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Signals and Systems, Final Exam Solutions (Draft) Spring 2007, Edited by bypeng 1. [12] A continuous-time LTI system with frequency response () Hj ω is constructed from two continuous-time LTI systems with frequency responses 1 and 2 , respectively. The straight-line approximations of the Bode magnitude plots of 1 and are shown in the following figure. All of the poles and zeros of 1 Hs and ( ) are on the real axis. a) [4] Specify 2 if 1 and 2 are connected in cascade form. b) [4] Specify 2 if 1 and 2 are connected in parallel form. c) [4] Specify 2 if 1 and 2 are connected in negative feedback form with 2 in the feedback loop. Solution: We know that 1) 1 (1 ) ( 8)( 40) Aj jj ωω + = ++ , and by 1 (0) 2 H = (since 6dB), 1 2 840 A = , 640 A = . 2) 2 (8 ) B j = + , any by (0) 0.1 H = (since 20dB ), 0.1 88 B = , 6.4 B = . Therefore: a) Cascade form: 12 ()() H j H j = , 2 2 1 ( ) 6.4 ( 8)( 40) 0.01( 40) ( ) ( 8) 640( 1) ( 8)( j j j j j j j + == = + + b) Parallel form: H j =+ , 21 2 2 22 2 2 6.4 640( () () () ( 8) ( 8)( 40) 6.4( 40) 640( 1)( 8) 640( ) 5753.6( ) 4864 ( 8) ( 40) ( 8) ( 40) 32 100( ) 899( ) 760 640( 0.94465)( 8.0453 5(8 ) (4 0 ) j H j j j j j j j ωωω + =− = + +− + + + + ≈− 2 5) ( 8) ( 40) c) Feedback form: 1 1( ) ( ) Hj Hj = + , 2 2 1 2 1 1 ( 8) ( 8)( 40) ( ) ( ) 6.4 640( ( 8) 100( ) 899( ) 760 ( 8)( 0.94465)( 8.04535) 640( 6.4( j j j j + = + ⎡⎤ + + ⎣⎦ =≈

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2. [12] Figure (a) below shows a model of amplitude modulation system using a pulse-train carrier. a) [4] Let () x t be a band-limited signal with ( ) 0 for / X jT ωω π =≥ , as shown above. Determine and sketch R j ω and Qj . b) [4] Find the maximum value of Δ such that wt xt = with an appropriate filter M j . c) [4] Determine and sketch the compensating filter ( ) M j such tha t = . Solution: a) We have [] 11 2 1 2 * * 2 kk Rj Xj Pj X j TT T T ππ δ ∞∞ =−∞ ⎤⎛ ⎛⎞ ==− = ⎜⎟ ⎢⎥ ⎝⎠ ⎦⎝ ∑∑ 2 2( ) s i n () ()() Rj Hj Δ == The sketch is as the following, where the dash-dot line is R j and the solid line is . b) We need 2 T > Δ to keep in the interval : −<< ⎩⎭ not distorted. 2 T Δ < . c) 2 2sin 0e l s e w h e r e l s e w h e r e T T Hj T T Mj Δ << ⎪⎪ ⎨⎨ , the sketch is as the following.
3. [8] Determine () X s and its region of convergence based on the following 5 descriptions about a signal x t , which is real, and its Laplace transform X s : i) (0) 4 X = ii) X s has exactly two poles iii) X s has no zeros in the finite s -plane iv) X s has a pole at 2 sj =− v) 3 t ext is absolutely integrable Solution:

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95ä¸‹ä¿¡è™Ÿèˆ‡ç³»çµ±

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