北京交通大学信号与系统期末考试试题

北 京 交 通 大 学 考 试 试 题

课程名称：信号与系统

姓名： 学号： 班级： 成绩：

一、填空题（每题3分，共30分）

1．=

+-⎰

∞

∞--dt t u t e t )1()42(δ .

2． x 1(n )={1,3,3；n = -1,0,1}, x 2(n )={1,4；n = -1,0}，determine x 1(n )*x 2(n ) = .

3．Consider sampling )10(Sa )(t t x =，determine the maximum of sampling interval T so that

there will be no aliasing =max T (s).

4．A LTI system has input )()sin()(t u t t x = and output )()sin cos ()(t u t t e t y t

+-=-，determine the impulse response of this system =)(t h .

5．A system has input )(1t x and output )(1t y . If the system has properties, then the

input and output pairs has the relationship: input is )3(3)2()(112-+-=t x t x t x , so output is )3(3)2()(112-+-=t y t y t y 。 6．The transfer function of a LTI system is

12

)(+=

s s H , the system belongs to type .

（high-pass, low-pass, band-pass or band-stop ?）

7．the FS of )(t x is )

(ωj X , the FS of is ω

ωj e j X -

)2(。

8. the impulse response of integrator is ，the impulse response of differentiator is , the impulse response of discrete-time time-shift operator is 。 9. ())2(8.0)(-=n u n x n

，determine )(Ω

j e

X = 。

10．)

73sin()72cos(1)(n n n x π

π++=，Sketch the magnitude spectra )(k X , where )(k X is

DTFS coefficients of x (n ).

二、简单计算题（每题6分，共30分）

1．x ( t ) is shown in following figure. Sketch x (2- 2t )u (-t ) and write out the brief steps.

t

t

2．)(t x and )(t h are shown in following figure. Suppose )(*)()(t h t x t y =. Sketch )(t y 。

t

3．)()2()( ),1()3()(21

1n u n h n u n h n n =-=-. Determine )(n h , the impulse response of this

system 。

x )

4．Determine a state-variable description for the system depicted in following figure. Write out A 、B 、C 、D 。

⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡++)()()()()1()1(212121n x n x n q n q n q n q B A ⎥

⎦⎤

⎢⎣⎡+⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡)()()()()()(212121n x n x n q n q n y n y D C

1y

2y 2x (n )

1x (n )

(n )

(n )

5．)(ωj H , frequency response of a

LTI system, is shown in following figure. Input is

)(),3cos(5)2cos(4)cos(32)(+∞<<-∞+++=t t t t t x . Determine the output y (t )。

三、计算题（40分） 1．（12分）A LTI system is described by following difference equation:

)()2(61

)1(65)(n x n y n y n y =-+--

input is )()(n u n x =，initial conditions are 0]2[,1]1[=-=-y y , compute in Z-domain : (1) Transfer function )(z H , impulse response )(n h ;

(2) )(n y zi , )(n y zs and )(n y . 2．（14分）A LTI system is described by following differential equation:

)(2)()(4)(5)(''"t x t x t y t y t y +=++，

input is )()(t u t x =，initial conditions are 4)0(',

2)0( ==--

y y , comput e in S-domain ：

(1) Transfer function )(s H . Is the system stable ? (2) )(t y zi , )(t y zs and )(t y ;

(3) Draw direct form implementation for the system.