Sampling (signal processing)

sampling ratesamplingsample ratesamplesampledsampling frequencysamplessamplersSampling (information theory)sound samplers
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal.wikipedia
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Aliasing

aliasaliasedtemporal aliasing
That fidelity is reduced when s(t) contains frequency components whose periodicity is smaller than two samples; or equivalently the ratio of cycles to samples exceeds ½ (see Aliasing).
In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or aliases of one another) when sampled.

Discrete time and continuous time

discrete timediscrete-timecontinuous-time
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal.
Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by sampling from a continuous-time signal.

Oversampling

oversampledoverachievingoversample
Although the use of oversampling can completely eliminate aperture error and aliasing by shifting them out of the pass band, this technique cannot be practically used above a few GHz, and may be prohibitively expensive at much lower frequencies.
In signal processing, oversampling is the process of sampling a signal at a sampling frequency significantly higher than the Nyquist rate.

Analog-to-digital converter

ADCanalog to digital converteranalog-to-digital conversion
In practice, the continuous signal is sampled using an analog-to-digital converter (ADC), a device with various physical limitations. One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for ADCs and DACs, but with modern oversampling sigma-delta converters this advantage is less important.
Furthermore, instead of continuously performing the conversion, an ADC does the conversion periodically, sampling the input, limiting the allowable bandwidth of the input signal.

Nyquist frequency

Nyquist limitNyquistN/2 different frequencies
The original signal is retrievable from a sequence of samples, up to the Nyquist limit, by passing the sequence of samples through a type of low pass filter called a reconstruction filter.
The Nyquist frequency, named after electronic engineer Harry Nyquist, is half of the sampling rate of a discrete signal processing system.

Nyquist–Shannon sampling theorem

sampling theoremNyquist-Shannon sampling theoremNyquist theorem
The approximately double-rate requirement is a consequence of the Nyquist theorem.
It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.

Whittaker–Shannon interpolation formula

interpolation/sampling theoryreconstructingsinc interpolation
The Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal lowpass filter whose input is a sequence of Dirac delta functions that are modulated (multiplied) by the sample values.
When the x[n] sequence represents time samples, at interval T, of a continuous function, the quantity f s = 1/T is known as the sample rate, and f s /2 is the corresponding Nyquist frequency.

Quantization (signal processing)

quantizationquantization errorquantized
An analog-to-digital converter (ADC) can be modeled as two processes: sampling and quantization.

Digital-to-analog converter

DACDACsD/A
One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for ADCs and DACs, but with modern oversampling sigma-delta converters this advantage is less important.
There are several DAC architectures; the suitability of a DAC for a particular application is determined by figures of merit including: resolution, maximum sampling frequency and others.

Low-pass filter

low-passlow pass filterlowpass filter
The original signal is retrievable from a sequence of samples, up to the Nyquist limit, by passing the sequence of samples through a type of low pass filter called a reconstruction filter. The Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal lowpass filter whose input is a sequence of Dirac delta functions that are modulated (multiplied) by the sample values.
A low-pass filter is used as an anti-aliasing filter prior to sampling and for reconstruction in digital-to-analog conversion.

Digital audio

digital musicdigitalaudio
Digital audio uses pulse-code modulation and digital signals for sound reproduction.
The ADC runs at a specified sampling rate and converts at a known bit resolution.

Anti-aliasing filter

anti-aliasinganti-aliasing (AA) filteroptical low-pass filter
The Audio Engineering Society recommends 48 kHz sampling rate for most applications but gives recognition to 44.1 kHz for Compact Disc (CD) and other consumer uses, 32 kHz for transmission-related applications, and 96 kHz for higher bandwidth or relaxed anti-aliasing filtering.
Since the theorem states that unambiguous reconstruction of the signal from its samples is possible when the power of frequencies above the Nyquist frequency is zero, a real anti-aliasing filter trades off between bandwidth and aliasing.

Dirac comb

Sampling functionimpulse traininfinite impulse train
When the time interval between adjacent samples is a constant (T), the sequence of delta functions is called a Dirac comb.
Owing to the Poisson summation formula, in signal processing, the Dirac comb allows modelling sampling by multiplication with it, but it also allows modelling periodization by convolution with it.

Digital Audio Tape

DATDATsDAT recorder
In appearance it is similar to a Compact Cassette, using 3.81 mm / 0.15" (commonly referred to as 4 mm) magnetic tape enclosed in a protective shell, but is roughly half the size at 73 mm × 54 mm × 10.5 mm. The recording is digital rather than analog. DAT can record at sampling rates equal to, as well as higher and lower than a CD (44.1, 48 or 32 kHz sampling rate respectively) at 16 bits quantization. If a comparable digital source is copied without returning to the analogue domain, then the DAT will produce an exact clone, unlike other digital media such as Digital Compact Cassette or non-Hi-MD MiniDisc, both of which use a lossy data reduction system.

44,100 Hz

44.1 kHz44.1kHz44,100 samples per second
In digital audio, 44,100 Hz (alternately represented as 44.1 kHz) is a common sampling frequency.

Signal processing

signal analysissignalsignal processor
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal.

DV

MiniDVDVCAMDVCPRO
Audio can be stored in either of two forms: 16-bit Linear PCM stereo at 48 kHz sampling rate (768 kbit/s per channel, 1.5 Mbit/s stereo), or four nonlinear 12-bit PCM channels at 32 kHz sampling rate (384 kbit/s per channel, 1.5 MBit/s for four channels).

Pulse-code modulation

PCMLPCMLinear PCM
Digital audio uses pulse-code modulation and digital signals for sound reproduction.
In a PCM stream, the amplitude of the analog signal is sampled regularly at uniform intervals, and each sample is quantized to the nearest value within a range of digital steps.

Dirac delta function

Dirac deltadelta functionimpulse
The Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal lowpass filter whose input is a sequence of Dirac delta functions that are modulated (multiplied) by the sample values.
A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the Shah distribution, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis.

Direct Stream Digital

DSDDSD-CD(DSD)
The signal is stored as delta-sigma modulated digital audio, a sequence of single-bit values at a sampling rate of 2.8224 MHz (64 times the CD audio sampling rate of 44.1 kHz, but only at 1⁄32768 of its 16-bit resolution).

Digital signal processing

DSPsignal processingdigital
However, digital signal processing operations can have very high dynamic range, consequently it is common to perform mixing and mastering operations at 32-bit precision and then convert to 16- or 24-bit for distribution.
The signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency.

Delta-sigma modulation

delta-sigmasigma-deltaSigma-delta modulation
One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for ADCs and DACs, but with modern oversampling sigma-delta converters this advantage is less important.
In a conventional ADC, an analog signal is sampled with a sampling frequency and subsequently quantized in a multi-level quantizer into a digital signal.

Compact disc

CDCDsCD single
The Audio Engineering Society recommends 48 kHz sampling rate for most applications but gives recognition to 44.1 kHz for Compact Disc (CD) and other consumer uses, 32 kHz for transmission-related applications, and 96 kHz for higher bandwidth or relaxed anti-aliasing filtering. When it is necessary to capture audio covering the entire 20–20,000 Hz range of human hearing, such as when recording music or many types of acoustic events, audio waveforms are typically sampled at 44.1 kHz (CD), 48 kHz, 88.2 kHz, or 96 kHz.
The format is a two-channel 16-bit PCM encoding at a 44.1 kHz sampling rate per channel.