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

Returns 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; \
   ...: x3 = sdr.zadoff_chu(N, 3)
   ...: 

In [2]: plt.figure(figsize=(8, 4)); \
   ...: sdr.plot.constellation(x3, linestyle="-", linewidth=0.5); \
   ...: plt.title(f"Root-3 Zadoff-Chu sequence of length {N}"); \
   ...: plt.tight_layout();
   ...:
../../_images/sdr_zadoff_chu_1.png

The periodic autocorrelation of a Zadoff-Chu sequence has sidelobes of magnitude 0.

# Perform periodic autocorrelation
In [3]: corr = np.correlate(np.roll(np.tile(x3, 2), -N//2), x3, mode="valid"); \
   ...: lag = np.arange(-N//2 + 1, N//2 + 2)
   ...: 

In [4]: plt.figure(figsize=(8, 4)); \
   ...: plt.plot(lag, np.abs(corr) / N); \
   ...: plt.ylim(0, 1); \
   ...: plt.xlabel("Lag"); \
   ...: plt.ylabel("Magnitude"); \
   ...: plt.title(f"Periodic autocorrelation of root-3 Zadoff-Chu sequence of length {N}"); \
   ...: plt.tight_layout();
   ...:
../../_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(N, 5)

In [6]: plt.figure(figsize=(8, 4)); \
   ...: sdr.plot.constellation(x5, linestyle="-", linewidth=0.5); \
   ...: plt.title(f"Root-5 Zadoff-Chu sequence of length {N}"); \
   ...: plt.tight_layout();
   ...:
../../_images/sdr_zadoff_chu_3.png

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

# Perform periodic cross correlation
In [7]: xcorr = np.correlate(np.roll(np.tile(x3, 2), -N//2), x5, mode="valid"); \
   ...: lag = np.arange(-N//2 + 1, N//2 + 2)
   ...: 

In [8]: plt.figure(figsize=(8, 4)); \
   ...: plt.plot(lag, np.abs(xcorr) / N); \
   ...: plt.ylim(0, 1); \
   ...: plt.xlabel("Lag"); \
   ...: plt.ylabel("Magnitude"); \
   ...: plt.title(f"Periodic cross correlation of root-3 and root-5 Zadoff-Chu sequences of length {N}"); \
   ...: plt.tight_layout();
   ...:
../../_images/sdr_zadoff_chu_4.png