Asymptotically Optimal Aperiodic Doppler Resilient Complementary Sequence Sets Via Generalized Quasi-Florentine Rectangles

Doppler-resilient complementary sequence (DRCS) sets play a vital role in modern communication and sensing systems, particularly in high-mobility environments. This work makes two primary contributions. First, we refine the definition of quasi-Florentine rectangles to a more general form,termed generalized quasi-Florentine rectangles, and propose a systematic method for their construction. Second, we propose several sets of aperiodic DRCS based on generalized quasi Florentine rectangles and Butson-type Hadamard matrices. The proposed aperiodic DRCS sets are shown to be asymptotically optimal with respect to the lower bound of aperiodic DRCS sets.

💡 Research Summary

The paper addresses the design of Doppler‑resilient complementary sequence (DRCS) sets for high‑mobility communication and sensing systems, focusing on the aperiodic case where both delay and Doppler shifts must be mitigated. Traditional complementary structures—Golay complementary pairs, complementary sequence sets (CS), and mutually orthogonal complementary sequence sets (MOCSS)—provide zero aperiodic autocorrelation but are limited by the “flock size” constraint, which caps the number of users that can be simultaneously supported. To overcome this limitation, recent work introduced quasi‑complementary sequence sets (QCSS) and Doppler‑resilient sequences (DRS), culminating in the concept of DRCS, where distinct sequences are transmitted on each pulse and their ambiguity functions (AF) are coherently summed at the receiver. This approach shifts performance evaluation from pure correlation metrics to the ambiguity function, which captures both delay and Doppler characteristics.

Existing constructions for aperiodic DRCS rely on quasi‑Florentine rectangles and Butson‑type Hadamard matrices. A quasi‑Florentine rectangle is a combinatorial object similar to a Florentine rectangle but missing exactly one element per row. While this structure yields DRCS sets that meet the lower bound on the maximum aperiodic AF for many sequence lengths N, it suffers from a severe row‑count limitation: for certain N the number of rows (the flock size) is only four, preventing optimal DRCS construction.

The authors’ primary contribution is the introduction of Generalized Quasi‑Florentine Rectangles (GQFR), which relax the “one‑missing‑element” restriction. In a GQFR each row contains n distinct symbols (2 ≤ n ≤ N) drawn from Z_N, and for any ordered pair of distinct symbols (a, b) and any step size 1 ≤ m < n, there exists at least one row where b appears exactly m positions to the right of a. When the step is considered modulo N, the rectangle becomes a generalized circular quasi‑Florentine rectangle. Special cases recover known structures: n = N yields a Florentine rectangle, n = N‑1 yields the classic quasi‑Florentine rectangle, and intermediate values of n produce new rectangles with many more rows while using fewer symbols per row.

Two constructive lemmas are provided:

1. From Florentine rectangles: By deleting the leftmost or rightmost c columns (0 ≤ c < N‑1) of an F(N) × N Florentine rectangle, one obtains a GQFR of size F(N) × (N‑c) with n = N‑c.
2. From quasi‑Florentine rectangles: By deleting the leftmost or rightmost (c‑1) columns (1 ≤ c < N‑1) of an F_Q,1(N) × (N‑1) quasi‑Florentine rectangle, one obtains a GQFR of size F_Q,1(N) × (N‑c) with n = N‑c.

Thus the number of rows (the flock size) can be preserved while the number of symbols per row is reduced, giving designers a new degree of freedom.

The second major ingredient is the use of Butson‑type Hadamard matrices B H(L, r), whose entries are r‑th roots of unity and satisfy B H·B H^H = L I. Seed matrices for small alphabet sizes (r ≤ 7) are listed, and Lemma 2 shows that Kronecker products of two such matrices produce a larger B H with order L₁L₂ and alphabet size lcm(r₁, r₂). This property enables the construction of large, structured matrices with controllable alphabet size, which is crucial for practical hardware implementation.

The DRCS set C consists of K blocks, each block containing M ≥ 2 sequences of length N. The aperiodic cross‑AF between two blocks is defined as the sum of the M individual cross‑AFs. The performance metric is the maximum magnitude of the aperiodic AF over a rectangular region Π = (−Z_x, Z_x) × (−Z_y, Z_y). Lemma 1 (from prior work) provides a lower bound:

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