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	Nyquist-Shannon sampling theorem (Site not responding. Last check: 2007-09-10)
		The theorem states that when converting from analog signal to digital (or otherwise sampling a signal at intervals) the sampling frequency must be greater than twice the highest frequency of the input in order to be able to reconstruct original perfectly from the sampled version.
		The theorem was first formulated by Harry Nyquist in 1928 ("Certain topics in telegraph theory") but was only formally proved by Claude E. Shannon in 1949 ("Communication in the presence noise").
		A well-known consequence of the sampling theorem that a signal cannot be both bandlimited and time-limited.
	www.freeglossary.com /Nyquist-Shannon (840 words)

	Nyquist Theorem Details
		Nyquist's theorem states that it is only necessary to take samples at a rate of slightly over two per cycle of the highest frequency component of the source analog signal.
		Nyquist's theorem states that, if we sample the complex waveform uniformly at a rate just a tad over twice the highest frequency component sine wave contained within, the conglomeration of samples thus obtained are sufficient information to reconstruct the waveform.
		Nyquist's frequency is the frequency that no part of the source material may attain or exceed, which is half of the sampling frequency.
	members.aol.com /ajaynejr/nyquist.htm (2368 words)

	Nyquist-Shannon sampling theorem : Nyquist sampling theorem
		The theorem states that, when converting from an analog signal to digital (or otherwise sampling a signal at discrete intervals), the sampling frequency must be greater than twice the highest frequency of the input signal in order to be able to reconstruct the original perfectly from the sampled version.
		The theorem was first formulated by Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), but was only formally proved by Claude E. Shannon in 1949 ("Communication in the presence of noise").
		The minimum sample frequency that allows reconstruction of the original signal, that is 2W samples per unit distance, is known as the Nyquist frequency, (or Nyquist rate).
	www.fastload.org /ny/Nyquist_sampling_theorem.html (335 words)

	techtutorials.net - Understanding Circuit Bandwidth
		The first was the Nyquist Theorem, which states that an analog signal can be accurately reproduced through sampling (periodically taking a sample of the analog form and quantizing it) if the sampling rate is at least twice the greatest frequency of the signal.
		According to the Nyquist Theorem, this would require that the analog signal be sampled 8,000 times each second (4,000 cycles per second, sampled twice per cycle).
		Nyquist specifies only how often to sample the signal, but not how to quantize the sample -- that is, how to represent the sampled information (e.g., the frequency).
	www.techtutorials.net /tutorials/networking/digital_circuits.shtml (851 words)

	Nyquist Sampling Theorem (Site not responding. Last check: 2007-09-10)
		Sampling a sound (using a microphone)) or image (using a CCD camera) can introduce a distortion into the signal called “aliasing.” Aliasing takes many forms; for example the apparent backward motion of wagon wheels in western movies, which occurs because the movie is actually a sequence of still frames.
		Nyquist showed that if the sampling rate is twice the highest frequency component, then aliasing does not occur.
		For CCD cameras, a sufficiently high pixel density (2X the highest spatial frequency in the image; at least 2 pixels across the full-width half maximum of a star image) will guarantee a distortion-free representation of the image.
	www.cyanogen.com /help/maxdslr/Nyquist_Sampling_Theorem.htm (129 words)

	Nyquist–Shannon sampling theorem - Wikipedia, the free encyclopedia
		Consequently, the theorem is directly applicable to time-dependent signals and is normally formulated in that context.
		To meet the requirements of the theorem, the signal must usually pass through a low-pass filter of appropriate cutoff frequency as part of the downsampling operation.
		The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon in 1949 ("Communication in the presence of noise").
	en.wikipedia.org /wiki/Nyquist-Shannon_sampling_theorem (5181 words)

	Nyquist Sampling (Site not responding. Last check: 2007-09-10)
		The Nyquist Sampling Theorem sets the fundamental limits on any system that digitizes signals, whether it be a CD player, a telecommunications system, or a CCD camera.
		The American physicist and electrical engineer Harry Nyquist proved that if the sampling is at least twice the highest spatial frequency component in the image, no distortion will occur and you can reconstruct an exact replica of the original image.
		The theorem really requires the image to be filtered (smoothed) to remove any higher frequency components; but in practice, the limitations of the optics and seeing take care of this for you.
	www.cyanogen.com /help/maximdl/Nyquist_Sampling.htm (279 words)

