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Topic: Nyquist Shannon interpolation formula

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In the News (Wed 16 Apr 14)

  Nyquist-Shannon sampling theorem
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).
Shannon, "Communication in the presence of noise," Proc.
www.ebroadcast.com.au /lookup/encyclopedia/ny/Nyquist's_theorem.html   (296 words)

 Nyquist-Shannon sampling theorem   (Site not responding. Last check: 2007-10-13)
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").
The minimum sample frequency that allows of the original signal that is 2W samples per unit distance is known the Nyquist frequency (or Nyquist rate).
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)

 A Chronology of Interpolation   (Site not responding. Last check: 2007-10-13)
This Lemma contains his general interpolation formula for unequal-interval data as well as the formula for equal-interval data, the latter of which is in fact a special case of the former and is equal to the one described earlier by Gregory, hence the Gregory-Newton formula.
Analyzing the frequency content of this interpolant, he observes that all constituents of period 2w, with w the distance between the abscissae, are absent.
In particular they show that the error introduced by a given interpolation or approximation kernel consists of two parts, one of which is dominant and is determined entirely by the spectrum of the function to be approximated and a certain error function, the latter of which depends only on the kernel.
imagescience.bigr.nl /meijering/research/chronology   (3092 words)

 Nyquist–Shannon sampling theorem - Wikipedia, the free encyclopedia
This procedure is represented by the Whittaker–Shannon interpolation formula.
The Nyquist rate is defined as twice the bandwidth of the continuous-time signal.
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)

 BeoWorld Community Forums -> Thought For The Day.............   (Site not responding. Last check: 2007-10-13)
The Shannon theorem does not, strictly speaking, affect the D/A process, what it does is tell you the minimum sampling frequency you need to use to capture a input signal with a given bandwidth and consequently what the cutoff frequency of your Low Pass Filter circuit needs to be to recover the original signal.
Firstly, in an ideal world, Shannon's interpolation theorem is a method of exactly calculating a value between sample points however it implies the use of the sinc(x) function which is infinite, therefore, in any real system, an approximation must be used which means the value calculated is never truly exact.
Thus Shannons interpolating theorem is actually used to enable digital filtering prior to D/A conversion to ease the cost/complexity of the output analogue Low pass filter.
www.beoworld.co.uk /forums/index.php?showtopic=5490   (2191 words)

 Interpolation   (Site not responding. Last check: 2007-10-13)
In the mathematical subfield of numerical analysis interpolation is a method of constructing new data points from a discrete set of known data points.
Furthermore, the interpolant is a polynomial and thus infinitely differentiable.
For instance, rational interpolation is interpolation by rational functions, and trigonometric interpolation is interpolation by trigonometric polynomials.
interpolation.kiwiki.homeip.net   (1032 words)

 Nyquist–Shannon sampling theorem - Wikipedia, the free encyclopedia
In addition to Claude Shannon and Harry Nyquist, it is also attributed to Whittaker and Kotelnikov, and sometimes simply referred to as the sampling theorem.
This procedure derives from the Nyquist-Shannon interpolation formula.
The theorem was first formulated by Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), but was only formally proven by Claude E. Shannon in 1949 ("Communication in the presence of noise").
www.physics.utah.edu /~springer/phys6620/lectures/lec04_20060124/Nyquist-Shannon_sampling_theorem.htm   (2932 words)

 Modeling off-axis vision II: the effect of spatial filtering and sampling by retinal neurons
This implies that the cone array and the array of bipolar interneurons have the same Nyquist limit, which means the neural image carried by the cone mosaic is effectively transferred to the input stage of ganglion cells without loss of spatial resolution.
Using these two formulas, we computed the Nyquist frequency of neural arrays from anatomical estimates of cell density measured along the horizontal meridian of the human retina.
We estimated the Nyquist frequency of the sub-population of P-type ganglion cells from measurements of total ganglion cell density by assuming that 80% of all human ganglion cells are P-cells, just as in the monkey.
research.opt.indiana.edu /Library/ModelOffAxisII/ModelOffAxisII.html   (16926 words)

