Comments on Nyquist rate in engineering application

The sampling theorem is an essential rule in digital signal processing (DSP). It concludes that a time-domain signal can be exactly represented by its time samplings under the condition that sampling rate must be equal or greater than twice of the max frequency of the time-domain signal. However, the resulting spectrum obtained by applying DFT to those samplings is a spectrum with distortions caused by the sinusoidal component with the max frequency. The paper here, by using a concept of position spectrum, studies the spectrum when the sampling rate is greater than twice of the max frequency and especially when the sampling rate equals to twice of the max frequency, i.e. Nyquist sampling rate under the condition that a finite sequence of samples is processed by DFT algorithm, demonstrates intuitively the manners, magnitudes of both amplitude and phase spectrum distortions, and points out that the spectrum of the sinusoidal component with maximum frequency will be heavily distorted. And finally comes to a conclusion that the samples, picked up isochronously from a sinusoidal or the max frequency component at Nyquist sampling rate, could not represent the original signal because of the aliasing.