ROSE HULMAN INSTITUTE OF TECHNOLOGY Department of Electrical and Computer Engineering ECE 300 Signals and Systems Spring 2007 Measurement of Fourier Coefficients Lab 06 by Bruce A Black with some tweaking by others Objectives To measure the Fourier coefficients of several waveforms and compare the measured values with theoretical values To become acquainted with the Agilent E4402B Spectrum Analyzer Equipment Agilent E4402B Spectrum Analyzer Agilent Function Generator Oscilloscope BNC T Connector Background Recently we learned to calculate the spectrum of a periodic signal by using the Fourier series We have in our lab spectrum analyzers that can display the spectrum of a signal in pseudo real time The Agilent E4402B Spectrum Analyzer SA can be used to view the power spectrum of any signal of frequency up to 3 GHz The SA displays a one sided spectrum in decibels dBs versus frequency In lab we will observe the spectra of sinusoids square and triangle waves and pulse trains but first we must learn how to convert the Fourier series coefficients that we calculate to the dB values displayed by the spectrum analyzer Recall that any periodic signal x t can be written as x t ae k k j 2 kf 0t where f 0 1 T0 Writing out a few terms gives x t a 2 e j 2 2 f0t a 1e j 2 f0t a0 a1e j 2 f0t a2 e j 2 2 f0t 1 a2 e j a2 e j 2 2 f0t a1 e j a1 e j 2 f0t a0 a1 e j a1 e j 2 f0t a2 e j a2 e j 2 2 f0t where we have used the fact that a k ak whenever x t is real valued Notice that aside from a0 the terms come in pairs actually complex conjugate pairs We can combine each positivefrequency term with its matching negative frequency term to obtain x t a0 2 a1 cos 2 f 0t a1 2 a2 cos 2 2 f 0t a2 2 A power spectrum for x t based on the Fourier series is shown in Fig 1 This is a two sided spectrum in which the power associated with the complex exponential at frequency kf 0 is seen Page 1 of 6 ECE 300 Signals and Systems to be ak 2 Spring 2007 and corresponds most directly with equation 1 The corresponding one sided power spectrum is shown in Fig 2 To make the one sided spectrum the powers associated with complex exponentials at frequencies kf 0 and kf 0 are added The result representing the average power in the sinusoid 2 ak cos 2 f k t ak is shown at frequency kf 0 and corresponds most directly with equation 2 Sx a2 a3 2 f 2 a1 a0 2 Sx 2 a2 a1 2 a0 2 2 2 a1 a3 f0 f 2 2 2 2 a3 f 0 2 a2 f0 Figure 1 Two Sided Power Spectrum 2 f 0 f0 Figure 2 One Sided Power Spectrum The SA displays a one sided spectrum as shown in Fig 2 but instead of showing the value of 2 2 ak at each frequency the spectrum analyzer shows average power in decibels with respect to a one millivolt RMS reference For the sinusoid at frequency kf 0 the average power in decibels is given by P Pk dB 10 log10 k Pref 2 where the power Pk represents the power spectrum coefficient 2 ak and the power Pref is the average power delivered to a one ohm resistor by a one millivolt RMS sinusoid We have 2 2 ak Pk dBmV 10 log dBmV 2 0 001 The units dBmV indicate that the reference for the decibels is a one millivolt RMS sinusoid 2 Note The spectrum analyzer will not display the DC term a0 even when one is present in the signal Instead it displays a large spike at zero frequency allowing for easy location of DC on the display Also because it is showing a power spectrum the spectrum analyzer does not measure or display the phase angles ak Further information on working with dBs is available in the document called Guide to dBs available on the class webpage It is suggested you read this before lab Page 2 of 6 ECE 300 Signals and Systems Spring 2007 Procedure Read the documents SA hints E4402B and Reading SA Display E4402B both available on the class webpage The waveforms we will be analyzing each having zero DC offset are a x1 t 0 1cos 2 100 103 t V b x2 t is a square wave of period 10 s and peak to peak amplitude 0 2 V c x3 t is a triangle wave of period 10 s and peak to peak amplitude 0 2 V Just to be sure there is no confusion regarding the waveforms they are displayed below a b c Getting Ready Before connecting any input to the spectrum analyzer be sure that there is no large DC offset on the signal in our case no DC offset should be contained in the signal The only sure way to know this is to properly examine the signal on the oscilloscope Be sure you understand the implications of the different input impedance of the scope and the spectrum analyzer before you begin this lab Summarize the proper measurement procedure on the final page of the lab Page 3 of 6 ECE 300 Signals and Systems Spring 2007 Calibrate the Spectrum Analyzer Calibrate the spectrum analyzer as described in the document SA hints E4402B Measuring the Spectrum 1 Use the function generator to generate a sinusoid of frequency 100 kHz and open circuit amplitude 0 1 V waveform a Use the oscilloscope to verify the amplitude Now observe the signal power spectrum on the spectrum analyzer Measure the power level and frequency Record your measurements in the table on the final page of this lab 2 Informally vary the frequency and the amplitude of the sinusoid and observe how the spectrum analyzer display changes 3 Use the function generator to generate a square wave of period 10 s and peak to peak amplitude 0 2 V waveform b Using the spectrum analyzer measure the level of the first nine harmonics Record your measurements in the table on the final page of this lab 4 Use the function generator to generate a triangle wave of period 10 s and peak to peak amplitude 0 2 V waveform c Use the spectrum analyzer to measure the level of the first nine harmonics Record your measurements in the table on the final page of this lab Note concerning Agilent FG amplitude readings The Agilent function generators have an interesting feature built into their displays The FG is a 50 output impedance device designed to deliver maximum power to a 50 load The FG by default will display the voltage amplitude delivered to a 50 load independent of what is actually connected By changing the display setting to High Z the FG display will display the voltage delivered to a high impedance load assuming a high impedance load is connected However the FG …
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