⚡ electrical engineering

Björck Sequences: Extension to Arbitrary Lengths, Correlation Analysis, and Applications to Wireless Systems

This paper proposes a Goldbach's conjecture-based framework to extend prime-length Björck CAZAC sequences to arbitrary lengths, demonstrating their effectiveness as robust reference signals for high-Doppler wireless environments while offering mitigation strategies for Doppler-induced misidentification.

Original authors: Harish K. Dureppagari, Chiranjib Saha, R. Michael Buehrer, Harpreet S. Dhillon

Published 2026-05-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Harish K. Dureppagari, Chiranjib Saha, R. Michael Buehrer, Harpreet S. Dhillon

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 you are trying to organize a massive orchestra where every musician needs a unique sheet of music. In the world of wireless communication (like your phone connecting to a cell tower or a satellite), these "sheet music" pieces are called sequences. They help devices find each other, sync up, and talk without getting confused by noise or other people talking at the same time.

For a long time, the best musicians (sequences) were only available in specific sizes—specifically, sizes that are "prime numbers" (like 11, 13, 17). But real-world wireless systems (like 5G and the upcoming 6G) need music sheets of any size (like 120 or 1000) to fit their specific hardware.

Here is the problem: If you try to force a 13-note song to fit a 120-note slot by just repeating the 13 notes over and over, the music gets messy. The notes start clashing, and the orchestra can't tell who is playing what. This causes interference and errors.

This paper introduces a clever new way to build these musical sequences for any size without the mess, and it tests a specific, high-performance type of music called Björck sequences to see if they are better than the current standard.

1. The "Goldbach" Construction: Building Any Size

The authors propose a new construction method based on a famous math idea called Goldbach's Conjecture.

The Analogy: Imagine you need to build a wall of exactly 120 bricks. You don't have a mold for 120. But you know that 120 can be made by adding two prime numbers together (e.g., 113 + 7).
The Method: Instead of just repeating a small pattern, the authors suggest taking a perfect 113-brick block and attaching a perfect 7-brick block to the end.
The Magic: By carefully arranging these blocks, they can create a 120-note sequence that keeps the "perfect harmony" (orthogonality) of the original prime numbers. Even better, they can mix and match these blocks to create many different versions of the music that don't interfere with each other, unlike the old "repetition" method which caused chaos.

2. The Star Performer: Björck Sequences

The paper focuses on a specific type of sequence called Björck sequences.

The Current Star (ZC Sequences): The industry currently uses "Zadoff-Chu" (ZC) sequences. They are great for normal ground-based towers (Terrestrial Networks) where the signal doesn't move much.
The New Contender (Björck): The authors argue that Björck sequences are the "Olympic athletes" of the wireless world. They have a special property called an Ambiguity Function.
- The Analogy: Imagine trying to hear a friend's voice in a noisy room.
  - ZC Sequences: If your friend is standing still, you hear them clearly. But if they start running fast (high speed/Doppler shift), their voice gets distorted, and you might think it's someone else or miss them entirely.
  - Björck Sequences: These are like a voice that stays crystal clear even if your friend is sprinting past you. They are much more robust against the "Doppler effect" (the change in pitch caused by high speed).

3. Testing the Waters: Ground vs. Sky

The authors tested these sequences in two scenarios:

Ground (Terrestrial Networks): Like a high-speed train. Here, the speed is fast, but not too fast. The results showed that Björck sequences perform just as well as the current ZC standard.
Sky (Non-Terrestrial Networks/LEO Satellites): This is the real test. Satellites orbit Earth at incredible speeds, creating massive Doppler shifts.
- The Result: In this high-speed environment, the old ZC sequences started to stumble, creating "ghost peaks" (false alarms) that confused the receiver. The Björck sequences, however, kept their cool, providing accurate timing and speed estimates even when the satellites were zooming by.

4. The Catch: The "Doppler Mix-Up"

There is one tricky behavior the authors found with Björck sequences.

The Problem: Because of how these sequences are mathematically built, a high-speed Doppler shift can sometimes make the receiver think a specific sequence is actually a different sequence. It's like a fast-moving car looking so much like a different car that you mistake them.
The Solution: The paper offers two ways to fix this:
1. The "Rough Guess" Strategy: Before trying to identify the exact sequence, the system makes a "rough guess" at the speed (Doppler) and corrects for it. Once the speed is roughly fixed, the sequence becomes easy to identify.
2. The "Social Distancing" Strategy: When assigning sequences to different satellites, the system ensures they are far enough apart in the "sequence family" so that even if the speed shifts, they never overlap or get confused.

Summary

This paper provides a mathematical "Lego kit" that allows engineers to build perfect wireless sequences of any length, not just prime numbers. It then proves that Björck sequences are a superior choice for the future of wireless communication, especially for satellite internet where devices are moving at thousands of miles per hour. While they have a slight quirk when moving very fast, the authors provide practical fixes to ensure they work reliably. This could mean faster, more reliable connections for 6G and satellite networks in the future.

