Fractional Fourier Transform Filtering Program

The Fractional Fourier Transform (FRFT) represents a generalized formulation of the conventional Fourier transform. It demonstrates superior energy concentration characteristics for Linear Frequency Modulated (LFM) signals, where both time and frequency domains can be interpreted as special instances of the FRFT domain. As shown in Figure 2 depicting LFM signal projections across various transform domains, we observe that LFM signal energy spreads across a broad frequency range, whereas it manifests as an impulse function in the optimal fractional Fourier domain. Furthermore, FRFT being a linear transformation ensures that the FRFT of combined signal and noise equals the superposition of their individual FRFT results. These two properties form the foundation for implementing LFM signal filtering in the FRFT domain. The core algorithm for detecting LFM signals with unknown parameters involves scanning through rotation angles as variables, computing fractional Fourier transforms of observed signals to construct two-dimensional energy distributions across parameter planes. In practical implementation, this typically requires: 1. Designing an angle sampling strategy covering the complete rotation range (0 to π/2) 2. Implementing efficient FRFT computation using discrete algorithms like Ozaktas method 3. Creating energy distribution maps through magnitude squared operations 4. Applying threshold-based peak detection algorithms for 2D search On this parameter plane, we can perform two-dimensional peak search by setting appropriate thresholds, enabling simultaneous LFM signal detection and parameter estimation. The computational efficiency can be enhanced through optimized FRFT algorithms and parallel processing techniques for large-scale signal analysis.