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decoding=\"async\" alt=\"\" height=\"163\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102531-6842c21b7a2b5.png\" width=\"225\" \/><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"166\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102531-6842c21b90523.png\" width=\"192\" \/><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"160\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102531-6842c21ba72cf.png\" width=\"191\" \/><\/p>\n<p>\u00a0 \u00a0 \u00a0 \u00a0 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style=\"text-align:center\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"187\" src=\"2025-06-06tu3u1ze2w4e.png\" width=\"221\" \/><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"190\" src=\"2025-06-0602maykctutb.png\" width=\"217\" \/><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"192\" src=\"2025-06-06kaofyhrecdd.png\" width=\"220\" \/><\/p>\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u4ece\u65f6\u57df\u53ef\u4ee5\u6e05\u6670\u7684\u770b\u51fa&#xff0c;\u4fe1\u53f7\u5206\u91cf\u65f6\u57df\u7a81\u53d8\u7684\u65f6\u95f4\u70b9&#xff0c;\u5373\u5404\u5206\u91cf\u4fe1\u53f7\u7684\u8d77\u59cb\u65f6\u95f4\u4ee5\u53ca\u7ec8\u6b62\u65f6\u95f4&#xff0c;\u4e0b\u56fe\u5c55\u793a\u4e86\u5bf9\u5e94\u7684\u9891\u8c31\u7ed3\u679c&#xff0c;\u53ef\u4ee5\u770b\u51fa\u9891\u8c31\u5206\u6790\u5e76\u4e0d\u80fd\u5f97\u5230\u4fe1\u53f7\u5206\u91cf\u7684\u65f6\u57df\u7279\u5f81\u3002<\/p>\n<p style=\"text-align:center\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"188\" src=\"2025-06-06res4jlj2to4.png\" width=\"220\" \/><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"189\" src=\"2025-06-06fa0hn0w4coy.png\" width=\"220\" \/><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"190\" src=\"2025-06-062quko02at13.png\" width=\"222\" \/><\/p>\n<p style=\"text-align:center\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"185\" src=\"2025-06-06os3i4ptjfot.png\" width=\"218\" \/><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"189\" src=\"2025-06-06nsk02w4mtzu.png\" width=\"219\" \/><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"192\" src=\"2025-06-06fvolvglphoe.png\" width=\"221\" \/><\/p>\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u5f53\u4fe1\u53f7\u5206\u91cf\u5b58\u5728\u65f6\u95f4\u6709\u4ea4\u53c9\u65f6&#xff0c;\u4fe1\u53f7\u65f6\u57df\u3001\u9891\u57df\u8868\u793a\u5982\u4e0b\u56fe\u6240\u793a&#xff0c;\u53ef\u4ee5\u770b\u51fa\u4fe1\u53f7\u7684\u8d77\u59cb\u65f6\u523b\u5f88\u96be\u901a\u8fc7\u65f6\u57df\u5206\u6790\u3001\u9891\u57df\u5206\u6790\u5f97\u5230&#xff0c;\u9700\u8981\u8054\u5408\u65f6\u57df-\u9891\u57df\u7279\u5f81\u624d\u80fd\u5f97\u5230&#xff0c;\u5373\u65f6\u9891\u5206\u6790\u3002<\/p>\n<p