Flag Sequence Set Design for Low-Complexity Delay-Doppler Estimation

TL;DR

The paper tackles low-complexity delay-Doppler estimation by designing Flag sequences with peak-curtain AAFs for arbitrary lengths, overcoming prime-length limits of prior work. It introduces Curtain sequence sets built from discrete chirps to achieve ideal curtain AAFs and zero/near-zero CAFs within the operation zone, enabling Flag sequence sets of any length. Two AWImSL-based optimization problems (asymmetric and symmetric reference designs) are solved via accelerated parallel majorization-minimization (AP-MM) algorithms, achieving significantly lower sidelobes and better peak isolation than the traditional HWS baseline. Numerical results show NWImSL, PMmSR, and estimation performance approaching CRLB/SB at high SNR, with asymmetric designs offering the best trade-off between design flexibility and detection accuracy, suggesting strong practical impact for radar, communications, and ISAC systems.

Abstract

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 Cramér-Rao lower bound and the sampling bound at high signal-to-noise power ratios.

Figures (7)

• Figure 1: (a) Ideal auto-AF (AAF) of Flag sequences and illustration of the Flag method in 2-step search; (b) Periodic AAF of a Heisenberg-Weil sequence (HWS) with a length of 37; (c) Periodic AAF of the Heisenberg sequence component (curtain) of the HWS shown in Fig. \ref{['AF_ideal_HWS']}(b); (d) Periodic AAF of the Weil sequence component (peak) of the HWS shown in Fig. \ref{['AF_ideal_HWS']}(b).
• Figure 2: Periodic AFs of the proposed Flag sequence set (asymmetric transmit sequences and receive reference sequences); (a) $\bm A_{\bm f_{1}^{s},\bm f_{1}^{r}}$; (b) $\bm A_{\bm f_{1}^{s},\bm f_{2}^{r}}$; (c) $\bm A_{\bm f_{1}^{s},\bm f_{3}^{r}}$; (d) $\bm A_{\bm f_{2}^{s},\bm f_{1}^{r}}$; (e) $\bm A_{\bm f_{2}^{s},\bm f_{2}^{r}}$; (f) $\bm A_{\bm f_{2}^{s},\bm f_{3}^{r}}$; (g) $\bm A_{\bm f_{3}^{s},\bm f_{1}^{r}}$; (h) $\bm A_{\bm f_{3}^{s},\bm f_{2}^{r}}$; (i) $\bm A_{\bm f_{3}^{s},\bm f_{3}^{r}}$.
• Figure 3: Periodic AAFs associated with 2 radar targets; (a) The HWS (with diagonal line, diagonal torus and $31$ as generator Fish2013); (b) The proposed Flag sequence (symmetric transmit sequence and receive reference sequence: $\bm f^{s} =\bm f^{r} =[0.0279 + 0.0215i; 0.0421 - 0.0096i\cdots]$).
• Figure 4: Evolution curves of the normalized Weighted Integrated masked Sidelobe Level (NWImSL) of our proposed algorithms.
• Figure 5: Receiver operating characteristic (ROC) curves of the proposed Flag sequences and the HWS Fish2013.

Theorems & Definitions (10)

• Theorem 1: Periodic AAF
• Remark 1
• Theorem 2: Aperiodic AAF
• Corollary 1: Low Periodic CAF
• Corollary 2: Zero Periodic CAF
• Corollary 3: Low Aperiodic CAF
• Corollary 4: Zero Aperiodic CAF
• Proposition 1
• Proposition 2
• Lemma 1