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：
题 号 一 二 三 四 五 六 七 八 九 十 总分 得 分 阅卷人
一、填空题（每题3分，共30分）
1．?? . 2． x1(n)={1,3,3；n= ?1,0,1}, x2(n)={1,4；n= ?1,0}，determine x1(n)*x2(n) = . 3．Consider sampling x(t)?Sa(10t)，determine the maximum of sampling interval T so that
there will be no aliasing Tmax? (s).
4．A LTI system has input x(t)?sin(t)u(t) and output y(t)?(e?cost?sint)u(t)，determine the impulse response of this system h(t)? .
5．A system has inputx1(t)and output y1(t). If the system has properties, then the input and output pairs has the relationship: input isx2(t)?x1(t?2)?3x1(t?3), so output is y2(t)?y1(t?2)?3y1(t?3)。 6．The transfer function of a LTI system is
（high-pass, low-pass, band-pass or band-stop ?）
?t??e?t?(2t?4)u(t?1)dt?H(s)?2s?1, the system belongs to type .
?j?7．the FS ofx(t)is X(j?), the FS of is X(j2?)e。
8. the impulse response of integrator is ，the impulse response of differentiator is , the impulse response of discrete-time time-shift operator is 。 9. x(n)??0.8?u(n?2)，determineX(enj?)= 。
2?3?x(n)?1?cos(n)?sin(n)7710．，Sketch the magnitude spectraX(k), where X(k)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.
x(t)x(2?2t)u(?t)1t?201
2．x(t)and h(t) are shown in following figure. Suppose y(t)?x(t)*h(t). Sketch y(t)。
0tx(t)12h(t)11-10t ?202t
3．h1(n)?(3)system。
n?1u(n?1), h2(n)?(2)nu(n). Determine h(n), the impulse response of this
x(n)h1(n)h2(n)++y(n)?
4．Determine a state-variable description for the system depicted in following figure. Write out A、B、C、D。
?q1(n?1)??q1(n)??x1(n)??y1(n)??q1(n)??x1(n)??A?B?C?D?q(n?1)??q(n)??x(n)??y(n)??q(n)??x(n)??2??2??2? ?2??2??2?
x1(n)?-z?12q1(n)?y1(n)x2(n)?-z?13q2(n)?y2(n)
5．H(j?), frequency response of a LTI system, is shown in following figure. Input is
x(t)?2?3cos(t)?4cos(2t)?5cos(3t),(???t???). Determine the output y(t)。
H(j?)8-404三、计算题（40分） 1．（12分）A LTI system is described by following difference equation:
?
51y(n?1)?y(n?2)?x(n)66
input is x(n)?u(n)，initial conditions arey[?1]?1,y[?2]?0, compute in Z-domain: (1) Transfer functionH(z), impulse response h(n);
y(n)?(2) yzi(n), yzs(n) and y(n). 2．（14分）A LTI system is described by following differential equation:
y\(t)?5y'(t)?4y(t)?x'(t)?2x(t)，
input is x(t)?u(t)，initial conditions arey(0)?2,(1) Transfer functionH(s). Is the system stable? (2) yzi(t), yzs(t) and y(t);
(3) Draw direct form implementation for the system.
?y' (0?)?4, compute in S-domain：
3．（14分）Consider amplitude modulation. A modulated signal is g(t)?x(t)cos?ct. In this
equation
x(t)?Ac[1?m(t)]where
Ac?1 and the modulating signal
m(t)?0.5cos?0t
3??2??10rad/s??2??10rad/s. c0 and
(1) Sketch g(t);
(2) Sketch G(j?), the FS of g(t);
（3）A LTI system has frequency response H(j?)?u(??1990)?u(??1990). If g(t) is the input, determine the output y(t).
Answer: 一、 1、e/2；2、{1，7，15，12；n=-2，-1，0，1}；3、10；
111x(t?)?t2； 4、h(t)?eu(t)； 5、LTI; 6、low-pass；7、22?2Tmax??0.64e?2j?X(e)?'h(n)??[n?1]h(t)??(t)h(t)?u(t)31?0.8e?j?； 8、1，2，；9、
10、X(k)?{14,0,7,?7j,?7j,7,0;k?0,1...6}
j?
二、1、 2、
x(2?2t)u(?t)y(t)40t
?3n?10?13t
3、h(n)?{h1(n)?h2(n)??(n)}?3u(n?1)*2nu(n)??(n)；
??20??1A???,B??00?3???4、0??1,C??1???11??0,D??1???00?0??
111yzs(n)?[?()n?()n?3]u(n)32331n21n()?()22335、y(t)?16?18cost?16cos2t
yzi(n)?三、1、
n?011h(n)?[3()n?2()n]u(n)23s?212H(s)?2h(t)?[e?t?e?4t]u(t)33s?5s?4，2、（1），稳定
111yzs(t)?(?e?t?e?4t)u(t),yzi(t)?4e?t?2e?4t236 （2）
（3） 1F(s)?s?1?5s?12?Y(s)?4
3、（1）g(t)波形如下：
(2)
G(j?) ?()2 (2?) ?()2 ?()2 (2?) ?()2 (3)y(t)=0.5cos(1980?t)
1980? 2000? 2020? ? 1980? 2000? 2020?
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