 Elements of an Electrical Communications System- Developer Zone - National Instruments
Nyquist set out to determine the optimum pulse shape that was bandlimited to W Hz and maximized the bit rate 1/T under the constraint that the pulse caused no intersymbol interference at the sampling times k/T, k = 0, ±1, ±2, ….
Nyquist's result is equivalent to a version of the sampling theorem for band-limited signals, which was later stated precisely by Shannon (1948).
Hartley's and Nyquist's results on the maximum transmission rate of digital information were precursors to the work of Shannon (1948a,b) who established the mathematical foundations for information theory and derived the fundamental limits for digital communication systems.
zone.ni.com /devzone/cda/ph/p/id/82   (3542 words)

 Multirate receive device and method using a single adaptive interpolation filter - Patent 5933467   (Site not responding. Last check: 2007-10-13)
Suitable interpolation filters are described by Bucket and Moeneclaey in the article "Symbol synchronizer performance affected by non-ideal interpolation in digital modems" ("Performance de la synchronisation symbolique affectee par une interpolation non-ideale dans les modems numeriques") ICC 94, p.
The interpolation filter 36 is controlled by a timing estimator module 38 connected to the outputs of the interpolation filter 36 and the decision module 37.
In this structure the choice of the oversampling ratio at the interpolator input is the result of a compromise between the complexity of the interpolator, that of the channel filter and the required level of performance.
www.freepatentsonline.com /5933467.html   (7056 words)

 Harry Nyquist   (Site not responding. Last check: 2007-10-13)
He emigrated to the USA in 1907 and entered the University of North Dakota in 1912.
In 1927 Nyquist determined that an analog signal should be sampled at regular intervals over time and at twice the frequency of its highest-frequency component in order to be converted into an adequate representation of the signal in digital form.
Nyquist published His results in the paper Certain topics in Telegraph Transmission Theory (1928).
harry-nyquist.iqnaut.net   (220 words)

 Sampling and Reconstruction of Periodic Signals | DSP-FPGA.com
In 1928, Henry Nyquist published an important paper titled "Certain Topics in Telegraph Transmission Theory." In it, he postulated a theorem which proposed that a sample of twice the highest signal frequency rate, captures the signal’s frequency content in a manner that enables it to be reconstructed without fear of aliasing.
This limitation has nothing to do with Nyquist's Theorem, but instead, is based upon the assumption made by Bell Labs that this frequency rate was good enough to make the voice intelligible (agreed it's not CD quality).
The sampling theorem was introduced by Nyquist in 1928 and later popularized by Shannon in 1949.
www.dsp-fpga.com /articles/neagoe_and_fugerer   (3277 words)

 Springer Online Reference Works
In fact, for band-limited functions the sampling theorem (including sampling of derivatives) is equivalent to the famous Poisson summation formula (Fourier analysis) and the Cauchy integral formula (complex analysis, cf.
Further, the approximate sampling theorem is equivalent to the general Poisson summation formula, the Euler–MacLaurin formula, the Abel–Plana summation formula (numerical mathematics), and to the basic functional equation for the Riemann zeta-function (number theory).
can also be regarded as a limiting case of the Lagrange interpolation formula (as the number of nodes tends to infinity), while the Gauss summation formula of special function theory is a particular case of Shannon's theorem.
eom.springer.de /s/s120120.htm   (802 words)

 Method and apparatus for registering color separation film - Patent 4849914
The accuracy of the interpolation is order h.sup.3, where h represents the size of the interpolation offset (a number between 0 and 1 in distance divided by the grid spacing).
If the magnitude of either of the 2-dimensional interpolation offsets dx, dy exceeds 1/2, then the central data point can be shifted to a neighboring point and a corresponding new offset can be taken from the moved data center to the desired point, that is less than or equal to 1/2 in magnitude.
The specialized interpolation formula is ##EQU3## where R is the interpolated response, R.sub.d is the interpolation response data, and dx.sub.i dy.sub.j are the interpolation offsets (.ltoreq.1/2 in magnitude).
www.freepatentsonline.com /4849914.html   (16822 words)