	PlanetMath: sampling theorem (Site not responding. Last check: 2007-09-10)
		The greyvalues of digitized one- or two-dimensional signals are typically generated by an analogue-to-digital converter (ADC), by sampling a continuous signal at fixed intervals (e.g.
		The sampling (or point sampling) theorem states that a band-limited analogue signal
		This is version 4 of sampling theorem, born on 2001-12-25, modified 2002-02-19.
	planetmath.org /encyclopedia/NyquistsTheorem.html (130 words)

	Sampling: What Nyquist Didn't Say, and What to Do About It
		The foremost difficulty with the Nyquist theorem is the requirement that the signal to be sampled be perfectly band limited.
		What the Nyquist theorem – absolutely and positively – does not say, is that you can design your system to operate right at the Nyquist rate, at least not with any reasonable chance of success.
		The quirk in the Nyquist theorem is that the bandwidth of the signal doesn't have to be contiguous – as long as you can chop it up into unambiguous pieces, you can spread that Nyquist bandwidth out as much as you want.
	www.wescottdesign.com /articles/Sampling/sampling.html (5106 words)

	Harry Nyquist: A Founding Father Of Digital Communications
		Nyquist arrived in the U.S. in 1907 and subsequently earned BSEE and MSEE degrees from the University of North Dakota in 1912 and 1915, respectively, and a PhD in physics from Yale University in 1917.
		According to the Sampling Theorem, an analog signal must be sampled at regular intervals over time and at twice the frequency of its highest-frequency component to be converted into an adequate representation of the signal in digital form.
		An outgrowth of Nyquist's work in feedback loops was the "Nyquist plot," which plots the magnitude and phase of a frequency response on orthogonal axes.
	www.elecdesign.com /Articles/ArticleID/11193/11193.html (1049 words)

	eFunda: Introduction to Nyquist Sampling Rate
		The sampling theorem is considered to have been articulated by Nyquist in 1928 and mathematically proven by Shannon in 1949.
		The sampling theorem clearly states what the sampling rate should be for a given range of frequencies.
		According to the sampling theorem, one should sample sound signals at least at 40 kHz in order for the reconstructed sound signal to be acceptable to the human ear.
	www.efunda.com /designstandards/sensors/methods/dsp_nyquist.cfm (380 words)

	Nyquist Theorem (Site not responding. Last check: 2007-09-10)
		The Nyquist Theorem is a principle that engineers follow in the digitization of analog signals.
		According to the Nyquist Theorem, the sampling frequency must be greater than 2fmax, or twice the highest analog frequency component.
		If the sampling rate is equal to or less than 2fmax, the original frequency components in the analog input signal may not be correctly represented in the digitized output.
	www.tdt.com /Sys3WebHelp/Nyquist_Theorem.htm (154 words)

	Clarkvision Photography - How much sampling?
		There is the well known Nyquist theorem that says if you sample 2 samples per cycle you get all the information possible.
		This theorem assumes the sampling is in phase with the signal, meaning one samples at the highs and lows of a sinusoidal signal.
		The Nyquist theorem ONLY applies to data and sampling that are in phase.
	www.clarkvision.com /imagedetail/sampling1.html (900 words)

	Bandwidth, Sample Rate, and Nyquist Theorem- Developer Zone - National Instruments
		The Nyquist Theorem explains the relationship between the sample rate and the frequency of the measured signal.
		The Nyquist theorem states that a signal must be sampled at a rate greater than twice the highest frequency component of the signal to accurately reconstruct the waveform; otherwise, the high-frequency content will alias at a frequency inside the spectrum of interest (passband).
		The dotted line indicates the aliased signal recorded by the ADC and is sampled as a 1 MHz signal instead of a 5 MHz signal.
	zone.ni.com /devzone/cda/tut/p/id/2709 (952 words)

	CSD - December 1998 - Building Blocks: The Shape of Things
		The second theorem, which specifically addresses the ISI and pulse shape issue, is a little less familiar and a little more intricate.
		The theorem states that impulses (in practice, samples), with a transmission rate of R symbols/sec can be filtered via an ideal brick-wall filter of bandwidth R /2 and still be recovered by an independent observation of each pulse.
		This theorem, as might be expected, is closely related to the sampling theorem, which states that a signal bandlimited to some low-pass bandwidth, B, can be uniquely encoded by its samples if they are taken at a rate of 1/2 B or greater (theoretical limit of 2 sym/sec/Hz).
	www.commsdesign.com /main/9812/9812building.htm (1932 words)

