Exploring Zadoff–Chu (ZC) Sequences and 3GPP Limitations

• Exploring Zadoff–Chu (ZC) Sequences and 3GPP Constraints

  A Zadoff–Chu (ZC) sequence is a complex-valued sequence with constant amplitude and zero cyclic auto-correlation, meaning that the sequence does not correlate with any of its cyclic shifts. When a ZC sequence is applied to a signal, the resulting waveform maintains constant amplitude. Furthermore, by applying cyclic shifts, it is possible to derive multiple orthogonal signals from a single root sequence, since different cyclically shifted ZC sequences remain mutually uncorrelated.

  These unique correlation properties make ZC sequences extremely valuable in wireless communication systems, particularly in scenarios that require robust synchronization, channel estimation, and multiple-access capabilities.

  Applications in Wireless Communications

  Several 3GPP-defined physical channels and signals use ZC sequences, including:

  • PSS (Primary Synchronization Signal)

  • PRACH (Random Access Channel)

  • PUCCH (Physical Uplink Control Channel)

  • PUSCH (Physical Uplink Shared Channel)

  • SRS (Sounding Reference Signal)

  Beyond ZC sequences, other orthogonal and error-correcting coding techniques commonly appear in modern systems, such as:

  • Walsh–Hadamard codes

  • LDPC (Low-Density Parity-Check) codes

  Zadoff–Chu Sequence Definition

  A ZC sequence of length N is defined using a root index $u$, where $u$ must be coprime with $N$. For a given sample index $k$, the sequence is expressed as follows.

  For odd N:

   x(k)=, 0≤N<k

   For even N:

   x(k)=,  0≤N<k

  Where:

• $N$ = sequence length

• $u$ = ZC root index (coprime with $N$)

• $k$ = sequence sample index

• $j=\sqrt{-1}$

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