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 ()xt can be written as ()0200, where 1jkftkkxtae fπ∞=−∞==∑T. Writing out a few terms gives ()00 0000 021 1222 2 2 222101222 2 2 2221012j ft j ft j ft j ftj ft j ft j ft j ftja ja ja jaxt ae ae a ae aeae e ae e a ae e ae eππ ππππ ππ−−−−−−−−=+ + ++ + +=+ + ++ + +(( ((""""0k (1) where we have used the fact that kaa∗−= whenever ()xt is real-valued. Notice that, aside from , the terms come in pairs (actually, complex conjugate pairs). We can combine each positive-frequency term with its matching negative frequency term to obtain 0a () ()()01 0 1 2 0 22cos2 2cos22xt a a ft a a ft aππ=+ + + + +((". (2) A power spectrum for ()xt 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 is seen 0kf Page 1 of 6ECE 300 Signals and Systems Spring 2007 to be 2ka (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 and 0kf0kf− are added. The result, representing the average power in the sinusoid ()2cos2kkaftπ+ (ka, is shown at frequency (and corresponds most directly with equation 0kf(2)). ()xSf f0f 0f− 20a 21a 21a 22a 22a 23a 23a 0 ()xSf f 0f 20a 212 a 222 a232 a0 Figure 1: Two-Sided Power Spectrum 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 22ka at each frequency, the spectrum analyzer shows average power in decibels with respect to a one millivolt RMS reference. For the sinusoid at frequency , the average power in decibels is given by 0kf 1010logkkdBrefPPP=, where the power Pk represents the power spectrum coefficient 22ka, and the power Pref is the average power delivered to a one-ohm resistor by a one millivolt RMS sinusoid. We have ()22210log dBmV0.001kkdBmVaP = . The units “dBmV” indicate that the reference for the decibels is a one millivolt RMS sinusoid.* Note: The spectrum analyzer will not display the DC term 20a 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 . ka( * 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 6ECE 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) ()()310.1cos 2 100 10 V=×xttπ b) ()2xt is a square wave of period 10 μs and peak-to-peak amplitude 0.2 V. c) ()3xt is a triangle wave of period 10 μs 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 6ECE 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 μs 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 μs 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 Ω output impedance device, designed to deliver maximum power to a 50 Ω load. The FG by default will display the voltage amplitude delivered to a 50 Ω load, independent of what is actually connected. By changing the display setting to
View Full Document
Unlocking...