# Bandlimiting

bandlimitedband-limitedband limitedband-limitband-limited signalbandlimited versus timelimitedlimited
Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.wikipedia
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### Nyquist rate

Nyquist sampling rateNyquist limitNyquist
This minimum sampling rate is called the Nyquist rate.
But only one of them is bandlimited to ½ f s cycles/second (hertz), which means that its Fourier transform, X(f), is 0 for all |f| ≥ ½ f s. The mathematical algorithms that are typically used to recreate a continuous function from samples create arbitrarily good approximations to this theoretical, but infinitely long, function.

### Nyquist–Shannon sampling theorem

sampling theoremNyquist-Shannon sampling theoremNyquist theorem
This result, usually attributed to Nyquist and Shannon, is known as the Nyquist–Shannon sampling theorem.
The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are bandlimited to a given bandwidth, such that no actual information is lost in the sampling process.

### Whittaker–Shannon interpolation formula

interpolation/sampling theoryreconstructingsinc interpolation
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.

### Uncertainty principle

Heisenberg uncertainty principleHeisenberg's uncertainty principleuncertainty relation
A similar relationship between duration in time and bandwidth in frequency also forms the mathematical basis for the uncertainty principle in quantum mechanics. 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.
The basic result, which follows from "Benedicks's theorem", below, is that a function cannot be both time limited and band limited (a function and its Fourier transform cannot both have bounded domain)—see bandlimited versus timelimited.

### Frequency domain

frequency-domainFourier spaceFourier domain
Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.

### Spectral density

frequency spectrumpower spectrumspectrum
Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.

### Frequency

frequenciesperiodperiodic
Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.

### Fourier transform

continuous Fourier transformFourierFourier transforms
A band-limited signal is one whose Fourier transform or spectral density has bounded support. Let's sample it faster than the Nyquist frequency, and compute respective Fourier transform and discrete-time Fourier transform.

### Support (mathematics)

supportcompact supportcompactly supported
A band-limited signal is one whose Fourier transform or spectral density has bounded support.

### Stochastic

stochasticsstochastic musicstochasticity
A bandlimited signal may be either random (stochastic) or non-random (deterministic).

### Determinism

deterministicdeterministcausal determinism
A bandlimited signal may be either random (stochastic) or non-random (deterministic).

### Fourier series

Fourier coefficientFourier expansionFourier coefficients
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.

### Sampling (signal processing)

sampling ratesamplingsample rate
A bandlimited signal can be fully reconstructed from its samples, provided that the sampling rate exceeds twice the maximum frequency in the bandlimited signal.

### Harry Nyquist

NyquistNyquist, Harry
This result, usually attributed to Nyquist and Shannon, is known as the Nyquist–Shannon sampling theorem.

### Claude Shannon

Claude E. ShannonShannonClaude Elwood Shannon
This result, usually attributed to Nyquist and Shannon, is known as the Nyquist–Shannon sampling theorem.

### Sine wave

sinusoidalsinusoidsine
An example of a simple deterministic bandlimited signal is a sinusoid of the form.

### Nyquist frequency

Nyquist limitNyquistN/2 different frequencies
Let's sample it faster than the Nyquist frequency, and compute respective Fourier transform and discrete-time Fourier transform.

### Discrete-time Fourier transform

convolution theoremDFTDTFT &sect; Properties
Let's sample it faster than the Nyquist frequency, and compute respective Fourier transform and discrete-time Fourier transform.

### Trigonometric polynomial

trigonometric polynomialstrigonometrictrigonometrical
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.

### Entire function

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.

### Bandwidth (signal processing)

bandwidthbandwidthssignal bandwidth
A similar relationship between duration in time and bandwidth in frequency also forms the mathematical basis for the uncertainty principle in quantum mechanics.

### Quantum mechanics

quantum physicsquantum mechanicalquantum theory
A similar relationship between duration in time and bandwidth in frequency also forms the mathematical basis for the uncertainty principle in quantum mechanics.

### Variance

sample variancepopulation variancevariability
In that setting, the "width" of the time domain and frequency domain functions are evaluated with a variance-like measure.

### Time–frequency analysis

time-frequency analysistime-frequency domainfrequency-time
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.

### Henry Landau

Landau, Henry
Henry Jacob Landau is an American mathematician known for his contributions to information theory, including the theory of bandlimited functions and on moment issues.