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 x t as k 1 k 1 x t a0 ak cos k o t bk sin k ot where a0 1 T0 T0 x t dt 2 x t cos k ot dt To T0 2 bk x t sin k ot dt To T0 T0 is the fundamental period ak 0 2 2 f 0 is the fundamental frequency T0 Page 1 of 1 ECE 300 Signals and Systems Winter 2008 A more useful way for us to write the Fourier series is using the complex or exponential form x t k ce k jk 0t k where 1 x t e jk 0t dt T To 0 If we use Euler s identity to expand this out we get ck ck 1 T0 T0 x t cos k 0t jsin k 0t dt or 1 1 x t cos k 0t dt j x t sin k 0t dt T0 T0 T0 T0 Equating terms with the trigonometric Fourier series we get ck c0 a0 ak b j k 2 2 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 ck x t c 2 e j 2 0t c 1e j 0t c0 c1e j 0t c2 e j 0t c2 e j c2 e j 2 0t c1 e j c1 e j 0t c0 c1 e j c1 e j 0t c2 e j a2 e j 2 0t where we have used the fact that c k ck whenever x t is real valued Notice that aside from c0 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 c0 2 c1 cos 0t c1 2 c2 cos 2 ot c2 From our study of power signals we known that for the periodic power signal x t A cos t 2 A 2 From the complex Fourier series representation we then know 1 2 P1 2 c1 2 c1 2 is the average power of x t at frequency 0 2 the average power is Pave Page 2 of 2 ECE 300 Signals and Systems P2 Winter 2008 1 2 2 c2 2 c2 2 is the average power of x t at frequency 2 0 2 and in general for k 0 1 2 Pk 2 ck 2 ck 2 is the average power of x t at frequency k 0 2 For k 0 the average power is P0 c02 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 2 but instead of showing the value of 2 ck at each frequency the spectrum analyzer shows average power in decibels with respect to a one millivolt RMS reference For the sinusoid at frequency k o the average power in decibels is given by P Pk dB 10 log10 k Pref 2 where the power P k represents the power spectrum coefficient 2 ck and the power P ref is the average power delivered to a one ohm resistor by a one millivolt RMS sinusoid We have 2 2 ck Pk dBmV 10 log10 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 c0 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 ck 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 3 ECE 300 Signals and Systems Winter 2008 Procedure 1 We have shown that if we have computed the ak and bk from the trigonometric Fourier series we can determine the ck as c0 a0 ck ak b j k 2 2 Now assume instead we have computed the ck and want to compute the ak and bk Show how we can do this Hint one simple way to do this is to use ck and its complex conjugate ck Write your answers on the answer sheet at the end of the lab 2a Use your trigonometric Fourier Series program from the homework to determine the trigonometric Fourier series representation for a square wave of period 10 s 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 0 0 2 0 N 0 The average power for each harmonic is given by ck 2 2 Since the fundamental frequency 0 is common to all of the frequency terms we often just plot the Fourier indices or harmonics 0 1 2 N To get the powers in terms of …
View Full Document
Unlocking...