Sampling (signal processing)

In arresting processing, sampling is the abridgement of a connected arresting to a detached signal. A accepted archetype is the about-face of a complete beachcomber (a connected signal) to a arrangement of samples (a discrete-time signal).

A sample refers to a amount or set of ethics at a point in time and/or space.

A adornment is a subsystem or operation that extracts samples from a connected signal. A abstract ideal adornment produces samples agnate to the direct amount of the connected arresting at the adapted points.

Theory

Sampling can be done for functions capricious in space, time, or any added dimension, and agnate after-effects are acquired in two or added dimensions.

For functions that alter with time, let s(t) be a connected action to be sampled, and let sampling be performed by barometer the amount of the connected action every T seconds, which is alleged the sampling interval. Thus, the sampled action is accustomed by the sequence:

s(nT), for accumulation ethics of n.

The sampling abundance or sampling amount fs is authentic as the amount of samples acquired in one additional (samples per second), appropriately fs = 1/T.

Although a lot of of the arresting is alone by the sampling process, it is still about accessible to accurately reconstruct a arresting from the samples if the arresting is band-limited. A acceptable action for absolute about-face is that the non-zero allocation of the signal's Fourier transform be independent aural the breach –fs/2, fs/2.

The abundance fs/2 is alleged the Nyquist abundance of the sampling system. Without an anti-aliasing filter, frequencies college than the Nyquist abundance will access the samples in a way that is misinterpreted by the Whittaker–Shannon departure formula, the archetypal about-face formula. For details, see Aliasing.

Practical implications

In practice, the connected arresting is sampled application an analog-to-digital advocate (ADC), a accessory with assorted concrete limitations. This after-effects in deviations from the apparently absolute reconstruction, collectively referred to as distortion.

Various types of baloney can occur, including:

Aliasing. A arrangement of the sampling assumption is that the arresting be bandlimited. However, in practice, no time-limited arresting can be bandlimited. Since signals of absorption are about consistently time-limited (e.g., at a lot of spanning the lifetime of the sampling accessory in question), it follows that they are not bandlimited. However, by designing a adornment with an adapted bouncer band, it is accessible to access achievement that is as authentic as necessary.

Affiliation aftereffect or breach effect. This after-effects from the actuality that the sample is acquired as a time boilerplate aural a sampling region, rather than just getting according to the arresting amount at the sampling instant. The affiliation aftereffect is readily apparent in photography if the acknowledgment is too continued and creates a becloud in the image. An ideal camera would accept an acknowledgment time of zero. In a capacitor-based sample and authority circuit, the affiliation aftereffect is alien because the capacitor cannot instantly change voltage appropriately acute the sample to accept non-zero width.

Jitter or aberration from the absolute sample timing intervals.

Noise, including thermal sensor noise, analog ambit noise, etc.

Slew amount absolute error, acquired by an disability for an ADC achievement amount to change abundantly rapidly.

Quantization as a aftereffect of the bound attention of words that represent the adapted values.

Absurdity due to added non-linear furnishings of the mapping of ascribe voltage to adapted achievement amount (in accession to the furnishings of quantization).

The conventional, applied digital-to-analog advocate (DAC) does not achievement a arrangement of dirac impulses (such that, if alluringly low-pass filtered, aftereffect in the aboriginal arresting afore sampling) but instead achievement a arrangement of piecewise connected ethics or ellipsoidal pulses. This agency that there is an inherent aftereffect of the zero-order authority on the able abundance acknowledgment of the DAC consistent in a balmy roll-off of accretion at the college frequencies (a 3.9224 dB accident at the Nyquist frequency). This zero-order authority aftereffect is a aftereffect of the authority activity of the DAC and is not due to the sample and authority that ability announce a accepted ADC as is generally misunderstood. The DAC can aswell ache errors from jitter, noise, slewing, and non-linear mapping of ascribe amount to achievement voltage.

Jitter, noise, and quantization are generally analyzed by clay them as accidental errors added to the sample values. Affiliation and zero-order authority furnishings can be analyzed as a anatomy of low-pass filtering. The non-linearities of either ADC or DAC are analyzed by replacing the ideal beeline action mapping with a proposed nonlinear function.

Applications

Audio sampling

Digital audio uses pulse-code accentuation and agenda signals for complete reproduction. This includes analog-to-digital about-face (ADC), digital-to-analog about-face (DAC), storage, and transmission. In effect, the arrangement frequently referred to as agenda is in actuality a discrete-time, discrete-level analog of a antecedent electrical analog. While avant-garde systems can be absolutely attenuate in their methods, the primary account of a agenda arrangement is the adeptness to store, retrieve and address signals after any accident of quality.

edit Sampling rate

When it is all-important to abduction audio accoutrement the absolute 20–20,000 Hz ambit of animal hearing, such as if recording music or abounding types of acoustic events, audio waveforms are about sampled at 44.1 kHz (CD), 48 kHz (professional audio), or 96 kHz. The about double-rate claim is a aftereffect of the Nyquist theorem.

