + k) = s l * (n + k)exp(− j2π f d (n + k − 1)T /N). To give a compact expression of A(s l , k) and C(s p , s q , k), we let s = exp(jφ) = [exp(jϕ 1 (1)), . . . , exp(jϕ L (N))] T , and define s′ d and J, J as follows Let Z p, q be an L × L square matrix, the (p, q)th element of which is 1, and others zero. Then we have Combining (1)-(3), and (6), we can obtain that Note that ideal orthogonal waveforms should be a group of sequences with the following three characteristics: (i) the autocorrelation functions of the constant-modulus sequences are impulse functions; (ii) the cross-correlation functions are zeros; (iii) the waveforms have good Doppler tolerance. However, ideal orthogonal waveforms do not exist. Thus, we try to design quasiorthogonal waveforms. Before presenting the optimisation model, we assume that the maximised Doppler shift is f d max . In order to improve the Doppler tolerance of the proposed method, the objective function must consider a Doppler shift range of . For simplicity, we discretise the Doppler shift interval into I bins. Thus, we formulate the waveform design problem as follows: where w is a weight factor to adjust the value of APSL and CCPL. We can also put an emphasis on the APSL by appropriately relaxing the performance of CCPL. To this end, we recast the following optimisation problem where σ is the threshold of CCPL.

Waveform optimisation via sequential cone programming
The optimisation problem in (9) is non-convex and is difficult to solve. To this end, we adopt the first-order Taylor expansion to approximate the non-convex constraints and propose a sequential cone programming to tackle this problem. Without loss of generality, we consider the ith Doppler shift as an example. We assume that the transmit waveform at the mth iteration is , thus the first-order Taylor expansion is given by where ∂A m (s l , k)/∂φ is gradient at the mth iteration, and Δφ m is the increment of the variable. In order to simplify the derivation, we define Then, the specific expression of the mth iteration is given in (12), where the (p, j) element of J l, l, k is denoted by a p, j . The derivation of Taylor approximation for cross-correlation is similar to (12). Due to space limitations, we omit it here Considering that we only use the first-order Taylor approximation for the constraints, the phase increment must satisfy (13) to overcome the error caused by the neglect of the high-order Taylor series where E l is LN × 1 vector, with the elements from N × (l − 1) + 1 to N × l ones, and others zero, δ l is the phase increment threshold of array element l. In addition, the phase is continuous in [0, 2π]. Thus, we have where D i is a 1 × LN dimension vector with the ith element being 1 and other elements being zero. Combining (11)-(14), we obtain the optimisation problem at the mth iteration: Note that (15) is an second-order cone programming (SOCP) problem, which can be solved effectively via primal-dual interior point method (PDIPM).
To summarise, the proposed method is listed in Algorithm 1 (see Fig. 1).
Discussion: The algorithm terminates if the maximum number of iterations is reached, or the change of the objective value between two iterations is less than a predefined value. In this paper, we choose the first one. Moreover, note that we have used the firstorder Taylor expansion to approximate (7) and (8). If the phase increment threshold is chosen to be too large, the algorithm performance cannot be improved and even degrades. Therefore, we set the threshold of phase increment as a linear function.

APSL and CCPL optimisation without Doppler shifts
Assume that the MIMO radar system transmit L = 4 orthogonal waveforms. Each of them consists of N = 40 sub-pulses. The weight factor is w = 1. We do not consider Doppler shifts in this subsection. Table 1 shows the phase values (in radian) for the designed sequences.
For comparison, the APSL and CPSL of Deng's codes [5] and the devised waveforms by the proposed method are shown in Tables 2 and 3, respectively.  From Tables 2 and 3, it can be seen that the performance of the waveforms designed by the proposed method is better than that of Deng's codes. Specifically, the APSL and the CPSL is about 1 dB lower and 2.5 dB lower than that of Deng's, respectively. This can be explained as follows. First, the phases of the designed waveforms belong to [0, 2π]. Thus, the feasibility region is larger than that in Deng's paper. Second, we adopt the first-order Taylor expansion to approximate the original non-linear and non-convex constraint. Fortunately, the approximation can be tackled by SOCP.
Moreover, from Figs. 2 and 3 as well as Table 3, we can notice that the waveforms designed by the proposed method can achieve identical values of APSL and CCPL. This is because we directly optimise the APSL and CCPL of the waveforms. In addition, the variable increment is controlled adaptively, which can improve the performance further.

APSL and CCPL optimisation with Doppler shifts
In this subsection, we assume that the phase increments of 2π f d T from the first sub-pulse to the last one, which are called max Doppler phase shift (in degree), belong to [ − 144, 144]. Other parameters are the same as those in section 4.1.
Figs. 4 and 5 show that the performance of APSL and CCPL of the designed waveforms is robust to the Doppler shifts. That is to say, our method can give a better performance when the target is moving.  To investigate the Doppler resilience of the waveforms designed by different methods, we plot the APSL and CCPL values of the designed waveforms versus the Doppler shift in Fig. 6.
From Fig. 6 we can observe that the waveforms designed by the proposed method achieve the lowest APSL and CCPL values in all Doppler shifts except zero. Remarkably, it can be seen that about the CCPL of the designed waveforms by the proposed algorithm is 5.5 and 4 dB than that of the random methods, and that of Yang's method [13], respectively. Moreover, the APSL of the waveforms designed by the proposed algorithm is about 4.5 dB lower than that of the random method, 1.5 dB lower than that of Yang's method. As a result, it can be concluded that the designed waveforms have better tolerance with the Doppler shifts. Fig. 7 plots the APSL of the waveforms optimised by the proposed method versus the maximum Doppler shifts when the CCPL is fixed. Comparing Fig. 7 with Fig. 6, we can observe that, by sacrificing the performance of CCPL by about 1 dB, we can achieve an improvement of APSL by about 4 dB. This is because that the feasible zone of (10) is determined by the constraint of CCPL. A larger feasible zone can be achieved with the relaxation of the CCPL. Thus, a lower APSL will be obtained.

Conclusion
We have proposed an algorithm to design Doppler-robust orthogonal waveforms. We have adopted the first-order Taylor approximation to transform the original non-convex problem into a sequence of SOCP problems. In addition, the threshold of phase increment is constrained as a linear function, which can further improve the performance. We have shown that the proposed algorithm can precisely control the APSL and CCPL. Moreover, by sacrificing the performance of CCPL slightly, we can obtain a lower APSL. Finally, the designed waveforms are robust to the target Doppler shifts and can be applied to detect moving targets with MIMO radar.

Acknowledgments
This work was supported in part by the National Natural Science