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

\begin{align*} x_u(n) = e^{-j\frac{\pi u n (n+1+2q)}{L}}, \quad n = 0, 1, \ldots, L-1, \end{align*}

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)