MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 003 Signals and Systems Spring 2004 Tutorial 4 Monday March 1 and Tuesday March 2 2004 Announcements Problem set 4 is due this Friday Quiz 1 will be held on Thursday March 11 7 30 9 30 p m in Walker Memorial The quiz will cover material in Chapters 1 3 of O W Lectures and Recitations through Feburary 27 Problem Sets 1 3 and that part of Problem Set 4 involving problems from Chapter 3 The TAs will jointly hold o ce hours from 2 8 p m on Wednesday March 10 and again from 10 a m 3 p m on Thursday March 11 A schedule will be posted on the 6 003 website A quiz review package will be available on the 6 003 website this Thursday TAs will hold two identical optional quiz review sessions on Monday March 8 and Tuesday March 9 7 30 9 30 p m in 34 101 Today s Agenda Frequency Response of LTI Systems Di erential and di erence equations Filtering Real systems Frequency response of cascaded systems CT Fourier Transform Synthesis and analysis equations Variations of the synthesis and analysis equations Rectangular pulse and sinc pair The multiplication and convolution properties 59 1 Frequency Response of LTI Systems In our Fourier series representation of periodic signals we set the CT variable s j so that est becomes ej t Likewise in DT we set z ej so that z n becomes ej n Then the eigenvalue of the LTI system corresponding to the eigenfunction ej t CT and ej n DT is the frequency response of a system H It is de ned through the impulse response h t CT and h n DT as The Frequency Response of LTI Systems in Terms of the Impulse Response H j H ej h t e j t dt CT h n e j n DT n From the eigenfunction property when these exponentials are the inputs of an LTI system the outputs are the same exponentials scaled by the frequency response of the system Now that we know how to write periodic input signals as the linear combination of complex exponentials by determining the Fourier series coe cients we can scale the coe cients appropriately according to the frequency response of the system to get the Fourier series coe cients of the output So if the inputs are periodic signals with Fourier series coe cients ak x t ak ejk 0 t CT ak ejk 0 n DT k x n k then the outputs are periodic signals with Fourier series coe cients bk H jk 0 ak for CT and bk H ejk 0 ak for DT y t ak H jk 0 ejk 0 t CT ak H ejk 0 ejk 0 n DT k y n k There is a caveat when speaking about the frequency response of LTI systems All stable systems have well de ned frequency responses for all frequencies However unstable systems generally do not have a frequency response 1 1 Di erential and di erence equations A large number of LTI systems that we study in real life are described by linear constant coe cient ordinary di erential CT and di erence DT equations LCCODEs so it would be helpful to develop techniques to analyze such systems As we found in problem set 2 nding the impulse response of such systems time domain analysis is a rather tedious procedure However it turns out that a frequency domain analysis is much more straightforward 60 Finding the Frequency Response of Di erential and Di erence Equations Suppose we are given a stable CT or DT system described by a di erential or di erence equation To nd the frequency response we do the following 1 Let x t ej t for CT or x n ej n for DT 2 Let y t H j ej t for CT or y n H ej ej n for DT 3 Plug x t and y t for CT or x n and y n for DT into the di erential or di erence equation 4 Solve for H j for CT or H ej for DT If we apply this method we get the following result The Frequency Response of Di erential and Di erence Equations Suppose we are given a stable CT system described by the following di erential equation dN dN 1 d y t a y t a1 y t a0 y t N 1 N N 1 dt dt dt dM dM 1 d bM M x t bM 1 M 1 x t b1 x t b0 x t dt dt dt aN Its frequency response is H j bM j M bM 1 j M 1 b1 j a0 aN j N aN 1 j N 1 a1 j a0 Similarly a stable DT system described by the following di erence equation aN y n N aN 1 y n N 1 a1 y n 1 a0 y n bM x n M bM 1 x n M 1 b1 x n 1 b0 x n has frequency response H ej bM e jM bM 1 e j M 1 b1 e j b0 aN e jN aN 1 e j N 1 a1 e j a0 61 1 2 Real systems Frequency Response of Real Systems For a real system namely a systems where the impulse response h t in CT and h n in DT is real the frequency response H is conjugate even magnitude and real part are even signals angle and imaginary parts are odd signals CT H j H j H j H j H ej H e j DT H ej H e j Thus when the input of the system is x t cos 0 t CT x n cos 0 n DT the output is y t H j 0 cos 0 t H j 0 y n H ej 0 cos 0 n H ej 0 62 CT DT Problem 4 1 Consider the stable CT LTI system described by see problem set 2 d2 3 5 d y t y t x t y t dt2 2 dt 2 a Find the frequency response of the system b Find the output when the inputs are i xi t cos 3 t ii xii t sin 3 4 t cos t 3 c Let xc t be an input signal with fundamental period T and Fourier series coe cients ak Write the Fourier series coe cients bk of the corresponding output signal yc t in terms of ak and the frequency response Assume that the fundamental period of yc t is also T Compare this method to nding the impulse response h t and convolving h t with the inputs to nd the outputs 63 Problem 4 2 Consider the stable DT LTI system described by see problem set 2 3 5 y n y n 1 y n 2 x n 2 2 a Find the frequency response of the system b Find the output when the inputs …
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