Sampling (signal processing)

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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-time 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 operator that extracts samples from continuous signal. A theoretical ideal sampler multiplies a continuous signal with a Dirac comb. This multiplication "picks out" values but the result is still continuous-valued. If this signal is then discretized (i.e., converted into a sequence) and quantized along all dimensions it becomes a discrete signal.

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[edit] Theory

See also: Nyquist–Shannon sampling theorem

For convenience, we will discuss signals which vary with time. However, the same results can be applied to signals varying in space or in any other dimension.

Let x(t) be a continuous signal which is to be sampled, and that sampling is performed by measuring the value of the continuous signal every T seconds. Thus, the sampled signal x[n] given by

x[n] = x(nT), with n = 0, 1, 2, 3, ...

The sampling frequency or sampling rate fs is defined as the number of samples obtained in one second, or fs = 1/T. The sampling rate is measured in hertz or in samples per second.

We can now ask: under what circumstances is it possible to reconstruct the original signal completely and exactly (perfect reconstruction)?

A partial answer is provided by the Nyquist–Shannon sampling theorem, which 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 one-half 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 fN 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 NfN + f and NfN – 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 anti-aliasing filter (usually a low-pass filter with cutoff near the Nyquist frequency) before conversion to the sampled discrete representation.

A more general statement of the Nyquist–Shannon sampling theorem says more or less that the signals with frequencies higher than the Nyquist frequency can be sampled without loss of information, provided their bandwidth (non-zero frequency band) is small enough to avoid ambiguity, and the bandlimits are known.

[edit] Sampling interval

The sampling interval is the interval T = 1/fs corresponding to the sampling frequency. [1]

[edit] Observation period

The observation period is the span of time during which a series of data samples are collected at regular intervals.[2] 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 sub-intervals.

[edit] Practical implications

In practice, the continuous signal is sampled using an analog-to-digital converter (ADC), a non-ideal 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 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.
  • 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 capacitor-based sample and hold circuit, the integration effect 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 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 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. The DAC can also suffer errors from jitter, noise, slewing, and non-linear 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 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.

[edit] Applications

[edit] Audio sampling

[edit] 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) or 48 kHz (professional audio). 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.[1] This is in contrast with laboratory experiments 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 low-pass filter design requirements for ADCs and DACs, but with modern oversampling sigma-delta converters this advantage is less important.

[edit] 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 [3]. 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. Very few analog to digital converters have signal to noise ratios (SNR) above 120 dB, which make useless the need of greater than 20 bit for the quantization process. In 24 bit converters, the 4 LSB has useless random values with no information. In a recording studio where multiple analog sources may be mixed together, 20 bit resolution is important for minimizing the noise floor; but the typical consumer is unlikely to see any benefit from 20-bit devices.

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[citation needed].

[edit] 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.

[edit] Video sampling

Standard-definition television (SDTV) uses either 720 by 480 pixels (US NTSC 525-line) or 704 by 576 pixels (UK PAL 625-line) for the visible picture area.

High-definition television (HDTV) is currently moving towards two 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.

[edit] IF/RF (bandpass) sampling

Plot of sample rates (y axis) versus the upper edge frequency (x axis) for a band of width 1; grays areas are combinations that are "allowed" in the sense that no two frequencies in the band alias to same frequency. The darker gray areas correspond to undersampling with the maximum value of n in the equations of this section.
Plot of sample rates (y axis) versus the upper edge frequency (x axis) for a band of width 1; grays areas are combinations that are "allowed" in the sense that no two frequencies in the band alias to same frequency. The darker gray areas correspond to undersampling with the maximum value of n in the equations of this section.

Real signals have Fourier spectra with symmetry about zero. That is, they have a negative-frequency spectrum that is a mirror image of the positive-frequency spectrum. Sampling effectively shifts both sides of the spectrum by multiples of the sampling frequency. The criterion to avoid aliasing is that none of these shifted copies of the spectrum overlap.

In the case of a bandpass (non-baseband) signal, with low and high band limits fL and fH respectively, the condition for an acceptable sample rate is that shifts of the bands from fL to fH and from –fH to –fL must not overlap when shifted by all integer multiples of sampling rate fs. This condition reduces to the constraint:[4][5]

\frac{2 f_H}{n} < f_s < \frac{2 f_L}{n - 1}, for some n satisfying: 1 \le n \le \left\lfloor \frac{f_H}{f_H-f_L} \right\rfloor

The highest n for which the condition is satisfied leads to the lowest possible sampling rates.

Important signals of this sort include a radio's intermediate-frequency (IF) or radio-frequency (RF) signal.

If n > 1, then the conditions result in what is sometimes referred to as undersampling, bandpass sampling, or using a sampling rate less than the Nyquist rate 2fH obtained from the upper bound of the spectrum. See aliasing for a simpler formulation of this Nyquist criterion that specifies the lower bound on sampling rate (but is incomplete because it does not specify the gaps above that bound, in which aliasing will occur). Alternatively, for the case of a given sampling frequency, simpler formulae for the constraints on the signal's spectral band are given below.

Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 44 MHz (n = 5) sampling. An anti-alias filter quite tight to the FM radio band is required, and there's not room for stations at nearby expansion channels such as 87.9 without aliasing.
Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 44 MHz (n = 5) sampling. An anti-alias filter quite tight to the FM radio band is required, and there's not room for stations at nearby expansion channels such as 87.9 without aliasing.
Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 56 MHz (n = 4) sampling, showing plenty of room for bandpass anti-aliasing filter transition bands. The baseband image is frequency-reversed in this case (even n).
Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 56 MHz (n = 4) sampling, showing plenty of room for bandpass anti-aliasing filter transition bands. The baseband image is frequency-reversed in this case (even n).
Example: Consider FM radio to illustrate the idea of undersampling.
In the US, FM radio operates on the frequency band from fL = 88 MHz to fH = 108 MHz. The bandwidth is given by
W = f_H - f_L = 108 \ \mathrm{MHz} - 88 \ \mathrm{MHz} = 20 \ \mathrm{MHz}
The sampling conditions are satisfied for
1 \le n \le \lfloor 5.4 \rfloor = \left\lfloor { 108 \ \mathrm{MHz} \over 20 \ \mathrm{MHz} } \right\rfloor
Therefore, n can be 1, 2, 3, 4, or 5.
The value n = 5 gives the lowest sampling frequencies interval 43.2\ \mathrm{MHz}<f_\mathrm{s}<44\ \mathrm{MHz} and this is a scenario of undersampling. In this case, the signal spectrum fits between and 2 and 2.5 times the sampling rate (higher than 86.4–88 MHz but lower than 108–110 MHz).
A lower value of n will also lead to a useful sampling rate. For example, using n = 4, the FM band spectrum fits easily between 1.5 and 2.0 times the sampling rate, for a sampling rate near 56 MHz (multiples of the Nyquist frequency being 28, 56, 84, 112, etc.). See the illustrations at the right.
When undersampling a real-world signal, the sampling circuit must be fast enough to capture the highest signal frequency of interest. Theoretically, each sample should be taken during an infinitesimally short interval, but this is not practically feasible. Instead, the sampling of the signal should be made in a short enough interval that it can represent the instantaneous value of the signal with the highest frequency. This means that in the FM radio example above, the sampling circuit must be able to capture a signal with a frequency of 108 MHz, not 43.2 MHz. Thus, the sampling frequency may be only a little bit greater than 43.2 MHz, but the input bandwidth of the system must be at least 108 MHz. Similarly, the accuracy of the sampling timing, or aperture uncertainty of the sampler, frequently the analog to digital converter, must be appropriate for the frequencies being sampled 108MHz, not the lower sample rate.
If the sampling theorem is interpreted as requiring twice the highest frequency, then the required sampling rate would be assumed to be greater than the Nyquist rate 216 MHz. While this does satisfy the last condition on the sampling rate, it is grossly oversampled.
Note that if a band is sampled with n > 1, then a band-pass filter is required for the anti-aliasing filter, instead of a lowpass filter.

As we have seen, the normal baseband condition for reversible sampling is that X(f) = 0 outside the open interval: \left(-\frac12f_\mathrm{s},\frac12f_\mathrm{s}\right),

and the reconstructive interpolation function, or lowpass filter impulse response, is \operatorname{sinc} \left( t/T \right) .

To accommodate undersampling, the bandpass condition is that X(f) = 0 outside the union of open positive and negative frequency bands

\left(-\frac{n}2f_\mathrm{s},-\frac{n-1}2f_\mathrm{s}\right) \cup\left(\frac{n-1}2f_\mathrm{s},\frac{n}2f_\mathrm{s}\right) for some positive integer n\,.
which includes the normal baseband condition as case n = 1 (except that where the intervals come together at 0 frequency, they can be closed).

The corresponding interpolation function is the bandpass filter given by this difference of lowpass impulse responses:

n\operatorname{sinc} \left(\frac{nt}T\right) - (n-1)\operatorname{sinc} \left( \frac{(n-1)t}T \right) .

On the other hand, reconstruction is not usually the goal with sampled IF or RF signals. Rather, the sample sequence can be treated as ordinary samples of the signal frequency-shifted to near baseband, and digital demodulation can proceed on that basis, recognizing the spectrum mirroring when n is even.

Further generalizations of undersampling for the case of signals with multiple bands are possible, and signals over multidimensional domains (space or space-time) and have been worked out in detail by Igor Kluvánek.

[edit] See also

[edit] References

  • 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.
  1. ^ Digital Pro Sound
  2. ^ http://world.std.com/~griesngr/intermod.ppt
  3. ^ MT-001: Taking the Mystery out of the Infamous Formula, "SNR=6.02N + 1.76dB," and Why You Should Care
  4. ^ Hiroshi Harada, Ramjee Prasad (2002). Simulation and Software Radio for Mobile Communications. Artech House. ISBN 1580530443. 
  5. ^ Angelo Ricotta. Undersampling SODAR Signals.

[edit] External links