薩克時頻轉換與其在通訊上的應用

Abstract

傳統的傅立葉轉換以其將信號轉換為對應頻譜的能力而聞名，但追蹤頻譜的時間演變卻存在挑戰。薩克時頻轉換是一種時頻分析的工具，可使信號從時間域轉換到時間-頻率域。與傅立葉轉換不同，薩克時頻轉換允許對信號頻率在時間上的動態變化進行更詳細的檢查。其線性特性是一個突出的特點，可防止在時頻分析中常見的干擾，如交叉項的出現。這種線性特性能夠精確地隔離信號，突顯了它在時頻信號處理應用中的重要性。

在我們探索薩克時頻轉換應用的過程中，我們發現了兩種不同的應用。首先，在利用完美序列集時，薩克時頻轉換使精確的時間-頻率分析成為可能，確保信號僅在其具有重要性的地方顯現，從而減輕了干擾問題。其次，在正交時頻空間的通信應用中，通過將薩克時頻轉換納入正交時頻空間系統，相較於傳統方法，我們不僅保持了相等的輸入-輸出關係，還減少了計算複雜度。

The Fourier transform is renowned for its ability to translate signals into their corresponding spectrum, yet tracking the temporal evolution of this spectrum presents challenges. Enter the Zak transform, a valuable tool facilitating the translation of signals from the time domain to the time-frequency domain. Unlike the Fourier transform, the Zak transform allows for a more detailed examination of dynamic changes in signal frequencies over time. Its linearity is a standout feature, preventing the emergence of interference, such as cross terms, commonly encountered in time-frequency analysis. This linearity, coupled with its ability to isolate signals precisely, underscores its significance in signal processing applications.

In our exploration of the Zak transform's applications, we've discovered its efficacy in two distinct contexts. Firstly, when leveraging perfect sequence sets, the Zak transform enables precise time-frequency analysis, ensuring that signals only manifest where they hold significance, thereby mitigating interference issues. Secondly, in communication applications like Orthogonal Time Frequency Space (OTFS), the Zak transform proves invaluable. By incorporating it into OTFS systems, we not only maintain equivalent input-output relationships but also reduce computational complexity compared to conventional methods.