15 Consider the following cases where we want to determine different types of responses. (a) The input to a LTI system isx(t) =u(t) – 2u(t – 1) + u(t – 2) and the Laplace transform of the output is given by (s + 2)(1 – e-s)2 s(s + 1)2 Y (s) = determine the impulse response of the system. (b) Without computing the inverse of the Laplace transform X (s) = s(s2 + 2s + 10) corresponding to a causal signal x(t), determine lim,→x(t). (c) The Laplace transform of the output of a LTI system is 1 Z(s) = s((s +2)² + 1) what would be the steady-state response zss(t)? (d) The Laplace transform of the output of a LTI system is

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15 Consider the following cases where we want to determine different types of responses.
(a) The input to a LTI system is x(t) =u(t) – 2u(t – 1) + u(t – 2) and the Laplace transform
of the output is given by
(s + 2)(1 – e¬s)²
s(s + 1)2
Y (s) =
determine the impulse response of the system.
(b) Without computing the inverse of the Laplace transform
1
X (s) =
s(s2
+ 2s + 10)
corresponding to a causal signal x(t), determine lim,→x(t).
(c) The Laplace transform of the output of a LTI system is
1
Z(s)=
s((s +2)² + 1)
what would be the steady-state response zss(1)?
(d) The Laplace transform of the output of a LTI system is
e-s
W (s) =
s((s – 2)² + 1)
how would you determine if there is a steady state or not? Explain.
(e) The Laplace transform of the output of a LTI system is
s +1
V (s) =
s((s+1)² + 1)
Determine the steady state and the transient responses corresponding to Y (s).
24
3.10 PROBLEMS
235
Answers: (a) H(s)= (s+2)/(s + 1)², ROC: 0 > 0; (b) lim;→x(1) = 0.1; (e) v,(t)=
-0.5e- cos(t)u(t) +0.5e¬' sin(t)u(t).
Transcribed Image Text:15 Consider the following cases where we want to determine different types of responses. (a) The input to a LTI system is x(t) =u(t) – 2u(t – 1) + u(t – 2) and the Laplace transform of the output is given by (s + 2)(1 – e¬s)² s(s + 1)2 Y (s) = determine the impulse response of the system. (b) Without computing the inverse of the Laplace transform 1 X (s) = s(s2 + 2s + 10) corresponding to a causal signal x(t), determine lim,→x(t). (c) The Laplace transform of the output of a LTI system is 1 Z(s)= s((s +2)² + 1) what would be the steady-state response zss(1)? (d) The Laplace transform of the output of a LTI system is e-s W (s) = s((s – 2)² + 1) how would you determine if there is a steady state or not? Explain. (e) The Laplace transform of the output of a LTI system is s +1 V (s) = s((s+1)² + 1) Determine the steady state and the transient responses corresponding to Y (s). 24 3.10 PROBLEMS 235 Answers: (a) H(s)= (s+2)/(s + 1)², ROC: 0 > 0; (b) lim;→x(1) = 0.1; (e) v,(t)= -0.5e- cos(t)u(t) +0.5e¬' sin(t)u(t).
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