ZadoffChu

Zadoff-Chu sequences, often abbreviated as ZC sequences, are a class of complex-valued sequences used in wireless communications for their favorable correlation properties. They are known for constant amplitude and zero autocorrelation (CAZAC), which makes them particularly suitable for timing and channel estimation tasks in noisy and multipath environments.

Mathematically, a Zadoff-Chu sequence of length N is defined for a root index q with gcd(q, N) = 1. If N is even, the sequence x[n] is x[n] = exp(-j π q n(n+1) / N) for n = 0, 1, ..., N−1. If N is odd, the sequence is x[n] = exp(-j π q n^2 / N) for the same range. In all cases, the magnitude |x[n]| is 1 for every element.

Key properties include constant magnitude and the CAZAC characteristic: the cyclic autocorrelation of a ZC sequence is zero for all non-zero cyclic shifts, which helps in precise timing synchronization and cell search. The sequences also exhibit favorable cross-correlation properties across different root indices, aiding multi-user multiplexing in communication systems.

Applications and usage: Zadoff-Chu sequences are widely used in cellular standards for synchronization signals and random access preambles. In LTE, the primary synchronization signal (PSS) uses ZC sequences of length 63 with three possible roots (commonly 25, 29, and 34), enabling robust cell search. They are also employed in various PRACH preambles and related signaling tasks. The concept originated from work by Solomon Zadoff and Ho-Ling Chu in the mid-20th century and remains a foundational tool in modern wireless design.