MATLAB Implementation for Fractional Fourier Transform (FRFT) Testing

The Fractional Fourier Transform (FRFT) testing program is primarily used to verify the appropriateness of the current transformation order. This testing is crucial for ensuring the accuracy of signal processing results, particularly in applications such as time-frequency analysis.

The core methodology of the testing program generally involves several key steps: First, appropriate test signals need to be prepared, commonly using linear frequency modulation (chirp) signals or Gaussian signals. These signals exhibit distinct characteristics in the fractional Fourier domain, making them ideal for testing FRFT implementations. The program sets different transformation order parameters, typically sampling values from 0 to 1 with fixed step sizes. Each order corresponds to a specific rotation angle in the time-frequency plane, with code implementations often using trigonometric functions to calculate the rotation matrix. After executing the FRFT transformation for each order, the energy concentration characteristics of the output results are analyzed. The appropriate transformation order should maximize the energy concentration of the signal in the fractional Fourier domain. This is typically implemented by computing the magnitude spectrum and identifying peak values. Quantitative metrics (such as energy peak ratios or entropy values) are used to evaluate the transformation effectiveness of different orders, thereby determining the optimal order. Code implementations often include functions to calculate these metrics automatically across all tested orders.

In practical testing, visualization techniques can assist in judgment. For example, plotting the magnitude spectra of transformation results under different orders allows observation of energy concentration variations. This approach is particularly suitable for preliminary evaluation of FRFT algorithm performance, with MATLAB implementations often using surface plots or heat maps to display the results across orders.

For algorithm verification, benchmark test cases with known optimal orders are typically included. The correctness of the implementation is confirmed by comparing the test program's output results with theoretical expectations. This testing method effectively validates both the accuracy of the FRFT implementation and its adaptability to different orders, with comprehensive test suites often incorporating edge cases and boundary conditions.