Björck Sequences: CAZAC & Function Spaces

• Björck sequences are constant amplitude zero autocorrelation sequences with unit-modulus entries and ideal autocorrelation and ambiguity properties.
• They serve as complete discrete invariants in Beurling–Björck spaces and are used as optimal reference signals in high-Doppler positioning and navigation systems.
• Their unique ambiguity function and low sidelobe levels improve delay–Doppler estimation and reduce positioning errors in challenging communication environments.

Björck sequences are a class of constant amplitude zero autocorrelation (CAZAC) sequences defined via explicit algebraic and analytic constructions, notable for their unit-modulus entries and ideal autocorrelation and ambiguity properties. They play a foundational role in two distinct domains: as complete discrete invariants in the functional analytic structure of Beurling–Björck spaces (a family of Gelfand–Shilov-type function spaces) and as optimal reference signal sequences in positioning, navigation, and timing (PNT) systems—particularly in environments with strong Doppler and interference such as low Earth orbit (LEO).

1. Mathematical Definition and Properties

For a prime length $P$, a Björck sequence $\{b_P(m)\}_{m=0}^{P-1}$ is defined by

$$b_{P}(m)=\exp\!\bigl(j\,\theta_{P}(m)\bigr),\quad m=0,1,\dots,P-1,$$

where the phase $\theta_P(m)$ is determined using the Legendre symbol and cases on $P \bmod 4$:

• $P\equiv1\pmod 4$:

$$\theta_{P}(m) = \left( \frac{m}{P} \right) \arccos \left( \frac{1}{1+\sqrt{P}} \right)$$

• $P\equiv3\pmod 4$:

$$\theta_{P}(m) = \begin{cases} \arccos \left( \frac{1-P}{1+P} \right), & \text{if } \left( \frac{m}{P} \right) = -1 \ 0, & \text{otherwise} \end{cases}$$

Björck sequences are CAZAC: their cyclic autocorrelation is exactly zero for all non-zero shifts: $$R[k]=\frac1P\sum_{m=0}^{P-1}b_P(m)\,b_P^*((m-k)\bmod P) = \begin{cases} 1, & k=0 \ 0, & k\neq0 \end{cases}$$ For non-prime $\{b_P(m)\}_{m=0}^{P-1}$0, novel constructions involve Goldbach splits ($\{b_P(m)\}_{m=0}^{P-1}$1, primes $\{b_P(m)\}_{m=0}^{P-1}$2), concatenating circulant blocks to maintain near-zero mean cross-correlation: for even $\{b_P(m)\}_{m=0}^{P-1}$3, $\{b_P(m)\}_{m=0}^{P-1}$4; for odd $\{b_P(m)\}_{m=0}^{P-1}$5, $\{b_P(m)\}_{m=0}^{P-1}$6 (Dureppagari et al., 31 May 2025).

2. Signal Processing Characteristics: Ambiguity, Doppler, and Interference

Björck sequences, when arranged in circulant matrices, produce $\{b_P(m)\}_{m=0}^{P-1}$7 orthogonal CAZAC sequences. The discrete ambiguity function

$\{b_P(m)\}_{m=0}^{P-1}$8

exhibits sidelobes $\{b_P(m)\}_{m=0}^{P-1}$9, smaller than those of Gold or Zadoff–Chu sequences, and does not suffer from Doppler-fold ambiguity. This property enables more accurate delay–Doppler estimation in high-Doppler environments. In LEO/NTN systems, this translates into improved resilience to interference and significantly lower positioning errors (Dureppagari et al., 31 May 2025).

A critical behavior arises: the Doppler shift can mimic a cyclic shift, leading to ambiguity between code index and Doppler. Two mitigation strategies address this:

• Coarse Doppler Pre-compensation: With an estimate $$b_{P}(m)=\exp\!\bigl(j\,\theta_{P}(m)\bigr),\quad m=0,1,\dots,P-1,$$0, pre-rotate the received signal and restrict the Doppler search window, resulting in a single ambiguity peak.
• Sequence Subset Selection: By ensuring sequence index spacing exceeds $$b_{P}(m)=\exp\!\bigl(j\,\theta_{P}(m)\bigr),\quad m=0,1,\dots,P-1,$$1, code-Doppler ambiguity is avoided for all sequences in use simultaneously.

