>>12585496Now, the shannon sampling theorem states that the minimum sampling frequency needed to perfectly reconstruct a signal must be larger than twice the maximum frequency of the signal. You can see this intuitively by considering a sinusoid of frequency (see image).
HOWEVER, recalling that frequencies are just imaginary numbers when we take the fourier transform, we can interpret the periodicity of aliasing as being equivalent to the periodicity of the complex numbers when represented using . So, we can think of sampling as being equivalent to the map , since maps the imaginary axis, i.e. the set of imaginary numbers (read: the set of frequencies) onto the unit circle, experiencing periodicity inversely proportional to the sampling frequency.
This entirely lines up with our definition of aliasing, since high frequencies would map down to lower frequencies by the periodicity of the exponential. And indeed, if you start looking at the relation between Laplace and Z transforms (which include not only the imaginary axis but the entire complex plane), you'd see that this map holds up for the rest of the complex plane as well.
