Sampling (signal processing): Difference between revisions

In

(a continuous signal) to a sequence of samples (a discrete-time signal).

A sample is a value or set of values at a point in time and/or space.

A sampler is a subsystem or operation that extracts samples from a

.

A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points.

Theory

See also: Nyquist–Shannon sampling theorem

Sampling can be done for functions varying in space, time, or any other dimension, and similar results are obtained 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.^[1]  Then the sampled function is given by the sequence:

s(nT),   for integer values of n.

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

Reconstructing a continuous function from samples is done by interpolation algorithms. The

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 whose periodicity is smaller than 2 samples; or equivalently the ratio of cycles to samples exceeds ½ (see

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 large
  order 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. In a capacitor-based sample and hold circuit, aperture error is introduced because the capacitor cannot instantly change voltage thus requiring the sample to have non-zero width.
• Jitter or deviation from the precise sample timing intervals.
• analog circuit
  noise, 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-linear
  effects 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 pass band, 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

.

Applications

Audio sampling

Digital audio uses pulse-code modulation 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.

Sampling rate

When it is necessary to capture audio covering the entire 20–20,000 Hz range of human hearing,^[4]  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.^[5]  The approximately double-rate requirement is a consequence of the

equipment manufacturers chose sampling rates in the region of 50 kHz for this reason.

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

  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 this advantage is less important.

The

A more complete list of common audio sample rates is:

Sampling rate	Use
8,000 Hz	sibilance; ess sounds like eff (/s/, /f /).
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	VoIP and VVoIP communication products.[13]
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.[14]
32,000 Hz	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 ]
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.
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.
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 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.

Bit depth

Audio is typically recorded at 8-, 16-, and 20-bit depth, which yield a theoretical maximum

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

Speech signals, i.e., signals intended to carry only human

sampling and quantization specifications.

Video sampling

This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources in this section. Unsourced material may be challenged and removed. (June 2007) (Learn how and when to remove this message)

625-line) for the visible picture area.

High-definition television (HDTV) uses 720p (progressive), 1080i (interlaced), and 1080p (progressive, also known as Full-HD).

In

. 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:

• 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

and other applications.

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.^[18]

Complex sampling

Complex sampling (I/Q sampling) is the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as

  When one waveform${\displaystyle ,{\hat {s}}(t),}$  is the Hilbert transform of the other waveform${\displaystyle ,s(t),\,}$  the complex-valued function,  ${\displaystyle s_{a}(t)\ {\stackrel {\text{def}}{=}}\ s(t)+j\cdot {\hat {s}}(t),}$  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 2]} More apparently, the equivalent baseband waveform,  ${\displaystyle s_{a}(t)\cdot e^{-j2\pi {\frac {B}{2}}t},}$  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 ${\displaystyle {\hat {s}}(t),}$  by processing the product sequence${\displaystyle ,\left[s(nT)\cdot e^{-j2\pi {\frac {B}{2}}Tn}\right],}$^{[note 3]}  through a digital lowpass filter whose cutoff frequency is B/2.^{[note 4]} 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

• Downsampling
• Upsampling
• Multidimensional sampling
• Sample rate conversion
• Digitizing
• Sample and hold
• Beta encoder
• Kell factor
• Bit rate

Notes

Citations

Further reading

• 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

• 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 Interactive presentation in a web-demo. Institute of Telecommunications, University of Stuttgart