 Knowledge Base for software by Scientific Volume Imaging B.V. Home of Huygens and FluVR, the premier choices for ...
The formula in the manual seems to imply that there are physical limits to one's ability to deconvolve that are solidly based on optics--not on photobleaching, detector sensitivity, numbers of photons emitted or the size of the bead.
The Nyquist rate (similar to the Shannon theorem) says that IF a signal is bandlimited, it is sufficient to sample it at twice the highest frequency.
The Shannon theorem says it doesn't matter whether you get the supersampled image during sampling or afterwards by interpolation, but it is more practical to get it during sampling, if only to improve the SNR situation.
support.svi.nl /faq.php?fid=9   (388 words)

 PSW Recording Forums: Reason In Audio => Is PCM a Tremolo Machine ?
The conclusion is drawn in the paper that Shannon's theorem requires an infinite sum and therefore renders the reconstruction process incomplete and inaccurate, thereby requiring higher sample frequencies.
Shannon's formula was for any waveform that has no frequency content above N. That means ANY waveform, not just stationary or cyclical waveforms.
It is not perfect evidence at all that Shannon is somehow broken or regulated to cyclical waveforms at all.
recforums.prosoundweb.com /index.php/m/8215/0   (2730 words)

 Hydrogenaudio Forums > Nyquist was wrong?!
This means that, the nearer the frequency is to the nyquist limit, the poorer the time resolution is (in terms of cycles taken for the sine wave to decay to some fraction of it's former amplitude) - but this isn't a fault - it's just the physics of bandlimitting the signal.
I'm not an expert in signal processing (yet), but i think the problem is cardinal (sinc) interpolation is practically impossible to do in real-time, since you should know the amplitude of samples which are after the ones you're processing (in the time domain).
Yes, the linear interpolation as shown in the article is a rather simplified one.
www.hydrogenaudio.org /forums/lofiversion/index.php/t8177.html   (9502 words)

 Citations: Numerical Methods Based on Sinc and Analytic Functions - STENGER (ResearchIndex)
After replacement of the cotangent with its Mittag Le er series, 8.2) becomes the barycentric formula for S h [9] S h (1 In this way we recover the sinc interpolant for more general functions on R (i.e.
A Chronology of Interpolation: From Ancient Astronomy to Modern..
[22] Thus, we reconstruct with the formula (1) where (2) The function S(x,k,h) is called the Sinc or the Cardinal function.
citeseer.ist.psu.edu /context/35475/0   (3798 words)

 AIPS++ Glossary
It is sometimes referred to as Whittaker's cardinal interpolation formula or the Whittaker-Shannon sampling series, having first been studied in detail by E. Whittaker in 1915 and later introduced into the literature of communications engineering by Shannon in 1949.
When we deal only with digitized images, the code that generates a slice must interpolate to obtain data along it, except when the slice is taken along a row or column of the original image.
The value in a cell can be calculated from a formula which may involve those in other cells, and is recalculated whenever a value on which it depends changes.
aips2.nrao.edu /docs/glossary/s.html   (3003 words)

 Sampling of Bandlimited Functions on Unions of Shifted Lattices (ResearchIndex)
Abstract: We consider Shannon sampling theory for sampling sets which are unions of shifted lattices.
An explicit reconstruction formula is given for sampling sets which are unions of two shifted lattices.
While explicit formulas for unions of more than two lattices are possible, it is more convenient...
citeseer.ist.psu.edu /behmard02sampling.html   (552 words)

 Interpolation   (Site not responding. Last check: 2007-10-13)
In engineering and science one often has a number of data points, as obtained by sampling or some experiment, and tries to construct a function which closely fits those data points - See curve fitting.
Of course when using the simple function to calculate new data points we usually do not receive the same result as when using the original function, but depending on the problem domain and the Interpolation method used the gain in simplicity might offset the error.
Like polynomial interpolation, spline Interpolation incurs a smaller error than linear Interpolation and the interpolant is smoother.
interpolation.iqnaut.net   (983 words)

