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Bandlimiting is the limiting of a signal's
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signa ...
representation or
spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, o ...
to zero above a certain finite
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
. A band-limited signal is one whose
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
or spectral density has bounded support. A bandlimited signal may be either random (
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
) or non-random (
deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
). In general, infinitely many terms are required in a continuous
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
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.


Sampling bandlimited signals

A bandlimited signal can be fully reconstructed from its samples, provided that the
sampling rate In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or spac ...
exceeds twice the maximum frequency in the bandlimited signal. This minimum sampling rate is called the
Nyquist rate In signal processing, the Nyquist rate, named after Harry Nyquist, is a value (in units of samples per second or hertz, Hz) equal to twice the highest frequency (bandwidth) of a given function or signal. When the function is digitized at a hig ...
. This result, usually attributed to Nyquist and Shannon, is known as the
Nyquist–Shannon sampling theorem The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that pe ...
. An example of a simple deterministic bandlimited signal is a
sinusoid A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...
of the form x(t) = \sin(2 \pi ft + \theta) \ . If this signal is sampled at a rate f_s =\frac > 2f so that we have the samples x(nT) \ , for all integers n, we can recover x(t) \ completely from these samples. Similarly, sums of sinusoids with different frequencies and phases are also bandlimited to the highest of their frequencies. The signal whose Fourier transform is shown in the figure is also bandlimited. Suppose x(t)\ is a signal whose Fourier transform is X(f)\ , the magnitude of which is shown in the figure. The highest frequency component in x(t)\ is B \ . As a result, the Nyquist rate is : R_N = 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)\ completely and exactly using the samples : x \ \stackrel\ x(nT) = x \left( \right) for all integers n \, and T \ \stackrel\ as long as :f_s > R_N \, The reconstruction of a signal from its samples can be accomplished using the
Whittaker–Shannon interpolation formula The Whittaker–Shannon interpolation formula or sinc interpolation is a method to construct a continuous-time bandlimited function from a sequence of real numbers. The formula dates back to the works of E. Borel in 1898, and E. T. Whittaker in ...
.


Bandlimited versus timelimited

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. 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 In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a characteristic of a sampler, which converts a continuous function or signal into a discrete sequence. In units of cycles per second ( Hz), it ...
, and compute respective
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
FT(f) = F_1(w) and
discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
DTFT(f) = F_2(w). According to properties of DTFT, F_2(w) = \sum_^ F_1(w+n f_x) , where f_x is the frequency used for discretization. If f is bandlimited, F_1 is zero outside of a certain interval, so with large enough f_x , F_2 will be zero in some intervals too, since individual supports of F_1 in sum of F_2 won't overlap. According to DTFT definition, F_2 is a sum of trigonometric functions, and since f(t) is time-limited, this sum will be finite, so F_2 will be actually a
trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
. 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. But this contradicts our earlier finding that F_2 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 Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
in frequency also forms the mathematical basis for the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. In that setting, the "width" of the time domain and frequency domain functions are evaluated with a
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
-like measure. Quantitatively, the uncertainty principle imposes the following condition on any real waveform: : W_B T_D \ge 1 where :W_B is a (suitably chosen) measure of bandwidth (in hertz), and :T_D is a (suitably chosen) measure of time duration (in seconds). In
time–frequency analysis In signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains ''simultaneously,'' using various time–frequency representations. Rather than viewing a 1-dimensional signal (a ...
, these limits are known as the ''
Gabor limit In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
,'' and are interpreted as a limit on the ''simultaneous'' time–frequency resolution one may achieve.


References

*{{cite book , author = William McC. Siebert , title = Circuits, Signals, and Systems , year = 1986 , location = Cambridge, MA , publisher = MIT Press Digital signal processing