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y(t)\\)\u3002\u53e6\u5916\uff0c\u7b80\u5355\u7684\u5c0f\u7cfb\u7edf\u53ef\u4ee5\u901a\u8fc7\u7efc\u5408\u8fd0\u7b97\uff0c\u7ec4\u5408\u6210\u66f4\u5927\u7684\u7cfb\u7edf\uff0c\u8fd9\u6837\u7684\u5206\u62c6\u601d\u60f3\u662f\u7814\u7a76\u590d\u6742\u7cfb\u7edf\u7684\u5173\u952e\u65b9\u6cd5\u3002<\/p>\n<p>\u3000\u3000\u65b9\u6846\u56fe\u53ef\u4ee5\u4e3a\u590d\u6742\u7cfb\u7edf\u63d0\u4f9b\u76f4\u89c2\u7684\u8868\u793a\uff0c\u56fe\u4e2d\u7684\u5b50\u7cfb\u7edf\u4e00\u822c\u7528\u65b9\u6846\u8868\u793a\uff0c\u50cf\u5e38\u7528\u7684\u79ef\u5206\u3001\u5fae\u5206\u4e00\u822c\u76f4\u63a5\u7528\\(\\int,D\\)\u8868\u793a\u3002\u66f4\u7b80\u5355\u7684\u4e58\u5e38\u6570\u7cfb\u7edf\uff0c\u53ef\u4ee5\u76f4\u63a5\u5728\u7bad\u5934\u4e0a\u6807\u51fa\u7cfb\u6570\\(a\\)\u3002\u52a0\u6cd5\u5668\\(\\bigoplus\\)\u662f\u65b9\u6846\u56fe\u4e2d\u7684\u5e38\u7528\u90e8\u4ef6\uff0c\u5b83\u4e00\u822c\u7528\u4f5c\u591a\u4e2a\u7cfb\u7edf\u7684<strong>\u5e76\u8054<\/strong>\u3002\u76f8\u5e94\u5730\uff0c\u590d\u5408\u7cfb\u7edf\\(S2(S1(x(t)))\\)\u79f0\u4e3a\u7cfb\u7edf\u7684<strong>\u4e32\u8054<\/strong>\u6216<strong>\u7ea7\u8054<\/strong>\u3002\u53f3\u56fe\u4e2d\u7531\u4e24\u4e2a\u5b50\u7cfb\u7edf\u548c\u4e00\u4e2a\u52a0\u6cd5\u5668\u7ec4\u6210\u7684\u7cfb\u7edf\u4e5f\u5f88\u5e38\u89c1\uff08D\u7cfb\u7edf\u7684\u8f93\u51fa\u4e00\u822c\u4e3a\u8d1f\u56e0\u5b50\uff09\uff0c\u5b83\u88ab\u79f0\u4e3a<strong>\u53cd\u9988\u4e92\u8054<\/strong>\u3002<\/p>\n<p style=\"text-align:center;\"><img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/395211625e0f42bfaf6ad695af56e409.jpg\" =\"\" =\"\" alt=\"\u3010\u4fe1\u53f7\u4e0e\u7cfb\u7edf\u301101 - \u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf (https:\/\/mushiming.com\/)  \u7b2c1\u5f20\" title=\"\u3010\u4fe1\u53f7\u4e0e\u7cfb\u7edf\u301101 - \u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf  \u7b2c1\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >&nbsp; &nbsp;<img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/e4a3f16666954f8db0fa4743406b5d02.jpg\" =\"\" =\"\" alt=\"\u3010\u4fe1\u53f7\u4e0e\u7cfb\u7edf\u301101 - \u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf (https:\/\/mushiming.com\/)  \u7b2c2\u5f20\" title=\"\u3010\u4fe1\u53f7\u4e0e\u7cfb\u7edf\u301101 - \u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf  \u7b2c2\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><\/p>\n<p>\u3000\u3000\u786e\u5b9a\u7cfb\u7edf\u7279\u6027\uff0c\u662f\u5206\u6790\u6216\u8bbe\u8ba1\u7cfb\u7edf\u4e4b\u524d\u5fc5\u987b\u505a\u7684\u4e8b\u60c5\u3002\u800c\u4e14\u9488\u5bf9\u666e\u904d\u7279\u6027\u7684\u7814\u7a76\uff0c\u624d\u66f4\u5bb9\u6613\u6709\u7406\u8bba\u7ed3\u679c\u548c\u5b9e\u7528\u4ef7\u503c\uff0c\u8fd9\u91cc\u5148\u4ecb\u7ecd\u51e0\u4e2a\u5e38\u89c1\u7684\u7cfb\u7edf\u7279\u6027\u3002\u9996\u5148\u5bf9\u4e00\u822c\u7684\u6709\u754c\u51fd\u6570\\(x(t)\\)\uff0c\u603b\u5e0c\u671b\u7cfb\u7edf\u7684\u8f93\u51fa\\(y(t)\\)\u4e5f\u662f\u6709\u754c\u7684\uff0c\u8fd9\u6837\u7684\u7cfb\u7edf\u79f0\u4e3a<strong>\u7a33\u5b9a<\/strong>\u7684\u3002\u4ee5\u65f6\u95f4\u4e3a\u81ea\u53d8\u91cf\u7684\u7cfb\u7edf\u4e2d\uff0c\u5f80\u5f80\\(y(t_0)\\)\u7684\u503c\u4e0e\\(x(t)\\)\u7684\\(t_0\\)\u65f6\u523b\u4e4b\u540e\u7684\u503c\u65e0\u5173\uff0c\u8fd9\u6837\u7684\u7cfb\u7edf\u53eb<strong>\u56e0\u679c\u7cfb\u7edf<\/strong>\u3002\u5982\u679c\\(y(t_0)\\)\u4ec5\u4ec5\u4e0e\\(x(t_0)\\)\u6709\u5173\uff0c\u5b83\u4e5f\u53eb<strong>\u65e0\u8bb0\u5fc6\u7cfb\u7edf<\/strong>\u3002\u4ee5\u4e0a\u51e0\u4e2a\u7cfb\u7edf\u6027\u8d28\u6bd4\u8f83\u7b26\u5408\u7ecf\u9a8c\u7279\u6027\uff0c\u4f46\u4e00\u5b9a\u7a0b\u5ea6\u4e0a\u9650\u5236\u4e86\u7cfb\u7edf\u7684\u5904\u7406\u8303\u56f4\uff0c\u5bf9\u66f4\u5e7f\u4e49\u7684\u7cfb\u7edf\u4e0d\u4e00\u5b9a\u6210\u7acb\uff0c\u6545\u4eca\u540e\u4e0d\u4f5c\u4e3a\u9ed8\u8ba4\u6027\u8d28\u3002\u771f\u6b63\u5bf9\u7cfb\u7edf\u5206\u6790\u6709\u7528\u7684\uff0c\u5176\u5b9e\u662f\u4e0b\u9762\u8981\u4ecb\u7ecd\u7684\u4e24\u4e2a\u6027\u8d28\u3002<\/p>\n<p>\u3000\u3000\u5bf9\u8f93\u5165\u4fe1\u53f7\u6700\u7b80\u5355\u7684\u53d8\u6362\uff0c\u65e0\u975e\u5c31\u662f\u65f6\u79fb\u548c\u53e0\u52a0\uff0c\u5f88\u591a\u91cd\u8981\u7cfb\u7edf\u5bf9\u8fd9\u4e9b\u53d8\u6362\u4ecd\u7136\u6709\u201c\u4e0d\u53d8\u201d\u7684\u8f93\u51fa\u3002\u9996\u5148\u6211\u4eec\u5e0c\u671b\u7cfb\u7edf\u7684\u8f93\u51fa\u4e0d\u53d7\u8f93\u5165\u4fe1\u53f7\u65f6\u79fb\u7684\u5f71\u54cd\uff0c\u5373\u5982\u679c\\(x(t)\\to y(t)\\)\uff0c\u90a3\u4e48\\(x(t-t_0)\\to y(t-t_0)\\)\u3002\u8f93\u51fa\u51fd\u6570\u968f\u8f93\u5165\u51fd\u6570\u4e00\u8d77\u65f6\u79fb\uff0c\u8fd9\u6837\u7684\u7cfb\u7edf\u79f0\u4e3a<strong>\u65f6\u4e0d\u53d8<\/strong>\u7684\u3002\u7ebf\u6027\u7ec4\u5408\u662f\u4fe1\u53f7\u7684\u6700\u7b80\u5355\u590d\u5408\uff0c\u5982\u679c\u7cfb\u7edf\u8fd8\u80fd\u6ee1\u8db3\u5f0f\uff082\uff09\uff0c\u5b83\u5c31\u88ab\u79f0\u4e3a<strong>\u7ebf\u6027\u7cfb\u7edf<\/strong>\u3002\u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf\u666e\u904d\u5b58\u5728\uff0c\u800c\u4e14\u5b83\u8fd8\u6709\u7740\u975e\u5e38\u5b8c\u5584\u7684\u7406\u8bba\u6210\u679c\uff0c\u56e0\u6b64\u672c\u7cfb\u5217\u628a<strong>\u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf\uff08LIT\uff09<\/strong>\u4f5c\u4e3a\u9ed8\u8ba4\u7cfb\u7edf\uff0c\u5e76\u900f\u5f7b\u8ba8\u8bba\u5b83\u7684\u7279\u6027\u3002\u4f46\u8981\u6ce8\u610f\uff1a\u5728\u5bf9\u964c\u751f\u7cfb\u7edf\u505a\u5206\u6790\u65f6\uff0c\u4e00\u5b9a\u8981\u5148\u786e\u8ba4\u5b83\u662fLIT\u7cfb\u7edf\u3002<\/p>\n<p>\\[x_1(t)\\to