Logo der Uni Stuttgart
Zadoff-Chu Sequences

Zadoff–Chu sequences have more special properties than CAZAC sequences. In addition to have a constant amplitude, for a fixed q, Zadoff–Chu sequences have zero periodic autocorrelation for all time shifts other than zero. The sequences are not orthogonal in case of the different values of $q$, however the cross-correlation between them is low. It is known that there are $N_{ZC}-1$ different sequences with periodic cross-correlation of $\frac{1}{\sqrt{N_{ZC}}}$ when the sequence length $N_{ZC}$ is selected as a prime number $(N_{ZC}\neq 1)$.

Zadoff–Chu sequences are used in the 3GPP LTE in the primary synchronization signal, random access preamble, uplink control channel, uplink traffic channel and sounding reference signals.

The formula of a Zadoff-Chu sequence (used in PUCCH (Physical uplink control channel) ) of length $N_{ZC}$ is:

$$ x_{q}(m)=\begin{Bmatrix} e^{-j\frac{\prod qm^{2}}{N_{ZC}}}, & if & N_{ZC} & is & even & 0\leq m\leq N_{ZC}-1\\ e^{-j\frac{\prod qm(m+1)}{N_{ZC}}}, & if & N_{ZC} & is & odd & 0\leq m\leq N_{ZC}-1\\ \end{Bmatrix} $$

The formula of a Zadoff-Chu sequence (used in PSS (Primary Synchronization Signal) ) of length $N_{ZC}$ is:

$$ x_{q}(m)=\begin{Bmatrix} e^{-j\frac{\prod qm(m+1)}{N_{ZC}}}, & 0\leq m\leq (N_{ZC}/2)-1\\ e^{-j\frac{\prod q(m+1)(m+2)}{N_{ZC}}}, & (N_{ZC}/2)-1\leq m\leq N_{ZC}-1\\ \end{Bmatrix} $$

where $q$ is the sequence index, which is relatively prime to $N_{ZC}$.

$N_{ZC}$ is equal to 63 for both of the sequences and for PSS q has the values of 25, 29 and 34 in implementation.