Bài giảng Signal & Systems - Chapter 2: Linear time-invariant systems
P2.9. (a) Show that if the response of an LTI system to f(t) is the
output y(t), then the response of the system to f’(t)=df(t)/dt is y’(t)?
(b) An LTI system has response y(t)=sinω0t to input f(t)=e-5tu(t).
Use the result of part (a) to aid in determine the impulse response
of this system?
P2.11. We are given a certain linear time-invariant system with impluse
response h0(t). We are told that when the input is f0(t) the output is y0(t),
which is sketched in Figure P2.11. We are then given the following set of
inputs f(t) to linear time-invariant systems with the indecated impulse
response h(t):
In each of these cases, determine whether or not we have enough
information to determine the output y(t) when the input is f(t) and the
system has impulse response h(t). If it is possible to determine y(t),
provide an accurate sketch of it with numerical values clearly indicated
on the graph.
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Nội dung text: Bài giảng Signal & Systems - Chapter 2: Linear time-invariant systems
- Ch-2: Linear time-invariant systems P2.1 . Determine and sketch the convolution of the following signals: t+1; 0≤ t ≤ 1 f(t)= 2-t; 1<t≤ 2 h(t)= δ(t+2)+2 δ(t+1) 0; elsewhere P2.2 . Suppose that ≤ ≤ 1; 0 t 1 t α ≤ f(t)= and h(t)=f(α ) where 0< 1 0; elsewhere a) Determine and sketch y(t)=f(t)*h(t)? b) If dy(t)/dt contains only three discontinuities, what is the value of α? Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-2: Linear time-invariant systems P2.3 . Let: f(t)=u(t− 3) − u(t − 5) and h(t)=e−3t u(t) a) Compute y(t)=f(t)*h(t) b) Compute g(t)=[df(t)/dt]*h(t) c) How is g(t) related to y(t) +∞ P2.4 . Let: h(t)=∆ (t) and f(t)=∑ δ (t-kT) k=- ∞ Determine and sketch y(t)=f(t)*h(t) for the following values of T: (a) T=4; (b) T=2; (c) T=3/2; and (d) T=1 P2.5 . Which of the following impulse response correspond(s) to stable LTI systems? −(1 − j2)t −t (a) h1 (t)=e u(t) (b) h2 (t)=e cos(2t)u(t) Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 1
- Ch-2: Linear time-invariant systems P2.9 . (a) Show that if the response of an LTI system to f(t) is the output y(t), then the response of the system to f’(t)=df(t)/dt is y’(t)? ω -5t (b) An LTI system has response y(t)=sin 0t to input f(t)=e u(t). Use the result of part (a) to aid in determine the impulse response of this system? P2.10 . Consider an LTI system and a signal f(t)=2e -3t u(t-1). If f(t)→ y(t) df(t)→ − − 2t and dt 3y(t)+e u(t) Determine the impulse response of this system. Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-2: Linear time-invariant systems P2.11 . We are given a certain linear time-invariant system with impluse response h 0(t). We are told that when the input is f 0(t) the output is y 0(t), which is sketched in Figure P2.11. We are then given the following set of inputs f(t) to linear time-invariant systems with the indecated impulse response h(t): (a) f(t)=2f0 (t); h(t)=h 0 (t) (b) f(t)=f0 (t)-f 0 (t-2); h(t)=h 0 (t) (d) f(t)=f0 (-t); h(t)=h 0 (t) (c) f(t)=f0 (t-2); h(t)=h 0 (t+1) (e) f(t)=f0 (-t); h(t)=h 0 (-t) ' ' (f) f(t)=f0 (t); h(t)=h 0 (t) In each of these cases, determine whether or not we have enough information to determine the output y(t) when the input is f(t) and the system has impulse response h(t). If it is possible to determine y(t), provide an accurate sketch of it with numerical values clearly indicated on the graph. Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 3
- Ch-2: Linear time-invariant systems P2.15 . Compute and sketch the impulse response h(t) of the following causal LTI, first-order diffrential system initially at rest: dy(t) df(t) 2+ 4y(t)=3 + 2f(t) dt dt P2.16 . Find the impulse response h(t) of the following causal LTI, second-order diffrential system initially at rest: d2 y(t) dy(t) df(t) +2 + y(t)= + f(t) dt2 dt dt Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 5
