An Optimization Design Method for Zero-Correlation Zone Sequences Based on Newton’s Method
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摘要: 具有零相关区的序列集在无线通信以及雷达中具有重要的应用。然而，现有的序列集大都基于数学解析构造方法得到，其参数如序列长度和序列数目等受到一些限制，不能灵活设定，这限制了其在实际场景中的应用。因此，研究具有灵活参数的序列集构造方法成为一个有意义的课题。为得到更多灵活参数的序列集，该文利用牛顿优化方法来进行零相关区序列设计研究，具体提出了非周期零相关区互补序列集和非周期零相关区序列集的优化设计方法，得到的序列集参数可灵活调节。最后，通过互补峰值旁瓣水平(CPSL)和加权峰值旁瓣水平(WPSL)评估了序列集性能。Abstract:
Objective Sequences with favorable correlation properties are widely applied in radar and communication systems. Sequence sets with zero or low correlation characteristics enhance radar resolution, target detection, imaging quality, and information acquisition, while also improving the omnidirectional transmission capability of massive multiple-input multiple-output (MIMO) systems. Designing aperiodic Zero Correlation Zone (ZCZ) sequence sets with excellent correlation performance is therefore critical for both wireless communication and radar applications. For example, aperiodic Z-Complementary Set (ZCS) sequence sets are often used in omnidirectional precoding for MIMO systems, whereas aperiodic ZCZ sequence sets are employed in integrated MIMO radar-communication systems. These ZCZ sequence sets are thus valuable across a range of system applications. However, most prior studies rely on analytical construction methods, which impose constraints on parameters such as sequence length and the number of sequences, thereby limiting design flexibility and practical applicability. This study proposes a numerical optimization approach for designing ZCS and aperiodic ZCZ sequence sets with improved correlation properties and greater parametric flexibility. The method minimizes the Complementary Peak Sidelobe Level (CPSL) and Weighted Peak Sidelobe Level (WPSL) using Newton’s method to achieve superior sequence performance. Methods This study proposes an optimization-based design method using Newton’s method to construct both aperiodic ZCS sequence sets and aperiodic ZCZ sequence sets with low sidelobe levels and flexible parameters. The optimization objective is first formulated using the CPSL and WPSL. The problem is then reformulated as an equivalent system of nonlinear equations, which is solved using Newton’s method. To reduce computation time, partial derivatives are approximated using numerical differentiation techniques. A loop iteration strategy is employed to address multiple constraints during the optimization process. To ensure algorithmic convergence, Armijo’s rule is used for step size selection, promoting stable descent of the objective function along the defined search direction. Results and Discussions The aperiodic ZCS sequence set is constructed using Newton’s method. As the number of sequences increases, the CPSL progressively decreases, falling below –300 dB when $M \geqslant 2$. The proposed method yields better sidelobe performance than the improved Iterative Twisted Approximation (ITORX) algorithm ( Fig. 1 ). The performance of ZCS sequences generated by both methods is evaluated under different ZCZ conditions. While both approaches achieve low CPSL, Newton’s method yields aidelobe levels closer to the ideal value (Fig. 2 ). Convergence behavior is assessed using CPSL and the number of iterations. The improved ITROX algorithm typically requires around20000 iterations to converge, with increasing iterations as ZCZ size grows. In contrast, Newton’s method achieves rapid convergence within approximately 10 iterations (Figs. 3 and4 ). The aperiodic ZCZ sequence set constructed using Newton’s