Zadoff–Chu sequence

A Zadoff–Chu sequence is a complex-valued mathematical sequence which, when applied to radio signals, gives rise to an electromagnetic signal of constant amplitude, whereby cyclicly shifted versions of the sequence comprising the signal do not cross-correlate with each other when the signal is recovered at the receiver. A generated Zadoff–Chu sequence that has not been shifted is known as a "root sequence".

The sequence then exhibits the useful property that cyclic-shifted versions of itself remain orthogonal to one another, provided, that is, that each cyclic shift, when viewed within the time domain of the signal, is greater than the combined propagation delay and multi-path delay-spread of that signal between the transmitter and receiver.

The complex value at each position (n) of each root Zadoff–Chu sequence ("u") given by

: $x\_u(n)=e^\{-jfrac\{pi\; un(n+1)\}\{N\_\; ext\{ZC\},\; ,$

where

: $0\; le\; n\; le\; N\_\; ext\{ZC\}-1,\; ,$

: $N\_\; ext\{ZC\}\; =\; ext\{length\; of\; sequence\}.,$

References