[img=342x230]17a55d942c5b590.png[/img]证明：如果把施加于系统的激励信号[tex=1.643x1.357]j6KcqgyprXeRIkkNI3dxdw==[/tex]按图那样分解为许多阶跃信号的叠加，设阶跃响应为[tex=1.643x1.357]n5qXYg45eJOfVXVn6Pm6Ng==[/tex]，[tex=1.643x1.357]1kzNatkaii6Vy8isc2LWlA==[/tex]的初始值为[tex=2.357x1.357]FkzHMrJWzy3HSl6z7fzzjQ==[/tex]，在[tex=0.714x1.143]XAJRHjyjbPRG+CPMSCNMPA==[/tex]时刻阶跃信号的幅度为[tex=3.071x1.357]NAUdFeUdLZs2JTxUSmcbLu7/aZTDFOQA8TwphtHqmX3877ERLb/4aaQ7+374CD3F[/tex]。试写出以阶跃响应的叠加取和而得到的系统响应近似式；证明，当取[tex=3.143x1.214]ivk7xrHHR5HWzeejNvtzH8TbhG8yX5VKatcWI71etpI=[/tex]的极限时，响应[tex=1.643x1.357]dNUh4YEmlbswOFiVA3wyRg==[/tex]的表示式为[tex=15.643x2.929]nZnK3QSQngllCLxpdIGFJixNOaRwmUt0i7r46ipcpNuc2GwcWoeMixVsCbnhOrUS0pos+PjrinhBLjqW8hkvGd8JKyXM8Pp/D+Qb4el0GwCWZSyMXiw1ixHywyRLjg72shIxTKT5gVRgRKaRSNC6BQ==[/tex][此式称为杜阿美尔积分]

2022-06-08

[img=342x230]17a55d942c5b590.png[/img]证明：如果把施加于系统的激励信号[tex=1.643x1.357]j6KcqgyprXeRIkkNI3dxdw==[/tex]按图那样分解为许多阶跃信号的叠加，设阶跃响应为[tex=1.643x1.357]n5qXYg45eJOfVXVn6Pm6Ng==[/tex]，[tex=1.643x1.357]1kzNatkaii6Vy8isc2LWlA==[/tex]的初始值为[tex=2.357x1.357]FkzHMrJWzy3HSl6z7fzzjQ==[/tex]，在[tex=0.714x1.143]XAJRHjyjbPRG+CPMSCNMPA==[/tex]时刻阶跃信号的幅度为[tex=3.071x1.357]NAUdFeUdLZs2JTxUSmcbLu7/aZTDFOQA8TwphtHqmX3877ERLb/4aaQ7+374CD3F[/tex]。试写出以阶跃响应的叠加取和而得到的系统响应近似式；证明，当取[tex=3.143x1.214]ivk7xrHHR5HWzeejNvtzH8TbhG8yX5VKatcWI71etpI=[/tex]的极限时，响应[tex=1.643x1.357]dNUh4YEmlbswOFiVA3wyRg==[/tex]的表示式为[tex=15.643x2.929]nZnK3QSQngllCLxpdIGFJixNOaRwmUt0i7r46ipcpNuc2GwcWoeMixVsCbnhOrUS0pos+PjrinhBLjqW8hkvGd8JKyXM8Pp/D+Qb4el0GwCWZSyMXiw1ixHywyRLjg72shIxTKT5gVRgRKaRSNC6BQ==[/tex][此式称为杜阿美尔积分]

答案：

解：当[tex=3.214x1.0]LxJ/oW/ITwkgXU/n5AuvUiIyFQVLkp7IrqAWhhPt1Q8=[/tex]时，可将信号[tex=1.643x1.357]LGoEqc4+Ga/yTFvu0VlvEg==[/tex]表示为[tex=16.643x2.643]5AGllWr6leKEjN919CdcJHT/qU7eTZpAACoUOt+8Bfn7uA2g99hPFIawAKjPcdKAQKy+5h1gZwkpo27TiTqyr6CYhBpz62jPq/x8z7L07jtBa4P0d1qHdyDjtxsGO5LPKk7Yg11gzIXZc5SHVvdBvA==[/tex]假设系统的冲激响应为[tex=1.786x1.357]TfR6nvfSU/rEPlumHtPUVA==[/tex]，则当系统的激励信号为阶跃信号[tex=1.643x1.357]j6KcqgyprXeRIkkNI3dxdw==[/tex]时，系统响应为[tex=22.071x7.214]qeiYnKXLEhyhuGRg8yLtrx1UM3HubiBdNgNoP9ChxhzGVtcsRoAFnCQlJDcv2JyPdsWWvzukgK1zRvHyvktbCCYLIzO6OaamFnEKc0Pb/Rq2OWlZp8OZZDbkOCsQbVNiVu6WEuJ2Hks67hhczbTG7Q5eTNuJvGI2T8PNgafsyo/BeY9A2F/UfSxG2kOMIJ/fDW48XqmU3iAJz5F9mE0I/WPP9d5LgoReU0cgf9fTjti44n1HIgeC4es/qOHjDZYbItKHnDr2cYkubft6+9J3JvafK/wmUsu8I5d9hYZQf5WN+eyM6BGFXRhH8RSdpVXxz7aa/lLFcxi0z1rZ5ME6zYmKHHsfHDOc+r8euVJy8dYyhmQSu5r6pb4tWnTWyeDm[/tex]题目已知阶跃响应为[tex=1.643x1.357]dVY93ll8VMQ0iaNbw05kQQ==[/tex]，因此有[tex=9.0x3.071]qeiYnKXLEhyhuGRg8yLtr27I/4ILJtEXLQD0+bXtUGrm1IU5MDYC7yxp/fnM1M0ZWaMJrxb9pafayGaf5KzAf4fcQalI60eh/Z+qz3oqCGgDk3CGRQJeEzq9QoEOCNW7[/tex]代入[tex=1.643x1.357]p6P234qupidP+NNJ+StGXQ==[/tex]的表示式可得[tex=15.643x2.786]EH14aRULmUGW/JpehXgYo6ftO0anLTMFiuQvoOJ79nJQN634nIGgKKD++0II6T2+C8FspXgpqsAqBQypji8e/CbOJoWishZXOCPOOetKeExOxZ78bRz53UO2n8Qde0Hq[/tex]

