Bandlimiting refers to a process which reduces the energy of a signal to an acceptably low level outside of a desired frequency range.
Bandlimiting is an essential part of many applications in signal processing and communications. Examples include controlling interference between radio frequency communications signals, and managing aliasing distortion associated with sampling for digital signal processing.
A bandlimited signal is, strictly speaking, a signal with zero energy outside of a defined frequency range. In practice, a signal is considered bandlimited if its energy outside of a frequency range is low enough to be considered negligible in a given application.
A bandlimited signal may be either random (stochastic) or non-random (deterministic).
In general, infinitely many terms are required in a continuous Fourier series representation of a signal, but if a finite number of Fourier series terms can be calculated from that signal, that signal is considered to be band-limited. In mathematic terminology, a bandlimited signal has a Fourier transform or spectral density with bounded support.
A bandlimited signal can be fully reconstructed from its samples, provided that the sampling rate exceeds twice the bandwidth of the signal. This minimum sampling rate is called the Nyquist rate associated with the Nyquist–Shannon sampling theorem.
Real world signals are not strictly bandlimited, and signals of interest typically have unwanted energy outside of the band of interest. Because of this, sampling functions and digital signal processing functions which change sample rates usually require bandlimiting filters to control the amount of aliasing distortion. Bandlimiting filters should be designed carefully to manage other distortions because they alter the signal of interest in both its frequency domain magnitude and phase, and its time domain properties.
An example of a simple deterministic bandlimited signal is a sinusoid of the form
x(t)=\sin(2\pift+\theta).
fs=\tfrac{1}{T}>2f
x(nT),
n
x(t)
The signal whose Fourier transform is shown in the figure is also bandlimited. Suppose
x(t)
X(f),
x(t)
B.
RN=2B
or twice the highest frequency component in the signal, as shown in the figure. According to the sampling theorem, it is possible to reconstruct
x(t)
x(nT)=x\left({n\overfs}\right)
n
T \stackrel{def
as long as
fs>RN
The reconstruction of a signal from its samples can be accomplished using the Whittaker–Shannon interpolation formula.
See main article: article. A bandlimited signal cannot be also timelimited. More precisely, a function and its Fourier transform cannot both have finite support unless it is identically zero. This fact can be proved using complex analysis and properties of the Fourier transform.
FT(f)=F1(w)
DTFT(f)=F2(w)
F2(w)=
+infty | |
\sum | |
n=-infty |
F1(w+nfx)
fx
F1
fx
F2
F1
F2
F2
F2
F2
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 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:
WBTD\ge1
where
WB
TD
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.