Sampling (signal processing)

 ( Digital Signal Processing ) 

In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or space; this definition differs from the term's usage in statistics, which refers to a set of such values.

A sampler is a subsystem or operation that extracts samples from a continuous signal. A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points.

The original signal can be reconstructed from a sequence of samples, up to the Nyquist limit, by passing the sequence of samples through a reconstruction filter.

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.

$s(nT)$, for integer values of $n$.

Reconstructing a continuous function from samples is done by interpolation algorithms. The Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal low-pass filter whose input is a sequence of Dirac delta functions that are modulated (multiplied) by the sample values. When the time interval between adjacent samples is a constant $(T)$, the sequence of delta functions is called a Dirac comb. Mathematically, the modulated Dirac comb is equivalent to the product of the comb function with $s(t)$. That mathematical abstraction is sometimes referred to as impulse sampling.

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 × $f\_s$ samples/second = $f\_s/2$, 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.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)

Various types of distortion can occur, including:

• Aliasing. Some amount of aliasing is inevitable because only theoretical, infinitely long, functions can have no frequency content above the Nyquist frequency. Aliasing can be made arbitrarily small by using a 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.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. 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.
• Noise, including thermal sensor noise, 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 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 . The non-linearities of either ADC or DAC are analyzed by replacing the ideal linear function mapping with a proposed nonlinear function.

When it is necessary to capture audio covering the entire 20–20,000 Hz range of auditory system 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.

There has been an industry trend towards sampling rates well beyond the basic requirements: such as 96 kHz and even 192 kHz Even though ultrasonic frequencies are inaudible to humans, recording and mixing at higher sampling rates is effective in eliminating the distortion that can be caused by foldback aliasing. Conversely, ultrasonic sounds may interact with and modulate the audible part of the frequency spectrum (intermodulation distortion), degrading the fidelity. 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 delta-sigma-converters this advantage is less important.

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. 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.

A more complete list of common audio sample rates is:

! Sampling rate
! Use
     

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

In digital video, the temporal sampling rate is defined as the rather the 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 game 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 . A common pixel sampling rate is:

• 13.5 MHz – CCIR 601, 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 chrominance video components shows up as a moiré pattern.

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 $\backslash hat\; s(t)$, by processing the product sequence, $\backslash lefts(nT)\backslash cdot$, through a digital low-pass filter whose cutoff frequency is $B/2$. 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.

• 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

• Matt Pharr, Wenzel Jakob and Greg Humphreys, Physically Based Rendering: From Theory to Implementation, 3rd ed., Morgan Kaufmann, November 2016. . The chapter on sampling ( available online) is nicely written with diagrams, core theory and code sample.

• 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