method exhibits autocorrelation and cross-correlation peak sidelobe levels below –300 dB within the ZCZ. Moreover, Newton’s method achieves the lowest WPSL, offering the best overall performance among all tested methods (Fig. 5 ). The smooth convergence curves further confirm the algorithm’s stability when applied to aperiodic ZCZ sequence construction (Fig. 6 ).Conclusions This study proposes an optimization-based algorithm for designing aperiodic ZCS and aperiodic ZCZ sequence sets using Newton’s method, aiming to address the limitations of fixed parameters and high peak sidelobe levels found in existing approaches. Two optimization problems are formulated by minimizing the WPSL and CPSL, respectively. To simplify computation, the optimization tasks are converted into systems of nonlinear equations, which are solved using Newton’s method. The Jacobian matrix is computed via numerical differentiation to reduce computational cost. A loop iteration strategy is introduced to meet multiple constraints in the construction of aperiodic ZCZ sequences. Simulation results confirm that the proposed method yields sequence sets with excellent correlation properties and flexible parameter configurations. By tuning the weighting coefficients, low sidelobe levels can be achieved in specific regions of interest, accommodating different application requirements. The combination of flexible design parameters and favorable correlation performance makes the proposed sequences suitable for a wider range of practical scenarios. -
Key words:
- Newton’s Method /
- aperiodic /
- Zero Correlation Zone(ZCZ) /
- Peak Sidelobe Level(PSL)
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1 基于牛顿法的ZCS优化算法
输入：序列个数$M$，序列长度$N$，零相关区长度$Z$，迭代终止
条件$\varepsilon $；输出：非周期零相关区互补集$(M,N,Z) - {\text{ZCS}}$； (1) 重复以下步骤 (2) 随机初始化$\{ {{{\boldsymbol{\theta}} }_1},{{{\boldsymbol{\theta}} }_2}, \cdots ,{{{\boldsymbol{\theta}} }_M}\} $； (3) ${\text{For }}d = 0:\infty {\text{ do}}$ (4) 根据式(8)计算$f({{{\boldsymbol{\theta}} }_1},{{{\boldsymbol{\theta}} }_2}, \cdots ,{{{\boldsymbol{\theta}} }_M})$； (5) ${\text{If CPSL}} \le \varepsilon {\text{ then}}$ (6) 返回${\boldsymbol{X}} = \{ {{\boldsymbol{x}}_1},{{\boldsymbol{x}}_2}, \cdots ,{{\boldsymbol{x}}_M}\} $，其中
${{\boldsymbol{x}}_i} = ({{{\mathrm{e}}}^{{{\mathrm{j}}}{{{\theta}} _{i,1}}}},{{\mathrm{{e}}}^{{{\mathrm{j}}}{{{\theta}} _{i,2}}}}, \cdots ,{{{\mathrm{e}}}^{{{\mathrm{j}}}{{{\theta}} _{i,N}}}})$；(7) ${\text{End If}}$ (8) 根据式(8)计算$f({{{\boldsymbol{\theta}} }_1},{{{\boldsymbol{\theta}} }_2}, \cdots ,{{{\boldsymbol{\theta}} }_M})$； (9) 根据式(9)计算$ J({{{\boldsymbol{\theta}} }_1},{{{\boldsymbol{\theta}} }_2}, \cdots ,{{{\boldsymbol{\theta}} }_M}) $； (10) 根据式(10)计算${\varPhi }({{{\boldsymbol{\theta}} }_1},{{{\boldsymbol{\theta}} }_2}, \cdots ,{{{\boldsymbol{\theta}} }_M})$； (11) 根据式(11)计算$\vartheta ({{{\boldsymbol{\theta}} }_1},{{{\boldsymbol{\theta}} }_2}, \cdots ,{{{\boldsymbol{\theta}} }_M})$； (12) 根据式(13)计算牛顿搜索方向${{\boldsymbol{v}}_{nt}}$； (13) 根据式(12)计算$ \{ {{\boldsymbol{\theta}} }_1^{k + 1},{{\boldsymbol{\theta}} }_2^{k + 1}, \cdots ,{{\boldsymbol{\theta}} }_M^{k + 1}\} $； (14) 步长设置为$\alpha = 1$； (15) ${\text{For }}k = 0:{k_{\max }}{\text{ do}}$ (16) ${\text{If}}$式(15)成立${\text{then}}$ (17) ${\text{break;}}$ (18) ${\text{End If}}$ (19) $\alpha = \alpha /2$； (20) 根据式(12)计算$ \{ {{\boldsymbol{\theta}} }_1^{k + 1},{{\boldsymbol{\theta}} }_2^{k + 1}, \cdots ,{{\boldsymbol{\theta}} }_M^{k + 1}\} $； (21) ${\text{End For}}$ (22)${\text{End For}}$ 
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2 基于牛顿法的非周期ZCZ序列集优化算法
