ECE 301 Division 1, Fall 2008 Instructor: Mimi Boutin Final Examination Instructions:

Transcription

1 ECE 30 Division, all 2008 Instructor: Mimi Boutin inal Examination Instructions:. Wait for the BEGIN signal before opening this booklet. In the meantime, read the instructions below and fill out the requested info. 2. When the end of the exam is announced, you must stop writing immediately. Anyone caught writing after the exam is over will get a grade of zero. 3. At the end of this document is a 5 page table of formulas and 4 pages of scratch paper. You may detach these once the exam begins provided you sign your name on top of each page and slide them back inside your exam before handing it in. 4. This is a closed book exam. The only personal items allowed are pens/pencils, erasers, your Purdue ID and something to drink. Anything else is strictly forbidden. 5. You must keep your eyes on your exam at all times. Looking around is strictly forbidden. 6. Please leave your Purdue ID out so the proctors may check your identity. Name: Signature: Itemized Scores Problem : Problem 6: Problem 2: Problem 7: Problem 3: Problem 8: Problem 4: Problem 5: Total:

2 (0 pts). Compute the energy and the power of the signal x[n] = + 3j e 2πjn. 2

3 (20 pts) 2. NOTE: EVEN THOUGH THIS PAGE CONTAINS MULTIPLE CHOICE QUESTIONS, IT WOULD BE UNWISE TO ATTEMPT TO CHEAT, AS THE QUESTIONS ARE PERMUTED IN THE 4 DIERENT VERSIONS O THIS TEST. a) Let x(t) and y(t) be the input and the output of a continuous-time system, respectively. Answer each of the questions below with either yes or no. (No justification needed). Yes No If y(t) = (t + 5)x(t), is the system causal? If y(t) = x(t 2 ), is the system causal? If y(t) = x(5 t), is the system memoryless? If y(t) = x(t/3), is the system stable? If y(t) = x(t/3), is the system linear? If y(t) = u(t) x(t), is the system LTI? b) Are the following statements true or false? (No justification needed.) True In a cascade of a time-scaling and a time-delay, the order of the systems does not matter. alse In a cascade of LTI systems, the order of the systems does not matter. If two linear systems have the same unit impulse response, then the two systems are the same. e 2πjt = ( e 2πj) t = t =. 3

4 (25 pts) 3. You have learned that a discrete-time LTI system can be specified in several ways, such as. an explicit input-output formula y[n] = f (x[n], n), 2. the unit impulse response of the system h[n], 3. the system s function H(z), with its ROC, 4. a difference equation. or each system below, you are given one of these representations. ind the others. (No justification needed.) System System 2 Explicit Input-output ormula y[n] = n k= x(k) h[n] 3 n u[ n] H(z), ROC Difference Equation (5 pts) 4. The Laplace transform of the unit impulse response of an LTI system is H(s) = s+3 s 2 +s 4. What is the system s response to the input x(t) = e2t (No justification needed. ) 4

5 (5 pts) 5. Let x(t) be a periodic signal with fundamental frequency ω 0. Without using the table of ourier transforms pairs, prove that the ourier transform of x(t) is X (ω) = 2πa k δ (ω kω 0 ) k= where the a k s are the ourier series coefficients of x(t). Note: if you write nothing on this page, you will automatically get 3 pts. 5

6 (5 pts) 6. Obtain the inverse Laplace transform of X(s) = s 2, ROC: Re(s) < s + 2 Note: if you write nothing on this page, you will automatically get 3 pts. 6

7 (5 pts) 7. Using the definition of the z-transform (and without otherwise using the table), compute the inverse z-transform of X(z) = z +, z > 4. 4z Note: if you write nothing on this page, you will automatically get 3 pts. 7

8 (25 pts) 8. Let x(t) be a pure imaginary and even signal with Nyquist rate equal to π. (3+2pts) a) Sketch the graph of a ourier transform that could be the ourier transform of x(t). (You will get two extra points if it has a cute shape!) Then base your answer to b) and c) on this graph. (0 pts) b) Sketch the graph of the ourier transform of x (t) = x(t) k= δ ( t 2k ) 3. (0 pts) c) Sketch the graph of the ourier transform of x 2 [n] = x( 2 3 n). 8

9 Table DT Signal Energy and Power E = CT Signal Energy and Power n= P = lim N 2N + x[n] 2 () N n= N x[n] 2 (2) E = P = lim T 2T x(t) 2 dt (3) T T x(t) 2 dt (4) ourier Series of CT Periodic Signals with period T x(t) = a k e jk( 2π T )t (5) a k = T k= T 0 x(t)e jk( 2π T )t dt (6) ourier Series of DT Periodic Signals with period N x[n] = N a k = N k=0 N a k e jk( 2π N )n n=0 x[n]e jk( 2π N )n (7) (8) 9

