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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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