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Comparative Analysis of Spread-Spectrum Codes for Fibre-Optic Distributed Acoustic Sensing

This paper numerically compares spread-spectrum codes for fibre-optic distributed acoustic sensing, demonstrating that high-diversity sequences significantly outperform low-diversity codes in phase noise resilience and estimation accuracy, particularly under conditions of increased laser linewidth.

Original authors: Maria Freire-Hermelo, Elie Awwad

Published 2026-08-10

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CC BY 4.0

Original authors: Maria Freire-Hermelo, Elie Awwad

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world's internet as a massive, invisible nervous system made of glass threads called fiber-optic cables. These cables carry our videos, messages, and data at the speed of light. But what if these same glass threads could also act as giant, super-sensitive ears? This is the magic of Distributed Acoustic Sensing (DAS). By sending laser light down a fiber, scientists can "listen" to vibrations along the entire length of the cable. If a truck drives over a road, a pipe leaks, or even if an earthquake rumbles nearby, the fiber shakes, and the laser detects it. This turns a simple data cable into a continuous sensor that can monitor everything from city infrastructure to earthquake zones.

However, there's a catch. To make these sensors super accurate, the laser light needs to be modulated with special patterns, or "codes," to distinguish the signal from the noise. Think of it like trying to hear a specific friend's voice in a crowded, noisy room. If you shout a simple "Hello," it gets lost. But if you shout a complex, unique song, your friend can pick it out easily. The problem is that lasers aren't perfect; they have a little bit of "jitter" or "phase noise," like a singer whose voice wavers slightly. This jitter can scramble the unique codes, making it hard to hear the signal clearly. The big question for engineers is: Which type of code song is the best at staying clear when the laser singer starts to wobble?

This paper dives into that question by acting like a musical judge for different types of coded sequences used in fiber-optic sensing. The researchers, Maria Freire-Hermelo and Élie Awwad, ran detailed computer simulations to test several famous "songs" (codes) to see which ones hold up best when the laser gets a bit shaky. They compared old favorites like m-sequences and Legendre sequences against more complex, modern options like Golay codes and CAZAC sequences (which include Zadoff-Chu and PS-MP types).

The study simulated a real-world scenario where the laser has a "linewidth" of 10 Hz (a measure of how much it jitters) and tested the codes over fiber lengths up to 20 kilometers. The results revealed a fascinating trade-off. While some codes are perfect at organizing the signal when everything is quiet, they fall apart when the laser starts jittering. The paper found that the secret weapon isn't just how "perfect" the code is, but how frequency diverse it is.

Imagine frequency diversity as how spread out a song's notes are. A code with "low diversity" is like a song that stays on just a few notes; if the laser jitters, those few notes get confused easily. A code with "high diversity" is like a song that sweeps across the entire piano keyboard, using many different frequencies at once. The simulations showed that these "wide-sweeping" codes are much tougher. For example, in a 1.5 km stretch of fiber with a 10 Hz laser linewidth, the high-diversity codes achieved a phase error of just 2.7 × 10⁻³ rad, while the low-diversity codes stumbled with an error of 5.3 × 10⁻³ rad. Even when the laser jitter increased to 100 Hz, the high-diversity codes still held their ground better than the others.

The paper suggests that while some codes (like Golay) are great for short distances, they can get confused by "echoes" if the fiber gets too long. Meanwhile, the CAZAC family of codes, particularly those with high frequency diversity, proved to be the most resilient against the laser's jitter, maintaining clearer signals over longer distances. The authors conclude that for the next generation of fiber-optic sensing—where we want to use the same cable for both internet data and sensing—the key is choosing codes that spread their energy widely across frequencies. This helps the system ignore the laser's wobbles and keep listening clearly, even when the conditions aren't perfect.

Technical Summary: Comparative Analysis of Spread-Spectrum Codes for Fibre-Optic Distributed Acoustic Sensing

Problem Statement
Distributed Acoustic Sensing (DAS) is a critical technology for continuous infrastructure monitoring, ranging from earthquake detection to optical network surveillance. While continuous-wave DAS systems utilizing spread-spectrum sequences offer advantages over pulsed techniques—such as reduced peak power, mitigated nonlinear crosstalk, and the potential for coexistence with data transmission (Optical-Fiber-based Integrated Sensing and Communication, or O-ISAC)—selecting the optimal coded sequence remains an open research problem. Previous studies have examined specific sequences like Legendre, m-sequences, and Golay codes, but comprehensive comparative analyses across different sequence types, particularly regarding their sensitivity to laser phase noise, are limited. Laser phase noise is a primary limitation in coherent $\Delta\phi$ -OTDR implementations, yet its specific impact on the performance of various spread-spectrum codes has not been fully characterized.

