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Encyclopedia > 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-time signal) to a sequence of samples (a discrete-time signal). Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... A continuous signal or a continuous time signal is a varying quantity (a signal) that can be, or is expressed, as a continuous function of an independent variable, usually time. ... Discrete sampled signal Digital signal A discrete signal or discrete-time signal is a time series, perhaps a signal that has been sampled from a continuous-time signal. ... This article is about compression waves. ... Continuous time occurs when time is sampled continuously. ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ... A sample refers to a value or set of values at a point in time and/or space. ... Discrete time is non-continuous time. ...


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. A continuous signal or a continuous time signal is a varying quantity (a signal) that can be, or is expressed, as a continuous function of an independent variable, usually time. ... In digital signal processing, an ideal sampler is a sampler that samples in an ideal fashion. ... In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions for some given period T. Some authors, notably Bracewell, refer to it as the Shah function (probably because its graph resembles the shape of the cyrillic letter sha Ш). From the orthogonality of the Fourier series... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... Discrete sampled signal Digital signal A discrete signal or discrete-time signal is a time series, perhaps a signal that has been sampled from a continuous-time signal. ...

Contents

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. The Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. ...


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] is 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. The sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ... MHZ redirects here. ...


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 Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. ... A bandlimited signal is a deterministic or stochastic signal (e. ... The Whittaker–Shannon interpolation formula dates back to works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935 in the formulation of the Nyquist–Shannon sampling theorem by C. E. Shannon in 1949. ...


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. The Nyquist frequency, named after Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system. ... Properly sampled image of brick wall. ... In digital signal processing, anti-aliasing is the technique of minimizing the distortion artifacts known as aliasing when representing a high-resolution signal at a lower resolution. ... A low-pass filter is a filter that passes low frequencies but attenuates (or reduces) frequencies higher than the cutoff frequency. ...


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.


Sampling interval

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


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. With regards to time, an interval is the duration between two events or occurrences of similar events. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... An earthquake is the result of a sudden release of stored energy in the Earths crust that creates seismic waves. ... For other uses, see Tsunami (disambiguation). ... A year (from Old English gēr) is the time between two recurrences of an event related to the orbit of the Earth around the Sun. ... A century (From the Latin cent, one hundred) is one hundred consecutive years. ...


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.


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. 4-channel stereo multiplexed analog-to-digital converter WM8775SEDS made by Wolfson Microelectronics placed on X-Fi Fatal1ty Pro sound card An analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic integrated circuit (i/c) that converts continuous signals to discrete digital numbers. ...


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. Properly sampled image of brick wall. ... A guard band is a small part of the radio spectrum in between radio bands, for the purpose of preventing interference. ... Photography [fÓ™tÉ‘grÓ™fi:],[foÊŠtÉ‘grÓ™fi:] is the process of recording pictures by means of capturing light on a light-sensitive medium, such as a film or electronic sensor. ... A photograph with an exposure time of 25 seconds A photograph of a night-time sky with an exposure time of 8 seconds In photography, exposure is the total amount of light allowed to fall on the photographic medium (photographic film or image sensor) during the process of taking a... See Capacitor (component) for a discussion of specific types. ... In electronics, a sample and hold circuit is used to interface real-world, changing analogue signals to a subsequent system such as an analog-to-digital converter. ... In telecommunication, jitter is an abrupt and unwanted variation of one or more signal characteristics, such as the interval between successive pulses, the amplitude of successive cycles, or the frequency or phase of successive cycles. ... In science, and especially in physics and telecommunication, noise is fluctuations in and the addition of external factors to the stream of target information (signal) being received at a detector. ... In electronics, the slew rate is a nonlinear effect in amplifiers. ... Quantized signal Digital signal In digital signal processing, quantization is the process of approximating a continuous range of values (or a very large set of possible discrete values) by a relatively-small set of discrete symbols or integer values. ... To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... In electronics, a digital-to-analog converter (DAC or D-to-A) is a device for converting a digital (usually binary) code to an analog signal (current, voltage or electric charge). ... The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ... In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of half-open intervals. ... The rectangular function (also known as the rectangle function or the normalized boxcar function) is defined as or in terms of the Heaviside step function The rectangular function is normalized: The Fourier transform of the rectangular function is where sinc is the sinc function. ... The Zero-order hold (ZOH) is a mathematical model of the practical reconstruction of sampled signals done by conventional digital-to-analog converters (DAC). ... The Nyquist frequency, named after Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system. ...


