The lowest sampling frequency not existing in theory

It was demonstrated that the lowest sampling frequency f min_sample doesn't exist in mathematics. With mathematical analysis, the principles of sampling and reconstructing a continuous signal were strictly calculated. We found that the spectra of sampling data at critical sampling frequency f critical_sample should overlap at the highest frequency f max of the continuous signal. The f critical_sample was defined as double of the f max , viz. f critical_sample =2 f max . As we know, the reconstructed signal will be distorted with this kind of overlapped spectra. Here, we will further illustrate the theoretical results. Aided with Fast Fourier Transform(FFT), the critical sampling and the process reconstructing continuous-time signal from it were discussed by spectroscopy. A symmetrical frequency-limited spectrum F (ω) was constructed with three modified rise-cosine pulses. Its corresponding time-domain signal f(t) was worked out theoretically. f(t) was sampled with δ T (t). By modifying T, the critical sampling signal was obtained. With FFT, the spectrum F d (ω)of the sampling signal was figured out. The calculated F d (ω) was compared with the constructed F(ω), and was analyzed for observing frequency alias. A cycle of F d (ω) for restoring the continuous signal could be obtained when F d (ω) was filtered by an ideal low-passed filter. With FFT, a continuous signal was reconstructed from it. As the results, the spectra of sampling data at the f critical_sample overlapped at the f max . The reconstructed signal distorted obviously. So, the lowest sampling frequency f min_sample doesn't exist. The sampling theorem couldn't include equal sign. It is unscientific to say that the f min_sample equal to double of the f max .