举一反三

内容

0
证明线性时不变系统有如下特性：即若系统在激励[tex=1.643x1.357]j6KcqgyprXeRIkkNI3dxdw==[/tex]作用下响应为[tex=1.643x1.357]p6P234qupidP+NNJ+StGXQ==[/tex]，则当激励为[tex=2.286x2.071]LZQ2Y9PJeU17iR0lBwfJeB2uPflqA2p7BrX7IyAf0n8=[/tex]时响应必为[tex=2.286x2.071]ptiyppJDfeBHrT5AAIKhkI24e+dIPefPpkAORLvuIMc=[/tex]。
1
判断下列命题是否为真：（1）[tex=3.643x1.357]/5abqJjwKZ1qr+6hsVFF5EBvfq3ggOFNlHMClz0h9nk=[/tex]（2）[tex=2.929x1.357]rGJpyjIjJpbcoBTWxP0Jiw==[/tex]（3）[tex=4.5x1.357]2wycHMoqU83MyEp17iBils58bR7YLuCTI2G9NVAdlfY=[/tex]（4）[tex=5.214x1.357]CTz2gu+IIm1GgNmYMGaduCRtA41wnW4WqwRWwEhq6aA=[/tex]（5）[tex=4.857x1.357]1DcE2BMMOaZhTuxR/mjgsboXxfg5ET59Dp4I/jjEDuw=[/tex]（6）[tex=4.643x1.357]BSryrsQYOvTP2hTWRu6t4nAuJwlSs4L9jaq70EpB+Us=[/tex]（7）若[tex=6.0x1.357]y0IZLUnBO88nR8WBZYvd7QXv5S1OMINV5cQNzPyiyAc=[/tex]，则[tex=3.429x1.357]1brfPwTkVVIX4GfoMIUskA==[/tex]（8）若[tex=7.643x1.357]MhLfJXZnhbXiB0x3oNtFzThV4Y1mJxe1VYr7PkJE/T6hmTD3WWp+UxbNwvUQ6DHk[/tex]，则[tex=4.143x1.357]LZUA94ISo1po5HWsOVeBCjo0rMvj7uw3bGw5HiZenrI=[/tex]
2
设信号[tex=1.643x1.357]Wfem9oxh0ZS7nZ3KGomKoQ==[/tex]的频谱 [tex=2.071x1.357]tibrCRwNLzDQQYea733HmQ==[/tex] 如图 (a)所示, 当该信号通过图(b)系统后, 证明 [tex=1.643x1.357]OiyUjvaNKwAtwHxCTVl2yg==[/tex] 恢复为 [tex=1.643x1.357]Wfem9oxh0ZS7nZ3KGomKoQ==[/tex] 。[img=891x270]17d4bb2b90fcfc0.png[/img]
3
已知某线性时不变系统在阶跃信号[tex=1.714x1.357]LQDjhWLbNBGk2tb5pbSofQ==[/tex]激励下产生的阶跃响应为[tex=8.286x1.571]ziPFOkuk+q98CioiN2ZM3upu6VMOcxgCZMyuMpqLL5CvWelaN0TkgdxpzR7V79ky[/tex],现观测到系统在输入信号[tex=1.643x1.357]QEos0FZVoxidwAY2Qi/JJQ==[/tex]激励下的零状态响应为[tex=9.429x1.571]HL7+rnwWfxkOKqzuPgRE6r+EPUPxV89+/yFH0tnMPttYRqiXHg2jLV1kZDE1FV4aMlBZqyxfpvNwczssURiubg==[/tex]，试确定输入信号[tex=1.643x1.357]QEos0FZVoxidwAY2Qi/JJQ==[/tex]。
4
若系统函数[tex=6.143x2.571]cievBNWeKmIhBUKM0S7X2jSAJRJPBk6BCB5NGyiPmIlRcCQD0/7iiYmnkJStr7wb[/tex]，激励为周期信号[tex=8.429x1.357]Btby3unMUACtlAOllga2OpFrfMxph9LgMfaQLTQRCL8=[/tex]，试求响应[tex=1.643x1.357]p6P234qupidP+NNJ+StGXQ==[/tex]，画出[tex=1.643x1.357]NjFDJ0ZjhSHd6w9dEk5nSQ==[/tex]，[tex=1.643x1.357]tqbEsKl6CL3fHqr4y3/ZGw==[/tex]波形，讨论经传输是否引起失真。