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. A low-pass filter is a filter that passes low frequencies but attenuates (or reduces) frequencies higher than the cutoff frequency. ... A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ...


Applications

Audio sampling

Sampling Rate

When it is necessary to capture audio covering the entire 18-20,000 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. The auditory system is the sensory system for the sense of hearing. ... A compact disc or CD is an optical disc used to store digital data, originally developed for storing digital audio. ... Professional audio, also pro audio, can be used a term to refer to both a type of audio equipment as well as a type of audio engineering application. ... The Nyquist-Shannon sampling theorem is the fundamental theorem in the field of information theory, in particular telecommunications. ...


Even a casual glance at professional audio magazines (e.g. Sound on Sound) will reveal 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 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 [1]. [2]. If intermodulation distortion is detected and preferred by listeners, then it is akin to the perceptual effects of harmonic distortion associated with valve amplification that is pleasing to some listeners. Ultrasound is sound with a frequency greater than the upper limit of human hearing, approximately 20 kilohertz. ... Intermodulation distortion: Nonlinear distortion characterized by the appearance, in the output of a device, of frequencies that are linear combinations of the fundamental frequencies and all harmonics present in the input signals. ... The total harmonic distortion, or THD, of a signal is a measurement of the harmonic distortion present and is defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental. ...


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. 4-channel stereo multiplexed analog-to-digital converter WM8775SEDS made by Wolfson Microelectronics placed on X-Fi Fatal1ty Pro sound card An analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic integrated circuit (i/c) that converts continuous signals to discrete digital numbers. ... In electronics, a digital-to-analog converter (DAC or D-to-A) is a device for converting a digital (usually binary) code to an analog signal (current, voltage or electric charge). ... The Delta-Sigma (ΔΣ) modulation is a kind of analog-to-digital or digital-to-analog conversion. ...


Resolution

Audio is typically recorded at 8-, 16-, and 20-bit resolution, 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. In trigonometry, an ideal sine wave is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C, where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C... DB or db or dB may stand for: Database, an organized collection of data Name of Person, Danda Beer DB connector, improper term for D-subminiature DB (car), a French automobile maker Decibel (dB), the ratio between two quantities, used in acoustics and electronics Denver Broncos, an NFL franchise Dubnium... An a-law algorithm is a standard companding algorithm, used in European digital communications systems to optimize, modify, the dynamic range of an analog signal for digitizing. ... In telecommunication, a mu-law algorithm (μ-law) is a standard analog signal compression or companding algorithm, used in digital communications systems of the North American and Japanese digital hierarchies, to optimize (in other words, modify) the dynamic range of an audio analog signal prior to digitizing. ... The total harmonic distortion, or THD, of a signal is a measurement of the harmonic distortion present and is defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental. ... Johnson-Nyquist noise (sometimes thermal noise, Johnson noise or Nyquist noise) is the noise generated by the equilibrium fluctuations of the electric current inside an electrical conductor, which happens without any applied voltage, due to the random thermal motion of the charge carriers (the electrons). ... In signal theory, the noise floor is the measure of the signal created from the sum of all the noise sources and unwanted signals within a measurement system. ...


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]. Programme levels are important in Audio if listeners to CDs, Radio and Television are to get the best experience. ...


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 0-4 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. Bold text This article does not cite any references or sources. ... In human language, a phoneme is the theoretical representation of a sound. ... In telecommunication, Telephony encompasses the general use of equipment to provide voice communication over distances. ... G.711 is an ITU-T standard for audio companding. ...


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. Standard-definition television or SDTV refers to television systems that have a lower resolution than HDTV systems. ... A pixel (a contraction of picture element) is one of the many tiny dots that make up the representation of a picture in a computers memory. ... NTSC is the analog television system in use in the United States, Canada, Japan, South Korea, the Philippines, Mexico, and some other countries, mostly in the Americas (see map). ... A pixel (a contraction of picture element) is one of the many tiny dots that make up the representation of a picture in a computers memory. ... For other uses, see PAL (disambiguation). ...


