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Topic: Convolution theorem

Related Topics

Dirichlet convolution

Convolution

Fourier transform

Continuous Fourier transform

Frequency domain

Discrete Fourier transform

Fast Fourier transform

Deconvolution

Fourier series

Frequency spectrum

Window function

Spectrogram

Time domain

Fourier optics

Digital signal processing

	Chapter 12: Properties of The Fourier Transform
		In words, this theorem states that the derivative of a convolution is equal to the convolution of either of the functions with the derivative of the other.
		Convolution is the mathematical operation which describes many physical processes in which the response or output of a system is the result of superposition of many individual responses.
		According to the convolution theorem, the prescribed convolution of input with impulse response is equivalent to multiplication of the input spectrum with the transfer function of the system.
	research.opt.indiana.edu /Library/FourierBook/ch12.html (3233 words)

	Convolution (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-12)
		In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g.
		A convolution is a kind of very general moving average, as one can see by taking one of the functions to be an indicator function of an interval.
		An out-of-focus photograph is the convolution of the sharp image with the blur circle formed by the iris diaphragm.
	convolution.kiwiki.homeip.net.cob-web.org:8888 (868 words)

	Convolution
		Convolution is, loosely spoken, a sliding weighted average of one function with another function providing the weights.
		The convolution of a function with an impulse function shifted by k is the function itself shifted by k.
		Important for our purposes is the convolution of a set of samples (located at integer positions for the sake of simplicity) with a continuous (finite) function (which we can regard as a reconstruction filter).
	www.cg.tuwien.ac.at /~theussl/DA/node23.html (284 words)

	Convolution Background (Site not responding. Last check: 2007-10-12)
		Convolution can be used to calculate the response of a system to arbitrary inputs by using the impulse response of a system.
		The superposition theorem states that the response of the system to the string of impulses is just the sum of the response to the individual impulses.
		Another excellent web-based demonstration of convolution is The Joy of Convolution at Johns Hopkins.
	www.swarthmore.edu /NatSci/echeeve1/Ref/Convolution/Convolution.html (911 words)

	The convolution theorem and its applications
		The correlation theorem is a result that applies to the correlation function, which is an integral that has a definition reminiscent of the convolution integral.
		The correlation theorem can be stated in words as follows: the Fourier tranform of a correlation integral is equal to the product of the complex conjugate of the Fourier transform of the first function and the Fourier transform of the second function.
		The only difference with the convolution theorem is in the presence of a complex conjugate, which reverses the phase and corresponds to the inversion of the argument u-x.
	www-structmed.cimr.cam.ac.uk /Course/Convolution/convolution.html (2266 words)

	Convolution
		In mathematics and in particular, functional analysis, the convolution (German: Faltung) is a mathematical operator which takes two functions $f$ and $g$ and produces a function as output that represents the amount of overlap of the two functions for each relative translation.
		Generalizing the above cases, the convolution can be defined for any two square-integrable functions defined on a locally compact topological group.
		In this case, it is also possible to give, for instance, a Convolution Theorem, however it is much more difficult to phrase and requires representation theory for these types of groups and the peter-weyl theorem[?] of Harmonic analysis.
	www.ebroadcast.com.au /lookup/encyclopedia/co/Convolution.html (458 words)

	Convolution theorem - Wikipedia, the free encyclopedia
		In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the point-wise product of Fourier transforms.
		Versions of the convolution theorem are true for various Fourier-related transforms.
		This theorem also holds for the Laplace transform and two-sided Laplace transform, and when suitably modified for the Mellin transform and Hartley transform (see Mellin inversion theorem).
	en.wikipedia.org /wiki/Convolution_theorem (304 words)

	The convolution theorem
		Theorem 2 (Convolution Theorem) The spectrum of the convolution of two functions in spatial domain is equivalent to the product of the transforms of both input signals, and vice versa, symbolically,
		This is especially useful if the circumstances require to transform the functions to the frequency domain anyhow, which results in a quite cheap way for convolution.
		Furthermore, the convolution theorem provides an excellent means to perform signal analysis in frequency domain.
	www.cg.tuwien.ac.at /~theussl/DA/node24.html (231 words)

