Spectra Analysis of Sampling and Reconstructing Continuous Signal Using Hamming Window Function

Hamming window function was applied to studying sampling theorem. A continuous band-limited spectrum function F(w) was constructed with Hamming window function. Its corresponding time-domain signal f(t) was worked out by inverse Fourier transform. f(t) was sampled with a comb function dT(t). By modifying the value of T, all kinds of sampling signals were produced, including critical, over and under sampling. With FFT, the frequency spectrum of each sampling signal was figured out. Each spectrum profile was analyzed. The process to reconstruct f(t) was suggested, and the reconstructed results from each of the three kinds of sampling signals were discussed. As the result, critical sampling frequency spectrum in FFT principal value sequence was aliasing at the middle point, over sampling's disconnected, and under sampling's overlapped. The original signal could be accurately reconstructed from over samplings, but couldn't from under one. Hamming window is a perfect model for analyzing and demonstrating the sampling theorem.