Machine learning has revolutionized biomedical signal analysis, particularly in electrocardiogram (ECG) classification. While convolutional neural networks (CNNs) excel at automatic feature extraction, the optimal integration of time- and frequency-domain information remains unresolved. This study introduces the Convolutional Fourier Analysis Network (CFAN), a novel architecture that unifies time-frequency analysis by embedding Fourier principles directly into CNN layers. We evaluate CFAN against four benchmarks - spectrogram-based 2D CNN (SPECT); 1D CNN (CNN1D); Fourier-based 1D CNN (FFT1D); and CNN1D with integrated Fourier Analysis Network (CNN1D-FAN) - across three ECG tasks: arrhythmia classification (MIT-BIH), identity recognition (ECG-ID), and apnea detection (Apnea-ECG). CFAN achieved state-of-the-art performance, surpassing all competing methods with accuracies of 98.95% (MIT-BIH), 96.83% (ECG-ID), and 95.01% (Apnea-ECG). Notably, on ECG-ID and Apnea-ECG, CFAN demonstrated statistically significant improvements over the second-best method (CNN1D-FAN, $p \leq 0.02$), further validating its superior performance. Key innovations include CONV-FAN blocks that combine sine, cosine and GELU activations in convolutional layers to capture periodic features and joint time-frequency learning without spectrogram conversion. Our results highlight CFAN's potential for broader biomedical and signal classification applications.