›
›

Bài tập Tín hiệu và hệ thống - Chương 1: Elementary of signal and system

Download
Ch-1: Elementary of signal and system  
P1.1. A continuous-time signal f(t) shown in Figure P1.1. Sketch  
an label carefully each of the following signals:  
(a) f(t −1)  
(b) f(2-t)  
(c) f(2t+1)  
t
(d) f(4− 2 )  
(f) f(t)[δ (t+ 32 )-δ (t − 23 )]  
(e) [f(t)+f(-t)]u(t)  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-1: Elementary of signal and system  
P1.2. The continuous-time signal f(t) depicted in Figure P1.2.  
Sketch an label carefully each of the following signals:  
(b) f( 2t − 2)  
(d) 4f( 4t )  
(a) f(t+3)  
(c) f(1− 2t)  
(e) f(t)u(t)+f( − t)u(t) (f) f( 2t )δ (t+1)  
1
2
(g) f(t)[u(t+1) − u(t −1)]  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
1
Ch-1: Elementary of signal and system  
P1.3. Consider the continuous-time signals f(t) and h(t) shown in  
Figure P1.3. Sketch an label carefully each of the following signals:  
(a) f(t)h(t+1)  
(b) f(t)h(−t)  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-1: Elementary of signal and system  
P1.4. Find the energies of the signals illustrated in Figure P1.4.  
Comment on the effect on energy of sign change, time shifting, or  
doubling of the signal?  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
2
Ch-1: Elementary of signal and system  
P1.5. (a) Find the energies of the pair signals x(t) and y(t)  
illustrated in Figure P1.5a and b. Sketch and find the energies of  
signals x(t)+y(t) and x(t)-y(t)? Can you make any conclusion from  
these results? (b) Repeat part (a) for signal pair illustrated in Figure  
P1.5c? Is the conclusion in part (a) still valid? Can you generalize  
condition of x(t) and y(t) that conclusion in part (a) always right?  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-1: Elementary of signal and system  
P1.6. Determine power of the following signals:  
f1(t)=C1cos(ω1t+θ1), f2(t)=C2cos(ω2t+θ2), and f1(t)+f2(t) in two  
cases: (a) ω2=ω2, and (b) ω1≠ω2?  
P1.7. Consider signal f(t) depicted in Figure P1.7. Find power of  
the following signals: f(t), -f(t), and 2f(t)?  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
3
Ch-1: Elementary of signal and system  
P1.8. Simplify the following expressions:  
sint  
t2 +2  
jω+2  
ω2 +9  
−t  
0
(a)  
(d)  
δ(t)  
(b)  
δ(ω)  
(c) e cos 3t − 60 δ(t)  
(
)
sin π (t − 2)   
(
)
1
sin(kω)  
2
δ(t −1) (e)  
δ(ω+3)  
(f)  
δ(ω)  
t2 +4  
jω+2  
ω
P1.9. Evaluate the following integrals:  
∞
∞
(a) ∞ δ(τ)f(t-τ)dτ  
(b)  
f(τ)δ(t-τ)dτ (c)  
δ(t)e-jωtdt  
∫
∫
∫
−∞  
−∞  
−∞  
∞
∞
∞
δ(t-2)sin πt dt (e) δ t+3 e-tdt  
(f)  
t3 +4 δ(1-t)dt  
(d)  
(g)  
( )  
(
)
(
)
∫
∫
∫
∞
−∞  
−∞  
−∞  
∞
e(x-1)cos (x-5) δ(x-3)dx  
π
2
(h)  
[ ]  
f 2-t δ(3-t)dt  
(
)
∫
∫
−∞  
−∞  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-1: Elementary of signal and system  
P1.10. Determine and sketch the even and odd parts of the signals  
depicted in Figure P1.10. Label your sketches carefully.  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
4
Ch-1: Elementary of signal and system  
P1.11. Determine the even and odd parts of the following signals:  
(a) u(t); (b) tu(t); (c) sin(ω0t)u(t); (d) cos(ω0t)u(t); (e) sinω0t; and  
(f) cosω0t?  
P1.12. For the systems describled by the equations below, with the  
input f(t) and output y(t), determine which of the system are linear  
and which are nonlinear.  
2
dy(t)  
dy(t)  
dt  
dy(t)  
dt  
