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Signals and Systems, Final Exam Solutions (Draft) Spring 2004, Edited by bypeng 1. (8) The output () yt of a causal LTI system is related to its input () x t by 3() dy t xt dt += . (a)(4) Determine the frequency response H j ω of the system. (b)(4) Find the group delay of this system. Solution: (a) From dy t dt , we have ()3 () () jYj Yj Xj ωωω + = , and then 1 Hj X jj == + . (b) The definition of the group delay is given by {} ( ) d H j d τ ωω ±) , and 1 () 1 t a n 33 =− = )) ) . Therefore, the group delay of this system is 11 22 3 3 t a n t a n 3 1 ( ) 9 dd τω −− ⎛⎞ =− = = ⋅ = ⎜⎟ ++ ⎝⎠ . Grading: (a) Finding 1 3 j = + gets 4 pts . Otherwise, writing X j = gets 2 pts . (b) Finding 2 3 9 = + gets 4 pts . Otherwise, writing d H j d ) and 1 () t a n 3 ) gets 3 pts , and any one of them costs 2 pts . 2. (10) For the discrete-time causal LTI system described by 2 [] 2c o s ()[ 1 ] [ 2 ] [] yn r r yn xn θ + = . (a)(6) Find the impulse response [] hn of this system. (b)(4) What is/are the conditions for the system to be stable? Solution: (a) 2 o s ] [ 2 ] r + = ()2 c o s ( ) j j j j Ye r e Ye re Ye Xe −+ = 1 1 ()1 2 c o s ( ) ( 1 ) ( 1 ) j j j j j j j He r e re r e e r = . If k π for any k Z , 2 1 (1 ) ) j j e ee re e re e θθ θω =+ 2 1 ( ) j jn jn n e e e run ) ) 2 1s i n [ ( 1 ) ] sin 1 j n j j n n j n j n n n j en e e e θ θ + = −= . If k = for some k Z , 2 1 ) j j re = ± [] ( 1 ) ( ) [] n n r un . (b) For both cases, we need 1 r < to make [] absolutely summable, and then to make the system stable. Grading: (a) Finding [] in both cases gets full 6 pts . Otherwise: finding 1 )(1 ) j re e re e = gets 2 pts . Finding 2 1 ) ) j j e re e re e in the first case gets 1 another pt , and finding sin[( 1) ] sin n n + = in the first case gets 1 more pt . Finding 2 1 ) j j re = ± in the second case gets 1 another pt and finding [] ( 1 ) n n in the second case gets 1 more pt . (b) Check the answer case by case.

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3. (14) A periodic triangle wave, given below, is to be sampled periodicly using impulse-train sampling. (a)(8) Draw the system diagram and show the required operations to sample and reconstruct it. (b)(6) Discuss performances of the system you designed in (a). Solution: There is no precise solution to this problem, but there is one key view point. One should mention that the triangle wave is NOT band-limited, so one cannot perfectly reconstruct (or can perfectly reconstruct with first-order hold interpolation and with probability 0) the triangular wave after the impulse-train sampling before knowing that the sampled signal is triangular. Grading: Check case by case. Mentioning the sampling theorem and the band-limitlessness of the triangular wave should get more pts.
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## This note was uploaded on 02/21/2012 for the course EE 101 taught by Professor 張捷力 during the Spring '07 term at National Taiwan University.

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92ä¸‹ä¿¡è™Ÿèˆ‡ç³»çµ±

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