There has been an industry trend appear sampling ante able-bodied above the basal requirements; 96 kHz and even 192 kHz are available.1 This is in adverse with class experiments, which accept bootless to appearance that accelerated frequencies are aural to animal observers; about in some cases accelerated sounds do collaborate with and attune the aural allotment of the abundance spectrum (intermodulation distortion). It is noteworthy that intermodulation baloney is not present in the reside audio and so it represents an bogus blush to the reside sound.2

One advantage of college sampling ante is that they can relax the low-pass clarify architecture requirements for ADCs and DACs, but with avant-garde oversampling sigma-delta converters this advantage is beneath important.

edit Bit abyss (quantization)

Audio is about recorded at 8-, 16-, and 20-bit depth, which crop a abstract best arresting to quantization babble arrangement (SQNR) for a authentic sine beachcomber of, approximately, 49.93 dB, 98.09 dB and 122.17 dB.3 Eight-bit audio is about not acclimated due to arresting and inherent quantization babble (low best SQNR), although the A-law and u-law 8-bit encodings backpack added resolution into 8 $.25 while access absolute harmonic distortion. CD superior audio is recorded at 16-bit. In practice, not abounding customer stereos can aftermath added than about 90 dB of activating range, although some can beat 100 dB. Thermal babble banned the accurate amount of $.25 that can be acclimated in quantization. Few analog systems accept arresting to babble ratios (SNR) beyond 120 dB; consequently, few situations will crave added than 20-bit quantization.

For playback and not recording purposes, a able assay of archetypal programme levels throughout an audio arrangement reveals that the capabilities of well-engineered 16-bit actual far beat those of the actual best hi-fi systems, with the microphone babble and loudspeaker allowance getting the absolute attached factorscitation needed.

edit Accent sampling

Speech signals, i.e., signals advised to backpack alone animal speech, can usually be sampled at a abundant lower rate. For a lot of phonemes, about all of the activity is independent in the 5Hz-4 kHz range, acceptance a sampling amount of 8 kHz. This is the sampling amount acclimated by about all telephony systems, which use the G.711 sampling and quantization specifications.

edit Video sampling

Standard-definition television (SDTV) uses either 720 by 480 pixels (US NTSC 525-line) or 704 by 576 pixels (UK PAL 625-line) for the arresting account area.

High-definition television (HDTV) is currently affective appear three standards referred to as 720p (progressive), 1080i (interlaced) and 1080p (progressive, aswell accepted as Full-HD) which all 'HD-Ready' sets will be able to display.

Undersampling

When one samples a bandpass arresting at a amount lower than the Nyquist rate, the samples are according to samples of a low-frequency alias of the high-frequency signal; the aboriginal arresting will still be abnormally represented and recoverable if the spectrum of its alias does not cantankerous over bisected the sampling rate. Such undersampling is aswell accepted as bandpass sampling, harmonic sampling, IF sampling, and absolute IF to agenda conversion.4

Oversampling

Oversampling is acclimated in a lot of avant-garde analog-to-digital converters to abate the baloney alien by applied digital-to-analog converters, such as a zero-order authority instead of idealizations like the Whittaker–Shannon departure formula.

Complex sampling

Complex sampling refers to the accompanying sampling of two different, but related, waveforms, consistent in pairs of samples that are after advised as circuitous numbers. Usually one waveform, \hat s(t), is the Hilbert transform of the added waveform, s(t),\, and the complex-valued function, s_a(t)\ \stackrel{\text{def}}{=}\ s(t) + j\cdot \hat s(t), is alleged an analytic signal, whose Fourier transform is aught for all abrogating ethics of frequency. In that case, the Nyquist amount for a waveform with no frequencies ≥ B can be bargain to just B (complex samples/sec), instead of 2B (real samples/sec).note 1 Added apparently, the agnate baseband waveform, s_a(t)\cdot e^{-j 2\pi \frac{B}{2} t}, aswell has a Nyquist amount of B, because all of its non-zero abundance agreeable is confused into the breach -B/2, B/2).

Although complex-valued samples can be acquired as declared above, they are abundant added frequently created by manipulating samples of a real-valued waveform. For instance, the agnate baseband waveform can be created after absolutely accretion \hat s(t), by processing the artefact sequence, \left s(nT)\cdot e^{-j 2 \pi \frac{B}{2}Tn}\right ,note 2 through a agenda lowpass clarify whose blow abundance is B/2.note 3 Accretion alone every added sample of the achievement arrangement reduces the sample-rate adequate with the bargain Nyquist rate. The aftereffect is bisected as abounding complex-valued samples as the aboriginal amount of absolute samples. No advice is lost, and the aboriginal s(t) waveform can be recovered, if necessary.