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komm.ZadoffChuSequence

Zadoff–Chu sequence. It is a periodic, complex sequence given by $$ z_{L,q}[n] = \mathrm{e}^{-\mathrm{j} \pi q n (n + 1) / L}, $$ where $L$ is the length (and period) of the sequence (which must be an odd integer) and $q \in [1:L)$ is called the root index of the sequence.

Zadoff–Chu sequences have the following properties:

  1. Constant amplitude: The magnitude of the sequence satisfies $$ |z_{L,q}[n]| = 1, \quad \forall n. $$

  2. Zero autocorrelation: If $q$ is coprime to $L$, then the cyclic autocorrelation of $z_{L,q}$ satisfies $$ \tilde{R}_{z_{L,q}}[\ell] = 0, \quad \forall \ell \neq 0 \mod L. $$

  3. Constant cross-correlation: If $|q' - q|$ is coprime to $L$, then the magnitude of the cyclic cross-correlation of $z_{L,q}$ and $z_{L,q'}$ satisfies $$ |\tilde{R}_{z_{L,q}, z_{L,q'}}[\ell]| = \sqrt{L}, \quad \forall \ell. $$

For more details, see And22.

Notes
  • Theses sequences are also called Frank–Zadoff–Chu sequences.

__init__()

Constructor for the class.

Parameters:

  • length (int) –

    The length $L$ of the Zadoff–Chu sequence. Must be an odd integer.

  • root_index (Optional[int]) –

    The root index $q$ of the Zadoff–Chu sequence. Must be in $[1:L)$. The default value is $1$.

Examples:

>>> zadoff_chu = ZadoffChuSequence(5, root_index=1)
>>> np.around(zadoff_chu.sequence, decimals=6)
array([ 1.      +0.j      ,  0.309017-0.951057j, -0.809017+0.587785j,  0.309017-0.951057j,  1.      +0.j      ])
>>> np.around(zadoff_chu.cyclic_autocorrelation(normalized=True), decimals=6)
array([ 1.+0.j, -0.-0.j, -0.-0.j,  0.+0.j, -0.+0.j])

length property

The length $L$ of the Zadoff–Chu sequence.

root_index property

The root index $q$ of the Zadoff–Chu sequence.