Bài giảng Signal & Systems - Chapter 6: Continuous-Time System Analysis Using the Laplace Transform

Nội dung text: Bài giảng Signal & Systems - Chapter 6: Continuous-Time System Analysis Using the Laplace Transform

1. Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.1 . Consider the signal f(t)=e -5t u(t-1) and denote its Laplace transform by F(s). a) Using analysis function, evaluate F(s) and specify its ROC. b) Determine the values of the finite numbers A and t 0 such that the -5t Laplace transform G(s) of g(t)=Ae u(-t-t0) has the same algebraic form as F(s). What is the ROC corresponding to G(s) P6.2 . Consider the signal f(t)=e -5t u(t)+e -βtu(t) and denote its Laplace transform by F(s). What are the constraints placed on the real and imaginary parts of β if the ROC of F(s) is Re{s}>-3? P6.3 . For the Laplace transform of et sin2t; t≤ 0 f(t)=  0; t>0 Indicate the location of its poles and its ROC Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.4 . How many signals have a Laplace transform that may be expressed as s− 1 (s+2)(s+3)(s2 +s+1) in its ROC? P6.5 . Given that at 1 e− u( t )↔ ; ROC: Re{s}>Re{-a} s+a determine the inverse Laplace transform of 2(s+ 2) F(s)= ; ROC:Re{s}>-3 s2 +7s+12 Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 1
2. Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.10 . Determine the function of time, f(t), for each of the following one-side Laplace transforms 2s+5 5 1 a) d) g) 4 s2 +5s+6 s2 (s+2) (s+1)(s+2) 3s+5 2s+1 s+1 b) e) h) s2 +4s+13 (s+1)(s2 +2s+2) s(s+2)2 (s 2 +4s+5) 2 (s+1 ) s+2 s3 c) f) i) s2 -s-6 s(s+1) 2 (s+1)2 (s 2 +2s+5) P6.11 . Determine the transfer function and step response of the system depicted in FigP6.11 1Ω 1H v (t) i 1F 1Ω v0 (t) FigP6.11 Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.12 . Determine transfer function of the system shown in FigP6.12(a), and (b) R C vi (t) C v0 (t) vi (t) R v0 (t) R 2 R 2 R1 R1 (a) (b) FigP6.12 P6.13 . The input f(t) and output y(t) of a causal LTI system are related through the block diagram representation shown in FigP6.13. a) Determine a differential equation relating y(t) and f(t). b) Is this system stable? Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 3
3. Ch-6: Continuous-Time System Analysis Using the Laplace Transform Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.17 . Show op-amp realization of the following transfer functions: -10 10 s+2 a) H(s)= b) H(s)= c) H(s)= s+5 s+5 s+5 P6.18 . Show two different op-amp realization of the transfer function: s+2 3 H(s)= =1- s+5 s+5 P6.19 . Show op-amp canonical realization of the following transfer functions: 3s+7 s2 +5s+2 a) H(s)= b) H(s)= s2 +4s+10 s2 +4s+13 Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 5