This paper is a compilation of well-known results about Zadoff-Chu sequences, including all proofs with a consistent mathematical notation, for easy reference. Moreover, for a Zadoff-Chu sequence $x_u[n]$ of prime length $N_{\text{ZC}}$ and root index $u$, a formula is derived that allows computing the first term (frequency zero) of its discrete Fourier transform, $X_u[0]$, with constant complexity independent of the sequence length, as opposed to accumulating all its $N_{\text{ZC}}$ terms. The formula stems from a famous result in analytic number theory and is an interesting complement to the fact that the discrete Fourier transform of a Zadoff-Chu sequence is itself a Zadoff-Chu sequence whose terms are scaled by $X_u[0]$. Finally, the paper concludes with a brief analysis of time-continuous signals derived from Zadoff-Chu sequences, especially those obtained by OFDM-modulating a Zadoff-Chu sequence.

翻译：本文汇编了关于Zadoff-Chu序列的已知结论，包含所有证明并采用一致的数学符号以便参考。此外，对于素数长度$N_{\text{ZC}}$、根指数$u$的Zadoff-Chu序列$x_u[n]$，推导出一个公式，可恒定复杂度（与序列长度无关）计算其离散傅里叶变换的首项（零频分量）$X_u[0]$，无需累加全部$N_{\text{ZC}}$项。该公式源于解析数论中的一个著名结果，并与“Zadoff-Chu序列的离散傅里叶变换仍是Zadoff-Chu序列且其项被$X_u[0]$缩放”这一事实形成有趣互补。最后，本文简要分析了由Zadoff-Chu序列导出的时间连续信号，特别是通过OFDM调制Zadoff-Chu序列所获得的信号。