y_1(t);\\;x_2(t)\\to y_2(t)\\;\\;\\Rightarrow\\;\\;ax_1(t)+bx_2(t)\\to ay_1(t)+by_2(t)\\tag{2}\\]<\/p>\n<h2>2. LIT\u7cfb\u7edf<\/h2>\n<h3>2.1 \u5355\u4f4d\u51b2\u6fc0\u51fd\u6570<\/h3>\n<p>\u3000\u3000\u4e3a\u4e86\u5f97\u5230LIT\u7cfb\u7edf\u7684\u4e00\u822c\u8868\u5f81\uff0c\u8fd8\u662f\u8981\u7ed3\u5408\u201c\u65f6\u4e0d\u53d8\u201d\u548c\u201c\u7ebf\u6027\u201d\u4e24\u5927\u7279\u6027\uff0c\u5bf9\u4efb\u610f\u8f93\u5165\\(x(t)\\)\u4f5c\u4e00\u4e2a\u7ebf\u6027\u62c6\u89e3\u3002\u8bbe\\(\\Delta(t-t_0)\\)\u662f\u4ec5\u5728\\(t_0\\)\u5904\u6709\u975e\u96f6\u503c\\(1\\)\u7684\u51fd\u6570\uff0c\u5219\\(x(t)\\)\u5c31\u662f\u6240\u6709\u51fd\u6570\\(x(t_0)\\Delta(t-t_0)\\)\u7684\u53e0\u52a0\uff08\\(t_0\\)\u53d6\u904d\u6240\u6709\u5b9e\u6570\uff09\u3002\u8fdb\u800c\u5982\u679c\\(\\Delta(t)\\)\u7684\u54cd\u5e94\u51fd\u6570\u4e3a\\(Y(t)\\)\uff0c\u90a3\u4e48\\(x(t)\\)\u7684\u54cd\u5e94\u51fd\u6570\\(y(t)\\)\u4fbf\u662f\u6240\u6709\u51fd\u6570\\(x(t_0)Y(t-t_0)\\)\u7684\u53e0\u52a0\u3002<\/p>\n<p>\u3000\u3000\u7531\u4e8e\\(t_0\\)\u53d6\u904d\u6240\u6709\u5b9e\u6570\uff0c\u201c\u7d2f\u52a0\u201d\u7684\u786e\u5207\u610f\u4e49\u4fbf\u662f\u79ef\u5206\uff0c\u4ee5\u4e0b\u5148\u628a\\(t_0\\)\u89c6\u4e3a\u8fde\u7eed\u53d8\u91cf\u5e76\u91cd\u65b0\u8bb0\u4e3a\\(\\tau\\)\u3002\u73b0\u5728\u9700\u8981\u6784\u9020\u4e00\u4e2a\u7b49\u4ef7\u4e8e\\(\\Delta(t-\\tau)\\)\u7684\u5fae\u5206\\(\\delta(t-\\tau)\\,\\text{d}\\tau\\)\uff0c\u4e5f\u5c31\u662f\u8bf4\u5fae\u5206\u4ec5\u5728\\(\\tau=t\\)\u5904\u6709\u975e\u96f6\u503c\\(1\\)\uff0c\u90a3\u5c31\u8981\u6c42\\(\\delta(t-\\tau)\\)\u5728\\(t\\)\u5904\u6709\u65e0\u7a77\u5927\u503c\u4e14\u79ef\u5206\uff08\u9762\u79ef\uff09\u4e3a\\(1\\)\u3002\u4e3a\u6b64\u9700\u8981\u5b9a\u4e49\u4e00\u4e2a\u6ee1\u8db3\u5f0f\uff083\uff09\u7684\u7279\u6b8a\u7684\u51fd\u6570\\(\\delta(t)\\)\uff0c\u5b83\u88ab\u79f0\u4e3a<strong>\u5355\u4f4d\u51b2\u6fc0\u51fd\u6570<\/strong>\uff0c\u5750\u6807\u7cfb\u4e2d\u5f80\u5f80\u4ee5\u4e00\u4e2a\u5355\u4f4d\u957f\u5ea6\u7684\u5411\u4e0a\u5411\u91cf\u8868\u793a\uff08\u5de6\u56fe\uff09\u3002\u5728\u6b64\u5b9a\u4e49\u4e0b\uff0c\\(x(t)\\)\u4fbf\u6709\u4e86\u5982\u5f0f\uff084\uff09\u79ef\u5206\u8868\u8fbe\u5f0f\u3002<\/p>\n<p