Three Different Fractional Fourier Transform Algorithms: Implementation Approaches and Comparisons

This article systematically explores three different fractional Fourier transform (FrFT) algorithms, providing detailed comparisons of their advantages, limitations, and application scenarios. We conduct in-depth analysis of each algorithm's mathematical principles and implementation methodologies, including key computational considerations such as discrete sampling strategies, eigenvalue decomposition approaches, and fast computation techniques. The discussion extends to practical applications in signal processing (time-frequency analysis, chirp signal detection), image processing (rotation-invariant pattern recognition), and data mining (feature extraction from non-stationary signals). Each algorithm's implementation challenges are addressed, covering computational complexity optimization and parameter selection criteria. Furthermore, we examine future development directions for FrFT algorithms, including potential hybrid approaches and hardware acceleration possibilities. The article concludes with novel research perspectives and practical recommendations for advancing FrFT algorithm development, emphasizing real-time implementation constraints and cross-domain applicability.