 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 discretetime 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.
Contents
Theory
 See also: Nyquist–Shannon sampling theorem
Sampling can be done for signals varying in space, time, or any other dimension, and similar results are obtained in two or more dimensions.
For signals that vary with time, let x(t) be a continuous signal to be sampled, and let sampling be performed by measuring the value of the continuous signal every T seconds, which is called the sampling interval. Thus, the sampled signal x[n] given by:
 x[n] = x(nT), with n = 0, 1, 2, 3, ...
The sampling frequency or sampling rate f_{s} is defined as the number of samples obtained in one second, or f_{s} = 1/T. The sampling rate is measured in hertz or in samples per second.
It is possible under some circumstances to reconstruct the original signal completely and exactly (perfect reconstruction).
The Nyquist–Shannon sampling theorem provides a sufficient (but not always necessary) condition under which perfect reconstruction is possible. The sampling theorem guarantees that bandlimited signals (i.e., signals which have a maximum frequency) can be reconstructed perfectly from their sampled version, if the sampling rate is more than twice the maximum frequency. Reconstruction in this case can be achieved using the Whittaker–Shannon interpolation formula.
The frequency equal to onehalf of the sampling rate is therefore a bound on the highest frequency that can be unambiguously represented by the sampled signal. This frequency (half the sampling rate) is called the Nyquist frequency of the sampling system. Frequencies above the Nyquist frequency f_{N} can be observed in the sampled signal, but their frequency is ambiguous. That is, a frequency component with frequency f cannot be distinguished from other components with frequencies Nf_{N} + f and Nf_{N} – f for nonzero integers N. This ambiguity is called aliasing. To handle this problem as gracefully as possible, most analog signals are filtered with an antialiasing filter (usually a lowpass filter with cutoff near the Nyquist frequency) before conversion to the sampled discrete representation.
Observation period
The observation period is the span of time during which a series of data samples are collected at regular intervals. More broadly, it can refer to any specific period during which a set of data points is gathered, regardless of whether or not the data is periodic in nature. Thus a researcher might study the incidence of earthquakes and tsunamis over a particular time period, such as a year or a century.
The observation period is simply the span of time during which the data is studied, regardless of whether data so gathered represents a set of discrete events having arbitrary timing within the interval, or whether the samples are explicitly bound to specified subintervals.
Practical implications
In practice, the continuous signal is sampled using an analogtodigital converter (ADC), a nonideal device with various physical limitations. This results in deviations from the theoretically perfect reconstruction capabilities, 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 timelimited signal can be bandlimited. Since signals of interest are almost always timelimited (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.
 Integration effect or aperture effect. This 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. The integration effect is readily noticeable in photography when the exposure is too long and creates a blur in the image. An ideal camera would have an exposure time of zero. In a capacitorbased sample and hold circuit, the integration effect is introduced because the capacitor cannot instantly change voltage thus requiring the sample to have nonzero 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 an inability for an ADC output value to change sufficiently rapidly.
 Quantization as a consequence of the finite precision of words that represent the converted values.
 Error due to other nonlinear effects of the mapping of input voltage to converted output value (in addition to the effects of quantization).
The conventional, practical digitaltoanalog converter (DAC) does not output a sequence of dirac impulses (such that, if ideally lowpass 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 zeroorder hold on the effective frequency response of the DAC resulting in a mild rolloff of gain at the higher frequencies (a 3.9224 dB loss at the Nyquist frequency). This zeroorder 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. The DAC can also suffer errors from jitter, noise, slewing, and nonlinear mapping of input value to output voltage.
Jitter, noise, and quantization are often analyzed by modeling them as random errors added to the sample values. Integration and zeroorder hold effects can be analyzed as a form of lowpass filtering. The nonlinearities of either ADC or DAC are analyzed by replacing the ideal linear function mapping with a proposed nonlinear function.
Applications
Audio sampling
Digital audio uses pulsecode modulation and digital signals for sound reproduction. This includes analogtodigital conversion (ADC), digitaltoanalog conversion (DAC), storage, and transmission. In effect, the system commonly referred to as digital is in fact a discretetime, discretelevel 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 doublerate 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.^{[1]} 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.^{[2]}
One advantage of higher sampling rates is that they can relax the lowpass filter design requirements for ADCs and DACs, but with modern oversampling sigmadelta converters this advantage is less important.
Bit depth (quantization)
Audio is typically recorded at 8, 16, and 20bit 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.^{[3]} Eightbit audio is generally not used due to prominent and inherent quantization noise (low maximum SQNR), although the Alaw and ulaw 8bit encodings pack more resolution into 8 bits while increase total harmonic distortion. CD quality audio is recorded at 16bit. 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 20bit quantization.
For playback and not recording purposes, a proper analysis of typical programme levels throughout an audio system reveals that the capabilities of wellengineered 16bit material far exceed those of the very best hifi systems, with the microphone noise and loudspeaker headroom being the real limiting factors^{[citation needed]}.
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 5Hz4 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
Standarddefinition television (SDTV) uses either 720 by 480 pixels (US NTSC 525line) or 704 by 576 pixels (UK PAL 625line) for the visible picture area.
Highdefinition television (HDTV) is currently moving towards three standards referred to as 720p (progressive), 1080i (interlaced) and 1080p (progressive, also known as FullHD) which all 'HDReady' sets will be able to display.
Undersampling
Main article: UndersamplingWhen one samples a bandpass signal at a rate lower than the Nyquist rate, the samples are equal to samples of a lowfrequency alias of the highfrequency signal; the original signal will still be uniquely represented and recoverable if the spectrum of its alias does not cross over half the sampling rate. Such undersampling is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF to digital conversion.^{[4]}
Oversampling
Oversampling is used in most modern analogtodigital converters to reduce the distortion introduced by practical digitaltoanalog converters, such as a zeroorder 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 complexvalued 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 nonzero frequency content is shifted into the interval [B/2, B/2).
Although complexvalued samples can be obtained as described above, they are much more commonly created by manipulating samples of a realvalued 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 samplerate commensurate with the reduced Nyquist rate. The result is half as many complexvalued samples as the original number of real samples. No information is lost, and the original s(t) waveform can be recovered, if necessary.
Notes
 ^ When the complex samplerate 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 presampled signal was analytic. Also see Aliasing#Complex_signal_representation
 ^ 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 realvalued coefficients. That is equivalent to separately filtering the sequences of real parts and imaginary parts and reforming complex pairs at the outputs.
See also
References
 Matt Pharr and Greg Humphreys, Physically Based Rendering: From Theory to Implementation, Morgan Kaufmann, July 2004. ISBN 012553180X. 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.
 ^ Digital Pro Sound
 ^ http://world.std.com/~griesngr/intermod.ppt
 ^ MT001: Taking the Mystery out of the Infamous Formula, "SNR=6.02N + 1.76dB," and Why You Should Care
 ^ Walt Kester (2003). Mixedsignal and DSP design techniques. Newnes. p. 20. ISBN 9780750676113. http://books.google.com/books?id=G8XyNItpy8AC&pg=PA20.
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