7.5. Zadoff–Chu Sequences#
If we allowed complex-valued symbols, then Zadoff-Chu sequences are popular choices for use as signature sequences.
An \(N\)-length root Zadoff-Chu sequence parameterized by \(M\) is defined by
\[\begin{equation*} x_M[n] = \begin{cases} \exp\left( -j \frac{M \pi n^2}{N} \right) & \text{ if } N \text{ is even} \\ \exp\left( -j \frac{M \pi n(n+1)}{N} \right)& \text{ if } N \text{ is odd} \end{cases} \end{equation*}\]where \(M\) and \(N\) are relative primes.
Clearly, different phases \(Tx_M, \ldots, T^{N-1}x_M\) are also Zadoff-Chu sequences and have the same periodic auto-correlation function as the root Zadoff-Chu sequence \(x_M\).
Properties of Zadoff–Chu sequences:
\(|x_M[n]| = 1\) for all \(n\).
\(\theta_{x_M}[k] = \begin{cases} N & \text{ if } k=0 \text{ mod } N\\ 0 & \text{ if } k\neq 0 \text{ mod } N. \end{cases}\)
Suppose that \(N\) is odd, \(M_1-M_2\) is relatively prime to \(N\), and that \(x_{M_1}\) and \(x_{M_2}\) are two \(N\)-length Zadoff-Chu sequences. Then \(\left|\theta_{x_{M_1}, x_{M_2}}[k]\right| = \sqrt{N}\).
Notice that Property 2 says that any Zadoff-Chu sequence has the ideal auto-correlation function.
From Property 8 of the cross-correlation function, given that the auto-correlation functions of any pair of sequences satisfy Property 2 above, the lower bound on the peak magnitude of the cross-correlation function between the two sequences is \(\sqrt{N}\). Hence, Property 3 above states that the two Zadoff-Chu sequences also give the best cross-correlation function performance.