Prove Time & frequency domain behaviour?

Question

Math proof that a limited time domain signal is infinite in the frequency domain & a limited band of frequencys signal that is infinite in freq domain, is unlimited in time domain. I suspect Fourier sereis and transforms are involved. Thax in advance.

Answer 1

I believe all you have to do is just apply the Fourier transform integral to each of the signals in question, and observe what you get. By limited time-domain signal you mean a finite-energy signal (integral of the squared signal is finite). Put that into the formula and you will get a transform that is periodic (hence band-unlimited). Similarly, if you put a bandlimited signal into the inverse Fourier integral formula (that is, its limits are finite), you will get a periodic time signal (time-unlimited). The general rule is:

Time Domain Frequency Domain
discrete periodic
periodic discrete
time-limited frequency-unlimited
frequency-limited time-unlimited
real even

and so on and so on. All of these may be seen by just putting signals fitting those descriptions into the appropriate formula, and observing the results.