[Paper Review] A Primer on Zadoff Chu Sequences

Jeffrey G. Andrews|arXiv (Cornell University)|Nov 10, 2022

Wireless Communication Networks Research4 citations

TL;DR

This paper provides a comprehensive overview of Zadoff-Chu (ZC) sequences, explaining their mathematical definition, key properties such as constant amplitude and zero cyclic autocorrelation, and their critical role in modern wireless systems like LTE and 5G NR. It demonstrates how ZC sequences enable robust initial access, synchronization, and channel estimation through optimal correlation properties and efficient generation via frequency-domain operations.

ABSTRACT

Zadoff-Chu (ZC) sequences are an important manifestation of spread spectrum in modern cellular systems, including LTE and 5G NR. They have to some extent displaced PN and Walsh sequences which were the mainstays of 3G cellular (WCDMA and cdma2000) and the 2G-era IS-95. ZC sequences are complex sequences with unit amplitude and particular phase shifts, as opposed to Walsh and PN codes which are real and binary valued, most commonly $\pm1$. ZC sequences have a number of remarkable and desirable properties that we define in the next section. Because of these properties, they are used for many key functions in current cellular systems, and are likely to be prevalent in future cellular systems as well. In LTE and 5G NR, they are widely used for a number of important initial access and overhead channel functions that are often overlooked by engineers who focus on data transmission. For example, ZC sequences are used for initial access in both the downlink (synchronization sequences) and uplink (random access premables). They are also used for transmitting uplink control information, and as pilot symbols for both uplink channel sounding and fine-grained channel estimation. It is not an exaggeration to say that most types of signals other than the data transmissions in modern cellular standards utilize ZC sequences. In this primer, we define ZC sequences and introduce their key properties, and provide some examples. We also discuss modified ZC sequences that are commonly used in practice, but are not, strictly speaking, ZC sequences. We also overview their uses in LTE and 5G.

Motivation & Objective

• To provide a foundational understanding of Zadoff-Chu (ZC) sequences for researchers and engineers working in wireless communications.
• To explain the mathematical structure and defining parameters of ZC sequences, including root index and sequence length.
• To highlight the critical role of ZC sequences in modern cellular systems such as LTE and 5G NR, especially in initial access and channel estimation.
• To clarify the conditions under which ZC sequences achieve optimal correlation properties, particularly when sequence length is prime.
• To demonstrate the practical utility of ZC sequences in real-world systems through specific use cases in synchronization, random access, and reference signal design.

Proposed method

• Defining ZC sequences using the formula $ s_q[n] = ext{exp}ig[-jrac{ au au}{ ext{mod}} ig] $, where $ q $ is the root index and $ ext{N}_{ ext{zc}} $ is the sequence length.
• Analyzing cyclic shifts and normalized cyclic autocorrelation to show that ZC sequences achieve zero sidelobes for all non-zero time shifts.
• Deriving the normalized cyclic cross-correlation between distinct ZC sequences as $ 1/ ext{N}_{ ext{zc}} $, which is optimal under the condition that $ |q_1 - q_2| $ is coprime to $ ext{N}_{ ext{zc}} $.
• Demonstrating that the DFT and IDFT of a ZC sequence yield another ZC sequence, enabling efficient frequency-domain generation.
• Applying these properties to real-world systems by mapping ZC sequences to specific physical layer functions in LTE and 5G NR.
• Validating the robustness of selected ZC sequences (e.g., $ ext{N}_{ ext{zc}} = 63 $, $ q = 29, 34, 25 $) under frequency and timing offsets, showing practical resilience.

Experimental results

Research questions

• RQ1 How do Zadoff-Chu sequences achieve optimal correlation properties, and under what conditions are these properties preserved?
• RQ2 Why are ZC sequences preferred over traditional PN and Walsh codes in modern cellular systems like LTE and 5G NR?
• RQ3 What is the role of ZC sequences in initial access procedures such as random access and synchronization in both downlink and uplink?
• RQ4 How does the DFT/IDFT property of ZC sequences enable efficient implementation in OFDMA and SC-FDMA waveforms?
• RQ5 Why is the sequence length $ ext{N}_{ ext{zc}} $ typically chosen as a prime number, and what happens when it is not?

Key findings

• Zadoff-Chu sequences are constant amplitude, zero autocorrelation (CAZAC) sequences, with all elements having unit magnitude and varying phase, resulting in a peak-to-average power ratio (PAPR) of 1.
• The normalized cyclic autocorrelation of a ZC sequence is $ ar{R}_{xx}[ au] = rac{1}{ ext{N}_{ ext{zc}}} R_{xx}[ au] $, which equals $ ext{N}_{ ext{zc}} imes ext{dirac}[ au] $, yielding zero correlation for all non-zero time shifts.
• For distinct ZC sequences of the same length $ ext{N}_{ ext{zc}} $, the normalized cyclic cross-correlation is exactly $ 1/ ext{N}_{ ext{zc}} $, which is optimal and achieved when $ ext{N}_{ ext{zc}} $ is prime and $ |q_1 - q_2| $ is coprime to $ ext{N}_{ ext{zc}} $.
• The DFT and IDFT of a ZC sequence produce another ZC sequence, enabling direct generation in the frequency domain, which is especially beneficial for OFDMA and SC-FDMA systems.
• In LTE, the primary synchronization sequence (PSS) uses $ ext{N}_{ ext{zc}} = 63 $ with $ q = 29, 34, 25 $, chosen for robustness under frequency and timing offsets, and $ q = 29 $ and $ q = 34 $ are complex conjugates, enabling single-correlator detection.
• In 5G NR, ZC sequences are used in PRACH (with $ ext{N}_{ ext{zc}} = 839 $ and $ 139 $), PUCCH (formats 0,1 with $ ext{N}_{ ext{zc}} = 11 $, extended to 12), and SRS/DM-RS (with $ ext{N}_{ ext{zc}} = 31 $, cyclically extended to 36 or used directly for DM-RS).