style=\"text-align:center;\"><img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/60d400989a2e4696b811270e20d5fd1b.jpg\" =\"\" =\"\" alt=\"\u3010\u4fe1\u53f7\u4e0e\u7cfb\u7edf\u301101 - \u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf (https:\/\/mushiming.com\/)  \u7b2c3\u5f20\" title=\"\u3010\u4fe1\u53f7\u4e0e\u7cfb\u7edf\u301101 - \u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf  \u7b2c3\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >&nbsp;&nbsp;<img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/7555fffaaac045339372c78e3aa33d81.jpg\" =\"\" =\"\" alt=\"\u3010\u4fe1\u53f7\u4e0e\u7cfb\u7edf\u301101 - \u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf (https:\/\/mushiming.com\/)  \u7b2c4\u5f20\" title=\"\u3010\u4fe1\u53f7\u4e0e\u7cfb\u7edf\u301101 - \u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf  \u7b2c4\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><\/p>\n<p>\\[\\delta(t)=\\left\\{\\begin{matrix}\\infty,&amp;(t=0)\\\\0,&amp;(t\\ne 0)\\end{matrix}\\right.,\\;\\;\\int_{-\\infty}^{\\infty}\\delta(t)\\,\\text{d}t=1\\tag{3}\\]<\/p>\n<p>\\[x(t)=\\int_{-\\infty}^{\\infty}x(\\tau)\\delta(t-\\tau)\\,\\text{d}\\tau\\tag{4}\\]<\/p>\n<p>\u3000\u3000\u6559\u6750\u4e0a\u4e00\u822c\u4f1a\u7528\u51fd\u6570\u5217\u7684\u6781\u9650\u6765\u89e3\u91ca\\(\\delta\\)\u51fd\u6570\uff0c\u4f46\u6211\u4eec\u5fc3\u91cc\u660e\u767d\uff0c\u5f0f\uff083\uff09\u7684\u5b9a\u4e49\u6709\u4e25\u91cd\u7684\u903b\u8f91\u7f3a\u9677\uff0c\u4e00\u4e2a\u503c\u4e3a\u65e0\u7a77\u5927\u7684\u51fd\u6570\u6839\u672c\u4e0d\u80fd\u7b97\u4f5c\u51fd\u6570\u3002\u8fd9\u6837\u7684\u95ee\u9898\u5728\u5de5\u79d1\u6559\u6750\u4e0a\u662f\u65e0\u6cd5\u89e3\u91ca\u6e05\u695a\u7684\uff0c\u56e0\u4e3a\u900f\u5f7b\u5730\u7406\u89e3\\(\\delta(t)\\)\u8fd8\u9700\u8981\u300a\u5b9e\u53d8\u51fd\u6570\u300b\u548c\u300a\u6cdb\u51fd\u5206\u6790\u300b\u7684\u77e5\u8bc6\u3002\u5728\u90a3\u91cc\u5b83\u53eb<strong>\u72c4\u62c9\u514b<\/strong>\\(\\delta\\)<strong>\u51fd\u6570<\/strong>\uff0c\u5b83\u662f\u4e00\u4e2a\u7ebf\u6027\u6cdb\u51fd\uff0c\u5177\u4f53\u8bf4\u662f\u6ee1\u8db3\\(\\delta(\\text{d}t)=\\text{d}H(t)\\)\u7684\u4e00\u4e2a\u6d4b\u5ea6\uff0c\u5176\u4e2d\\(H(t)\\)\u662f\u53f3\u56fe\u7684<strong>Heviside\u9636\u68af\u51fd\u6570<\/strong>\u3002<\/p>\n<p>\u3000\u3000\\(\\delta(t)\\)\u7684\u4e25\u683c\u5b9a\u4e49\u662f\u6ee1\u8db3\u5f0f\uff084\uff09\u7684\u6cdb\u51fd\uff0c\u6ce8\u610f\u8fd9\u91cc\u7684\u51fd\u6570\u5df2\u7ecf\u4e0d\u662f\u4e00\u822c\u610f\u4e49\u7684\u51fd\u6570\uff0c\u751a\u81f3\u79ef\u5206\uff08\u52d2\u8d1d\u683c\u79ef\u5206