2
(c)  
+3ty(t)=t f(t)  
+2y(t)=f(t)  
(a)  
(d)  
(g)  
+2y(t)=f 2 (t) (b)  
dt  
t
dy(t)  
dt  
+y2 (t)=f(t)  
(f) y(t)= f(τ)dτ  
(e) 3y(t)+2=f(t)  
∫
−∞  
dy(t)  
dt  
df(t)  
+2y(t)=f(t)  
dt  
dy(t)  
dt  
df(t)  
(h)  
+ sint y(t)=  
(
+2f(t)  
)
dt  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-1: Elementary of signal and system  
P1.13. For the systems describled by the equations below, with the  
input f(t) and output y(t), determine which of the system are time-  
invariant parameter systems and which are time-varying parameter  
systems.  
(b) y(t)=f(-t)  
5
(c) y(t)=f(at)  
(a) y(t)=f(t-2)  
2
df(t)  
(e) y(t)= f(τ)dτ  
(d) y(t)=tf(t-2)  
(f) y(t)=  
∫
dt  
-5  
P1.14. For the systems describled by the equations below, with the  
input f(t) and output y(t), determine which of the system are causal  
and which are noncausal.  
(a) y(t)=f(t-2)  
(b) y(t)=f(-t)  
(c) y(t)=f(at);a>1  
(d) y(t)=f(at);a<1  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
5
Ch-1: Elementary of signal and system  
P1.15. Consider a continuous-time system with input f(t) and  
output y(t) related by: y(t)=f(t)[sin(t)]. (a) Is the system causal?  
(b) Is the system linear?  
P1.16. In particular, a system may or may not be (1) Memoryless,  
(2) Time-invariant, (3) Linear, (4) Causal, (5) Stable. Determine  
which of these properties hold and which do not hold for each of  
the following continuous-time systems. Justify your answers. In  
each example, y(t) denotes the system output and f(t) is the system  
input.  
2t  
(c) y(t)= f(τ )dτ  
(a) y(t)=f(t − 2)+f(2− t) (b) y(t)=[cos(3t)]f(t)  
∫
−∞  
0
(t<0)  
f(t)+f(t − 2); (t ≥ 0)  
(g) y(t)= df(t)  
0
[f(t)<0]  
(d) y(t)=  
(e) y(t)=  
f(t)+f(t − 2); [f(t) ≥ 0]  
(f) y(t)=f( 3t )  
dt  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-1: Elementary of signal and system  
P1.17. Consider the system shown in Figure P1.17a. Here the  
square root produces the positive square root. (a) Find an explicit  
relationship between y(t) and f(t)? (b) Is the system linear? (c) Is it  
time-invariant? (d) What is the response y(t) when the input is  
shown in Figure P1.17b?  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
6
Ch-1: Elementary of signal and system  
P1.18. Suppose that a LTI system has the following output y(t)  
when the input is the unit step f(t)=u(t): y(t)=e-tu(t)+u(-1-t).  
Determine and sketch the response of the system to the input f(t)  
shown in Figure P1.18  
P1.19. A particular linear system has the property that the response  
to tk is cos(kt). What is the response of this system to the input:  
f(t) = π + 6t2 − 47t5 + et6  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-1: Elementary of signal and system  
P1.20. A continuous-time linear system with input f(t) and output  
y(t) yields the following input-output pairs:  
and  
f(t) = ej2t → y(t) = ej3t  
f(t) = e− j2t → y(t) = e− j3t  
a) If f1(t)=cos(2t), determine the corresponding output y1(t) for  
the system?  
b) If f2(t)=cos(2(t-1/2)), determine the corresponding output y2(t)  
for the system?  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
7
pdf 7 trang Thùy Anh 12480 Free
Download
Bạn đang xem tài liệu "Bài tập Tín hiệu và hệ thống - Chương 1: Elementary of signal and system", để tải tài liệu gốc về máy hãy click vào nút Download ở trên

File đính kèm:

  • pdfbai_tap_tin_hieu_va_he_thong_chuong_1_elementary_of_signal_a.pdf