Efficient DFT of Zadoff-Chu Sequences using lmFH Pattern

TL;DR

This paper introduces a novel method for efficiently computing the DFT of Zadoff-Chu sequences by leveraging their inherent micro-frequency hopping pattern, simplifying calculations and revealing new structural insights.

Contribution

It presents an intuitive visualization of DFT/IDFT computation for ZC sequences using the lmFH pattern and introduces an alternative cumulative sum calculation via the Generalized Quadratic Gauss Sum.

Findings

DFT of ZC sequences can be represented as an lmFH symbol with frequency shift and phase offset.

The cumulative sum of ZC sequences can be computed using the Generalized Quadratic Gauss Sum.

The proposed method simplifies DFT computation for ZC sequences.

Abstract

Having established that Zadoff-Chu (ZC) sequences are inherently linear micro-frequency hopping (lmFH) symbols, this paper first presents an intuitive and visual exposition of the computation of the DFT and IDFT of ZC sequences using the lmFH pattern. This yields interesting results. Subsequently, an alternative form for computing the cumulative sum of ZC sequences using the Generalized Quadratic Gauss Sum is introduced. Furthermore, building on the micro-frequency hopping (mFH) concept, this paper shows that the DFT of ZC sequences can be transformed into an lmFH symbol with frequency shift and phase offset. Therefore, the DFT of ZC sequences can be computed via cumulative frequency points, similar to the computation of normal mFH symbols.