# 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 § 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

**entireHadamard productorder**

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.