MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 003 Signals and Systems Spring 2009 Final Exam Review Packet Final Date Time Location Coverage Notes Wednesday May 20 2009 1 30 PM 4 30 PM Johnson All material covered in the course The exam is closed book except for three 8 5 11 two sided sheet of notes No calculators are allowed We will provide copies of all the tables in O W that we used this term namely the Tables of CT and DT Fourier Series properties on pages 206 and 221 of O W the Tables of CT Fourier transform pairs and properties on pages 328 and 329 the Tables of DT Fourier transform pairs and properties on pages 391 and 392 the Tables of Laplace transform pairs and properties on pages 691 and 692 and the Tables of z transform pairs and properties on pages 775 and 776 Marathon Office Hours The TAs will jointly hold office hours in the week of May 11 A schedule will be posted on course website Final Exam Review Session The TAs will hold an optional final review session on the following date time and location Review session Friday May 15 1 30 4 30 PM in 34 101 Practice Problems The attached set of problems should provide you with ample opportunity to exercise your understanding of and facility with the material covered on this quiz These problems will be covered in the quiz review session The solutions to these problems will be posted on the 6 003 website on the same day as the review session A practice final exam from a previous semester will be available online Both of these sets of problems should help to spark questions you might want to discuss with the TAs during their office hours 1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 003 Signals and Systems Spring 2009 Final Exam Review Problems Problem 1 The purpose of this problem is to test your understanding of continuous time convolution A CT LTI system has input x t impulse response h t and output y t as shown below Note that the scales for x t are not necessarily the same as for h t and y t x t h t A y t 1 12 9 6 3 T T t 6 4 2 2 4 6 t 6 4 2 2 4 6 t Determine 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 sampling Suppose that we have two discrete time signals x1 n and x2 n that we wish to transmit simultaneously using frequency division multiplexing The problem is that each of the signals fills the entire frequency band In particular suppose that X1 ej and X2 ej the DTFTs of x1 n and x2 n are as shown below X1 ej 2 0 2 2 X2 ej 2 0 2 To perform the frequency division multiplexing a system with the following structure is proposed x1 n z1 n Insert Zeros Lowpass Filter y n x2 n z2 n Insert Zeros Highpass Filter where the lowpass and highpass filters H1 ej and H2 ej respectively are as shown below H1 ej 1 2 2 0 2 2 2 H2 ej 1 2 2 0 2 The signals z1 n and z2 n are obtained by inserting zeros between successive values of x1 n and x2 n respectively This can be mathematically expressed as follows x1 n n even 2 z1 n 0 n odd x2 n n even 2 z2 n 0 n odd a Sketch the DTFTs of z1 n z2 n and y n b Suppose that y n is passed through another lowpass filter whose frequency response is H1 ej given above This is illustrated in the figure below It is claimed that x1 n can be recovered from the filter output w n Show that the claim is valid and describe how x1 n can be recovered y n H1 ej 3 w n Problem 3 ulation The purpose of this problem is to test your understanding of continuous time modxm t y t Nonlinear No Memory Filter H j z t w t cos c t In the modulator shown above the modulating signal xm t and a sinusoid at the intended carrier frequency are added to produce y t xm t cos c t which is then passed through a non linear device to yield z t 5y t y 2 t a Assume xm t is a real even function having the spectrum shown below where W c Xm j 1 W W Make a carefully labeled sketch of Z j over the range 3 c 3 c b Describe the frequency response H j of the filter such that w t has the form of xm t double sideband amplitude modulated with carrier on a carrier at c Problem 4 The purpose of this problem is to test your understanding of continuous time modulation We would like to transmit the signal x t with the Fourier transform depicted on the left side of the figure below Unfortunately the only available communications channels have limited bandwidth Specifically each such channel can be viewed as an LTI system with frequency response H j depicted on the right side of the figure below 2W W X j H j 1 1 W 2W W W Fortunately we have two such channels at our disposal and thus it is possible to design systems S1 and S2 depicted below so that z t x t 4 H j x t S1 S2 z t H j Both S1 and S2 can be constructed using 1 signal generators that can produce signals of the form cos 0 t at any fixed frequency 0 2 multipliers and adders 3 ideal filters Specify the designs of S1 and S2 Problem 5 The purpose of this problem is to test your understanding of the Laplace transform An LTI system with system function H s has input x t and output y t It is known that When x t e t u t then y t K e 3t u t et u 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 8 3 for 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 is X z 1 3z 1 1 3z 1 2z 2 for 1 z 2 Problem 7 The purpose of this problem is to test your understanding of discrete time system functions The system function of a DT LTI system is z4 z 4 a4 H z 5 where a is real and positive It is known that X h n n and that the unit sample response of the system h n 0 for all n …
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