2010年信号与系统A卷

X ( j ) X ( s )
s j
then （8）
.(T or F)
If x(t) is real and if X(s) has a pole （极点）at s = s0 , then X(s) also has a pole at the point
s - s 0 . (T or F)
a k 0 for | k | 1 .

2 0
| x (t ) | dt 1
2
Specify two different signals that satisfy these conditions. (Parseval’s Relation:
1 T

T
x (t ) dt
2

k
Suppose x 1 ( t ) and x 2 ( t ) are band limited to 1 and 2 , respectively, that is
X 1 ( j ) 0 , | | 1 , X 2 ( j ) 0 , | | 2 . Under the sampling theorem, the minimum

ak
2
)
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3.3 (15 points) Consider a continuous-time ideal lowpass filter S whose frequency response is
H

j

1， 0，
c c
The input to this filter is a signal x t the following questions: a. Calculate X ( j ) . b. c. d.
j t 0
continuous.(T or F) （3） （4） （5） An LTI system with frequency function H ( j ) e has linear phase(线性相位). (T or F)
est is the eigenfunction(特征函数) of LTI systems.(T or F) The continuous-time Fourier transform of a real-even(实且偶的) signal is a real-even
s in 0 t
t
, and the output of the filter is y ( t ) . Please answer
When 0 c ，determine y ( t ) . When 0 c ，determine y ( t ) . In the case b or c, which will result in distortion(失真) in output？
（9）
A discrete-time LTI system is stable if and only if the ROC of its system function H(z) includes
the unit circle of z-plane. (T or F)
2 Blank Filling (20 points)
X ( j )
, and the phase
spectrum(相位谱) is （4）
.
(3 points) （Compressing,
Байду номын сангаас
Expansion in the time domain corresponding to
Expansion ）in the frequency domain. (2 points) （5） （6）
（1） ( t 1) * ( t ) .(2 points)
第 1 页 共 3页
（2）
If x 1 [ n ] {1, 2 , 3 }, x 2 [ n ] {1, 2 , 3 , 4 } , and y [ n ] x 1 [ n ] x 2 [ n ] , then the maximum
1 3
.
第 3 页 共 3页
1 True or False Question (10 points)
（1） （2）
x [ n ] sin ( 3 n / 4 ) is aperiodic(非周期的) .( T or F )
The Fourier transform of an aperiodic discrete-time signal is periodic (T or F ) and
x(t) 2 4 1/s –2 Figure 1 1/s -6 y(t)
3.5 (10 points) Find the inverse z-transform （z 反变换）of X ( z ) , where
X (z) 3
5 6
z
1 1
1
1 4
z
1
1
1 3
z

,
z
x(t ) e
2 t
u ( t ), a 0
an d
h(t ) u(t ) .
Calculating the output of the system y(t) by use of convolution integral equation（卷积公式）, but not Fourier Transform or Laplace Transform. 3.2 (15 points) Suppose we are given the following information about a signal x(t): a. b. c. d. x(t) is real and odd. x(t) is periodic （周期）with period T = 2 and has Fourier coefficients a k .
If x ( t ) X ( j ) , then x ( t ) c o s 0 t
F
F
. (2 points)
If x ( t ) X ( j ) , then the inverse Furious transform（傅里叶反变换） of
F
X [ j ( 0 )] is
3.4 (15 points) The input x(t) and output y(t) of a causal（因果）LTI system are related through the block-diagram representation（方框图表示） shown in Figure 1. a. b. c. Determine the system function H(s) and its ROC. Determine a differential equation（微分方程） relating y(t) and x(t). Is the system stable（稳定）?
. (2 points)
n
（7）
Consider an LTI system with unit impulse response h [ n ] a u [ n ], 0 a 1 . The . The system is a
frequency response of the system is (highpass, lowpass or bandpass) filter. (4 points) （8）
sampling frequency (最小采样频率) for signal x 1 ( t ) x 2 ( t ) is
.(3 points)
3. Calculation (70 points)
3.1 (15 points) Let x(t) be the input to an LTI system with unit impulse response h(t), where

length of y[n] is （3）
. (2 points)
If the spectrum-density （谱密度）function of an aperiodic continues-time signal x(t) is
1 1 j , then the magnitude spectrum(幅度谱) is
四川大学期末考试试题（闭卷）A 卷
（2009——2010 学年第二学期）
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function. (T or F) （6） The discrete-time Fourier transform of a real-even signal is an imaginary-odd(纯虚且奇的)
function. (T or F) （7） If the ROC (收敛域)of the Laplace transform X(s) includes the unit circle of s-plane 平面） （s ,