输入：序列个数$M$，序列长度$N$，零相关区长度$Z$，迭代终止
条件$\varepsilon $；重复以下步骤 (1) 随机初始化$\{ {{{\boldsymbol{\theta}} }_1},{{{\boldsymbol{\theta}} }_2}, \cdots ,{{{\boldsymbol{\theta}} }_M}\} $，取值在$[0,2{\pi }]$均匀分布； (2) ${\text{For }}d = 0:\infty {\text{ do}}$ (3) 第1部分优化序列集的互相关水平； (4) $\{ {{{\boldsymbol{\theta}} }_i},{{{\boldsymbol{\theta}} }_j}\} ,1 \le i \le M,1 \le j \le M,i \ne j$； (5) ${\text{For }}d = 0:\infty {\text{ do}}$ (6) 根据式(20)计算$f({{{\boldsymbol{\theta}} }_i},{{{\boldsymbol{\theta}} }_j})$； (7) ${\text{If max(}}\mathop {{\text{max}}}\limits_{i,j,i \ne j} |{\rho _{{{\boldsymbol{x}}_i},{{\boldsymbol{x}}_j}}}(\tau )|/N{\text{)}} \le \varepsilon {\text{ then}}$ (8) 输出${{\boldsymbol{x}}_i} = ({{{\mathrm{e}}}^{{{\mathrm{j}}}{{{\theta}} _{i,1}}}},{{{\mathrm{e}}}^{{{\mathrm{j}}}{{{\theta}} _{i,2}}}}, \cdots ,{{{\mathrm{e}}}^{{{\mathrm{j}}}{{{\theta}} _{i,N}}}}) $,
${{\boldsymbol{x}}_j} = ({{{\mathrm{e}}}^{{{\mathrm{j}}}{{{\theta}} _{j,1}}}},{{{\mathrm{e}}}^{{{\mathrm{j}}}{{{\theta}} _{j,2}}}}, \cdots ,{{{\mathrm{e}}}^{{{\mathrm{j}}}{{{\theta}} _{j,N}}}})$；(9) ${\text{End If}}$ (10) 计算式(20)、式(9)、式(10)、式(11)、式(13)和式(12)； (11) ${\text{For }}k = 0:{k_{\max }}{\text{ do}}$ (12) 计算式(15)确保优化过程的单调性； (13) 根据式(12)计算$\{ {{\boldsymbol{\theta}} }_i^{k + 1},{{\boldsymbol{\theta}} }_j^{k + 1}\} $； (14) ${\text{End For}}$ (15) ${\text{End For}}$ (16) 第2部分优化序列集的自相关水平(见算法1)； (17) 根据式(21)计算WPSL； (18) ${\text{If WPSL}} \le \varepsilon {\text{ then}}$ (19) 返回${\boldsymbol{X}} = \{ {{\boldsymbol{x}}_1},{{\boldsymbol{x}}_2}, \cdots ,{{\boldsymbol{x}}_M}\} $，其中
${{\boldsymbol{x}}_i} = ({{{\mathrm{e}}}^{{{\mathrm{j}}}{{{\theta}} _{i,1}}}},{{{\mathrm{e}}}^{{{\mathrm{j}}}{{{\theta}} _{i,2}}}}, \cdots ,{{{\mathrm{e}}}^{{{\mathrm{j}}}{{{\theta}} _{i,N}}}})$；(20) ${\text{End If}}$ (21) ${\text{End For}}$
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[1] PEZESHKI A, CALDERBANK A R, MORAN W, et al. Doppler resilient Golay complementary waveforms[J]. IEEE Transactions on Information Theory, 2008, 54(9): 4254–4266. doi: 10.1109/TIT.2008.928292. [2] GIRNYK M A and PETERSSON S O. Efficient cell-specific beamforming for large antenna arrays[J]. IEEE Transactions on Communications, 2021, 69(12): 8429–8442. doi: 10.1109/TCOMM.2021.3113755. [3] HUANG Wenjie, NAGHSH M M, LIN Ronghao, et al. Doppler sensitive discrete-phase sequence set design for MIMO radar[J]. IEEE Transactions on Aerospace and Electronic Systems, 2020, 56(6): 4209–4223. doi: 10.1109/TAES.2020.2988623. [4] PETERSSON S O and GIRNYK M A. Energy-efficient design of broad beams for massive MIMO systems[J]. IEEE Transactions on Vehicular Technology, 2022, 71(11): 11772–11785. doi: 10.1109/TVT.2022.3192245. [5] ZHOU Zhengchun, LIU Bing, SHEN Bingsheng, et al. Doppler-resilient waveform design in integrated MIMO radar-communication systems[J]. Science China Information Sciences, 2024, 67(1): 112301. doi: 10.1007/S11432-022-3733-2. [6] 王佳欢, 范平志, 时巧, 等. 一种具有多普勒容忍性的通感一体化波形设计[J]. 雷达学报, 2023, 12(2): 275–286. doi: 10.12000/JR22155.WANG Jiahuan, FAN Pingzhi, SHI Qiao, et al. Doppler resilient integrated sensing and communication waveforms design[J]. Journal of Radars, 2023, 12(2): 275–286. doi: 10.12000/JR22155. [7] LI Fengjie, JIANG Yi, DU Cheng, et al. Construction of Golay complementary matrices and its applications to MIMO omnidirectional transmission[J]. IEEE Transactions on Signal Processing, 2021, 69: 2100–2113. doi: 10.1109/TSP.2021.3067467. [8] MEN Xinyu and LI Yubo. Omnidirectional precoding designs based on 2-D Golay complementary array pairs for massive MIMO systems with fully-connected