sdr.zadoff_chu_sequence(length: int, root: int, shift: int = 0) → NDArray[complex128]

Generates the root-\(u\) Zadoff-Chu sequence of length \(N\).

Parameters:
length: int

The length \(N\) of the Zadoff-Chu sequence.

root: int

The root \(0 < u < N\) of the Zadoff-Chu sequence. The root must be relatively prime to the length, i.e., \(\gcd(u, N) = 1\).

shift: int = 0

The shift \(q \in \mathbb{Z}\) of the Zadoff-Chu sequence. When \(q \ne 0\), the returned sequence is a cyclic shift of the root-\(u\) Zadoff-Chu sequence.

Returns:

The root-\(u\) Zadoff-Chu sequence of length \(N\).

Notes

The root-\(u\) Zadoff-Chu sequence with length \(N\) and shift \(q\) is defined as

\[x_u[n] = \exp \left( -j \frac{\pi u n (n + c_{f} + 2q)}{N} \right) ,\]

where \(c_{f} = N \mod 2\).

References

Examples

Create a root-3 Zadoff-Chu sequence \(x_3[n]\) with length 139.

In [1]: N = 139

In [2]: x3 = sdr.zadoff_chu_sequence(N, 3)

In [3]: plt.figure(); \
   ...: sdr.plot.constellation(x3, linestyle="-", linewidth=0.5); \
   ...: plt.title(f"Root-3 Zadoff-Chu sequence of length {N}");
   ...:
../../_images/sdr_zadoff_chu_1.png

The periodic auto-correlation of a Zadoff-Chu sequence has sidelobes with magnitude 0.

In [4]: plt.figure(); \
   ...: sdr.plot.correlation(x3, x3, mode="circular"); \
   ...: plt.ylim(0, N);
   ...:
../../_images/sdr_zadoff_chu_2.png

Create a root-5 Zadoff-Chu sequence \(x_5[n]\) with length 139.

In [5]: x5 = sdr.zadoff_chu_sequence(N, 5)

In [6]: plt.figure(); \
   ...: sdr.plot.constellation(x5, linestyle="-", linewidth=0.5); \
   ...: plt.title(f"Root-5 Zadoff-Chu sequence of length {N}");
   ...:
../../_images/sdr_zadoff_chu_3.png

The periodic cross-correlation of two prime-length Zadoff-Chu sequences with different roots has sidelobes with magnitude \(1 / \sqrt{N}\).

In [7]: plt.figure(); \
   ...: sdr.plot.correlation(x3, x5, mode="circular"); \
   ...: plt.ylim(0, N);
   ...:
../../_images/sdr_zadoff_chu_4.png