Sampling (signal processing)
In
A sampler is a subsystem or operation that extracts samples from a
The original signal can be reconstructed from a sequence of samples, up to the
Theory
Functions of space, time, or any other dimension can be sampled, and similarly in two or more dimensions.
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 or sampling period.[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 the average number of samples obtained in one second, thus fs = 1/T, with the unit samples per second, sometimes referred to as hertz, for example e.g. 48 kHz is 48,000 samples per second.
Reconstructing a continuous function from samples is done by interpolation algorithms. The
Most sampled signals are not simply stored and reconstructed. The fidelity of a theoretical reconstruction is a common measure of the effectiveness of sampling. That fidelity is reduced when s(t) contains frequency components whose cycle length (period) is less than 2 sample intervals (see Aliasing). The corresponding frequency limit, in cycles per second (hertz), is 0.5 cycle/sample × fs samples/second = fs/2, known as the Nyquist frequency of the sampler. Therefore, s(t) is usually the output of a low-pass 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]
Practical considerations
In practice, the continuous signal is sampled using an analog-to-digital converter (ADC), a device with various physical limitations. This results in deviations from the theoretically perfect reconstruction, collectively referred to as distortion.
Various types of distortion can occur, including:
- sufficiently largeorder of the anti-aliasing filter.
- Aperture error results from the fact that the sample is obtained as a time average within a sampling region, rather than just being equal to the signal value at the sampling instant.[4] In a capacitor-based sample and hold circuit, aperture errors are introduced by multiple mechanisms. For example, the capacitor cannot instantly track the input signal and the capacitor can not instantly be isolated from the input signal.
- Jitter or deviation from the precise sample timing intervals.
- analog circuitnoise, etc.
- Slew rate limit error, caused by the inability of the ADC input value to change sufficiently rapidly.
- Quantization as a consequence of the finite precision of words that represent the converted values.
- Error due to other non-lineareffects of the mapping of input voltage to converted output value (in addition to the effects of quantization).
Although the use of oversampling can completely eliminate aperture error and aliasing by shifting them out of the passband, this technique cannot be practically used above a few GHz, and may be prohibitively expensive at much lower frequencies. Furthermore, while oversampling can reduce quantization error and non-linearity, it cannot eliminate these entirely. Consequently, practical ADCs at audio frequencies typically do not exhibit aliasing, aperture error, and are not limited by quantization error. Instead, analog noise dominates. At RF and microwave frequencies where oversampling is impractical and filters are expensive, aperture error, quantization error and aliasing can be significant limitations.
Jitter, noise, and quantization are often analyzed by modeling them as random errors added to the sample values. Integration and zero-order hold effects can be analyzed as a form of low-pass filtering. The non-linearities of either ADC or DAC are analyzed by replacing the ideal linear function mapping with a proposed nonlinear function[disambiguation needed].
Applications
Audio sampling
Digital audio uses pulse-code modulation (PCM) and digital signals for sound reproduction. This includes analog-to-digital conversion (ADC), digital-to-analog conversion (DAC), storage, and transmission. In effect, the system commonly referred to as digital is in fact a discrete-time, discrete-level analog of a previous electrical analog. While modern systems can be quite subtle in their methods, the primary usefulness of a digital system is the ability to store, retrieve and transmit signals without any loss of quality.
When it is necessary to capture audio covering the entire 20–20,000 Hz range of
There has been an industry trend towards sampling rates well beyond the basic requirements: such as 96 kHz and even 192 kHz
The Audio Engineering Society recommends 48 kHz sampling rate for most applications but gives recognition to 44.1 kHz for CD and other consumer uses, 32 kHz for transmission-related applications, and 96 kHz for higher bandwidth or relaxed anti-aliasing filtering.[9] Both Lavry Engineering and J. Robert Stuart state that the ideal sampling rate would be about 60 kHz, but since this is not a standard frequency, recommend 88.2 or 96 kHz for recording purposes.[10][11][12][13]
A more complete list of common audio sample rates is:
Sampling rate | Use |
---|---|
8,000 Hz | /)). |
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.[citation needed] |
16,000 Hz | ] |
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 AM Radio.[15]
|
32,000 Hz | DVCAM with four 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.[16] Suitable for digitizing FM radio.[citation needed ]
|
37,800 Hz | CD-XA audio |
44,056 Hz | Used by digital audio locked to frames per second ).
|
44,100 Hz | 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 3 samples per line, 588 lines per frame, 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 reconstruct frequencies up to 22 kHz 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. (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.
|
64,000 Hz | Uncommonly used, but supported by some hardware[17][18] and software.[19][20] |
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 | 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. Four times the frequency of 44.1 kHz.
|
192,000 Hz | 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 Direct Stream Digital (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 2× the rate of the SACD. Used in some professional DSD recorders. |
11,289,600 Hz | Quad-Rate DSD, 1-bit Direct Stream Digital at 4× the rate of the SACD. Used in some uncommon professional DSD recorders. |
22,579,200 Hz | Octuple-Rate DSD, 1-bit Direct Stream Digital at 8× the rate of the SACD. Used in rare experimental DSD recorders. Also known as DSD512. |
45,158,400 Hz | Sexdecuple-Rate DSD, 1-bit Direct Stream Digital at 16× the rate of the SACD. Used in rare experimental DSD recorders. Also known as DSD1024.[B] |
Bit depth
Audio is typically recorded at 8-, 16-, and 24-bit depth, which yield a theoretical maximum
Speech sampling
Speech signals, i.e., signals intended to carry only human
Video sampling
This section needs additional citations for verification. (June 2007) |
High-definition television (HDTV) uses 720p (progressive), 1080i (interlaced), and 1080p (progressive, also known as Full-HD).
