Gabor Transform in Time-Frequency Analysis

This program implements Gabor transform for time-frequency analysis, demonstrating effective frequency separation performance. In the Gabor transform formulation, constants a and b define critical parameters: a represents the grid interval length in the time domain, while b determines the frequency resolution span. The expansion coefficients correspond to the one-dimensional signal x(t), where h(t) serves as the mother function. The basis functions are generated by applying shifting and modulation operations to h(t), creating a complete set for signal decomposition. Gabor transform represents a fundamental method for signal analysis in the joint time-frequency domain, enabling decomposition of signals into constituent frequency components while preserving temporal and spectral information. This transformation finds extensive applications across signal processing, image analysis, and pattern recognition domains. From an implementation perspective, the algorithm typically involves windowed Fourier analysis with Gaussian windows, where the code would calculate convolution operations between the signal and modulated window functions at different time-frequency grid points.