	Fourier Analysis
		Thus, convolution is a way of combining two functions, in a sense using each one to blur the other, making all possible relative shifts between the two functions when computing the integral of their product to obtain the corresponding output values.
		Convolution is extremely important because it is one basis of describing how any linear system h(t) acts on any input s(t) to generate the corresponding output r(t).
		Filtering is a linear operation implemented by the convolution of an image f(x,y) with filter kernel(s) g(x,y), and the resulting output ``image" h(x,y) normally then undergoes non-linear operations of various kinds for image segmentation, motion detection, texture classification, pattern recognition, and image understanding.
	www.cl.cam.ac.uk /Teaching/2000/ContMaths/JGD-notes/node11.html (1462 words)

	CHAPTER-5
		The convolution theorem says that the FT of a convolution of two functions is proportional to the products of the individual Fourier transforms, and vice versa.
		The convolution theorem tells us that this is a sinc function at the frequency of the sine wave.
		With the convolution theorem it can be seen that the convolution of an NMR spectrum with a Lorentzian function is the same as the Fourier Transform of multiplying the time domain signal by an exponentially decaying function.
	www.cis.rit.edu /htbooks/nmr/chap-5/chap-5.htm (1199 words)

	Convolution - Wikipedia, the free encyclopedia
		This use of periodic domains is sometimes called a cyclic, circular or periodic convolution.
		Generalizing the above cases, the convolution can be defined for any two integrable functions defined on a locally compact topological group (see convolutions on groups below).
		also the probability distribution of the sum of two independent random variables is the convolution of each of their distributions.
	en.wikipedia.org /wiki/Convolution (878 words)

	CONVOLUTION (Site not responding. Last check: 2007-10-12)
		Convolution represents one of the most fundamental operations of time series analysis and one of the most physically meaningful.
		Convolution is this process of linearly modifying a signal by a filter.
		Convolution is most easily understood by examining the convolution of discrete functions.
	www.higp.hawaii.edu /~cecily/courses/gg313/DA_book/node100.html (582 words)

	PlanetMath: multiplicative function
		monoids and, because of the fundamental theorem of arithmetic, is completely determined by its restriction to prime numbers.
		A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic.
		This shows that the multiplicative functions with the convolution form an abelian group with the identity element
	planetmath.org /encyclopedia/MultiplicativeFunction.html (565 words)

	Convolution Theorem
		This is perhaps the most important single Fourier theorem of all.
		It is the basis of a large number of FFT applications.
		Since an FFT provides a fast Fourier transform, it also provides fast convolution, thanks to the convolution theorem.
	ccrma.stanford.edu /~jos/mdft/Convolution_Theorem.html (447 words)

	4.3 Discrete Convolution (Site not responding. Last check: 2007-10-12)
		Assuming both s(t) and h(t) are digital functions with a sampling interval of unity, the convolution operation is defined as
		Convolution of these two discrete signals equals 2, 4, -4, 0, 2 1/2, -1, i.e.
		Action convolution of discrete functions (2, -2, 1) and (1, 3, 1/2, -1) yields (2, 4, -4, 0, 2 1/2, -1).
	www-rohan.sdsu.edu /~jiracek/digital/filtering/discreteconvolution.html (153 words)

	A Systems Approach to the Convolution Theorem (Site not responding. Last check: 2007-10-12)
		Convolution is a powerful tool for determining the output of a system to any input.
		The Convolution Theorem is developed here in a completely mathematical way.
		On this page we will derive the convolution theorem: If the input to a system is x(t), and the impulse response of that system is h(t), then we can determine the output of the system, y(t), from the integral:
	www.swarthmore.edu /NatSci/echeeve1/Ref/Convolution/SysConvolve.html (270 words)

