信号与系统2012-2013试题参考答案

transform P( ) and sketch the amplitude spectrum P( ) .

Answer: X ( )



x (t )e jt dt
j t e dt
2

j j 1 (e 2 e 2 ) sin c j 2
1 t 3t (e e ), t 0 4
(c) Y ( s ) H ( s) X ( s)
1 1 1 1 1 1 1 ( ) . . ( s 3)( s 1) s 3 s 4 s 1 12 s 3
1 1 1 y (t ) et e3t , t 0 3 4 12
3.
A linear time-invariant continuous-time system has transfer function
H (s)
s 2 16 s 7 s 12
2
Compute the steady-state and transient response resulting from the input x(t ) 2 cos 4t , t 0 , with zero initial conditions. (where (cos t )u (t ) s s2 2 )
y (t ) 2 H (0) 2 H (50) cos(50t H (0)

2
H (50))
10 10 1 ； H (50) 0 j 10 j 50 10
y (t ) 2 0.3922 cos(50t 1.974)
6.
Given the input x(t ) 4 2 cos(10t / 4) 3cos(30t / 2) . Find the output y(t) to each of the following

ANSWER THE FOLLOWING QUESTIONS (40
1, for t / 2 0, for all other t
A Signal is p(t )
Use the definition
X ( )

x(t )e
jt
dt
to compute the Fourier
y (t ) 6e 3t 8e 4t , t 0
An ideal lowpass discrete-time filter has frequency response H () given by
1, H ( ) 0, 0 Biblioteka 4.4
4



p2 B ( 2 k )
cos 0 n
k


[ ( 0 2k ) ( 0 2k )] )
ANSWER:
H ()
k
p ( 2k ) h[n] 4 sin c 4 , n 0, 1, 2, 3,...
PART 3 COMPUTE AND ANSWER THE FOLLOWING QUESTIONS (30 points)
1. Suppose that x(t ) u (t ) 2u (t 1) u (t 2) , and that (t ) is the pulse (t ) u (t ) u (t 1) . Compute the convolution of x(t ) and (t ) .
Answer:
4.
A continuous-time signals x(t) has Fourier transform X ( )
1 j b 1 jt b
Where b is a constant. Determine the Fourier transform V ( ) for (t ) Answer:
(a)Compute the transfer function H(s) of the system;
Answer: (a) H ( s) (b) H ( s)
1 s 2s 3
2
1 1 1 1 ( ); ( s 3)( s 1) 4 s 1 s 3
h(t )
T
0.5s
0.5s
Let x(t ) 2 cos(50 t ) and T=0.01sec.
(a) Draw X s ( ) , where xs (t ) x(t ) p (t ) , determining if aliasing occurs.
(b) Determine the expression for y(t)
一 、PART 1
CHOOSING THE BEST ANSWER (30 points)
(Arrange the answers for all questions into the following Table.) Question number Answer PART 2 1. 1 C 2 B 3 B 4 C 5 C 6 B 7 C 8 A 9 A 10 A points)
Give out the pole values and then determine whether or not the following linear time-invariant discrete-time systems is stable, marginally stable, or unstable: (a) H ( z ) (b) H ( z )
ANSWER: 2 s 200 T
x[n]
Ideal Reconstruction
y(t )
The output y(t) of the ideal reconstruction can be found by sending the sampled signal
xs (t ) x(t ) p(t ) through an ideal lowpass filter with the frequency response function
5.
Consider the system with the frequency response given by H ( ) determine the output for the signal x(t ) 2 2 cos(50t / 2) .
10 j 10
Answer:
Answer:
X (s)
2s s 16
2
Y ( s) H ( s) X ( s) c1 c 2 s3 s4
( s 2 16) 2s 2s . 2 2 s 7 s 12 s 16 ( s 4)( s 3)
c1
2s 2s 6 ; c2 8; ( s 4) s 3 ( s 3) s 4
2

1
n
x[n] cos
5.

8
n,

8


4
, y[n] x[n] cos

8
n, n 0, 1, 2, 3,...
Consider the following sampling and reconstruction configuration:
x(t )
Sample at T
Answer:
for
for t0 x(t ) * v (t ) 0
t [1, 2]
1 t t 1 1
；
for
t [0,1]
t 0
x(t ) * v(t ) (1)d t
；
x(t ) * v(t ) (1)d (1)d 1 (t 1) (1)(t 1) 2t 3
2
2.
Determine whether the continuous-time system is causal or noncausal, has memory or is memoryless, linear or nonlinear, time varying or time invariant, give out your reasons.
(a) Determine the unit-pulse response h[n] of the filter. (b) Compute the response y[n] when the input x[n] cos( (where B B sin c( n)
n
8
).


k
z4 z 2 1.5 z 0.5
1 ( z 0.5)( z 0.5)( z 1.5)
Answer:
(a) z1,2
1.5 0.5 , z1 0.5, z2 1 , the system is marginally stable. 2
(b) z1 0.5, z2 0.5, z3 1.5 , the system is unstable.
s4 s2 7s 3 2s 3 s 2 2 s 12
Answer:
(a) p1,2 (b) p1,2 8. 7 35 , p1,2 0 , the system is stable 2 2 j 52 , the two poles are both in the open left half region, so the system is stable. 2
filters:
(a)
(b)
Answer:
(a) y (t ) 4 2cos(10t / 4)