Technical Summary: Björck Sequences: Extension to Arbitrary Lengths, Correlation Analysis, and Applications to Wireless Systems

Problem Statement
Constant Amplitude Zero Autocorrelation (CAZAC) sequences, such as Zadoff–Chu (ZC) sequences, are fundamental to modern wireless systems (e.g., LTE, 5G NR) for synchronization and random access. However, these sequences are inherently defined for prime lengths. Practical systems, particularly Orthogonal Frequency Division Multiplexing (OFDM) based networks, require sequence lengths that are typically non-prime (e.g., multiples of 12). Current industry practice adapts prime-length sequences to arbitrary lengths using repetition-based extensions. This approach disrupts the underlying sequence structure, leading to a loss of orthogonality among cyclically shifted versions and increased zero-lag cross-correlation (inner product) between sequences with different root indices. Furthermore, existing analytical constructions for CAZAC sequences are restricted to specific length classes, and numerical extensions often lack structural guarantees. These limitations are exacerbated in Non-Terrestrial Networks (NTNs), particularly Low Earth Orbit (LEO) systems, where large propagation delays and high Doppler shifts degrade the correlation properties of standard ZC sequences, causing spurious peaks and estimation errors.

Methodology
The authors propose a generic sequence construction framework to extend prime-length CAZAC sequences to arbitrary lengths while preserving key correlation properties. The core of this methodology relies on Goldbach's conjecture:

Even Lengths ( $N$ ): The sequence length $N$ is expressed as the sum of two primes, $N = Q_1 + Q_2$ . The construction involves appending a prime-length sequence of length $Q_2$ to a prime-length sequence of length $Q_1$ .
Odd Lengths ( $N$ ): The length is expressed as the sum of three primes, $N = Q_1 + Q_2 + Q_3$ , following a similar concatenation logic.

The framework supports two primary extension modes:

Cyclically Shifted Versions: Extending a base sequence and its cyclic shifts.
Different Root Indices: Extending sequences generated from different root indices (where applicable, such as in ZC sequences; for Björck sequences, distinct sequences are generated via cyclic shifts).

The paper provides a rigorous analytical characterization of the resulting correlation behavior. It derives expressions for the inner product (zero-lag cross-correlation) and the Root Mean Square (RMS) of both periodic and aperiodic cross-correlations. The analysis quantifies the trade-off between the total number of available sequences and the number of mutually orthogonal sequences based on the choice of prime pairs $(Q_1, Q_2)$ .

Key Contributions

Generic Construction Framework: The paper introduces a unified method to extend prime-length CAZAC sequences to arbitrary lengths using Goldbach's conjecture. This framework preserves orthogonality among a subset of cyclic shifts and maintains favorable zero-lag cross-correlation properties across different root indices.
Correlation Analysis: The authors analytically characterize the inner-product behavior, demonstrating that:
- For cyclic shifts, perfect orthogonality can be preserved if the appended segments do not overlap, limiting the number of orthogonal sequences to $\min\{Q_1, Q_2\}$ .
- For different root indices, the normalized inner product scales as $O(1/\sqrt{N})$ , similar to prime-length sequences, regardless of the specific prime decomposition.
- The RMS of periodic and aperiodic cross-correlations scales favorably, preserving the $1/\sqrt{N}$ behavior typical of CAZAC sequences.
Application to Björck Sequences: The framework is applied specifically to Björck sequences, a class of CAZAC sequences known for favorable ambiguity function properties. The study evaluates Björck sequences as reference signal candidates for both Terrestrial Networks (TNs) and Non-Terrestrial Networks (NTNs).
Doppler-Dependent Behavior and Mitigation: The authors identify an inherent Doppler-dependent behavior in Björck sequences where large Doppler shifts can cause sequence misidentification (ambiguity between cyclic shifts). Two mitigation strategies are proposed:
- Leveraging a coarse Doppler estimate prior to detection to narrow the search space.
- Selecting appropriately spaced subsets of orthogonal sequences to ensure frequency separation exceeds the maximum Doppler uncertainty.

Results

Interference Resilience: In terrestrial scenarios with overlapping signals, the proposed construction significantly outperforms conventional repetition-based extensions. Specifically, configurations with larger sets of mutually orthogonal sequences (e.g., using prime pairs like 101 and 19 for $N=120$ ) achieve higher probability of detection and lower time-offset estimation errors under high interference (low SINR).
TN vs. NTN Performance:
- In low-Doppler TN environments, Björck sequences exhibit performance comparable to ZC sequences for time and frequency offset estimation.
- In high-Doppler NTN environments (e.g., LEO satellites with Doppler shifts up to 45 kHz), Björck sequences demonstrate superior robustness compared to ZC sequences. They maintain consistent estimation accuracy and exhibit sharper correlation peaks with reduced sidelobes, whereas ZC sequences suffer from distorted ambiguity functions and spurious peaks.
Trade-offs: The study confirms a fundamental trade-off: maximizing the number of available sequences often requires accepting some overlap in appended segments, which increases the inner product. Conversely, enforcing strict non-overlapping conditions preserves perfect orthogonality but limits the total number of usable sequences.

Significance
The paper establishes Björck sequences as a strong alternative to ZC sequences for reference signal design in next-generation wireless systems, particularly for high-mobility and NTN scenarios. By providing a generic framework to extend these sequences to arbitrary lengths, the work addresses a critical gap in practical deployment where prime-length constraints are incompatible with OFDM subcarrier allocations. The identification of Doppler-dependent misidentification and the proposal of practical mitigation strategies further enhance the viability of Björck sequences in high-Doppler environments. Overall, the work enables the practical design of arbitrary-length CAZAC sequences with controlled correlation properties, offering a potential replacement for ZC sequences in future 6G and NTN standards.

1. The "Goldbach" Construction: Building Any Size

2. The Star Performer: Björck Sequences

3. Testing the Waters: Ground vs. Sky

4. The Catch: The "Doppler Mix-Up"

Summary

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