Methodology
The authors conducted a numerical simulation study to evaluate and compare the performance of five distinct coded sequences within a coherent DAS framework:

m-sequence: A maximum-length pseudorandom binary sequence (PRBS).
Legendre sequence: A sequence based on quadratic residues, defined for odd prime lengths.
Golay complementary sequences: Orthogonal pairs used for dual-polarization probing.
PS-MP CAZAC (Perfect-Square Minimum-Phase Constant Amplitude Zero Auto-Correlation): Complex-valued sequences with perfect autocorrelation.
Zadoff-Chu (ZC) CAZAC: Complex-valued sequences where frequency diversity is tunable via a root index $q$ .

The study utilized a simulation framework incorporating a continuous-wave laser modulated by these sequences, propagating through a Single-Mode Fiber (SMF) modeled with Rayleigh backscattering. The system included a dual-polarization coherent receiver sharing the transmitter laser. Key impairments modeled included Additive White Gaussian Noise (AWGN) and laser phase noise, the latter modeled as a Wiener process with varying linewidths (0 Hz, 10 Hz, and 100 Hz).

Performance was assessed using two primary metrics:

Relative determinant error: Measuring the accuracy of the estimated channel matrix determinant.
Phase error: Calculating the deviation in the estimated phase of the channel determinant.

Simulations were run over fiber lengths ranging from 1 km to 20 km, with specific attention paid to the trade-offs between sequence length, correlation properties, and frequency diversity.

Key Contributions and Results
The paper provides a detailed correlation analysis and system-level performance comparison, yielding the following findings:

Correlation and Frequency Diversity: In ideal conditions (no phase noise), CAZAC (PS-MP and ZC) and Golay sequences demonstrated perfect or near-perfect autocorrelation with zero sidelobes, whereas Legendre and m-sequences exhibited near-ideal but imperfect autocorrelation with visible sidelobes. Spectrogram analysis revealed that Golay, Legendre, and m-sequences possess high frequency diversity (broad energy spread), while PS-MP and ZC CAZAC sequences exhibit frequency-swept behavior. Specifically, the frequency diversity of ZC sequences is tunable via the root index $q$ , allowing them to span from minimal to maximal diversity.
Impact of Laser Phase Noise: The introduction of laser phase noise significantly degraded the autocorrelation properties of all sequences, but the degree of degradation was highly dependent on frequency diversity.
- Sequences with low frequency diversity (e.g., PS-MP CAZAC and ZC CAZAC with minimum $q$ ) suffered the most, developing high sidelobes and significant estimation errors even at moderate linewidths (10 Hz).
- Sequences with high frequency diversity (e.g., Golay, Legendre, m-sequence, and ZC CAZAC with high $q$ ) maintained lower sidelobes and better estimation accuracy under the same noise conditions.
Quantitative Performance:
- In a 1.5 km fiber scenario with a 10 Hz laser linewidth, high-diversity sequences achieved a phase error of $2.7 \times 10^{-3}$ rad, compared to $5.3 \times 10^{-3}$ rad for low-diversity codes.
- At a 100 Hz linewidth, this disparity widened, with high-diversity sequences achieving $8.5 \times 10^{-3}$ rad phase error versus $1.7 \times 10^{-2}$ rad for low-diversity sequences.
- Golay sequences performed well in AWGN-only scenarios but degraded sharply beyond their maximum alias-free range (approx. 8.4 km) due to non-zero autocorrelation zones, whereas PS-MP and ZC CAZAC sequences maintained near-ideal channel estimation up to 16.8 km.

Significance and Claims
The authors conclude that while perfect autocorrelation sequences (like CAZAC) are theoretically ideal for channel estimation, their practical utility in DAS is heavily constrained by laser phase noise unless they possess high frequency diversity. The study establishes that frequency diversity is a critical design parameter that often outweighs pure correlation quality in noisy environments.

The findings highlight a fundamental trade-off for next-generation O-ISAC systems: designers must balance correlation quality, phase noise resilience, and implementation constraints. The paper asserts that for systems operating with non-ideal lasers, selecting sequences with high frequency diversity is essential for maintaining accurate phase estimation and extending system reliability. The authors suggest that future work should focus on phase noise compensation techniques to further mitigate these limitations, rather than relying solely on sequence selection.

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