	Sampling Theory
		Next, the sampling theorem which states that any signal can be perfectly reconstructed, in principle, from uniformly spaced samples of that signal, provided that the sampling rate is higher than twice the highest frequency present in the signal —; Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theorem.html')">sampling theorem is proved.
		The sampling theorem provides that a properly bandlimited continuous-time signal can be sampled and reconstructed from its samples without error, in principle.
		An early derivation of the sampling theorem is often cited as a 1928 paper by Harold Nyquist, and Claude Shannon is credited with reviving interest in the sampling theorem after World War II when computers became public.
	ccrma.stanford.edu /~jos/mdft/Sampling_Theory.html (239 words)

	Chapter Five: Principles of Digital Audio
		He presented the concept of sampling amplitudes at a specific rate, as described on the previous page, and most importantly determined that the sampling rate would need to be at least twice the highest frequency to be reproduced.
		According to this Theorem, the highest reproducible frequency of a digital system will be less than one-half the sampling rate.
		The frequency they produced is predictable, in that they are mirrored the same distance below the Nyquist frequency as the original was above it, at the original amplitude.
	www.indiana.edu /~emusic/etext/digital_audio/chapter5_nyquist.shtml (621 words)

	Turning Nyquist upside down by undersampling (Site not responding. Last check: 2007-09-10)
		In the 1920s, these gentlemen created the now-well-known Nyquist theorem, which states that when sampling a signal at discrete intervals, the sampling must be greater than twice the highest frequency of the input signal.
		In my discussions with engineers, I use the Nyquist theorem to explain the accuracy of sampling systems in which the bandwidth of the signal of interest is less than twice the sampling frequency of the converter.
		This situation is usually an engineer's initial exposure to the Nyquist theorem, in which signals with frequencies greater than one-half of the converter's sampling rate can come back to haunt you.
	www.ferret.com.au /articles/13/0C03A513.asp (685 words)

	Walker Stumbles Over Shannon
		W is the Nyquist bandwidth, defined as the minimum bandwidth that can be used to pass the signal.
		Nyquist's theorem is best known today for its application to digital audio systems: in order to reproduce all of the frequencies in a band limited analog signal, the sampling rate must be at least twice the signal bandwidth.
		Indeed, he doesn't even understand the Nyquist theorem, which was the foundation for Shannon's work 20 years later on the limits of coding.
	www.ka9q.net /vmsk/shannon.html (1543 words)

	The Nyquist Theorem (Site not responding. Last check: 2007-09-10)
		Nyquist's theorem says that if we sample a signal in which the highest frequency we wish to reproduce correctly is f then we must sample the signal at a minimum frequency of 2f.
		A signal which is sampled below this minimum rate is undersampled, and the effect of doing this is to generate spurious contributions of lower frequencies in the reconstructed signal.
		Since aliased components are indistinguishable from components of the same frequency which were present in the original signal, their occurrence must be prevented by a combination of sampling at a sufficiently high frequency and filtering out from the original signal all frequencies higher than the Nyquist frequency.
	www.inf.fu-berlin.de /inst/ag-tech/teaching/WS0203/19546-U/nyquist.html (336 words)

	Anti-Aliasing in Computer Graphics
		The Nyquist Theorem states that the sampling rate must be twice the frequency of the signal or an effect known as aliasing occurs.
		The top wave is sampled at a frequency within the Nyquist limit and the digital sample produces the original frequency.
		The bottom wave on the other hand is sampled beyond the Nyquist limit and the resulting sampled wave has a much lower frequency than the source.
	www.cc.gatech.edu /classes/AY2001/cs4451_spring/projects/Five (1033 words)

	Nyquist Theorem
		Earlier you should have been exposed to the concepts behind sampling and the sampling theorem.
		While learning about these ideas, you should have begun to notice that if we sample at too low of a rate, there is a chance that our original signal will not be uniquely defined by our sampled signal.
		theorem 1: Nyquist Theorem ("Fundamental Theorem of DSP")
	cnx.org /content/m10791/latest (305 words)

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