ZC

The Zadoff-Chu (ZC) sequence¹ is named after Solomon A. Zadoff and D. C. Chu, which is a constant amplitude zero auto-correlation (CAZAC) sequence² with its cylically shifted versions are zero correlated, i.e. orthogonal to each other. A ZC sequence without cyclic shift is termed a root sequence, which can be expressed as

where $u$ is the index of the root sequence, $L$ is the length and $q\in \mathbb{Z}$.

• The auto-correlation of a ZC sequence is delta function, i.e. zero correlated.
• ZC sequences with odd length are periodic, i.e. $x_u(n+L)=x_u(n)$.
• For a ZC sequence with odd length, its DFT is another ZC sequence.
• The cross-correlation between two prime length ZC sequences, $x_{u_1}(n)$ and $x_{u_2}(n)$, is constant $\sqrt{L}$, provided that $u_1-u_2$ is relative prime to $L$.

ZC sequences are widely used in 3GPP LTE/LTE-Advanced system³

• Primary synchronization signal (PSS)
• Random access preamble
• Uplink demodulation reference signal (DMRS)
• Sounding reference signal (SRS)