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The reconstruction can only occur by means of a filter that cancels out all spectral images except for the one directlycoming from the original continuous-time signal. In other words, the canceled images are those having frequencycomponents higher than the Nyquist frequency defined as F s 2 . The condition required by the sampling theorem is equivalent to saying that no overlaps between spectral images are allowed. Ifsuch superimpositions were present, it wouldn't be possible to design a filter that eliminates the copies of the original spectrum. In case of overlapping, a filterthat eliminates all frequency components higher than the Nyquist frequency would produce a signal that is affected by aliasing . The concept of aliasing is well illustrated in the Aliasing Applet , where a continuous-time sinusoid is subject to sampling. If the frequency ofthe sinusoid is too high as compared to the sampling rate, we see that the the waveform that is reconstructed from samples is not theoriginal sinusoid, as it has a much lower frequency. We all have familiarity with aliasing as it shows up in moving images, forinstance when the wagon wheels in western movies start spinning backward. In that case, the sampling rate is given by the frame rate , or number of pictures per second, and has to be related with the spinning velocity of the wheels. This is one of several stroboscopic phenomena.

In the case of sound, in order to become aware of the consequences of the 2 π periodicity of discrete-time signal spectra (see [link] ) and of violations of the condition of the sampling theorem, we examine a simple case.Let us consider a sound that is generated by a sum of sinusoids that are harmonics (i.e., integer multiples) of a fundamental. The spectrumof such sound would display peaks corresponding to the fundamental frequency and to its integer multiples.Just to give a concrete example, imagine working at the sampling rate of 44100 Hz and summing 10 sinusoids. From the sampling theorem we know that, in our case, we can represent without aliasing all frequencycomponents up to 22050 Hz. So, in order to avoid aliasing, the fundamental frequency should be lowerthan 2205 Hz. The Processing (with Beads library) code reported in table [link] implements a generator of sounds formed by 10 harmonic sinusoids. To produce such sounds it is necessary to click on a point of the display window. The x coordinate would vary with thefundamental frequency, and the window will show the spectral peaks corresponding to the generated harmonics. When we click on a pointwhose x coordinate is larger than 1 10 of the window width, we still see ten spectral peaks. Otherwise, we violate the sampling theorem andaliasing will enter our representation.

Aliasing test: Applet to experience the effect of aliasing onsounds obtained by summation of 10 sinusoids in harmonic ratio import beads.*; // import the beads library import beads.Buffer;import beads.BufferFactory; AudioContext ac;PowerSpectrum ps; WavePlayer wavetableSynthesizer;Glide frequencyGlide; Envelope gainEnvelope;Gain synthGain; int L = 16384; // buffer sizeint H = 10; //number of harmonics float freq = 10.00; // fundamental frequency [Hz]Buffer dSB; void setup() {size(1024,200); frameRate(20);ac = new AudioContext(); // initialize AudioContext and create bufferfrequencyGlide = new Glide(ac, 200, 10); // initial freq, and transition time dSB = new DiscreteSummationBuffer().generateBuffer(L, H, 0.5);wavetableSynthesizer = new WavePlayer(ac, frequencyGlide, dSB);gainEnvelope = new Envelope(ac, 0.0); // standard gain control of AudioContext synthGain = new Gain(ac, 1, gainEnvelope);synthGain.addInput(wavetableSynthesizer); ac.out.addInput(synthGain);// Short-Time Fourier AnalysisShortFrameSegmenter sfs = new ShortFrameSegmenter(ac); sfs.addInput(ac.out);FFT fft = new FFT(); sfs.addListener(fft);ps = new PowerSpectrum(); fft.addListener(ps);ac.out.addDependent(sfs); ac.start(); // start audio processinggainEnvelope.addSegment(0.8, 50); // attack envelope }void mouseReleased(){ println("mouseX = " + mouseX);} void draw(){ background(0);text("click and move the pointer", 800, 20);frequencyGlide.setValue(float(mouseX)/width*22050/10); // set the fundamental frequency // the 10 factor is empirically foundfloat[] features = ps.getFeatures(); // from Beads analysis library// It will contain the PowerSpectrum: // array with the power of 256 spectral bands.if (features != null) { // if any features are returned for (int x = 0; x<width; x++){ int featureIndex = (x * features.length) / width;int barHeight = Math.min((int)(features[featureIndex] * 0.05 *height), height - 1); stroke(255);line(x, height, x, height - barHeight); }} }public class DiscreteSummationBuffer extends BufferFactory { public Buffer generateBuffer(int bufferSize) { //Beads generic bufferreturn generateBuffer(bufferSize, 10, 0.9f); //default values }public Buffer generateBuffer(int bufferSize, int numberOfHarmonics, float amplitude) {Buffer b = new Buffer(bufferSize); double amplitudeCoefficient = amplitude / (2.0 * (double)numberOfHarmonics);double theta = 0.0; for (int k = 0; k<= numberOfHarmonics; k++) { //additive synthesis for (int i = 0; i<b.buf.length; i++) { b.buf[i]= b.buf[i] + (float)Math.sin(i*2*Math.PI*freq*k/b.buf.length)/20;} }return b; }public String getName() { //mandatory method implementation return "DiscreteSummation";} }

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
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
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
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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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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