Exploration of Mathematical Constructs in Fourier Transform Techniques for Biomedical Signal Interpretation

Abstract

This study introduces a comprehensive mathematical framework for Fourier transform-based analysis of biomedical signals including ECG, EEG, and EMG, addressing the limitations of traditional time-domain methods in capturing frequency-domain physiological signatures essential for early diagnosis of cardiovascular, neurological, and neuromuscular disorders. Methodologically, it derives explicit analytical expressions for continuous-time Fourier transform (CTFT), discrete-time Fourier transform (DTFT), discrete Fourier transform (DFT), short-time Fourier transform (STFT), and spectrograms tailored to bio signal characteristics, encompassing ECG P-QRS-T spectral modeling (5–50 Hz bands), EEG canonical band powers (delta 0.5–4 Hz to gamma 30–100 Hz), EMG mean/median frequencies (MNF/MDF), noise models (baseline wander 0.05–0.5 Hz, powerline 50/60 Hz), and time–frequency representations for non-stationary analysis. The proposed work establishes a modular processing pipeline with standardized preprocessing (anti-aliasing, windowing via Hann/Hamming/Blackman), feature extraction (PSD via Welch’s method, spectral moments, spectro-temporal metrics), and validation on benchmark datasets, enabling seamless integration with machine learning classifiers for tasks like arrhythmia detection, seizure identification, and fatigue assessment. Expected outcomes include improved diagnostic accuracy through interpretable frequency-domain biomarkers, a reusable library of mathematical formulations for reproducible research, enhanced real-time monitoring capabilities for wearables, and foundational advancements toward hybrid AI-time-frequency models that bridge classical signal processing with modern deep learning paradigms.