ROSE-HULMAN INSTITUTE OF TECHNOLOGY Department of Electrical and Computer Engineering ECE 300 Signals and Systems Winter 2008 Computation /Measurement of Average Power in Periodic Signals Lab 05 by Bruce A. Black with some tweaking by others Objectives • To predict the average power in different frequency components of a periodic signal using the Fourier series representation of the periodic signal. • To measure the average power in different frequency components of a periodic signal. • To compare the predicted average power with the measured average power. • To become acquainted with the Agilent E4402B Spectrum Analyzer. Equipment Agilent E4402B Spectrum Analyzer, AgilentFunction Generator, Oscilloscope Background Recently we learned to represent 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 spectrum of a signal is a representation of the average power in each frequency component of the signal. 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, but first we must learn how to convert the Fourier series coefficients that we calculate to the dB values displayed by the spectrum analyzer. Using a trigonometric Fourier series, we can represent any periodic function()xt as 011() cos( ) sin( )ko kkkoxta a kt b ktωω∞∞===+ +∑∑ where 0001()TaxtT=∫dt 02()cos( )koToaxtktTω=∫dt 02()sin( )koTobxtktTω=∫dt 0T is the fundamental period 00022fTπωπ== is the fundamental frequency Page 1 of 1ECE 300 Signals and Systems Winter 2008 A more useful way for us to write the Fourier series is using the complex, or exponential form, 0()kjk tkkxtceω=∞=−∞=∑ where 001()jk tTokcxteTω−=∫dt If we use Euler’s identity to expand this out, we get 00001()[ ( ) ( )]kTxt cos k t jsin k t dtcTωω=−∫ or 00000011() ( ) () ( )kTTxtcosk tdt j xtsinkctTTωω=−∫∫dt Equating terms with the trigonometric Fourier series we get 0022kkkcaabcj==− Hence, from the trigonometric Fourier series we can also determine the coefficients for the complex (exponential) Fourier series. . Writing out a few terms of the exponential Fourier series gives ()021 100000 022210122 221012j t jt jt jtj t jt jt j tjc jc jc jaxt ce ce c ce cece e ce e c ce e ce eωω ωωωω ω−−−−−−−−=+ + ++ + +=+ + ++ + +(( ((""""0ωkc where we have used the fact that kc∗−= 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 0c ()()()01 1 2 202 cos 2 cos 2oxt c c t c c t cωω=+ + + + +((". From our study of power signals, we known that for the periodic power signal () cos( )xt A tωθ=+ the average power is 22avePA= From the complex Fourier series representation, we then know []1122112| | 2| |2cP==cis the average power of ()xt at frequency 0ω Page 2 of 2ECE 300 Signals and Systems Winter 2008 []2222212| | 2| |2cP==cis the average power of ( )xt at frequency 02ω and in general, for 0k ≠[]2212| | 2| |2kkcP==kcis the average power of ()xt at frequency 0kω For the average power is . 0k =002Pc= We can make a plot of the average power in each frequency, which is called a single sided (or one sided) power spectrum. The spectrum analyzer displays the one-sided spectrum of a signal, but instead of showing the value of 22kc at each frequency, the spectrum analyzer shows average power in decibels with respect to a one millivolt RMS reference. For the sinusoid at frequencyokω, the average power in decibels is given by 1010logkkdBrefPPP=, where the power Pk represents the power spectrum coefficient22kc , and the power Pref is the average power delivered to a one-ohm resistor by a one millivolt RMS sinusoid. We have ()1220210log dBmV0.001kkdBmVcP= . 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 20c 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. kc( * 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 3 of 3ECE 300 Signals and Systems Winter 2008 Procedure 1. We have shown that, if we have computed the and from the trigonometric Fourier series, we can determine the as kakbkc0022kkkcaabcj==− Now assume instead we have computed the and want to compute the and . Show how we can do this . Hint: one simple way to do this is to use and its complex conjugate, . Write your answers on the answer sheet at the end of the lab. kckakbkc*kc2a. Use your trigonometric Fourier Series program from the homework to determine the trigonometric Fourier series representation for a square wave of period 10 μs and peak-to-peak amplitude 0.2 V. The waveform we want to use is displayed below Note: The function we use for numerical integration, quadl, has a default tolerance. This tolerance level is used so the function can determine when its estimate of the integral is “good enough”. When dealing with either small valued functions, or very short intervals, you may need to change this tolerance. For the signal in this lab, you will need to change the tolerance to get the proper results (use help quadl). 2b. Plot the trigonometric Fourier series representation and the single-sided power spectrum in dBmV. Use 9 terms. To plot the single sided power spectrum, we just plot the average power terms versus the corresponding frequency 0002...N0ωωω. The average power for each harmonic is given by 22kc . Since the fundamental frequency 0ωis common to all of the frequency terms, we often just plot the Fourier indices, or harmonics, 01. To get the powers in terms of dB, you should use the log10 command to compute the base 10 logarithm. You should also use the subplot command so you can plot both a signal and its Fourier
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