\uff09\u4e5f\u4e0d\u662f\u4e00\u822c\u610f\u4e49\u7684\u79ef\u5206\uff08\u9ece\u66fc\u79ef\u5206\uff09\u3002\u53e6\u5916\uff0c\u8fd8\u53ef\u4ee5\u628a\\(\\delta(t)\\)\u770b\u6210\u4e00\u4e2a\u5e7f\u4e49\u7684\u5206\u5e03\u51fd\u6570\uff0c\u8fd9\u5176\u5b9e\u662f\u5bf9\u672c\u8bfe\u7a0b\u975e\u5e38\u9002\u5408\u7684\u89d2\u5ea6\u3002\u76f8\u6bd4\u4e00\u822c\u51fd\u6570\uff0c\u5b83\u4eec\u88ab\u79f0\u4e3a<strong>\u5947\u5f02\u51fd\u6570<\/strong>\uff0c\u6570\u5b66\u4e0a\u5df2\u7ecf\u4e25\u683c\u8bc1\u660e\uff0c\u5947\u5f02\u51fd\u6570\u548c\u5e38\u89c4\u51fd\u6570\u4e4b\u95f4\u53ef\u4ee5\u6b63\u5e38\u8fdb\u884c\u56db\u5219\u8fd0\u7b97\u4ee5\u53ca\u5fae\u79ef\u5206\uff0c\u800c\u4e14\u8ba1\u7b97\u7684\u7ed3\u679c\u4e0e\u76f4\u89c2\u7684\u7406\u89e3\u662f\u4e00\u81f4\u7684\uff0c\u4ee5\u540e\u53ef\u4ee5\u653e\u5fc3\u4f7f\u7528\u3002<\/p>\n<h3>2.2 LIT\u7684\u5206\u89e3<\/h3>\n<p>\u3000\u3000\u6709\u4e86\u5f0f\uff084\uff09\u5bf9\u8f93\u5165\u4fe1\u53f7\\(x(t)\\)\u5206\u89e3\uff0c\u5c31\u53ef\u4ee5\u7ee7\u7eed\u8ba8\u8bba\u7cfb\u7edf\u7684\u8f93\u51fa\u3002\u5728\u5fae\u5206\u610f\u4e49\u4e0b\uff0c\u628a\u54cd\u5e94\\(Y(t-\\tau)\\)\u5199\u4f5c\\(h(t-\\tau)\\,\\text{d}\\tau\\)\uff0c\u90a3\u4e48\u54cd\u5e94\u51fd\u6570\\(y(t)\\)\u5c31\u662f\u5f0f\uff085\uff09\uff0c\u5b83\u5c31\u662fLIT\u7cfb\u7edf\u7684\u8868\u5f81\u51fd\u6570\u3002\u867d\u7136\u4e00\u822c\u628a\\(h(t)\\)\u79f0\u4e3a<strong>\u5355\u4f4d\u51b2\u6fc0\u54cd\u5e94<\/strong>\uff0c\u4f46\u5f62\u5f0f\\(h(t-\\tau)\\,\\text{d}\\tau\\)\u624d\u662f\u771f\u6b63\u610f\u4e49\u4e0a\u7684\u54cd\u5e94\u51fd\u6570\u3002\u5c31\u597d\u6bd4\u628a\\(\\delta(t-\\tau)\\,\\text{d}\\tau\\)\u89c6\u4e3a\u771f\u6b63\u7684\u8f93\u5165\u4fe1\u53f7\uff0c\u800c\\(\\delta(t-\\tau),\\,h(t-\\tau)\\)\u53ea\u662f\u54cd\u5e94\u7cfb\u6570\u800c\u5df2\u3002\u7406\u6e05\u8fd9\u4e2a\u5173\u7cfb\uff0c\u5728\u4eca\u540e\u7684\u63a8\u5bfc\u4e2d\u624d\u4e0d\u4f1a\u56f0\u60d1\u3002<\/p>\n<p>\\[x(t)\\,\\to\\, y(t)=\\int_{-\\infty}^{\\infty}x(\\tau)h(t-\\tau)\\text{d}\\tau\\tag{5}\\]<\/p>\n<p>\u3000\u3000\u521a\u624d\u4e00\u76f4\u662f\u628a\u5f0f\uff085\uff09\u770b\u6210\u51fd\u6570\u7684\u53e0\u52a0\uff0c\u5982\u679c\u5728\u67d0\u4e2a\u56fa\u5b9a\u7684\u65f6\u95f4\u70b9\\(t=t_0\\)\u89c2\u5bdf\u8fd9\u4e2a\u5f0f\u5b50\uff0c\\(h(t_0-\\tau)\\)\u663e\u7136\u8868\u793a\u65f6\u95f4\u70b9\\(\\tau\\)\u7684\u5355\u4f4d\u51b2\u6fc0\u5728\\(t_0\\)\u65f6\u523b\u7684\u54cd\u5e94\u7cfb\u6570\uff0c\u800c\\(y(t_0)\\)\u5c31\u662f\u6240\u6709\u65f6\u95f4\u7684\u54cd\u5e94\u53e0\u52a0\u3002\u5728\u8fd9\u6837\u7684\u7406\u89e3\u4e0b\uff0c\u