A signal or function is bandlimited if it contains no Energy at frequencies higher than some bandlimit or bandwidth,B. A signal that is bandlimited is constrained in how rapidly it changes in time, and therefore how much detail it can convey in an interval of time. The sampling theorem asserts that the uniformly spaced discrete samples are a complete representation of the signal if this bandwidth is less than half the sampling rate.
To formalize these concepts, let $ x(t)\, $ represent a continuous-time signal and $ X(w)\, $ be the continuous Fourier transform of that signal (which exists if $ x(t)\, $ is square-integrable)
- $ X(w)=\ \int_{-\infty}^{\infty} x(t) \ e^{- i 2 \pi w t} \ dt. \ $
The signal $ x(t)\, $ is bandlimited to a one-sided baseband bandwidth,B, if
- $ X(w) = 0 \quad $ for all $ |w| > B \, $
or, equivalently, the support of X supp(X)⊆[-B,B]. Then the sufficient condition for exact reconstructability from samples at a uniform sampling rate $ w_s\, $ (in samples per unit time) is
- $ w_s > 2 B,\, $
or equivalently
- $ B < { w_s \over 2 }.\, $
$ 2 B\, $ is called the Nyquist rate and is a property of the bandlimited signal, while $ w_s /2\, $ is called the Nyquist frequency and is a property of this sampling system.
The time interval between successive samples is referred to as the sampling interval
- $ T\ =\ \frac{1}{w_s},\, $
and the samples of $ x(t)\, $ are denoted by
- $ x(nT) \quad n\in\mathbb{Z}\, $(integers).
The sampling theorem leads to a procedure for reconstructing the original $ x(t)\ $ from the samples and states sufficient conditions for such a reconstruction to be exact.
