# Sampling rate

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Analog signal (light blue) with a sampled signal (red) with a fixed spacing or sampling rate.

The sampling rate, sample rate, or sampling frequency (${\displaystyle f_{\mathrm {s} }}$) defines the number of samples per unit of time (usually seconds) taken from a continuous signal to make a discrete signal. For time-domain signals, the unit for sampling rate is hertz (inverse seconds, 1/s, s−1), sometimes noted as Sa/s or S/s (samples per second). The reciprocal of the sampling frequency is the sampling period or sampling interval, which is the time between samples.[1]

## Sampling theorem

The Nyquist–Shannon sampling theorem states that perfect reconstruction of a signal is possible when the sampling frequency is greater than twice the maximum frequency of the signal being sampled, or equivalently, when the Nyquist frequency (half the sample rate) exceeds the highest frequency of the signal being sampled. If lower sampling rates are used, the original signal's information may not be completely recoverable from the sampled signal.[2] For example, if a signal has an upper band limit of 100 Hz, a sampling frequency greater than 200 Hz will avoid aliasing and would theoretically allow perfect reconstruction.

The full range of human hearing is between 20 Hz and 20 kHz.[3] The minimum sampling rate that satisfies the sampling theorem for this full bandwidth is 40 kHz. The 44.1 kHz sampling rate used for Compact Disc was chosen for this and other technical reasons.

## Oversampling

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In some cases it is desirable to have a sampling frequency more than twice the desired system bandwidth so that a steep digital filter and a less steep analog anti-aliasing filter can be used in exchange for a steep analog anti-aliasing filter. The reason for wanting a less steep analog anti-aliasing filter is that the digital filter is not subject to any component variations thus always giving the filter response (filtering function) that the designer has chosen. This process is known as oversampling.[4]

## Undersampling

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Conversely, one may sample below the Nyquist rate. For a baseband signal (one that has components from 0 to the band limit), this introduces aliasing, but for a passband signal (one that does not have low frequency components), there are no low frequency signals for the aliases of high frequency signals to collide with, and thus one can sample a high frequency (but narrow bandwidth) signal at a much lower sample rate than the Nyquist rate.

## Audio

In digital audio the most common sampling rates are 44.1 kHz, 48 kHz, 88.2 kHz, 96 kHz and 192 kHz.[5] Lower sampling rates have the benefit of smaller data size and easier storage and transport. Because of the Nyquist–Shannon sampling theorem, sampling rates higher than about 50 kHz to 60 kHz cannot supply more usable information for human listeners. Early professional audio equipment manufacturers chose sampling rates in the region of 50 kHz for this reason. 88.2 kHz and 96 kHz are often used in modern professional audio equipment, along with 44.1 kHz and 48 kHz. Higher rates such as 192 kHz are prone to ultrasonic artifacts causing audible intermodulation distortion, and inaccurate sampling caused by too much speed.[6] The Audio Engineering Society recommends 48 kHz sample rate for most applications but gives recognition to 44.1 kHz for Compact Disc and other consumer uses, 32 kHz for transmission-related application and 96 kHz for higher bandwidth or relaxed anti-aliasing filtering.[7]

A more complete list of common audio sample rates is:

Sampling rate Use
8,000 Hz Telephone and encrypted walkie-talkie, wireless intercom[8][9] and wireless microphone[10] transmission; adequate for human speech but without sibilance; ess sounds like eff (Template:IPAslink, Template:IPAslink).
11,025 Hz One quarter the sampling rate of audio CDs; used for lower-quality PCM, MPEG audio and for audio analysis of subwoofer bandpasses.{{ safesubst:#invoke:Unsubst $B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} 16,000 Hz Wideband frequency extension over standard telephone narrowband 8,000 Hz. Used in most modern VoIP and VVoIP communication products.[11] 22,050 Hz One half the sampling rate of audio CDs; used for lower-quality PCM and MPEG audio and for audio analysis of low frequency energy. Suitable for digitizing early 20th century audio formats such as 78s.[12] 32,000 Hz miniDV digital video camcorder, video tapes with extra channels of audio (e.g. DVCAM with 4 Channels of audio), DAT (LP mode), Germany's Digitales Satellitenradio, NICAM digital audio, used alongside analogue television sound in some countries. High-quality digital wireless microphones.[13] Suitable for digitizing FM radio.{{ safesubst:#invoke:Unsubst$B=