6211\u4eec\u5c31\u6bd4\u8f83\u5bb9\u6613\u8ba8\u8bba\u56e0\u679c\u6027\u7cfb\u7edf\u7684\u8f93\u51fa\u51fd\u6570\u3002\u56e0\u679c\u7cfb\u7edf\u6bcf\u4e2a\u65f6\u523b\u7684\u4fe1\u53f7\u53ea\u80fd\u5bf9\u4e4b\u540e\u7684\u8f93\u51fa\u9020\u6210\u5f71\u54cd\uff0c\u5728\u51fd\u6570\u4e0a\u7684\u8868\u73b0\u5c31\u662f\uff1a\u5f53\\(\\tau&gt;t_0\\)\u65f6\\(h(t_0-\\tau)=0\\)\uff08\u4e5f\u53ef\u4ee5\u8bf4\uff0c\u5f53\\(t_0&lt;0\\)\u65f6\\(h(t_0)=0\\)\uff09\u3002\u4f7f\u7528\u6362\u5143\u6cd5\u6574\u7406\u5f0f\uff085\uff09\uff0c\u4fbf\u5f97\u5230\u56e0\u679cLIT\u7cfb\u7edf\u7684\u8868\u5f81\u51fd\u6570\u5982\u5f0f\uff086\uff09\u6240\u793a\u3002\u7279\u522b\u5730\uff0c\u65e0\u8bb0\u5fc6\u7cfb\u7edf\u7684\u5355\u4f4d\u51b2\u6fc0\u54cd\u5e94\u53ef\u4ee5\u8ba1\u4f5c\\(h(t)=k\\delta(t)\\)\uff0c\u7cfb\u7edf\u8868\u5f81\u51fd\u6570\u7b80\u5316\u4e3a\\(y(t)=kx(t)\\)\uff0c\u8fd9\u5c31\u662f\u6211\u4eec\u76f4\u89c9\u4e0a\u8ba4\u4e3a\u201c\u5e94\u8be5\u662f\u201d\u7684\u90a3\u4e2a\u7ebf\u6027\u7cfb\u7edf\u3002<\/p>\n<p>\\[x(t)\\,\\to\\, y(t)=\\int_0^{\\infty}x(t-\\tau)h(\\tau)\\text{d}\\tau\\tag{6}\\]<\/p>\n<p>\u3000\u3000\u5bf9\u4e8e\u4e00\u4e2a\u7a33\u5b9a\u7cfb\u7edf\uff0c\u8981\u6c42\\(x(t)\\)\u6709\u9650\u65f6\\(y(t)\\)\u4e5f\u6709\u9650\uff0c\u7528\u53cd\u8bc1\u6cd5\u53ef\u77e5\\(h(t)\\)\u662f<strong>\u7edd\u5bf9\u53ef\u79ef<\/strong>\u7684\uff08\u5f0f\uff087\uff09\uff09\u3002\u53cd\u4e4b\u5982\u679c\u5f0f\uff087\uff09\u6210\u7acb\uff0c\u4e5f\u4e0d\u96be\u8bc1\u660e\\(y(t)\\)\u6709\u754c\uff08\u53d6\\(|x(t)|\\)\u7684\u4e0a\u9650\uff09\uff0c\u4ece\u800c\u6211\u4eec\u77e5\u9053\uff1a\u5355\u4f4d\u51b2\u6fc0\u54cd\u5e94\u7edd\u5bf9\u53ef\u79ef\u662f\u7cfb\u7edf\u7a33\u5b9a\u7684\u5145\u8981\u6761\u4ef6\u3002\u7a33\u5b9a\u7cfb\u7edf\u7684\\(h(t)\\)\u4e00\u822c\u8868\u73b0\u4e3a\u6e10\u8fdb\u8d8b\u4e8e\\(0\\)\uff0c\u5982\u679c\u5b83\u5728\u6709\u9650\u65f6\u95f4\u540e\u53ea\u6709\\(0\\)\u503c\uff0c\u8fd9\u6837\u7684\u54cd\u5e94\u4e5f\u53eb<strong>\u6709\u9650\u51b2\u6fc0\u54cd\u5e94<\/strong>\uff08<strong>FIR<\/strong>\uff09\uff0c\u8fd9\u79cd\u7cfb\u7edf\u4e5f\u6bd4\u8f83\u5e38\u89c1\u3002\u53e6\u5916\uff0c\\(h(t)\\)\u6536\u655b\u4e8e0\u7684\u901f\u5ea6\uff0c\u4e00\u5b9a\u7a0b\u5ea6\u4e0a\u4f53\u73b0\u4e86\u7cfb\u7edf\u54cd\u5e94\u7684\u5feb\u6162\u3002<\/p>\n<p>\\[\\int_{-\\infty}^{+\\infty}|h(\\tau)|\\,\\text{d}\\tau&lt;\\infty\\tag{7}\\]<\/p>\n<h3>2.3 \u5377\u79ef<\/h3>\n<p>\u3000\u3000LIT\u7684\u8868\u5f81\u51fd\u6570\uff085\uff09\u7531\u4e24\u4e2a\u51fd\u6570\u751f\u6210\u53e6\u4e00\u4e2a\u51fd\u6570\uff0c\u8fd9\u6837\u51fd\u6570\u95f4\u7684\u201c\u8fd0\u7b97\u201d\u5728\u5b9e\u53d8\u51fd\u6570\u91cc\u53eb<strong>\u7b97\u5b50<\/strong>\uff0c\u800c\u5f0f\uff085\uff09\u8fd9\u6837\u7684\u7b97\u5b50\u88ab\u79f0\u4e3a<strong>\u5377\u79ef<\/strong>\u3002\u5377\u79ef\u5c31\u597d\u50cf\u662f\u628a\u5176\u4e2d\u4e00\u4e2a\u51fd\u6570\u53cd\u8f6c\uff0c\u7136\u540e\u79fb\u52a8\u7740\u548c\u53e6\u4e00\u4e2a\u51fd\u6570\u79ef\u5206\uff0c\u660e\u767d\u5f0f\uff085\uff09\u7684\u7269\u7406\u610f\u4e49\u540e\uff0c\u4e5f\u80fd\u7c7b\u63a8\u8fd9\u79cd\u7b97\u5b50\u7684\u4f5c\u7528\u4e86\u3002\u4e00\u822c\u628a\u51fd\u6570\u7684\u5377\u79ef\u8ba1\u4f5c\\(x(t)*y(t)\\)\uff0c\u7531\u5f0f\uff085\uff09\u4e2d\\(x(t),h(t)\\)\u89d2\u8272\u7684\u5bf9\u79f0\u6027\uff0c\u4e0d\u96be\u5f97\u77e5\u5377\u79ef\u6ee1\u8db3<strong>\u4ea4\u6362\u5f8b<\/strong>\uff08\u5f0f\uff088\uff09\uff09\u3002<\/p>\n<p>\\[x(t)*y(t)=y(t)*x(t)\\tag{8}\\]<\/p>\n<p>\u3000\u3000\u53e6\u5916\u4e5f\u4e0d\u96be\u8bc1\u660e\uff08\u5b9e\u53d8\u51fd\u6570\u8bfe\u7a0b\u4e2d\u6709\u8be6\u7ec6\u8bc1\u660e\uff0c\u4e5f\u5e76\u4e0d\u96be\uff09\uff0c\u5377\u79ef\u8fd8\u6ee1\u8db3\u5f0f\uff089\uff09\uff0810\uff09\u7684<strong>\u5206\u914d\u7387<\/strong>\u548c<strong>\u7ed3\u5408\u5f8b<\/strong>\uff08\u7ed3\u5408\u5f8b\u672c\u8d28\u4e0a\u8ddf\u77e9\u9635\u4e58\u6cd5\u7684\u7ed3\u5408\u5f8b\u76f8\u540c\uff09\u3002\u65e2\u7136LIT\u7684\u8868\u5f81\u51fd\u6570\u662f\u4e00\u4e2a\u5377\u79ef\\(x(t)*h(t)\\)\uff0c\u518d\u6839\u636e\u5377\u79ef\u7684\u8fd0\u7b97\u5f8b\uff0c\u5c31\u80fd\u5f97\u5230LIT\u7cfb\u7edf\u7684\u4e00\u4e9b\u4e3b\u8981\u6027\u8d28\u3002\u6bd4\u5982\u7cfb\u7edf\\(h_1(t),h_2(t)\\)\u5e76\u8054\u540e\u7684\u7cfb\u7edf\u6ee1\u8db3\uff1a\u5355\u4f4d\u51b2\u6fc0\u54cd\u5e94\u4e3a\\(h_1(t)+h_2(t)\\)\uff1b\u800c\u4e32\u8054\uff08\u7ea7\u8054\uff09\u540e\u7684\u7cfb\u7edf\u6ee1\u8db3\uff1a\u5355\u4f4d\u51b2\u6fc0\u54cd\u5e94\u4e3a\\(h_1(t)*h_2(t)\\)\uff0c\u4e14\u4e32\u8054\u7684\u5b50\u7cfb\u7edf\u53ef\u4ee5\u4efb\u610f\u8c03\u6362\u987a\u5e8f\u3002\u53cd\u4e4b\u5982\u679c\u7cfb\u7edf\u7684\u5355\u4f4d\u51b2\u6fc0\u54cd\u5e94\u53ef\u62c6\u89e3\u4e3a\u591a\u4e2a\u7b80\u5355\u7684\u54cd\u5e94\uff0c\u5219\u53ef\u4ee5\u65b9\u4fbf\u5730\u5c06\u7cfb\u7edf\u62c6\u89e3\u4e3a\u5b50\u7cfb\u7edf\u3002<\/p>\n<p>\\[x(t)*[y(t)+z(t)]=x(t)*y(t)+x(t)*z(t)\\tag{9}\\]<\/p>\n<p>\\[x(t)*[y(t)*z(t)]=[x(t)*y(t)]*z(t)\\tag{10}\\]<\/p>\n<p>\u3000\u3000\u4e0d\u96be\u9a8c\u8bc1\uff0c\u5fae\u5206\u3001\u79ef\u5206\u672c\u8eab\u5c31\u662fLIT\u7cfb\u7edf\uff0c\u6839\u636e\u5377\u79ef\u7684\u7ed3\u5408\u5f8b\u4fbf\u6709\u5f0f\uff0811\uff09\uff0812\uff09\u6210\u7acb\uff0c\u5b83\u4eec\u662f\u5377\u79ef\u548c\u5fae\u79ef\u5206\u8fd0\u7b97\u7684<strong>\u4e92\u6362\u6cd5\u5219<\/strong>\u3002\u4e3a\u4e86\u663e\u5f0f\u8868\u793a\u5fae\u79ef\u5206\u7684\u5355\u4f4d\u51b2\u6fc0\u54cd\u5e94\uff0c\u5148\u8003\u5bdf\u6052\u5b9a\u53d8\u6362\\(x(t)\\to