92下信號與系çµ&plusmn

92下信號與系統

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 5

92下信號與系統

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online