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Bài giảng Tín hiệu và hệ thống - Chương 2: Linear time-invariant systems

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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)+2δ(t+1)  
0;  
elsewhere  
P2.2. Suppose that  
1; 0 ≤ t ≤1  
t
α
and h(t)=f( ) where 0<α ≤1  
f(t)=  
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  
f(t)=u(t −3) − u(t −5) and h(t)=e−3tu(t)  
P2.3. Let:  
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)  
+∞  
h(t)=∆(t) and f(t)= δ (t-kT)  
∑
P2.4. Let:  
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?  
(a) h1(t)=e−(1−j2)tu(t) (b) h2 (t)=e−tcos(2t)u(t)  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
1
Ch-2: Linear time-invariant systems  
P2.6. The following are the impulse response of continuous-time  
LTI systems. Determine whether each systems is causal and/or  
stable. Justify your answer.  
(a) h(t)=e-4t u(t − 2) (b) h(t)=e-6tu(3−t) (c) h(t)=e-2tu(t+50)  
(d) h(t)=e2tu(-1-t) (e) h(t)=e6|t|  
(g) h(t)=[2e-t -e(t-100)/100 ]u(t)  
(f) h(t)=te-tu(t)  
P2.7. Consider a system whose input f(t) and output y(t) satisfy the  
first-order differential equation  
dy(t) +2y(t)=f(t)  
dt  
The system also satisfies the condition of initial rest.  
a) Determine the system output y1(t) when the input is f1(t)=e-3tu(t)  
b) Determine the system output y2(t) when the input is f2(t)=e-2tu(t)  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-2: Linear time-invariant systems  
P2.8. Consider an LTI system with input and output related through  
the equation  
t
y(t)=  
e
−(t−τ )f(τ -2)dτ  
∫
−∞  
(a) What is the impulse response h(t) for this system?  
(b) Determine the response of the system when the input f(t) is as  
shown in Figure P2.8  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
2
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)?  
(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?  
P2.10. Consider an LTI system and a signal f(t)=2e-3tu(t-1). If  
f(t) → y(t)  
df(t)  
and  
→ −3y(t)+e−2tu(t)  
dt  
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 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):  
(a) f(t)=2f0 (t); h(t)=h0 (t)  
(d) f(t)=f0 (-t); h(t)=h0 (t)  
(c) f(t)=f0 (t-2); h(t)=h0 (t+1)  
(e) f(t)=f0 (-t); h(t)=h0 (-t)  
(f) f(t)=f0' (t); h(t)=h'0 (t)  
(b) f(t)=f0 (t)-f0 (t-2); h(t)=h0 (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  
P2.12. The input signal of the LTI system shown in Figure P2.12 is  
the following:  
f(t)=u(t)-u(t-2)+δ (t+1)  
The impulse responses of the subsystems are h1(t)=e-tu(t), and  
h2(t)=e-2tu(t).  
a) Compute the impulse response h(t) of overall system  
b) Find an equivalent system (same impulse response) configured  
as a parallel interconnection of two LTI systems.  
c) Sketch the input signal f(t). Compute the output signal y(t)  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-2: Linear time-invariant systems  
P2.13. Consider the following second-order, causal differential  
system initially at rest:  
d2y(t)  
dt2  
dy(t)  
dt  
+ 5  
+ 6y(t)=f(t)  
Calculate the impulse response h(t) of this system  
P2.14. Consider the following second-order, causal differential  
system initially at rest:  
d2y(t) dy(t)  
+
+ y(t)=f(t)  
dt2  
dt  
Calculate the impulse response h(t) of this system  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
4
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)  
dt  
df(t)  
dt  
2
+ 4y(t)=3  
+ 2f(t)  
P2.16. Find the impulse response h(t) of the following causal LTI,  
second-order diffrential system initially at rest:  
d2y(t)  
dt2  
dy(t)  
dt  
df(t)  
dt  
+ 2  
+ y(t)=  
+ f(t)  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
5
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