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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

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
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_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
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Sadik
How can Commensal Bacteria change into pathogen?
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all
Tesfaye
by fussion
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what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
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what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
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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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