The lowest sampling frequency not existing in theory

Yizhong Song; Zhimin Zhao

doi:10.1117/12.805614

2 February 2009 The lowest sampling frequency not existing in theory

Yizhong Song, Zhimin Zhao

Proceedings Volume 7160, 2008 International Conference on Optical Instruments and Technology: Optoelectronic Measurement Technology and Applications; 71601G (2009) https://doi.org/10.1117/12.805614
Event: International Conference of Optical Instrument and Technology, 2008, Beijing, China

Abstract

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}=2f_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.

Citation Download Citation

Yizhong Song and Zhimin Zhao "The lowest sampling frequency not existing in theory", Proc. SPIE 7160, 2008 International Conference on Optical Instruments and Technology: Optoelectronic Measurement Technology and Applications, 71601G (2 February 2009); https://doi.org/10.1117/12.805614

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KEYWORDS

Detection theory

Optical filters

Fourier transforms

Signal processing

Electronic filtering

Mathematics

Electronics

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