MASSACHUSETTS INSTITUTE OF TECHNOLOGYDepartment of Electrical Engineering and Computer Science6.003: Signals and Systems — Spring 2009Final Exam Review PacketFinal Date: Wednesday, May 20, 2009Time: 1.30 PM–4.30 PM.Location: JohnsonCoverage: All material covered in the course.Notes: The exam is closed book except for three 8.5′′× 11′′two-sided sheet of notes. Nocalculators are allowed. We will provide copies of all the tables in O&W that weused this term, namely, the Tables of CT and DT Fourier Series properties on pages206 and 221 of O&W, the Tables of CT Fourier transf orm pairs and properties onpages 328 and 329, the Tables of DT Fourier transform pairs and properties onpages 391 and 392, the Tables of Laplace transform pairs and properties on pages691 and 692, and the Tables of z-transform pairs and properties on pages 775 and776.Marathon Office Hours: The TAs will jointly hold office hours in the week of May 11. A sched-ule will be posted on course website.Final Exam Review Session: The TAs will hold an optional fi nal r eview session on the followingdate, time and location:Review s ession: Friday, May 15, 1:30–4:30 PM in 34-101.Practice Problems: The attached set of problems should pr ovide you with ample opportun ity toexercise your understanding of and facility with the material covered on this quiz. These problemswill be covered in the quiz review session. The solutions to these problems will be posted on the6.003 website on the same day as the review session. A practice final exam from a pr evious semesterwill be available online. Both of these sets of problems should help to spark questions you m ightwant to discuss with the TAs during their office hours.1MASSACHUSETTS INSTITUTE OF TECHNOLOGYDepartment of Electrical Engineering and Computer Science6.003: Signals and Systems — Spring 2009Final Exam Review ProblemsProblem #1 The purpose of this problem is to test your understanding of continuous-time con-volution.A CT LTI system has input x(t), impulse response h(t), and output y(t) as shown below. Notethat the scales for x(t) are not necessarily the same as for h(t) and y(t).Atx(t) h(t) y(t)tt136912-6 -4 -2 2 4 6 2 4 6-6 -2-4-T TDetermine the values of the parameters of x(t): A and T.Problem #2 The purpose of this problem is to test your understanding of discrete-time sam-pling.Suppose that we have two discrete-time signals, x1[n] and x2[n], that we wish to transmit simul-taneously using frequency-division multiplexing. The pr oblem is that each of the signals fills theentire frequency band. In particular, suppose that X1(ejω) and X2(ejω), the DTFTs of x1[n] andx2[n], are as shown below.ω0π2π−π−2πjω1X (e )jω2X (e )ω0π2π−π−2π2To perform the frequen cy-division multiplexing, a system with the following structure is proposed,x [n]1z [n]1x [n]2z [n]2y[n]InsertZerosZerosInsertLowpassFilterHighpassFilterwhere th e lowpass and highpass filters, H1(ejω) and H2(ejω) respectively, are as shown below.jω1H (e )jωH (e )2ωπ2π−π−2πωπ2π−π−2π11π/20π/20−π/2−π/2The signals z1[n] and z2[n] are obtained by inserting zeros between successive values of x1[n] andx2[n], respectively. This can be mathematically expressed as follows,z1[n] =x1n2 n even0 n od dz2[n] =x2n2 n even0 n od d(a) Sketch the DTFTs of z1[n], z2[n], and y[n ].(b) Suppose that y[n] is passed through another lowp ass filter whose f requency response is H1(ejω)given above. This is illustrated in the figure below. It is claimed that x1[n] can be recoveredfrom the filter output w[n]. Show that the claim is valid, and describe how x1[n] can berecovered.y[n] w[n]jω1H (e )3Problem #3 The purpose of this problem is to test your understanding of continuous-time mod-ulation.cos( t)ωcx (t)mH(j )ωz(t)y(t) w(t)FilterNo MemoryNonlinearIn the modulator shown above, the modulating signal xm(t) and a sinusoid at the intended carrierfrequency are addedto produce y(t) = xm(t) + cos(ωct), which is then passed throu gh a non-lineardevice to yieldz(t) = 5y(t) + y2(t).(a) Assume xm(t) is a real, even function having the spectrum shown below, where W ≪ ωc.X (j )ωm-W W1ωMake a carefully labeled sketch of Z(jω) over the range −3ωc< ω < 3ωc.(b) Describe the frequency response H(jω) of the filter such that w(t) has the form of xm(t)double-sideband amplitude-modulated (with carrier) on a carrier at ωc.Problem #4 The purpose of this problem is to test your understanding of continuous-time mod-ulation.We would like to transm it the signal x(t) with the Fourier transform depicted on the left side of thefigure below. Unfortunately, the only available communications chan nels have limited bandwidth.Specifically, each such channel can be viewed as an LTI system with frequency response H(jω)depicted on the right side of the fi gu re below.X(j )ωH(j )ω-W Wω11ω-W W-2W 2WFortunately, we have two such channels at our disposal, and thus, it is possible to design systemsS1and S2, depicted below , so that z(t) = x(t).4ωH(j )H(j )ωS1S2x(t) z(t)Both S1and S2can be constructed using:(1) signal generators that can produce signals of the form cos(ω0t) at any fixed frequency ω0;(2) multipliers and adders;(3) ideal filters.Specify the designs of S1and S2.Problem #5 The purpose of this problem is to test your understanding of the Laplace transform.An LTI system with system f unction H(s) has inpu t x(t) and outpu t y(t). It is known that:• When x(t) = e−tu(t), theny(t) = Ke−3tu(t) + etu(−t),where K is a constant that you will need to determine to solve the problem.• When x(t) = 1 for all t, then y(t) =83for all t.Find H(s) including its region of convergence (ROC).Problem #6 The purpose of this problem is to test your understanding of z-transforms.Determine the DT signal x[n] given that the z-transform isX(z) =1 + 3z−11 + 3z−1+ 2z−2,for 1 < |z| < 2.Problem #7 The purpose of this problem is to test your understanding of discrete-time systemfunctions.The s ystem function of a DT LTI system isH(z) =z4z4− a4,5where a is real and positive. It is known that∞Xn=−∞|h[n]| < ∞and that the unit-sample response of the system h[n] = 0 for all n < N for some value of N.(a) Sketch the pole/zero diagram of H(z).(b) Determine the region of convergence of H(z) consistent with the information given.(c) Determine the range of a that is consistent with the information given.Problem #8 The purpose of this problem is to test
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