Editing Bandlimiting (section)

=== Proof ===
Assume that a signal f(t) which has finite support in both domains and is not identically zero exists. Let's sample it faster than the [[Nyquist frequency]], and compute respective [[Fourier transform]] <math> FT(f) = F_1(w) </math> and [[discrete-time Fourier transform]] <math> DTFT(f) = F_2(w)</math>. According to properties of DTFT, <math> F_2(w) = \sum_{n=-\infty}^{+\infty} F_1(w+n f_x) </math>, where <math>f_x</math> is the frequency used for [[discretization]]. If f is bandlimited, <math> F_1 </math> is zero outside of a certain interval, so with large enough <math> f_x </math>, <math> F_2 </math> will be zero in some intervals too, since individual [[Support (mathematics)|supports]] of <math> F_1 </math> in sum of <math> F_2 </math> won't overlap. According to DTFT definition, <math> F_2 </math> is a sum of [[trigonometric functions]], and since f(t) is time-limited, this sum will be finite, so <math> F_2 </math> will be actually a [[trigonometric polynomial]]. All trigonometric polynomials are [[Entire function|holomorphic on a whole complex plane]], and there is a simple theorem in complex analysis that says that [[Zero (complex analysis)|all zeros of non-constant holomorphic function are isolated]]. But this contradicts our earlier finding that <math> F_2 </math> has intervals full of zeros, because points in such intervals are not isolated. Thus the only time- and bandwidth-limited signal is a constant zero.

One important consequence of this result is that it is impossible to generate a truly bandlimited signal in any real-world situation, because a bandlimited signal would require infinite time to transmit.  All real-world signals are, by necessity, ''timelimited'', which means that they ''cannot'' be bandlimited.  Nevertheless, the concept of a bandlimited signal is a useful idealization for theoretical and analytical purposes.  Furthermore, it is possible to approximate a bandlimited signal to any arbitrary level of accuracy desired.

A similar relationship between duration in time and [[Bandwidth (signal processing)|bandwidth]] in frequency also forms the mathematical basis for the [[uncertainty principle]] in [[quantum mechanics]]. In that setting, the "width" of the time domain and frequency domain functions are evaluated with a [[variance]]-like measure. Quantitatively, the uncertainty principle imposes the following condition on any real waveform:

:<math> W_B T_D \ge 1 </math>

where

:<math>W_B</math> is a (suitably chosen) measure of bandwidth (in hertz), and

:<math>T_D</math> is a (suitably chosen) measure of time duration (in seconds).

In [[time–frequency analysis]], these limits are known as the ''[[Gabor limit]],'' and are interpreted as a limit on the ''simultaneous'' time–frequency resolution one may achieve.