 More on Digital Signal Processing
Frequencies above the Nyquist frequency N can be observed in the digital signal, but their frequency is ambiguous.
To handle this problem as gracefully as possible, most analog signals are filtered with an anti-aliasing filter (usually a low-pass filter) at the Nyquist frequency before conversion to the digital representation.
Within the limitations of the sampling theorem, the original signal can be completely reconstructed (to within the resolution of the sample values) from the set of ideal samples by expanding each sample into a signal component constructed from the sinc function, using the Nyquist-Shannon interpolation formula.
www.artilifes.com /digital-signal-processing.htm   (1192 words)

 The Future of Anti Aliasing? Will the Quincunx-Like Blur Technologies Return? - Page 3 - Beyond3D Forum
The box filter also damages phase information; for the example of a line that is <0.5 pixel thick, if you move the line slowly, this loss of phase information causes the movement of the line to appear jumpy.
Please correct me if I'm wrong here, but it seems that a simple application of Nyquist theorem implies that to eliminate aliasing, one should implement a filter at half the spatial frequency of the pixel pitch, in other words, 2 pixels wide.
Aliasing in Nyquist frequency sense is a different beast from aliasing in an image processing sense, I think it's far more important to realise the differences than the similarities.
www.beyond3d.com /forum/showthread.php?p=760924   (3830 words)

 Mother of Tone - The CD Format
As Nyquist seems to have been more interested in data transmission than in high-fidelity, we should not wonder, that his statement just defines a maxiumum data-rate of a communications channel.
The first prerequisite of Shannon's sampling theorem is that the Fourier transform or the input signal is zero for all signal frequencies above half of the sampling frequency.
Oversampling is an attempt to make use of Shannon's interpolation formula, in order to get the beat frequencies (alias) out of the sampled signal (that we interpret as correctly recorded samples).
www.altmann.haan.de /byob/cd.htm   (2448 words)

 [No title]
Second 1/3 of midterm given as a take-home exam due on 18 Nov. 17 Nov. Intro to discreteness.
Quaternions: definitions and applications for 3D rotation, and with uniform magnification, when 2nd axis is in rotated frame, quaternion interpolation.
Homogenous coordinates and their use for translation, rotation, and nonuniform and uniform magnification.
www.cs.unc.edu /Courses/comp235-f04/lectures/235LectureOutline.doc   (311 words)

 Hydrogenaudio Forums > 'Normalization' of PCM audio - subjectively benign?
Mathematical interpolation is what allows them to be 'reconstituted' and the aliasing removed by a typical DAC (at the expense of other aspects of fidelity).
It seems Adobe Audition is the only one able to show the waveform as is obtained by bandlimited interpolation rather than joining the dots, so I suggest Mr Rockfan to download some demo of this software and create various signals and see what happens.
This approach only corresponds to a reconstruction filter whose impulse response is a good deal closer to the normalized sinc function (see Whittaker—Shannon interpolation formula) than your stair step thingy.
www.hydrogenaudio.org /forums/lofiversion/index.php/t47827-100.html   (5511 words)

 Imatest - How to test lenses with Imatest
This indicates that some interpolation was required to obtain the final result, and that you should be aware that results may not be as accurate or repeatable as they would be for larger ROIs.
Their contents are described in MTF (Sharpness) plot, Chromatic Aberration, Noise, and Shannon Capacity plot, and Multiple ROI (Region of Interest) plot.
No such simple formula is available for edge responses.
www.imatest.com /docs/lens_testing.html   (4578 words)

 low-pass filter   (Site not responding. Last check: 2007-10-13)
However, this filter is not realizable because the sinc function requires infinite time, which means the filter would need to predict the future.
The perfect low-pass filter is used in the Nyquist-Shannon interpolation formula in conjunction with the Nyquist-Shannon sampling theorem to reconstruct a digital signal from a continuous signal.
Real filters approximate the ideal filter by delaying the signal for a small period of time, allowing them to "see" a little bit into the future.
www.33beat.com /low-pass_filter.html   (1203 words)

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