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Consider the signal x that is zero everywhere but at the instants 1 , 0 , and 1 where it has values 1 , 0.5 , and 0.25 , respectively. At every instant n , x n can be expressed as 1 δ n 1 0.5 δ n 0.25 δ n 1 . By linearity, the output can be obtained by composition of carefully translated and weighted impulseresponses: y n 1 h n 1 0.5 h n 0.25 h n 1 .

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To generalize the example [link] we can define the operation of convolution .

Convolution of two signals h and x
y n h * x n m x m h n m

The operation of convolution can be fully understood by the explicit construction of some examples ofconvolution product. The module Discrete-Time Convolution gives the graphic construction of an examples and it offers pointers toother examples.


The properties of the convolution operation are well illustrated in themodule Properties of Convolution . The most interesting of such properties is the extension:


If x n is extended over M 1 samples, and h n is extended over M 2 samples, then the convolution product y n is extended over M 1 M 2 1 samples.

Therefore, the signal convolution product is longer than both the input signal and the impulse response.

Another interesting property is the commutativity of the convolutionproduct, such that the input signal and the impulse response can change their roles without affecting the outputsignal.

Frequency response and filtering

The Fourier Transform of the impulse response is called Frequency Response and it is represented with H ω . The Fourier transform of the system output is obtained by multiplication of the Fourier transform of theinput with the frequency response, i.e., Y ω H ω X ω .

The frequency response shapes, in a multiplicative fashion, the input-signal spectrum or, in other words, it performs some filtering by emphasizing some frequency components and attenuating some others. A filtering can alsooperate on the phases of the spectral components, by delaying them of different amounts.

Filtering can be performed in the time domain (or space domain), by the operation of convolution, or in the frequency domain by multiplication of the frequency response.

Take the impulse response that is zero everywhere but at the instants 1 , 0 , and 1 where it has values 1 , 0.5 , and 0.25 , respectively. Redefine the filtering operation filtra() of the Sound Chooser presented in the module Media Representation in Processing . In this case filtering is operated in the time domain by convolution.

void filtra(float[]DATAF, float[] DATA, float WC, float RO) {//WC and R0 are useless, here kept only to avoid rewriting other //parts of codefor(int i = 2; i<DATA.length-1; i++){ DATAF[i]= DATA[i+1] + 0.5*DATA[i]+ 0.25*DATA[i-1];} }
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The notion of causality is quite intuitive and it corresponds to the experience of stimulating a system andgetting back a response only at future time instants. For discrete-time LTI systems, this happens when the impulseresponse is zero for negative (discrete) time instants. Causal LTI systems can produce with no appreciabledelay an output signal sample-by-sample , because the convolution operator acts only on present andpast values of the input signal. In [link] the impulse response is not causal, but this is not a problem because the whole input signal isalready available, and the filter can process the whole block of samples.

2d filtering

The notions of impulse response, convolution, frequency response, and filtering naturally extend from 1D to 2D, thusgiving the fundamental concepts of image processing.

Convolution of two 2D signals (images)
y m n h * x m n k l x k l h m k n l
If x is the image that we are considering, it is easy to realize that convolution isperformed by multiplication and translation in space of a convolution mask or kernel h (it is the impulse response of the processing system). As in the 1D casefiltering could be interpreted as a combination of contiguous samples (where the extension of such clusterdepends on the extension of the filter impulse response) that is repeated in time, sample by sample. So, in 2D spacefiltering can be interpreted as a combination of contiguous samples (pixels) in a cluster, whose extension is given bythe convolution mask. The so-called memory of 1-D systems becomes in 2-D a sort of distance effect .

As in the 1D case, the Fourier transform of the impulse response iscalled Frequency response and it is indicated by H ω X ω Y . The Fourier transform of the system output is obtained by Fourier-transforming the input and multiplying theresult by the frequency response. Y ω X ω Y H ω X ω Y X ω X ω Y .

Consider the Processing code of the blurring example and find the lines that implement the convolution operation.

for(int y=0; y<height; y++) { for(int x=0; x<width/2; x++) { float sum = 0;for(int k=-n2; k<=n2; k++) { for(int j=-m2; j<=m2; j++) { // Reflect x-j to not exceed array boundaryint xp = x-j; int yp = y-k;//... omissis ... //auxiliary code to deal with image boundariessum = sum + kernel[j+m2][k+n2]* red(get(xp, yp)); }} output[x][y] = int(sum);} }
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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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I think
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what is the stm
Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
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what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Smarajit Reply
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Source:  OpenStax, Media processing in processing. OpenStax CNX. Nov 10, 2010 Download for free at http://cnx.org/content/col10268/1.14
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