ZC sequences and application in LTE

Zadoff-chu sequence is a polyphase sequence which is widely used in LTE for Primary synchronization signal,PRACH, PUCCH DMRS,PUSCH DMRS and sounding reference signals(SRS). This is because ZC sequence has the following properties.

  1. The Auto correlation of a prime length ZC sequence with a cyclic shifted version of itself has a zero auto correlation. It means that the auto correlation is nonzero only at one instant which corresponds to the cyclic shift. This also means that two generated sequences are orthogonal to each other. In communication systems use of orthogonal sequence is wide spread and using this property of ZC sequence, orthogonal sequences can be easily generated; just by cyclically shifting a ZC sequence.
  2. Another important property of ZC sequence is its circular crosscorrelation property. It can be stated as follows: “The absolute value of the cyclic crosscorrelation function between any two ZC sequences is constant and equal to 1/sqrt(N_ZC), if |u1u2|is relatively prime with respect to N_ZC.” Where, u1, and u2 are root indices and N_ZC is sequence length.

Attached are octave scripts that illustrate these properties.