10 CT ourier Transform.T. : X (ω) = Inverse.T.: x(t) = Z x(t)e jωt dt (9) Z X (ω)e jωt dω 2π (0) Properties of CT ourier Transform Let x(t) be a continuous-time signal and denote by X (ω) its ourier transform. continuous-time signal and denote by Y(ω) its ourier transform. Let y(t) be another Signal Linearity: ax(t) + by(t) ax (ω) + by(ω) () Time Shifting: x(t t 0 ) e jωt 0 X (ω) (2) requency Shifting: e jω0t x(t) X (ω ω 0 ) (3) Time and requency Scaling: x(at) ω a X a (4) Multiplication: x(t)y(t) X (ω) Y(ω) 2π (5) Convolution: x(t) y(t) X (ω)y(ω) (6) Differentiation in Time: T d x(t) dt jωx (ω) (7) Some CT ourier Transform Pairs e jω 0t 2πδ(ω ω0 ) (8) sin W t πt u(t + T ) u(t T ) δ(t) e at u(t), Re{a} > 0 te at u(t), Re{a} > 0 2πδ(ω) (9) u(ω + W ) u(ω W ) (20) 2 sin(ωt ) (2) ω (22) a + jω (23) (a + jω) 2 (24) 0

11 DT ourier Transform Let x[n] be a discrete-time signal and denote by X(ω) its ourier transform..t.:x (ω) = Inverse.T.: x[n] = X x[n]e jωn (25) n= Z X (ω)e jωn dω (26) 2π 2π Properties of DT ourier Transform Let x(t) be a signal and denote by X (ω) its ourier transform. Let y(t) be another signal and denote by Y(ω) its ourier transform. Signal.T. Linearity: ax[n] + by[n] ax (ω) + by(ω) (27) Time Shifting: x[n n 0 ] e jωn 0 X (ω) (28) requency Shifting: e jω0n x[n] X (ω ω 0 ) (29) Time Reversal: x[ n] X ( ω) (30) Multiplication: x[n]y[n] X (ω) Y(ω) 2π (3) Convolution: x[n] y[n] X (ω)y(ω) (32) Differencing in Time: x[n] x[n ] ( e jω )X (ω) (33) Some DT ourier Transform Pairs N X k=0 a k e jk( 2π N )n 2π X k= e jω 0n 2π X a k δ(ω 2πk N ) (34) δ(ω ω 0 2πl) (35) l= X 2π δ(ω 2πl) (36) l= j, sin W n πn, 0 < W < π X (ω) = δ[n] u[n] α n u[n], α < (n + )α n u[n], α < 0 ω < W 0, π ω > W X (ω)periodic with period 2π (37) (38) e jω + π X δ(ω 2πk) k= (39) αe jω (40) ( αe jω ) 2 (4)

12 Laplace Transform X(s) = Z x(t)e st dt (42) Properties of Laplace Transform Let x(t), x (t) and x 2 (t) be three CT signals and denote by X(s), X (s) and X 2 (s) their respective Laplace transform. Let R be the ROC of X(s), let R be the ROC of X (z) and let R 2 be the ROC of X 2 (s). Signal L.T. ROC Linearity: ax (t) + bx 2 (t) ax (s) + bx 2 (s) At least R R 2 (43) Time Shifting: x(t t 0 ) e st 0 X(s) R (44) Shifting in s: e s 0t x(t) X(s s 0 ) R + s 0 (45) Conjugation: x (t) X (s ) R (46) Time Scaling: x(at) s a X a ar (47) Convolution: x (t) x 2 (t) X (s)x 2 (s) At least R R 2 (48) Differentiation in Time: Differentiation in s: Integration : d x(t) dt sx(s) At least R (49) dx(s) tx(t) R (50) ds Z t x(τ)dτ X(s) At least R Re{s} > 0 (5) s Some Laplace Transform Pairs Signal LT ROC e αt u(t) e αt u( t) δ(t) all s (52) s + α s + α Re{s} > α (53) Re{s} < α (54) 2

13 z-transform X(z) = X n= x[n]z n (55) Properties of z-transform Let x[n], x [n] and x 2 [n] be three DT signals and denote by X(z), X (z) and X 2 (z) their respective z-transform. Let R be the ROC of X(z), let R be the ROC of X (z) and let R 2 be the ROC of X 2 (z). Signal z-t. ROC Linearity: ax [n] + bx 2 [n] ax (z) + bx 2 (z) At least R R 2 (56) Time Shifting: x[n n 0 ] z n 0 X(z) R, but perhaps adding/deleting z = 0 (57) Time Shifting: x[ n] X(z ) R (58) Scaling in z: e jω0n x[n] X(e jω 0 z) R (59) Conjugation: x [n] X (z ) R (60) Convolution: x [n] x 2 [n] X (z)x 2 (z) At least R R 2 (6) Some z-transform Pairs Signal LT ROC u[n] z z > (62) u[ n ] z z < (63) α n u[n] αz z > α (64) α n u[ n ] αz z < α (65) δ[n] all z (66) 3

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