93下信號與系çµ&plusmn

93下信號與系統

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Signals and Systems, Final Exam Solutions Spring 2005, Edited by bypeng 1. [21 points] Consider a continuous-time system with impulse response 3 1 () () 2 tt ht t e ut eut δ −− =+ + a) [3] Determine the transfer function of the inverse system of h 1 . b) [3] Is the inverse system causal and stable? c) [3] Sketch the asymptotic approximation to the gain and phase of the Bode plot for the transfer function of the inverse system. d) [3] Draw a direct-form representation of the inverse system using a minimum number of integrators and multipliers. e) [3] Repeat Step d) and draw a parallel-form representation of the inverse system. f) [3] Repeat Step d) and draw a cascade-form representation of the inverse system. g) [3] Suppose the continuous-time system 1 is cascaded with a sampler, a converter that converts an impulse train to a sequence, and an discrete-time LTI system 2 [] hn , as shown in the following block diagram. Determine the frequency response 2 j He ω such that wn n = when the input x t is a unit impulse. h 1 ( t ) Conversion of impulse train to a sequence h 2 [ n ] w [ n ] y [ n ] y ( t ) x ( t ) n pt tn T +∞ =−∞ = Solution: a) We need the transfer function of the inverse system 1 I Hs satisfying 11 () () 1 I HsHs = , and we have 2 1 1 2 (3 )(1 ) (1 ) 2(3 ) 10 7 (2 )(5 ) () 1 3 1 (3 ) (3 ) (3 ) ss s s s s s s s s ++ + + + + + + + = = = + + so 1 (3 ) (2 )(5 ) = ; There are three possible ROCs Re[ ] 5 s < − ; 5R e [ ] 2 s << ; Re[ ] 2 s >− . But we need the ROC includes Re[ ] 1 s , so the ROC is Re[ ] 2 s . b) Since 1 I is rational and the ROC is right-hand-sided, the inverse system is causal. Since 1 I has all poles in the left half side of the s-plane, the inverse system is stable. c) The sketch of the Bode plot is as the following figure. [Note that the dashed line is the actual response while the solid line is the approximation using the technique in Section 6.5/9.4 in the textbook. One should use the technique in Section 6.5/9.4 to get the full credit.]
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d) 2 1 2 34 () 10 7 I ss Hs ++ = e) 8 1 33 1 () 1 25 I =+ + + + f) 1 31 I ⎛⎞ = ⎜⎟ ⎝⎠ . One instance of the cascade-form is as following figure. (The multiplier “1” should be omitted since we need a minimum number of integrators and multipliers; but the existence of this block makes no loss of points.) g) When x tt δ = , 1 y th t = , and then 1 [ ] () () y ny n Th n T = = , and 1 12 1 2 j kk Ye Y j H j TT T T π ∞∞ Ω =−∞ Ω− == ∑∑ . For <Ω< , 1 j j j Tj j ΩΩ Ω = , and by 2 1 jj j He Ye We Ω = = , we have 2 j j j He T j j Ω = for −<Ω < and with period 2 . 2. [15 points] Consider the sinusoidal modulating signal 00 cos ( ) x tA t ω = with amplitude 0 A and frequency 0 . The carrier wave is ( ) cc ct A t = with amplitude c A and frequency c . a) [3] Suppose the signal is transmitted through double-sideband/suppressed carrier (DSB/SC) modulation. What is the mathematical expression of the modulated signal yt ? b) [3] Show that y t can be expressed as 11 22 y t A t = + and determine the value of , , and A .
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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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93下信號與系統

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