Bandlimiting
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 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.
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
The signal whose Fourier transform is shown in the figure is also bandlimited.
As a result, the Nyquist rate is or twice the highest frequency component in the signal, as shown in the figure.
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.
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
All trigonometric polynomials are holomorphic on a whole complex plane, and there is a simple theorem in complex analysis that says that all zeros of non-constant holomorphic function are 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 in frequency also forms the mathematical basis for the uncertainty principle in quantum mechanics.
