哈工大版信号与系统第1章习题

1、2021/3/2711-1 绘出下列信号波形绘出下列信号波形1. tu(t)(ttut02. (t-1)u(t)3. tu(t-1)() 1(tutt01) 1( ttut0114. (t-1)u(t-1) 1() 1(tutt012021/3/2721-31-3、写出如图所示各波形的函数表达式、写出如图所示各波形的函数表达式(a)(tft0221)()() 2() 2(21)()() 2() 2(21)(tuttututtuttututtf2021/3/273(b)2() 1()()(tutututf)2()()2(21)()2()2(21)(tututtututtf)(tft0211220

2、21/3/274(c)(tft0TE)()(2sin)(TtututTEtf2021/3/2751-41-4、试证明、试证明cost,cos2t,cost,cos2t,cosntcosnt是在区间是在区间(0,2(0,2 ) )的正交函数集的正交函数集解解:根据两函数正交条件根据两函数正交条件:2020coscos)()(jtdtitdttgtgji20)cos()cos(21dttjitji)(0)sin()sin(2120jijitjijitji2021/3/276202202cos)(itdtdttgi20 12cos21dtit2022sin21tiit因此因此,函数集在函数集在(0,

3、2 )区间上是正交函数集区间上是正交函数集2021/3/2771-7、利用冲激信号的抽样性质、利用冲激信号的抽样性质,求下列表示式的函数求下列表示式的函数值值 dttttf)()(. 20)() 0(00tftfdttt)4)(3(. 21431-8、利用冲激信号的抽样性质、利用冲激信号的抽样性质,求下列表示式的函数求下列表示式的函数值值 2021/3/2781-9 已知已知f(t)的波形如图，试画的波形如图，试画出出)2 ()(1tftg和和) 32()(2tftg的波形的波形)(tft011231解解:1、g1(t)的波形的波形:)( tf t011123反褶反褶:2021/3/279)2

4、(tft011121移位移位:f(2-t)=f-(t-2)2、g2(t)=-f(2t-3)的波形的波形:1)2( tft01123)(tft011231展缩展缩:2021/3/2710移位移位:f(2t-3)=f2(t-1.5)1) 32(tft01123沿横轴反褶沿横轴反褶:-f(2t-3)1) 32(tft011232021/3/27111-10 已知已知f(t)的波形如图的波形如图,试画出试画出 下列函数的波形下列函数的波形:1. f(3t)2. f(t/3)u(3-t)dttdf)(. 3tdf)(. 40tf(t)23112021/3/27120tf(t)12311. 解解:0tf(

5、3t)12312. 解解:0tf(t/3)12314567892021/3/27130tf(t/3)u(3-t)12313. 解解:将将f(t)表示为函数形式表示为函数形式)3() 1()()(tutRtRtf所以所以,)3() 1()()(ttututf0tf(t)1231(1)0tf(t)12312021/3/27144. 解解:将将f(t)表示为函数形式表示为函数形式) 3() 1()()(tutRtRtf所以所以:ttduRRdf)3() 1()()(tttddd3101) 1(当当t0时时0)(tdf当当 0t1时时221)(tdft0tf(t)23112021/3/2715当当 1

6、=t3时时tttdddf10) 1()(21 t当当 3=t 时时ttttddddf3101) 1()(5 . 20tf(t)23112021/3/2716tdf)(10.532t2.52021/3/27171-14、计算卷积、计算卷积)()(21tftf) 1()()()(. 221tututftfdtfftftf)()()()(2121 dtutuuu) 1()() 1()(dtuudtuu) 1()()()(dtuudtuu) 1() 1()() 1(解解:2021/3/2718)2(1) 1(1) 1(1)(1111100tudtudtudtudtttt)2()2() 1() 1()

7、1() 1()(tuttuttutttu)2()2() 1() 1(2)(tuttutttu2021/3/2719) 1() 1()(,cos)(. 421tttfttf解解:) 1(cos) 1(cos)()(21ttttftf) 1(cos) 1(costt2021/3/27201-17、用图解法求卷积波形、用图解法求卷积波形)(1tft0a1)(2tft0b21解解:1、变量替换、变量替换:将将t换为换为 ,并将并将f2( )反褶反褶:)(1f0a1)(2f0b21)(2f0b212021/3/2721)(1f0a1)(2f0b21)(1f0a1)(1f0a1)(1f0a1)()(21f

8、f0)3 ()(21ff0)2 (2f0b21)(2tf0b112bt)()(21tff012btat)(2tf0b21t)()(21tff012/abt2) 1( tabt2021/3/2722)()(21tff0t2abt1、当、当0t1时时,)1()(221)(tututabttg)1()(42tutuabt2021/3/27232、当、当1t2时时,)2() 1()2) 1(221)(tututababttg)2() 1(42tutuababt)()(21tff012abt2)1 (abt t2021/3/27243、当、当2t3时时,)3() 2()2(1 2) 1(21)(tutu

9、ttababtg)3()2(42432tutuabtabtab)()(21tff01ab2)1 (abt 2t2021/3/27251、当、当0t1时时,)1()(4)(2tutuabttg2、当、当1t2时时,)2() 1(42)(tutuababttg3、当、当2t3时时,)3() 2(2443)(2tutuabtabtabtg)(tgt0ab1234ab2021/3/2726)(1tft0a1)(2tft0b111-17 b解解:1、变量替换、变量替换:将将t换为换为 ,并将并将f2( )反褶反褶:)(1f0a1)(2f0b11)(2f0b112021/3/2727)(2tf0b11 t

10、1 tt)(1f0a1)(tgt0ab1234ab12021/3/27281-17 c)(2tft011)(1tft0111)(1f0111)(2f011)(2f011解解:1、变量替换、变量替换:将将t换为换为 ,并将并将f2( )反褶反褶:2021/3/2729)(1f0111)()(21tff01t)(1f0111t1t)()(21tff01t)(1f0111t1t)()(21tff01t01t21) 1(21)(ttg10t5 . 0)(21tttg2021/3/2730)(tgt0112214321t23)1 (21)(ttg)(1f0111t1t)()(21tff01t2021/3/27311-