基於傅立葉級數之演算法應用於心電圖信號壓縮

Abstract

在實際信號處理中，信號通常存在於有限的頻寬內，因此使用傳統方法展開需要大量的展開項數來精準模擬信號，這不但會增加計算複雜度並可能產生不必要的頻率干擾。為克服這些限制，本研究引入了新的修正基底，以改善傳統傅立葉級數展開需要無限多項才能完整表示信號的問題，使得信號在有限頻寬情況下也能達到信號的完備性。 本研究提出一種新的方法，使用傳統傅立葉級數的基底之中插入與之頻率錯開的相依基底，以實現對信號的完整展開。相較於傳統方法需要展開無限項才能達到完備性的情況，本研究成功地降低了所需展開的項數，使其誤差小於10-5，並同時確保了信號的完整性。此外，我們使用PTB Diagnostic ECG Database作為資料庫，將該方法應用於心電圖(ECG)信號分析，以評估其在實際情況下的應用。 在實驗中，我們首先對PTB Diagnostic ECG Database中的心電圖信號進行預處理，然後使用我們所提出的展開方法進行信號展開，通過比較展開結果與原始信號之間的誤差來評估此方法的效能。結果表明，我們的方法在心電圖信號分析方面表現出色。相較於傳統方法，我們的方法能夠更準確地還原信號，並提供更可靠的分析結果。這些結果證明了我們所提出的方法在處理心電圖信號方面的應用潛力。 綜上所述，本研究提出的新的修正基底方法在傳統傅立葉級數展開的基礎上進行了改進，通過引入修正基底，在有限頻帶情況下能夠實現信號的完整表示。我們的方法在減少展開項數的同時，保持了信號的完備性和準確性。這項研究為心電圖信號分析提供了新的方法和思路，不僅提高訊號展開的效率和準確性，在實際應用方面也有助於醫學診斷和疾病判讀領域的進一步研究。

In practical signal processing, signals typically exist within a limited bandwidth. Therefore, using traditional methods for signal expansion requires a large number of expansion terms to accurately simulate the signal. This not only increases computational complexity but may also introduce unnecessary frequency interference. To overcome these limitations, this study introduces a new modified basis to improve the problem of traditional Fourier series expansion requiring an infinite number of terms to fully represent the signal, enabling the signal to achieve completeness even within a limited bandwidth. This study proposes a novel approach that utilizes the traditional Fourier basis and inserts orthogonal and dependent bases to achieve a complete expansion of the signal. Compared to the traditional method that requires an infinite number of terms for completeness, this study successfully reduces the required number of expansions with an error smaller than 10-5 while ensuring the integrity of the signal. Furthermore, the PTB Diagnostic ECG Database is used as the dataset to apply this method to electrocardiogram (ECG) signal analysis and evaluate its practical application. In the experiments, we first preprocess the electrocardiogram signals from the PTB Diagnostic ECG Database, and then employ our proposed expansion method to expand the signals. The performance of this method is evaluated by comparing the expansion results with the original signals. The results demonstrate the excellent performance of our method in electrocardiogram signal analysis. Compared to traditional methods, our approach is able to more accurately reconstruct the signals and provide more reliable analysis results. These findings demonstrate the potential of our proposed method in handling electrocardiogram signals. In conclusion, this study presents a novel modified basis method that improves upon traditional Fourier series expansion by introducing modified bases. This method enables the complete representation of signals within a limited bandwidth. Our approach reduces the number of expansion terms while preserving the integrity and accuracy of the signal. This research provides new methods and insights for electrocardiogram signal analysis, enhancing the efficiency and accuracy of signal expansion, and contributing to further research in medical diagnosis and disease interpretation.