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MSU PHY 440 - Digital Synthesis and Analysis of Periodic Complex Waves

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Digital synthesis, Page 1Digital Synthesis and Analysis of Periodic Complex WavesIntroductionIn this lab you will synthesize periodic waveforms by addition of sine waves. Awaveform is generated by using a computer to add up the Fourier series using theLabView program “Arbitrary Waveform Generator.vi”. This program has the capabilityof generating two separate outputs but, you will use only one. The other will be neededto help you assemble a complex waveform. Once the waveform is assembled to yoursatisfaction on Channel 0, it will be output via the DAC component on the LabViewboard. You will observe analog waveforms on the oscilloscope and you will also hear thewaveforms through headphones.The frequency of the output wave is determined by setting the number ofwaveforms in the construction window which contains 1000 points and by specifyingthe number of points to read every second.Wave GenerationThe Sine WaveUse the library feature of the program to set up a sine wave with amplitude 6 V, 2waveforms in the window and an acquisition rate of 100,000 points/second. What is thefrequency of the resulting wave? Verify your conclusion by making a measurementwith the oscilloscope. Also listen to the tone on the earphones.The TriangleMake an approximate 200 Hz triangle by adding a third harmonic to the sineabove with amplitude chosen from the Fourier series expansion for a triangle:x(t) =8p21n2n=1,3,5 ...•Âcos(2pnt / T )Add five more harmonics and observe the improvement of the triangle shape.The Square WaveMake an approximation to a 200 Hz square wave by using two components, thefirst and third harmonics, as given in the series below:x(t) =4p1nn=1,3,5 ...•Âsin(2pnt / T )Observe the oscillations. Now add eight more terms one at a time. Observe thatthe oscillations do not go away but tend to be concentrated in the region of thediscontinuity. In fact, with a finite number of terms, the oscillations at the discontinuitynever go away. This effect is known as the “Gibbs phenomenon”.Other wavesAdd up the first 4 terms of the series for the half-wave rectified sine:x(t) =1p+12cos(2pt / T ) -2p(-1)n2n= 2,4 ,6 ...•Â1n2-1cos(2pnt / T)Digital synthesis, Page 2Waveforms and pitchUsing the built-in library generate a sawtooth wave with frequency 200 Hz.Compare the pitch with the sine oscillator. You should find that the pitch of the saw isclose to the pitch of the 200 Hz sine.The case of the missing fundamentalThe Fourier expansion of a sawtooth wave with unity amplitude, starting at x = -1when t = 0, is given by:x(t) = -2p1nn=1•Âsin(2pnt / T )• Create a sawtooth wave with a missing fundamental frequency. The simplest way todo this is to construct a sawtooth wave of amplitude A and of 0° phase in onechannel, the fundamental with amplitude A*2/p and 0° phase in the other and thenadd them. The choice of sawtooth expansion phase ensures that we subtract thefundamental.• Observe that the periodicity of the waveform is still 1/200 Hz i.e. 1/200 s. Explainhow this can be.• Study the pitch of the waveform with the missing fundamental. You should find thatit is still 200 Hz. And yet, there is no spectral energy at 200 Hz. This observationplayed an important role in developing theories of human pitch perception.Crest factorThe shape of the waveform of a tone depends on the amplitude and the phases ofthe components. The power spectrum depends only on the amplitudes. Therefore, onecan change the waveform shape while leaving the power spectrum the same by changingthe phases of the components.The crest factor is the maximum value of the waveform, divided by the RMSvalue. In communications practice, there is an advantage to keeping the crest factor low.Below we consider a waveform having the first three harmonics, all with the sameamplitude.• Show that the largest possible crest factor is obtained by choosing phases so as toadd up cosine waves.• Show that the crest factor for three cosines of equal amplitude is √6.• Generate this wave. Observe it and listen to it.• The smallest crest factor can be obtained by reversing the sign of the third harmonic.(It is not obvious why should be so, but it is so.) Generate this wave. Compare itsshape and its sound with the wave from part (c)Components above the Nyquist frequency and fold-over distortionDigital Synthesis creates components with high frequencies that are not desired inthe output. The purpose of a reconstruction filter is to remove them. The generation of asine tone is the simplest illustration. Suppose we want to generate a 5,000-Hz sine, usingthe 20,000 sample rate. In fact we generate quite a complex spectrum. Not only do weget 5,000 Hz. We get 20,000 ± 5,000. We also get (2 * 20,000) ± 5,000, and so on.Digital synthesis, Page 3Explain why this means that the reconstruction filter ought to cutoff below a frequency,which is half the sample rate. Half the sample rate is a frequency known as the


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MSU PHY 440 - Digital Synthesis and Analysis of Periodic Complex Waves

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