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44,056 Hz Used by digital audio locked to NTSC color video signals (245 lines by 3 samples by 59.94 fields per second = 29.97 frames per second).
44,100 Hz Audio CD, also most commonly used with MPEG-1 audio (VCD, SVCD, MP3). Originally chosen by Sony because it could be recorded on modified video equipment running at either 25 frames per second (PAL) or 30 frame/s (using an NTSC monochrome video recorder) and cover the 20 kHz bandwidth thought necessary to match professional analog recording equipment of the time. A PCM adaptor would fit digital audio samples into the analog video channel of, for example, PAL video tapes using 588 lines by 3 samples by 25 frames per second.
47,250 Hz world's first commercial PCM sound recorder by Nippon Columbia (Denon)
48,000 Hz The standard audio sampling rate used by professional digital video equipment such as tape recorders, video servers, vision mixers and so on. This rate was chosen because it could deliver a 22 kHz frequency response and work with 29.97 frames per second NTSC video - as well as 25 frame/s, 30 frame/s and 24 frame/s systems. With 29.97 frame/s systems it is necessary to handle 1601.6 audio samples per frame delivering an integer number of audio samples only every fifth video frame.[7] Also used for sound with consumer video formats like DV, digital TV, DVD, and films. The professional Serial Digital Interface (SDI) and High-definition Serial Digital Interface (HD-SDI) used to connect broadcast television equipment together uses this audio sampling frequency. Most professional audio gear uses 48 kHz sampling, including mixing consoles, and digital recording devices.
50,000 Hz First commercial digital audio recorders from the late 70s from 3M and Soundstream.
50,400 Hz Sampling rate used by the Mitsubishi X-80 digital audio recorder.
88,200 Hz Sampling rate used by some professional recording equipment when the destination is CD (multiples of 44,100 Hz). Some pro audio gear uses (or is able to select) 88.2 kHz sampling, including mixers, EQs, compressors, reverb, crossovers and recording devices.
96,000 Hz DVD-Audio, some LPCM DVD tracks, BD-ROM (Blu-ray Disc) audio tracks, HD DVD (High-Definition DVD) audio tracks. Some professional recording and production equipment is able to select 96 kHz sampling. This sampling frequency is twice the 48 kHz standard commonly used with audio on professional equipment.
176,400 Hz Sampling rate used by HDCD recorders and other professional applications for CD production.
192,000 Hz DVD-Audio, some LPCM DVD tracks, BD-ROM (Blu-ray Disc) audio tracks, and HD DVD (High-Definition DVD) audio tracks, High-Definition audio recording devices and audio editing software. This sampling frequency is four times the 48 kHz standard commonly used with audio on professional video equipment.
352,800 Hz Digital eXtreme Definition, used for recording and editing Super Audio CDs, as 1-bit DSD is not suited for editing. Eight times the frequency of 44.1 kHz.
2,822,400 Hz SACD, 1-bit delta-sigma modulation process known as Direct Stream Digital, co-developed by Sony and Philips.
5,644,800 Hz Double-Rate DSD, 1-bit Direct Stream Digital at 2x the rate of the SACD. Used in some professional DSD recorders.

## Video systems

In digital video, the temporal sampling rate is defined the frame rate – or rather the field rate – rather than the notional pixel clock. The image sampling frequency is the repetition rate of the sensor integration period. Since the integration period may be significantly shorter than the time between repetitions, the sampling frequency can be different from the inverse of the sample time:

• 50 Hz – PAL video
• 60 / 1.001 Hz ~= 59.94 Hz – NTSC video

Video digital-to-analog converters operate in the megahertz range (from ~3 MHz for low quality composite video scalers in early games consoles, to 250 MHz or more for the highest-resolution VGA output).

When analog video is converted to digital video, a different sampling process occurs, this time at the pixel frequency, corresponding to a spatial sampling rate along scan lines. A common pixel sampling rate is:

Spatial sampling in the other direction is determined by the spacing of scan lines in the raster. The sampling rates and resolutions in both spatial directions can be measured in units of lines per picture height.

Spatial aliasing of high-frequency luma or chroma video components shows up as a moiré pattern.

## References

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2. C. E. Shannon, "Communication in the presence of noise", Proc. Institute of Radio Engineers, vol. 37, no.1, pp. 10–21, Jan. 1949. Reprint as classic paper in: Proc. IEEE, Vol. 86, No. 2, (Feb 1998)
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