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# MIT 6 003 - Signals and Systems

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MASSACHUSETTS INSTITUTE OF TECHNOLOGYDepartment of Electrical Engineering and Computer Science6.003: Signals and Systems — Spring 2004Tutorial 4Monday, March 1 and Tuesday, March 2, 2004Announcements• 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 willcover 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 oﬃce hours from 2–8 p.m. on Wednesday, March 10 and again from 10a.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 identicaloptional 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– Diﬀerential and diﬀerence 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 properties591 Frequency Response of LTI SystemsIn our Fourier series representation of periodic signals, we set the CT variable s = jω, so that estbecomesejωt. Likewise, in DT, we set z = ejω, so that znbecomes ejωn. Then, the eigenvalue of the LTI systemcorresponding to the eigenfunction ejωt(CT) and ejωn(DT) is the frequency response of a system H. It isdeﬁned through the impulse response h(t) (CT) and h[n](DT)as:The Frequency Response of LTI Systems in Terms of the ImpulseResponse:H(jω)=+∞−∞h(t)e−jωtdt (CT)H(ejω)=+∞n=−∞h[n]e−jωn(DT)From the eigenfunction property, when these exponentials are the inputs of an LTI system, the outputsare the same exponentials scaled by the frequency response of the system. Now that we know how to writeperiodic input signals as the linear combination of complex exponentials by determining the Fourier seriescoeﬃcients, we can scale the coeﬃcients appropriately according to the frequency response of the systemto get the Fourier series coeﬃcients of the output. So, if the inputs are periodic signals with Fourier seriescoeﬃcients ak:x(t)=kakejkω0t(CT)x[n]=kakejkω0n(DT)then the outputs are periodic signals with Fourier series coeﬃcients bk= H(jkω0)akfor CT and bk=H(ejkω0)akfor DT:y(t)=kakH(jkω0)ejkω0t(CT)y[n]=kakH(ejkω0)ejkω0n(DT)There is a caveat when speaking about the frequency response of LTI systems. All stable systems havewell-deﬁned frequency responses for all frequencies. However, unstable systems generally do not have afrequency response.1.1 Diﬀerential and diﬀerence equationsA large number of LTI systems that we study in real life are described by linear constant-coeﬃcientordinary diﬀerential (CT) and diﬀerence (DT) equations (LCCODEs), so it would be helpful to developtechniques to analyze such systems. As we found in problem set 2, ﬁnding the impulse response of suchsystems (time-domain analysis) is a rather tedious procedure. However, it turns out that a frequency-domainanalysis is much more straightforward:60Finding the Frequency Response of Diﬀerential and Diﬀerence Equa-tions:Suppose we are given a stable CT or DT system described by a diﬀerentialor diﬀerence equation. To ﬁnd the frequency response, we do the following:1. Let x(t)=ejωtfor CT or x[n]=ejωnfor DT.2. Let y(t)=H(jω)ejωtfor CT or y[n]=H(ejω)ejωnfor DT.3. Plug x(t)andy(t) for CT or x[n]andy[n] for DT into the diﬀerential ordiﬀerence equation.4. Solve for H(jω) for CT or H(ejω)forDT.If we apply this method, we get the following result.The Frequency Response of Diﬀerential and Diﬀerence Equations:Suppose we are given a stable CT system described by the following diﬀer-ential equation:aNdNdtNy(t)+aN−1dN−1dtN−1y(t)+···+ a1ddty(t)+a0y(t)= bMdMdtMx(t)+bM−1dM−1dtM−1x(t)+···+ b1ddtx(t)+b0x(t).Its frequency response is:H(jω)=bM(jω)M+ bM−1(jω)M−1+ ···+ b1(jω)+a0aN(jω)N+ aN−1(jω)N−1+ ···+ a1(jω)+a0Similarly, a stable DT system described by the following diﬀerence equation:aNy[n − N]+aN−1y[n − (N − 1)] + ···+ a1y[n − 1] + a0y[n]= bMx[n − M]+bM−1x[n − (M − 1)] + ···+ b1x[n − 1] + b0x[n],has frequency response:H(ejω)=bMe−jM ω+ bM−1e−j(M −1)ω+ ···+ b1e−jω+ b0aNe−jN ω+ aN−1e−j(N −1)ω+ ···+ a1e−jω+ a0611.2 Real systemsFrequency Response of Real Systems:For a real system, namely, a systems where the impulse response h(t)inCT(and h[n] in DT) is real, the frequency response H is conjugate even (magnitudeand real part are even signals, angle and imaginary parts are odd signals):CT:|H(jω)| = |H(−jω)|,∠H(jω)=−∠H(−jω).DT:|H(ejω)| = |H(e−jω)|,∠H(ejω)=−∠H(e−jω).Thus, when the input of the system is:x(t)=cos(ω0t) (CT),x[n]=cos(ω0n)(DT),the output is:y(t)=|H(jω0)|cos(ω0t + ∠H(jω0)) (CT),y[n]=|H(ejω0)|cos(ω0n + ∠H(ejω0)) (DT).62Problem 4.1Consider the stable CT LTI system described by (see problem set 2):d2dt2y(t)+52ddty(t) −32y(t)=x(t).(a) Find the frequency response of the system.(b) Find the output when the inputs are:(i) xi(t) = cos(3πt).(ii) xii(t)=sin(3π4t)+cos(πt +π3).(c) Let xc(t) be an input signal with fundamental period T and Fourier series coeﬃcients ak. Write theFourier series coeﬃcients bkof the corresponding output signal yc(t) in terms of akand the frequencyresponse. Assume that the fundamental period of yc(t) is also T .Compare this method to ﬁnding the impulse response h(t) and convolving h(t) with the inputs to ﬁnd theoutputs.63Problem 4.2Consider the stable DT LTI system described by (see problem set 2):y[n]+52y[n − 1] −32y[n − 2] = x[n].(a) Find the frequency response of the system.(b) Find the output when the inputs are:(i) xi[n]=cos(π3n).(ii) xi[n]=sin(3π4n)+cos(π4n +π3).(c) Let xc[n] be an input signal with fundamental period N and Fourier series coeﬃcients ak. Write theFourier series coeﬃcients bkof the corresponding output signal yc[n] in terms of akand the frequencyresponse. Assume that the fundamental period of yc[n] is also N .Compare this method to ﬁnding the impulse response h[n] and convolving h[n] with the inputs to ﬁnd theoutputs.64Problem 4.3(From 6.003 Quiz 1, Fall

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