3. Björck Sequences in Beurling–Björck and Gelfand–Shilov Spaces

Within functional analysis, Björck sequences arise as the complete discrete invariants in the sequence-space representations of the Beurling–Björck spaces $$b_{P}(m)=\exp\!\bigl(j\,\theta_{P}(m)\bigr),\quad m=0,1,\dots,P-1,$$2 and $$b_{P}(m)=\exp\!\bigl(j\,\theta_{P}(m)\bigr),\quad m=0,1,\dots,P-1,$$3, spaces of rapidly decaying smooth functions defined by weight functions $$b_{P}(m)=\exp\!\bigl(j\,\theta_{P}(m)\bigr),\quad m=0,1,\dots,P-1,$$4. In this context:

• A function $$b_{P}(m)=\exp\!\bigl(j\,\theta_{P}(m)\bigr),\quad m=0,1,\dots,P-1,$$5 in $$b_{P}(m)=\exp\!\bigl(j\,\theta_{P}(m)\bigr),\quad m=0,1,\dots,P-1,$$6 is fully described by sequences of inner products $$b_{P}(m)=\exp\!\bigl(j\,\theta_{P}(m)\bigr),\quad m=0,1,\dots,P-1,$$7 (Gabor expansion) or $$b_{P}(m)=\exp\!\bigl(j\,\theta_{P}(m)\bigr),\quad m=0,1,\dots,P-1,$$8 (Wilson expansion), both of which yield Björck-type CAZAC sequences that topologically classify the function (Debrouwere et al., 7 Aug 2025).
• These invariants allow isomorphic identification of the space with power-series-type sequence spaces $$b_{P}(m)=\exp\!\bigl(j\,\theta_{P}(m)\bigr),\quad m=0,1,\dots,P-1,$$9 or $\theta_P(m)$0.

Notably, the classical Gelfand–Shilov spaces $\theta_P(m)$1 (for indices $\theta_P(m)$2) correspond as $\theta_P(m)$3, and their invariant representation is $\theta_P(m)$4. This provides both analytic and explicit, constructive classifications for large classes of function spaces.

4. Sequence Construction for Arbitrary Lengths and Large Systems

While prime-length CAZAC properties are ideal, practical systems require Björck sequences of arbitrary $\theta_P(m)$5. The Goldbach split construction for even $\theta_P(m)$6 and analogous splits for odd $\theta_P(m)$7 minimize the mean cross-correlation of the resulting sets, permitting scalable application to large PNT and communications systems (Dureppagari et al., 31 May 2025).

For LEO mega-constellations, a spatial reuse framework tessellates the ground into hexagonal PNT-cells, each with an assigned set of unique sequence identifiers. Cell coloring with reuse factor $\theta_P(m)$8 ensures non-interfering resource allocation, with pairwise separation in the sequence space guaranteed by the underlying CAZAC features and Doppler-aware subset selection.

5. Quantitative Performance and Comparative Evaluation

Empirical comparisons under LEO PNT conditions yield a consistent performance advantage for Björck-based PRS schemes. In high-Doppler/strong-interference scenarios:

• Median time-of-arrival (TOA) error using Björck sequences is reduced by up to a factor of two compared to Gold codes, e.g., at 0 dB SNR: Gold (100 ns) vs. Björck (70 ns) for 15 kHz × 12 symbol PRS, and 50 ns for Björck at single-symbol 1.25 kHz SCS.
• Fifty-percentile position error is halved for Björck (80 m for 1 MHz, 30 m for 5 MHz BW) compared with Gold (120 m, 60 m) under worst-case interference geometry. Further reductions are seen with partial PRS overlap and larger bandwidths (Dureppagari et al., 31 May 2025).

These benefits are directly attributable to the lower ambiguity sidelobes and inherent resilience of the Björck construction.

Table: Summary of Median TOA and Position Errors

Configuration	Median TOA Error (0 dB SNR)	Median Position Error (1 MHz BW)
Gold (15kHz × 12 sym)	100 ns	120 m
Björck (15kHz × 12)	70 ns	80 m
Björck (1.25kHz × 1)	50 ns	30 m

6. Role in Modern Harmonic Analysis and Signal Design

The dual occurrence of Björck sequences in both abstract functional analysis (sequence space invariants for ultradifferentiable function spaces) and concrete engineering (robust CAZAC signal design for Doppler-rich wireless environments) highlights their centrality in problems demanding both idealized mathematical invariance and practical implementation feasibility. In the analytic setting, they provide complete discrete invariants classifying Gelfand–Shilov and Beurling–Björck spaces up to isomorphism, enabling explicit constructive bases and sequence space representations (Debrouwere et al., 7 Aug 2025). In applications, their Doppler/ambiguity behavior and cross-correlation structure permit robust, scalable time–frequency referencing in modern communication and navigation systems.

7. Classification Theorems and Isomorphic Structures

The explicit sequence space isomorphisms yield precise criteria for equivalence of Beurling–Björck (or Gelfand–Shilov) spaces: $\theta_P(m)$9 and hence reduces the analytic classification of such spaces to the asymptotic behavior of their Björck sequence parameters (Debrouwere et al., 7 Aug 2025). For the special case of Gelfand–Shilov spaces: $P \bmod 4$0 with critical indices $P \bmod 4$1, unifying smoothness and decay properties through discrete CAZAC invariants.

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Björck sequences thus form a mathematically rigorous, algorithmically explicit, and empirically validated bridge between the theory of ultradifferentiable function spaces and the design of signals for robust time–frequency localization under physical constraints prevalent in advanced communication and navigation systems (Dureppagari et al., 31 May 2025, Debrouwere et al., 7 Aug 2025).