x(t)\\)\uff0c\u6839\u636e\u5f0f\uff084\uff09\u53ef\u77e5\u5b83\u7684\u5355\u4f4d\u51b2\u6fc0\u54cd\u5e94\u5c31\u662f\\(\\delta(t)\\)\u3002\u5229\u7528\u7ed3\u5408\u5f8b\u53ef\u77e5\uff0c\u5fae\u3001\u79ef\u5206\u7cfb\u7edf\u7684\u5355\u4f4d\u51b2\u6fc0\u54cd\u5e94\u5c31\u662f\\(\\delta(t)\\)\u7684\u5fae\u3001\u79ef\u5206\u3002\u628a\\(\\delta(t)\\)\u8ba1\u4f5c\\(u_0(t)\\)\uff0c\u5b83\u7684\uff08\u9ad8\u9636\uff09\u5fae\u79ef\u5206\u8ba1\u4f5c\\(u_n(t)\\)\uff08\u5f0f\uff0813\uff09\uff0c\\(n\\leqslant 0\\)\u65f6\u90fd\u662f\u5947\u5f02\u51fd\u6570\uff09\u3002\u5176\u4e2d\\(u_{-1}(t)\\)\u5c31\u662f\u4e0a\u9762\u63d0\u5230\u7684Heviside\u9636\u68af\u51fd\u6570\uff0c\u5b83\u4e5f\u88ab\u79f0\u4e3a<strong>\u9636\u8dc3\u54cd\u5e94\u51fd\u6570<\/strong>\uff0c\u7b80\u8ba1\u4f5c\\(u(t)\\)\u3002\u518d\u4e00\u6b21\u4f7f\u7528\u7ed3\u5408\u5f8b\uff0c\u5bb9\u6613\u6709\u5f0f\uff0814\uff09\u7684\u6027\u8d28\u3002<\/p>\n<p>\\[\\dfrac{\\text{d}}{\\text{d}t}[x(t)*y(t)]=x(t)*\\dfrac{\\text{d}y(t)}{\\text{d}t}\\tag{11}\\]<\/p>\n<p>\\[\\int_{-\\infty}^tx(\\tau)*y(\\tau)\\,\\text{d}\\tau=x(t)*\\int_{-\\infty}^ty(\\tau)\\,\\text{d}\\tau\\tag{12}\\]<\/p>\n<p>\\[u_{n}(t)=\\dfrac{\\text{d}^n\\delta(t)}{\\text{d}t^n};\\;\\;u_{-n}(t)=\\int_{-\\infty}^t\\dots\\int_{-\\infty}^\\tau\\delta(\\tau)\\,\\text{d}\\tau\\tag{13}\\]<\/p>\n<p>\\[u_m(t)*u_n(t)=u_{m+n}(t),\\;\\;(m,n\\in\\Bbb{Z})\\tag{14}\\]<\/p>\n<p>&nbsp;<\/p>\n<p>\u672c\u7cfb\u5217<strong>\u76ee\u5f55<\/strong><\/p>\n<p>\u3000\u3000\u3000\u300001 - \u7ebf\u6027\u65f6\u4e0d\u53d8\u7cfb\u7edf<\/p>\n<p>\u3000\u3000\u3000\u300002 - \u5085\u91cc\u53f6\u53d8\u6362<\/p>\n<p>\u3000\u3000\u3000\u300003 - \u7cfb\u7edf\u51fd\u6570\u7684\u6027\u8d28<\/p>\n<p>\u3000\u3000\u3000\u300004 - \u79bb\u6563\u65f6\u95f4\u4fe1\u53f7<\/p>\n<p>\u3000\u3000\u3000\u300005 - \u6ee4\u6ce2\u3001\u91c7\u6837\u548c\u901a\u4fe1<\/p>\n<p>\u3000\u3000\u3000\u300006 - \u6709\u7406\u7cfb\u7edf<\/p>\n<p>&nbsp;\u535a\u5ba2<strong>\u603b\u76ee\u5f55<\/strong>\u5728<strong>\u8fd9\u91cc<\/strong><\/p>\n<p>&nbsp;<\/p>\n<hr>\n<p>\u3010\u524d\u5e8f\u5b66\u79d1\u3011\u5fae\u79ef\u5206\u3001\u5085\u91cc\u53f6\u5206\u6790\u3001\u5e38\u5fae\u5206\u65b9\u7a0b<\/p>\n<p>\u3010\u53c2\u8003\u8d44\u6599\u3011<\/p>\n<p>[1] 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