Sampling (signal processing)
In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave (a continuous signal) to a sequence of samples (a discrete-time signal).
A sample refers to 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 continuous signal.
A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points.
- 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. Thus, 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.
Reconstructing a continuous function from samples is done by interpolation algorithms. The Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal lowpass 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 purely mathematical function is often loosely referred to as the sampled signal.
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 Hz, which is known as the Nyquist frequency of the sampler. 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. For details, see Aliasing.
Practical implications 
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:
- Aliasing. A precondition of the sampling theorem is that the signal be bandlimited. However, in practice, no time-limited signal can be bandlimited. Since signals of interest are almost always time-limited (e.g., at most spanning the lifetime of the sampling device in question), it follows that they are not bandlimited. However, by designing a sampler with an appropriate guard band, it is possible to obtain output that is as accurate as necessary.
- 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.
- 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).
The conventional, practical digital-to-analog converter (DAC) does not output a sequence of dirac impulses (such that, if ideally low-pass filtered, result in the original signal before sampling) but instead output a sequence of piecewise constant values or rectangular pulses. This means that there is an inherent effect of the zero-order hold on the effective frequency response of the DAC resulting in a mild roll-off of gain at the higher frequencies (a 3.9224 dB loss at the Nyquist frequency). This zero-order hold effect is a consequence of the hold action of the DAC and is not due to the sample and hold that might precede a conventional ADC as is often misunderstood.
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.
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, such as when recording music or many types of acoustic events, audio waveforms are typically sampled at 44.1 kHz (CD), 48 kHz (professional audio), or 96 kHz. The approximately double-rate requirement is a consequence of the Nyquist theorem.
There has been an industry trend towards sampling rates well beyond the basic requirements; 96 kHz and even 192 kHz are available. 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). It is noteworthy that intermodulation distortion is not present in the live audio and so it represents an artificial coloration to the live sound.
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.
Bit depth (quantization) 
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. Eight-bit audio is generally not used due to prominent and inherent quantization noise (low maximum SQNR), although the A-law and u-law 8-bit encodings pack more resolution into 8 bits while increase total harmonic distortion. CD quality audio is recorded at 16-bit. In practice, not many consumer stereos can produce more than about 90 dB of dynamic range, although some can exceed 100 dB. 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; consequently, few situations will require more than 20-bit quantization.
For playback and not recording purposes, a proper analysis of typical programme levels throughout an audio system reveals that the capabilities of well-engineered 16-bit material far exceed those of the very best hi-fi systems, with the microphone noise and loudspeaker headroom being the real limiting factors.
Speech sampling 
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 
High-definition television (HDTV) is currently moving towards three standards referred to as 720p (progressive), 1080i (interlaced) and 1080p (progressive, also known as Full-HD) which all 'HD-Ready' sets will be able to display.
When a bandpass signal is sampled slower than its Nyquist rate, the samples are indistinguishable from samples of a low-frequency alias of the high-frequency signal. That is often done purposefully in such a way that the lowest-frequency alias satisfies the Nyquist criterion, because the bandpass signal is still uniquely represented and recoverable. Such undersampling is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF to digital conversion.
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.
Complex sampling 
Complex sampling refers to the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as complex numbers. Usually one waveform is the Hilbert transform of the other waveform and the complex-valued function, 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 1] More apparently, the equivalent baseband waveform, 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 much more commonly 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[note 2] through a digital lowpass filter whose cutoff frequency is B/2.[note 3] 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.
- 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.
See also 
- Beta encoder
- Kell factor
- Digital signal processing
- Multidimensional sampling
- 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.
- Shannon, Claude E., Communications in the presence of noise, Proc. IRE, vol. 37, pp. 10–21, Jan. 1949.