High-definition television (HDTV) is currently moving towards two standards referred to as 720p (progressive) and 1080i (interlaced), which all 'HD-Ready' sets will be able to display. Projection screen in a home theater, displaying a high-definition television image. ...


Video reconstruction filtering

Most TV sets do not achieve basic SDTV quality, because they do not reconstruct the vertically sampled image properly. Digital video produces a 2-dimensional set of samples of each frame, which requires a 2-dimensional 'brick-wall' reconstruction filter for proper reproduction of the image. CRT displays produce a raster scan of horizontal lines, and the digital signal is low-pass filtered along the horizontal lines, giving good resolution of vertical lines without aliasing, but reconstruction is not usually attempted vertically, so that the resulting picture contains very visible artifacts (loss of resolution, staircasing effects, fringing pattern, sampling harmonics, and other adverse effects).


Proper 2-dimensional reconstruction requires a final display with many more pixels than the signal format,[citation needed] and modern HDTV sets can provide this, producing much better resolution pictures than even a top studio monitor can from SDTV signals (though they are not so good regarding grey-level accuracy, especially near black level).


As with audio, this theoretical need for reconstruction is not commonly realised, though it was recognised by the BBC who then backed off from broadcasting HDTV but started to record programmes in HDTV. The British Broadcasting Corporation, which is usually known as the BBC, is the largest broadcasting corporation in the world in terms of audience numbers, employing 26,000 staff in the United Kingdom alone and with a budget of more than GB£4 billion. ...


To get a true HDTV image you really need a 'super HDTV' display, with at least twice as many pixels again (3840 x 2160).[citation needed] Worth bearing in mind though not currently practical. Nevertheless, HDTV does a very significant increase in resolution over SDTV when both are compared on a HDTV set, the higher Nyquist frequency bringing improvements despite the fact that the image is not properly reconstructed on currently available displays.


IF/RF (bandpass) sampling

Plot of allowed sample rates (gray areas) versus the upper edge frequency for a band of width W = 1. The darker gray areas correspond to the condition with n = 0 in the equations of this section.
Plot of allowed sample rates (gray areas) versus the upper edge frequency for a band of width W = 1. The darker gray areas correspond to the condition with n = 0 in the equations of this section.

For sampling a non-baseband signal, such as a radio's intermediate-frequency (IF) or radio-frequency (RF) signal, the Nyquist–Shannon conditions to avoid aliasing can be restated as follows. Let 0 < fL < fH be the lower and higher boundaries of a frequency band and W = fH − fL be the bandwidth. Then there is a non-negative integer N with Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Baseband is an adjective that describes signals and systems whose range of frequencies is measured from 0 to a maximum bandwidth or highest signal frequency; it is sometimes used as a noun for a band of frequencies starting at 0. ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ...

N < { f_mathrm{L} over W } < (N+1) < { f_mathrm{H} over W }

In addition, we define the remainder r as

r = f_mathrm{L}-NWin[0,f_mathrm{L}].

Any real-valued signal x(t) with a spectrum limited to this frequency band, that is with

X(f)= mathcal{F} { x }(f) = 0 for |f| , outside the interval [f_mathrm{L},f_mathrm{H}] ,,

is uniquely determined by its samples obtained at a sampling rate of fs, if this sampling rate satisfies one of the following conditions: The sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ...

  • 2left(W+frac{nW+r}{N-n+1}right) < f_mathrm{s} < 2left(W+frac{nW+r}{N-n}right) for one value of n = { 0, 1, ..., N-1 }
OR the usual Nyquist condition:
  • 2f_mathrm{H} < f_mathrm{s},.

If N > 0, then the first 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. Properly sampled image of brick wall. ...

Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 44 MHz (N–n = 4) 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–n = 4) 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–n = 3) sampling, showing plenty of room for bandpass anti-aliasing filter transition bands. The baseband image is frequency-reversed in this case (odd N–n).
Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 56 MHz (N–n = 3) sampling, showing plenty of room for bandpass anti-aliasing filter transition bands. The baseband image is frequency-reversed in this case (odd N–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
N < 4.4 = { 88 mathrm{MHz} over 20 mathrm{MHz} } < N+1
Therefore
N=4, r=8 MHz and n = 0,1,2,3.
The value n = 0 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 86.4–108 but lower than 88-110 MHz).
A lower value of N will also lead to a useful sampling rate, equivalent to picking a nonzero n. For example, using N–n = 3, 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 a nonzero N, 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) Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... FM radio is a broadcast technology invented by Edwin Howard Armstrong that uses frequency modulation to provide high-fidelity sound over broadcast radio. ... MegaHertz (MHz) is the name given to one million (106) Hertz, a measure of frequency. ... The frequency axis of this symbolic diagram would be logarithmically scaled. ... An anti-aliasing filter is commonly used in conjuction with digital signal processing and is a filter to restrict the bandwidth to approximately satisfy the Shannon-Nyquist sampling theorem. ... Baseband is an adjective that describes signals and systems whose range of frequencies is measured from 0 to a maximum bandwidth or highest signal frequency; it is sometimes used as a noun for a band of frequencies starting at 0. ...


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


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

left(-frac{N+1}2f_mathrm{s},-frac{N}2f_mathrm{s}right) cupleft(frac{N}2f_mathrm{s},frac{N+1}2f_mathrm{s}right) for some nonnegative integer N,.
which includes the normal baseband condition as case N=0 (except that where the intervals come together at 0 frequency, they can be closed).

And the corresponding interpolation function is the bandpass filter given by this difference of lowpass impulse responses: Baseband is an adjective that describes signals and systems whose range of frequencies is measured from 0 to a maximum bandwidth or highest signal frequency; it is sometimes used as a noun for a band of frequencies starting at 0. ...

(N+1)operatorname{sinc} left(frac{(N+1)t}Tright) - Noperatorname{sinc} left( frac{Nt}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.


See also

The Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. ... Quantized signal Digital signal In digital signal processing, quantization is the process of approximating a continuous range of values (or a very large set of possible discrete values) by a relatively-small set of discrete symbols or integer values. ... Kell factor is a number used to determine the effective resolution of a discrete display device. ... Properly sampled image of brick wall. ... In digital signal processing, anti-aliasing is the technique of minimizing the distortion artifacts known as aliasing when representing a high-resolution signal at a lower resolution. ... An anti-aliasing filter is commonly used in conjuction with digital signal processing and is a filter to restrict the bandwidth to approximately satisfy the Shannon-Nyquist sampling theorem. ... Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ... The sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ... Downsampling (or subsampling) is the process of reducing the sampling rate of a signal. ... Upsampling is the process of increasing the sampling rate of a signal. ... In signal processing, oversampling is the process of sampling a signal with a sampling frequency significantly higher than twice the bandwidth or highest frequency of the signal being sampled. ...

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. ^ http://www.digitalprosound.com/Htm/SoapBox/soap2_Apogee.htm
  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

External links

  • Nyquist sampling in digital microscopy

  Results from FactBites:
 
InformIT: Understanding Digital Signal Processing: Periodic Sampling > ALIASING: SIGNAL AMBIGUITY IN THE FREQUENCY ... (0 words)
Periodic sampling, the process of representing a continuous signal with a sequence of discrete data values, pervades the field of digital signal processing.
There is a frequency-domain ambiguity associated with discrete-time signal samples that does not exist in the continuous signal world, and we can appreciate the effects of this uncertainty by understanding the sampled nature of discrete data.
Figure 2-3 Shark's tooth pattern: (a) aliasing at multiples of the sampling frequency; (b) aliasing of the 7-kHz sinewave to 1 kHz, 13 kHz, and 19 kHz.
The SpectrumWare Approach to Wireless Signal Processing (7237 words)
However, given the rates at which the sampling and processing technologies are improving, we are convinced that the bandwidth and variety of computationally accessible signals, as well as the complexity of their processing, will expand.
Temporal decoupling allows the samples to be processed in large blocks, and a whole new range of algorithms that leverage statistical behavior may be brought to bear on the problem space.
Furthermore, this sample processing may be performed in bursts as dictated by the behavior of the application program and the time-slicing of the processor among applications.
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