and partially-connected structures[J]. IEEE Transactions on Communications, 2024, 72(5): 3019–3034. doi: 10.1109/TCOMM.2024.3351749. [9] PENG Xiuping, WANG Yu, and LIU Zilong. New constructions of 2-D Golay complementary array sets with highly flexible array sizes for massive MIMO omni-directional transmission[J]. IEEE Signal Processing Letters, 2024, 31: 3129–3133. doi: 10.1109/LSP.2024.3490403. [10] ZHAO Youqi, PAI Chengyu, HUANG Zhenming, et al. Two-dimensional Golay complementary array sets with arbitrary lengths for omnidirectional MIMO transmission[J]. IEEE Transactions on Communications, 2024, 72(2): 1135–1145. doi: 10.1109/TCOMM.2023.3326498. [11] DAS S, BUDISIN S, MAJHI S, et al. A multiplier-free generator for polyphase complete complementary codes[J]. IEEE Transactions on Signal Processing, 2018, 66(5): 1184–1196. doi: 10.1109/TSP.2017.2780050. [12] ARASU K T, DING Cunsheng, HELLESETH T, et al. Almost difference sets and their sequences with optimal autocorrelation[J]. IEEE Transactions on Information Theory, 2001, 47(7): 2934–2943. doi: 10.1109/18.959271. [13] ADHIKARY A R, FENG Yanghe, ZHOU Zhengchun, et al. Asymptotically optimal and near-optimal aperiodic quasi-complementary sequence sets based on Florentine rectangles[J]. IEEE Transactions on Communications, 2022, 70(3): 1475–1485. doi: 10.1109/TCOMM.2021.3132364. [14] SOLTANALIAN M and STOICA P. Computational design of sequences with good correlation properties[J]. IEEE Transactions on Signal Processing, 2012, 60(5): 2180–2193. doi: 10.1109/TSP.2012.2186134. [15] SOLTANALIAN M, NAGHSH M M, and STOICA P. A fast algorithm for designing complementary sets of sequences[J]. Signal Processing, 2013, 93(7): 2096–2102. doi: 10.1016/j.sigpro.2013.02.008. [16] STOICA P, HE Hao, and LI Jian. New algorithms for designing unimodular sequences with good correlation properties[J]. IEEE Transactions on Signal Processing, 2009, 57(4): 1415–1425. doi: 10.1109/TSP.2009.2012562. [17] SONG Junxiao, BABU P, and PALOMAR D P. Sequence set design with good correlation properties via majorization-minimization[J]. IEEE Transactions on Signal Processing, 2016, 64(11): 2866–2879. doi: 10.1109/TSP.2016.2535312. [18] GU Zhi, ADHIKARY A R, ZHOU Zhengchun, et al. A computational design of unimodular complementary and Z-complementary sets[C]. 2023 IEEE International Symposium on Information Theory (ISIT), Taipei, China, 2023: 1419–1424. doi: 10.1109/ISIT54713.2023.10206582. [19] RAMEZANI P, GIRNYK M A, and BJÖRNSON E. On broad-beam reflection for dual-polarized RIS-assisted MIMO systems[J]. IEEE Transactions on Wireless Communications, 2025, 24(2): 1264–1277. doi: 10.1109/TWC.2024.3507182. [20] BADEN J M, O'DONNELL B, and SCHMIEDER L. Multiobjective sequence design via gradient descent methods[J]. IEEE Transactions on Aerospace and Electronic Systems, 2018, 54(3): 1237–1252. doi: 10.1109/TAES.2017.2780538. [21] ALAEE-KERAHROODI M, MODARRES-HASHEMI M, and NAGHSH M M. Designing sets of binary sequences for MIMO radar systems[J]. IEEE Transactions on Signal Processing, 2019, 67(13): 3347–3360. doi: 10.1109/TSP.2019.2914878. [22] RAEI E, ALAEE-KERAHROODI M BABU P, et al. Generalized waveform design for sidelobe reduction in MIMO radar systems[J]. Signal Processing, 2023, 206: 108914. doi: 10.1016/J.SIGPRO.2022.108914. [23] ZEBARDAST H, FARHANG M, and SHEIKHI A. Minimum PSL sequence set design for MIMO radars via manifold-based optimization[J]. IEEE Sensors Journal, 2024, 24(13): 20981–20988. doi: 10.1109/JSEN.2024.3398780. -
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