	Convolution and Autocorrelation (Site not responding. Last check: 2007-10-12)
		Here the difference between autocorrelation and convolution is illustrated by considering the following function which is a unit ramp cut off at t = 1.
		and the relation between the fourier transform of f(t) and the transform of the convolution is shown below.
		From the convolution theorem, the transform of the convolution of 2 functions is the product of the transform of each function.
	cnyack.homestead.com /files/aconv/convau1.htm (276 words)

	psfmatch
		The psf matching function is computed directly from the reference and input image data using the objects specified in psfdata, the data regions specified by dnx, dny, pnx, and pny, and the convolution theorem.
		PSFMATCH computes the convolution kernel required to match the point-spread functions of the input images input to the point-spread functions of the reference images reference using either the image data or pre-computed psfs and the convolution theorem.
		If convolution = "image", the matching function is computed directly from the input and reference image data using the objects listed in psfdata and the convolution theorem as described in the ALGORITHMS section.
	stsdas.stsci.edu /cgi-bin/gethelp.cgi?psfmatch (2969 words)

	Convolution Theorem, Transfer Functions, and Filtering
		This can be modeled by an operation known as a convolution of two functions where one function is the image and the other is the neighborhood weighting function or kernel.
		Convolution in either domain is equivalent to multiplication in the other.
		Terminology: a convolution function in the spatial domain is a kernel; a pointwise multiplication function in the frequency domain is a filter.
	rivit.cs.byu.edu /morse/550-F95/node12.html (1007 words)

	Kevin Cowtan's Book of Fourier, University of York, UK (Site not responding. Last check: 2007-10-12)
		The convolution theorem is one of the most important relationships in Fourier theory, and in its application to x-ray crystallography.
		To convolute two functions, the first function must be superimposed on the second at every possible position, and multiplied by the value of the second function at that point.
		If we convolute it with a delta-function somewhere else, then the duck is moved to that point.
	www.ysbl.york.ac.uk /~cowtan/fourier/convthry.html (270 words)

	Part I: Fourier Transforms and Sampling
		Convolution is a linear process, so g(t) must be a linear function of f(t) to be expressed by equation (1b).
		The convolution theorem states that the Fourier transform of g(t) is
		The convolution theorem also proves that a signal that is finite in time has an infinite spectrum: the response can always be expressed as a convolution with a sinc function, which extends the spectrum to infinity.
	www.silcom.com /~aludwig/Signal_processing/Signal_processing.htm (2926 words)

	Interactive guide to diffraction .. (Site not responding. Last check: 2007-10-12)
		The real space electron density distribution of a single atom (left image) is convoluted with a finite train of five point scatterers (middle image) to form a finite sized crystal (right image).
		The Fourier transform of the finite train of point scatterers in itself is an example of the inverse to the convolution theorem, the multiplication theorem.
		As a result each reciprocal lattice point is convoluted by the Fourier transform of the box function which results in the subsidiary maxima.
	www.uni-duesseldorf.de /WWW/MathNat/AC2/teaching/conv_a.html (274 words)

	Image Processing Khorosware: Convolution Theorem II (Site not responding. Last check: 2007-10-12)
		But if linear convolution is sought then the functions need to be extended (zero padded) in order to avoid wraparound errors.
		The dimensions of the original image "gull.viff" are 256x256, and the dimensions of the convolution kernel, Laplacian filter, are 3x3.
		In order to perform Linear convolution between the original image and filter kernel, we must first determine new dimensions to avoid the wraparound effect.
	www.cs.ioc.ee /~khoros2/linear/convolution-fft/front-page.html (398 words)

	Signal Analysis Review
		The Fourier transform F(jw) of the convolution f(t) of two functions f1(t) and f2(t) equals the product of the Fourier transforms F1(jw) and F2(jw) of these two functions.
		The convolution integral, as expressed in Eqn.(1), holds for all cases as long as the system is linear and time-invariant.
		Carrying this idea farther, it is possible to write down the laws of a convolution algebra with similarities to those for ordinary multiplication.
	www.neurophys.wisc.edu /www/comp/docs/not012.html (1922 words)

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