Fractional Fourier Transform of Linear Frequency Modulated Signals

The Fractional Fourier Transform (FrFT) provides a fully operational method for analyzing Linear Frequency Modulated (LFM) signals. By implementing Pei's sampling algorithm, this approach enables LFM signals that display broadband properties in the conventional Fourier domain to demonstrate narrowband characteristics when transformed to the appropriate fractional Fourier domain. This transformation technique allows for more detailed analysis of LFM signal spectral properties, facilitating deeper signal investigation and understanding. In code implementation, the FrFT typically involves computing the transformation kernel through discrete sampling based on the rotation angle parameter. The Pei algorithm efficiently handles the discretization process by properly sampling both time and frequency domains, ensuring accurate representation of signal energy concentration in the optimal fractional domain. Key computational steps include parameter calculation for the transformation matrix, eigenvalue decomposition, and proper scaling to maintain energy conservation during domain rotation.