In
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:
- 13.5 MHz – D1 video
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.
3D sampling
The process of
Undersampling
When a
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.[23]
Complex sampling
Complex sampling (or I/Q sampling) is the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as
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 by processing the product sequence[E] through a digital low-pass filter whose cutoff frequency is B/2.[F] 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
- Crystal oscillator frequencies
- Downsampling
- Upsampling
- Multidimensional sampling
- In-phase and quadrature components and I/Q data
- Sample rate conversion
- Digitizing
- Sample and hold
- Beta encoder
- Kell factor
- Bit rate
- Normalized frequency
Notes
- sample size" in statistics.
- ^ Even higher DSD sampling rates exist, but the benefits of those are likely imperceptible, and the size of those files would be humongous.
- ^ Sample-pairs are also sometimes viewed as points on a constellation diagram.
- ^ 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.
- ^ When s(t) is sampled at the Nyquist frequency (1/T = 2B), the product sequence simplifies to
- ^ The sequence of complex numbers is convolved with the impulse response of a filter with real-valued coefficients. That is equivalent to separately filtering the sequences of real parts and imaginary parts and reforming complex pairs at the outputs.
References
- ISBN 0412083914.
- ISBN 9788120338593.
- 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) Archived 2010-02-08 at the Wayback Machine
- ^ H.O. Johansson and C. Svensson, "Time resolution of NMOS sampling switches", IEEE J. Solid-State Circuits Volume: 33, Issue: 2, pp. 237–245, Feb 1998.
- ^ D'Ambrose, Christoper; Choudhary, Rizwan (2003). Elert, Glenn (ed.). "Frequency range of human hearing". The Physics Factbook. Retrieved 2022-01-22.
- ISBN 978-0240812731.
- ^ "Digital Pro Sound". Retrieved 8 January 2014.
- ^ 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.
in many cases, we can hear the sound of higher sample rates not because they are more transparent, but because they are less so. They can actually introduce unintended distortion in the audible spectrum
- ^ 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
- ^ Lavry, Dan (May 3, 2012). "The Optimal Sample Rate for Quality Audio" (PDF). Lavry Engineering Inc.
Although 60 KHz would be closer to the ideal; given the existing standards, 88.2 KHz and 96 KHz are closest to the optimal sample rate.
- ^ Lavry, Dan. "The Optimal Sample Rate for Quality Audio". Gearslutz. Retrieved 2018-11-10.
I am trying to accommodate all ears, and there are reports of few people that can actually hear slightly above 20KHz. I do think that 48KHz is pretty good compromise, but 88.2 or 96KHz yields some additional margin.
- ^ Lavry, Dan. "To mix at 96k or not?". Gearslutz. Retrieved 2018-11-10.
Nowdays there are a number of good designers and ear people that find 60-70KHz sample rate to be the optimal rate for the ear. It is fast enough to include what we can hear, yet slow enough to do it pretty accurately.
- CiteSeerX 10.1.1.501.6731.
both psychoacoustic analysis and experience tell us that the minimum rectangular channel necessary to ensure transparency uses linear PCM with 18.2-bit samples at 58kHz. ... there are strong arguments for maintaining integer relationships with existing sampling rates – which suggests that 88.2kHz or 96kHz should be adopted.
- ^ "Cisco VoIP Phones, Networking and Accessories - VoIP Supply".
- ^ "The restoration procedure – part 1". Restoring78s.co.uk. 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.
- ^ "Zaxcom digital wireless transmitters". Zaxcom.com. Archived from the original on 2011-02-09. Retrieved 2011-01-18.
- ^ "RME: Hammerfall DSP 9632". www.rme-audio.de. Retrieved 2018-12-18.
Supported sample frequencies: Internally 32, 44.1, 48, 64, 88.2, 96, 176.4, 192 kHz.
- ^ "SX-S30DAB | Pioneer". www.pioneer-audiovisual.eu. Retrieved 2018-12-18.
Supported sampling rates: 44.1 kHz, 48 kHz, 64 kHz, 88.2 kHz, 96 kHz, 176.4 kHz, 192 kHz
- ^ Cristina Bachmann, Heiko Bischoff; Schütte, Benjamin. "Customize Sample Rate Menu". Steinberg WaveLab Pro. Retrieved 2018-12-18.
Common Sample Rates: 64 000 Hz
- ^ "M Track 2x2M Cubase Pro 9 can ́t change Sample Rate". M-Audio. Retrieved 2018-12-18.
[Screenshot of Cubase]
- ^ "MT-001: Taking the Mystery out of the Infamous Formula, "SNR=6.02N + 1.76dB," and Why You Should Care" (PDF).
- ^
Walt Kester (2003). Mixed-signal and DSP design techniques. Newnes. p. 20. ISBN 978-0-7506-7611-3. Retrieved 8 January 2014.
- ISBN 1563962837.
Further reading
- Matt Pharr, Wenzel Jakob and Greg Humphreys, Physically Based Rendering: From Theory to Implementation, 3rd ed., Morgan Kaufmann, November 2016. ISBN 978-0128006450. The chapter on sampling (available online) is nicely written with diagrams, core theory and code sample.
External links
- Journal devoted to Sampling Theory
- I/Q Data for Dummies – a page trying to answer the question Why I/Q Data?
- Sampling of analog signals – an interactive presentation in a web-demo at the Institute of Telecommunications, University of Stuttgart