Fourier Analysis and Synthesis of Complex WavesIntroductionIn this lab we will study some aspects of digital synthesis of wave forms and Fourieranalysis of waves to extract their frequency components. We will touch on thequestions of noise spectra and analyze the noise spectrum produced by a zener diode.Digital SynthesisIn this part of the lab we will synthesize periodic waveforms by the addition of sinewaves. We will use the LabView program “Arbitrary Waveform Generator.vi” tobuild up waveforms in one of its windows by adding components assembled in theother. When the final assembly is complete on Channel 0, it will be converted to acontinuous stream of pulses to be observed by an oscilloscope or alternately to beheard through earphones.The frequency of the output wave is determined by setting the number of waveformsin the construction windows which contains 1,000 points and by specifying thenumber of points to read each second.Wave GenerationThe Sine WaveUse the library feature of the program to set up a sine wave withamplitude 6 V, phase angle of 90°, 2 waveforms in the windowand an acquisition rate of 100,000 points/second. What is thefrequency of the resulting wave? Verify your conclusion bymaking a measurement with the oscilloscope. Also listen to thetone on the earphones.The TriangleMake an approximate 200 Hz triangle by adding a third harmonicto the sine above with amplitude chosen from the Fourier seriesexpansion 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 twocomponents, the first and third harmonics, as given in the seriesbelow:x(t) =4p1nn=1,3,5 ...•Âsin(2pnt / T )Observe the oscillations. Now add eight more terms one at a time.Observe that the oscillations do not go away but tend to beconcentrated in the region of the discontinuity. In fact, with afinite number of terms, the oscillations at the discontinuity nevergo away. This effect is known as the “Gibbs phenomenon”.Crest factorThe shape of the waveform of a tone depends on the amplitude andthe phases of the components. The power spectrum depends onlyon the amplitudes. Therefore, one can change the waveform shapewhile leaving the power spectrum the same by changing the phasesof the components.The crest factor is the maximum value of the waveform, divided bythe RMS value. In communications practice, there is an advantageto keeping the crest factor low.Below we consider a waveform having the first three harmonics,all with the same amplitude.Show that the largest possible crest factor is obtained by choosing phases so as to addup 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-overdistortionDigital Synthesis creates components with high frequencies thatare not desired in the output. The purpose of a reconstruction filteris to remove them. The generation of a sine tone is the simplestillustration. Suppose we want to generate a 5,000-Hz sine, usingthe 20,000 sample rate. In fact we generate quite a complexspectrum. Not only do we get 5,000 Hz. We get 20,000 ± 5,000.We also get (2 * 20,000) ± 5,000, and so on.Explain why this means that the reconstruction filter ought to cutoff below afrequency, which is half the sample rate. Half the sample rate is a frequency knownas the “Nyquist frequency”.Analog to digital conversion and the FFTA LabView program, “Acquire&FFT_Nscans.vi”, is available onyour pc which (1) captures a waveform and digitizes it using ananalog-to-digital converter (ADC), (2) takes the fast Fouriertransform (FFT) of a specified number samples of the waveform,and (3) displays the Fourier transform on the monitor. For thenumber of samples always use a number that is a power of 2, i.e.2n, because the FFT program works much more efficiently on sucha sample.Digitize low-frequency waves from the function generator, sine, triangle, and square.Observe the power spectra on the monitor and compare with expected power spectra.(Note: Because there is no anti-aliasing filter in front of the ADC, some componentsof the triangle and square are bound to be above the Nyquist frequency (componentfrequencies greater than half the sample rate) and lead to aliasing. But if thefundamental frequency is low, the components should be small.)Digitize a sine wave with frequency f0 greater than half the sample rate fSR Comparethe frequency of the FFT peak with the calculated frequency of the alias, fSR – f0.NoiseThe best way to build a noise generator is to use a zener diode,reverse biased, and a high-gain amplifier. Figure 1 below showshow.The strength and quality of the noise depends upon the diode andupon the current limiting resistor R1. As a rule, high-voltagezeners work best. Because the power supply is 15 volts, a zenerwith breakdown voltage between 10 and 13 volts is recommended.Different zeners behave differently as noise sources. Even if theycome from the same production lot they behave differently. Thenoise output from each zener is maximized by a different value ofR1. Some zeners don’t make much noise for any value of R1.Some zeners make a lot of noise but only for a rather specific valueof R1. Other zeners make a lot of noise for a wide range of valuesof R1. In the end, if you want to make a noise source, buy tenhigh-voltage zeners and select the best one.Figure 1: Noise Generation Circuit.+15 V1M10K0.047mF1N4740R1+For this lab you will receive a 10-volt zener, type 1N4740. Theoptimum value of R1 is in the range 30 to 50 Wk. Try a couple ofresistors in this range selecting the one that gives the most noise.Make the noise source using a TL084 quad op amp. Observe the noise on theoscilloscope. Can you use the oscilloscope to estimate a typical frequency of the noise?(The expected answer is NO.)The Color of NoiseWhite noise has equal power per unit frequency. That is, thespectral density measured in watts per Hertz is the same for allfrequencies. The zener diode noise source makes noise that iswhite over a very broad frequency range.Show that noise cannot be absolutely
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