<< Chapter < Page Chapter >> Page >

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

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Media processing in processing. OpenStax CNX. Nov 10, 2010 Download for free at http://cnx.org/content/col10268/1.14
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Media processing in processing' conversation and receive update notifications?

Ask