Sampling (signal processing)

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See also: Nyquist–Shannon sampling theorem


For functions that vary with time, let s(t) be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every T seconds, which is called the sampling interval.[1]  Then the sampled function is given by the sequence:

s(nT),   for integer values of n.

The sampling frequency or sampling rate, fs, is defined as the number of samples obtained in one second (samples per second), thus fs = 1/T.


Most sampled signals are not simply stored and reconstructed. But the fidelity of a theoretical reconstruction is a customary measure of the effectiveness of sampling. That fidelity is reduced when s(t) contains frequency components higher than fs/2, which is known as the Nyquist frequency of the sampler.[note 1]  Therefore s(t) is usually the output of a lowpass filter, functionally known as an anti-aliasing filter. Without an anti-aliasing filter, frequencies higher than the Nyquist frequency will influence the samples in a way that is misinterpreted by the interpolation process.[3] For details, see Aliasing.

Practical considerations[edit]


Various types of distortion can occur, including:

  • Jitter or deviation from the precise sample timing intervals.




Audio sampling[edit]


Sampling rate[edit]


There has been an industry trend towards sampling rates well beyond the basic requirements: such as 96 kHz and even 192 kHz[6]  This is in contrast with laboratory experiments, which have failed to show that ultrasonic frequencies are audible to human observers; however in some cases ultrasonic sounds do interact with and modulate the audible part of the frequency spectrum (intermodulation distortion).[7]  It is noteworthy that intermodulation distortion is not present in the live audio and so it represents an artificial coloration to the live sound.[8]  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.

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.[9]

A more complete list of common audio sample rates is:

Sampling rate Use
8,000 Hz Telephone and encrypted walkie-talkie, wireless intercom[10][11] and wireless microphone[12] transmission; adequate for human speech but without sibilance; ess sounds like eff (/s/, /f/).
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.[15] Suitable for digitizing FM radio.[citation needed]
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).
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.
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.

Bit depth[edit]

See also: Audio bit depth

Audio is typically recorded at 8-, 16-, and 20-bit depth, which yield a theoretical maximum Signal-to-quantization-noise ratio (SQNR) for a pure sine wave of, approximately, 49.93 dB, 98.09 dB and 122.17 dB.[16] CD quality audio uses 16-bit samples. Thermal noise limits the true number of bits that can be used in quantization. Few analog systems have signal to noise ratios (SNR) exceeding 120 dB. 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.

Speech sampling[edit]

Speech signals, i.e., signals intended to carry only human speech, can usually be sampled at a much lower rate. For most phonemes, almost all of the energy is contained in the 5Hz-4 kHz range, allowing a sampling rate of 8 kHz. This is the sampling rate used by nearly all telephony systems, which use the G.711 sampling and quantization specifications.

Video sampling[edit]




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






3D sampling[edit]


Main article: Undersampling



Main article: Oversampling

Oversampling is used in most modern analog-to-digital converters to reduce the distortion introduced by practical digital-to-analog converters, such as a zero-order hold instead of idealizations like the Whittaker–Shannon interpolation formula.[18]

Complex sampling[edit]

Complex sampling (I/Q sampling) refers to the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as complex numbers.[note 2]  When one waveform, \hat s(t),  is the Hilbert transform of the other waveform, s(t),\,  the complex-valued function,  s_a(t)\ \stackrel{\text{def}}{=}\ s(t) + j\cdot \hat s(t),  is called an analytic signal,  whose Fourier transform is zero for all negative values of frequency. In that case, the Nyquist rate for a waveform with no frequencies ≥ B can be reduced to just B (complex samples/sec), instead of 2B (real samples/sec).[note 3] More apparently, the equivalent baseband waveform,  s_a(t)\cdot e^{-j 2\pi \frac{B}{2} t},  also has a Nyquist rate of B, because all of its non-zero frequency content is shifted into the interval [-B/2, B/2).

Although complex-valued samples can be obtained as described above, they are also created by manipulating samples of a real-valued waveform. For instance, the equivalent baseband waveform can be created without explicitly computing \hat s(t),  by processing the product sequence, \left [s(nT)\cdot e^{-j 2 \pi \frac{B}{2}Tn}\right ],[note 4]  through a digital lowpass filter whose cutoff frequency is B/2.[note 5] Computing only every other sample of the output sequence reduces the sample-rate commensurate with the reduced Nyquist rate. The result is half as many complex-valued samples as the original number of real samples. No information is lost, and the original s(t) waveform can be recovered, if necessary.

See also[edit]


  1. ^ In this context, the factor of ½ has units of cycles per sample, as explained at Aliasing#Sampling_sinusoidal_functions. Therefore, with fs in samples per second, the Nyquist frequency has units of Hz.
  2. ^ Sample-pairs are also sometimes viewed as points on a constellation diagram.
  3. ^ When the complex sample-rate is B, a frequency component at 0.6 B, for instance, will have an alias at −0.4 B, which is unambiguous because of the constraint that the pre-sampled signal was analytic. Also see Aliasing#Complex_sinusoids
  4. ^ When s(t) is sampled at the Nyquist frequency (1/T = 2B), the product sequence simplifies to \left [s(nT)\cdot (-j)^n\right ].


  6. ^ "Digital Pro Sound". Retrieved 8 January 2014. 
  7. ^ Colletti, Justin (February 4, 2013). "The Science of Sample Rates (When Higher Is Better—And When It Isn’t)". Trust Me I'm A Scientist. Retrieved February 6, 2013. 
  8. ^ David Griesinger. "Perception of mid frequency and high frequency intermodulation distortion in loudspeakers, and its relationship to high-definition audio" (Powerpoint presentation). Archived from the original on 2008-05-01. 
  9. ^ a b AES5-2008: AES recommended practice for professional digital audio - Preferred sampling frequencies for applications employing pulse-code modulation, Audio Engineering Society, 2008, retrieved 2010-01-18 
  10. ^ HME DX200 encrypted wireless intercom[dead link]
  11. ^ "Telex BTR-1 encrypted wireless intercom". Retrieved 2011-01-18. 
  12. ^ "Telex SAFE-1000 wireless microphone". Retrieved 2011-01-18. 
  13. ^[unreliable source?]
  14. ^ "The restoration procedure - part 1". Archived from the original on 2009-09-14. Retrieved 2011-01-18. "For most records a sample rate of 22050 in stereo is adequate. An exception is likely to be recordings made in the second half of the century, which may need a sample rate of 44100." 
  15. ^ "Zaxcom digital wireless transmitters". Retrieved 2011-01-18. 
  16. ^ "MT-001: Taking the Mystery out of the Infamous Formula, "SNR=6.02N + 1.76dB," and Why You Should Care". 

Further reading[edit]

  • Matt Pharr and Greg Humphreys, Physically Based Rendering: From Theory to Implementation, Morgan Kaufmann, July 2004. ISBN 0-12-553180-X. The chapter on sampling (available online) is nicely written with diagrams, core theory and code sample.

External links[edit]