This paper studies Flag sequences for low-complexity delay-Doppler estimation by exploiting their distinctive peak-curtain ambiguity functions (AFs). Unlike the existing Flag sequence designs that are limited to prime lengths and periodic auto-AFs, we aim to design Flag sequence sets of arbitrary lengths with low (nontrivial) periodic/aperiodic auto- and cross-AFs. Since every Flag sequence consists of a Curtain sequence and a Peak sequence, we first investigate the algebraic design of Curtain sequence sets of arbitrary lengths. Our proposed design gives rise to novel Curtain sequence sets with ideal curtain auto-AFs and zero/near-zero cross-AFs within the delay-Doppler zone of operation. Leveraging these Curtain sequence sets, two optimization problems are formulated to minimize the Weighted Integrated masked Sidelobe Level (WImSL) of the Flag sequence set. Accelerated Parallel Partially Majorization-Minimization Algorithms are proposed to jointly optimize the transmit Flag sequences and symmetric/asymmetric reference sequences stored in the receiver. Simulations demonstrate that our proposed Flag sequences lead to improved WImSL and peak-to-max-masked-sidelobe ratio compared with the existing Flag sequences. Additionally, our Flag sequences under the Flag method exhibit Mean Squared Errors that approach the Cram\'er-Rao Lower Bound and the Sampling Bound at high signal-to-noise power ratios.

翻译：本文研究利用Flag序列独特的峰值-帷幕模糊函数(AF)特性实现低复杂度时延多普勒估计。与现有仅限于素数长度和周期自模糊函数的Flag序列设计不同，本文旨在设计任意长度的Flag序列集，使其具有低（非平凡）周期/非周期自模糊函数与互模糊函数。由于每个Flag序列均由帷幕序列和峰值序列构成，我们首先研究任意长度帷幕序列集的代数设计方法。所提出的设计产生了新型帷幕序列集，其在时延多普勒工作区域内具有理想的帷幕自模糊函数和零/近零互模糊函数。基于这些帷幕序列集，本文构建了两个优化问题以最小化Flag序列集的加权积分掩蔽旁瓣电平。提出了加速并行部分主最小化算法，用于联合优化发射端Flag序列与接收端存储的对称/非对称参考序列。仿真结果表明，与现有Flag序列相比，本文提出的Flag序列在加权积分掩蔽旁瓣电平和峰值-最大掩蔽旁瓣比方面均有改善。此外，采用Flag方法的所提序列在高信噪比条件下，其均方误差逼近克拉美-罗下界与采样界。