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^{-jfrac{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)