style=\"text-align:center\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"172\" src=\"2025-06-06ufe0bserwgh.png\" width=\"239\" \/><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"174\" src=\"2025-06-06sye1hw2orp0.png\" width=\"203\" \/><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"174\" src=\"2025-06-06o12akyhe2ty.png\" width=\"203\" \/><\/p>\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u7ed3\u8bba&#xff1a;\u65f6\u57df\u5206\u6790\u53ef\u4ee5\u5f97\u5230\u4fe1\u53f7\u7684\u65f6\u57df\u7279\u5f81&#xff0c;\u5c24\u5176\u662f\u4fe1\u53f7\u65f6\u57df\u7a81\u53d8\u7279\u5f81&#xff1b;\u9891\u57df\u5206\u6790\u53ef\u4ee5\u5f97\u5230\u4fe1\u53f7\u7684\u9891\u7387\u7279\u5f81&#xff0c;\u5c24\u5176\u662f\u4fe1\u53f7\u5206\u91cf\u7684\u9891\u7387\u6210\u5206\u3002\u5f53\u5206\u91cf\u4fe1\u53f7\u5b58\u5728\u65f6\u9891\u6df7\u53e0\u65f6&#xff0c;\u8054\u5408\u65f6\u57df\u5206\u6790\u548c\u9891\u57df\u5206\u6790\u65e0\u6cd5\u5b8c\u5168\u89e3\u6790\u6df7\u5408\u4fe1\u53f7\u7684\u65f6\u57df\u7279\u5f81\u4ee5\u53ca\u9891\u57df\u7279\u5f81\u3002<\/p>\n<h2 id=\"%E4%BA%8C%E3%80%81%E7%BA%BF%E6%80%A7%E6%97%B6%E9%A2%91%E5%88%86%E6%9E%90\">\u4e8c\u3001\u7ebf\u6027\u65f6\u9891\u5206\u6790<\/h2>\n<h3 id=\"2.1%E3%80%81%E7%9F%AD%E6%97%B6%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2%EF%BC%88%E5%88%86%E8%BE%A8%E7%8E%87%E5%9B%BA%E5%AE%9A%EF%BC%89\">2.1\u3001\u77ed\u65f6\u5085\u91cc\u53f6\u53d8\u6362&#xff08;\u5206\u8fa8\u7387\u56fa\u5b9a&#xff09;<\/h3>\n<p style=\"text-align:center\">\u00a0 \u00a0 \u00a0 \u00a0<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"160\" src=\"2025-06-06icujk5h4ydh.png\" width=\"138\" \/><\/p>\n<p>\u00a0 \u00a0 \u00a0 \u00a0\u6301\u7eed\u65f6\u95f4<img decoding=\"async\" alt=\"T\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a24ffe.png\" \/>\u7684\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u7684\u77ed\u65f6\u5085\u91cc\u53f6\u53d8\u6362\u4e3a\u00a0<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"X_{\\\\text{STFT}}\\\\left ( f,\\\\tau\\\\right )&#061;\\\\int_{0}^{T}x\\\\left ( t \\\\right )w\\\\left ( t-\\\\tau \\\\right )e^{-j2\\\\pi ft}dt\\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: (2)\" class=\"mathcode\" src=\"2025-06-06otm1mqcoppg.png\" \/><\/p>\n<p>\u00a0 \u00a0 \u00a0 \u79bb\u6563\u5316\u7ed3\u679c<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"X_{\\\\text{STFT}}\\\\left (m\\\\Delta f,nT_{hop}\\\\right )&#061;\\\\int_{0}^{T}x\\\\left ( t \\\\right )w\\\\left ( t-nT_{hop} \\\\right )e^{-j2\\\\pi m\\\\Delta ft}dt\" class=\"mathcode\" src=\"2025-06-06xjpmxehz431.png\" \/><\/p>\n<p>\u00a0 \u00a0 \u00a0 \u5f53\u7a97\u51fd\u6570<img decoding=\"async\" alt=\"w\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-061qeweo4dwue.png\" \/>\u4e3a\u77e9\u5f62\u7a97\u65f6&#xff0c;\u77ed\u65f6\u5085\u91cc\u53f6\u53d8\u6362\u7ed3\u679c\u53d7\u7a97\u957f\u5ea6&#xff0c;\u4ee5\u53ca\u7a97\u79fb\u52a8\u8ddd\u79bb\u5f71\u54cd\u3002<\/p>\n<h3 id=\"2.2%E3%80%81%E5%B0%8F%E6%B3%A2%E5%8F%98%E6%8D%A2%EF%BC%88%E5%A4%9A%E5%88%86%E8%BE%A8%E7%8E%87%EF%BC%89\">2.2\u3001\u5c0f\u6ce2\u53d8\u6362&#xff08;\u591a\u5206\u8fa8\u7387&#xff09;<\/h3>\n<p style=\"text-align:center\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"201\" src=\"2025-06-06mipxml4gmnc.png\" width=\"229\" \/><\/p>\n<p>\u6301\u7eed\u65f6\u95f4<img decoding=\"async\" alt=\"T\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a24ffe.png\" \/>\u7684\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u7684\u8fde\u7eed\u5c0f\u6ce2\u53d8\u6362\u4e3a\u00a0<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"X_{\\\\text{CWT}}\\\\left ( a,b\\\\right )&#061;\\\\int_{0}^{T}x\\\\left ( t \\\\right )\\\\Psi ^{*}\\\\left ( t;a,b \\\\right )dt\\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: (3)\" class=\"mathcode\" src=\"2025-06-06tlapmmduhkl.png\" \/><\/p>\n<p>\u5176\u4e2d<img decoding=\"async\" alt=\"a&gt;0\" class=\"mathcode\" src=\"2025-06-06xemekrkt4t5.png\" \/>\u8868\u793a\u7f29\u653e\u56e0\u5b50&#xff0c;<img decoding=\"async\" alt=\"b\" class=\"mathcode\" src=\"2025-06-06m0ebwxnwib0.png\" \/>\u8868\u793a\u65f6\u5ef6\u56e0\u5b50&#xff0c;\u5f0f\u4e2d\u7cfb\u5217\u51fd\u6570<img decoding=\"async\" alt=\"\\\\Psi \\\\left ( t ;a,b\\\\right )&#061;\\\\frac{1}{\\\\sqrt{a}}\\\\Psi\\\\left ( \\\\frac{t-b}{a} \\\\right )\" class=\"mathcode\" src=\"2025-06-061gmh05qxfqe.png\" \/>\u79f0\u4e3a\u5c0f\u6ce2\u51fd\u6570&#xff0c;\u5b83\u662f\u7531\u6bcd\u51fd\u6570<img decoding=\"async\" alt=\"\\\\Psi \\\\left ( t \\\\right )\" src=\"2025-06-06wwjtu0cvdae.png\" \/>\u7ecf\u8fc7\u4e0d\u540c\u7684\u65f6\u95f4\u5c3a\u5ea6\u4f38\u7f29\u548c\u4e0d\u540c\u7684\u65f6\u95f4\u5e73\u79fb\u5f97\u5230\u7684\u3002\u4e0b\u56fe\u5c55\u793a\u4e86\u77ed\u65f6\u5085\u91cc\u53f6\u53d8\u6362\u548c\u5c0f\u6ce2\u53d8\u6362\u7684\u65f6\u9891\u56fe&#xff0c;\u53ef\u4ee5\u5f88\u6e05\u695a\u7684\u770b\u51fa&#xff0c;\u5c0f\u6ce2\u53d8\u6362\u9891\u7387\u8d8a\u4f4e&#xff0c;\u9891\u7387\u5206\u8fa8\u7387\u8d8a\u597d&#xff0c;\u5e76\u4e14\u76f8\u6bd4\u4e8e\u77ed\u65f6\u5085\u91cc\u53f6\u53d8\u6362&#xff0c;\u5177\u6709\u66f4\u597d\u7684\u65f6\u95f4\u5206\u8fa8\u6027\u80fd\u3002<\/p>\n<h3 id=\"2.3%E3%80%81S%E5%8F%98%E6%8D%A2%EF%BC%88Stockwell%E5%8F%98%E6%8D%A2%EF%BC%8C%E5%A4%9A%E5%88%86%E8%BE%A8%E7%8E%87%EF%BC%89\">2.3\u3001S\u53d8\u6362&#xff08;Stockwell\u53d8\u6362&#xff0c;\u591a\u5206\u8fa8\u7387&#xff09;<\/h3>\n<p class=\"img-center\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"194\" src=\"2025-06-06qttz3tjmmup.png\" width=\"211\" \/><\/p>\n<p>\u6301\u7eed\u65f6\u95f4<img decoding=\"async\" alt=\"T\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a24ffe.png\" \/>\u7684\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u7684S\u53d8\u6362\u4e3a<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"X_{\\\\text{S}}\\\\left ( \\\\tau ,f \\\\right )&#061;\\\\int_{0}^{T}x\\\\left ( t \\\\right )\\\\frac{\\\\left | f \\\\right |}{\\\\sqrt{2\\\\pi}}e^{-\\\\frac{\\\\left ( t-\\\\tau \\\\right )^{2}f^{2}}{2}}e^{-j2\\\\pi f t}dt\\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: (4)\" class=\"mathcode\" src=\"2025-06-06h5zg44v3442.png\" \/><\/p>\n<\/p>\n<h3 id=\"2.4%E3%80%81%E5%88%86%E6%95%B0%E9%98%B6%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2\">2.4\u3001\u5206\u6570\u9636\u5085\u91cc\u53f6\u53d8\u6362<\/h3>\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u6301\u7eed\u65f6\u95f4<img decoding=\"async\" alt=\"T\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a24ffe.png\" \/>\u7684\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u7684\u5206\u6570\u9636\u5085\u91cc\u53f6\u53d8\u6362\u4e3a\u00a0\u00a0<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"X_{\\\\text{FrFT}}\\\\left ( \\\\alpha ,u \\\\right )&#061;A_{\\\\alpha }e^{j\\\\pi \\\\left (\\\\cot \\\\alpha \\\\right ) u^{2}}\\\\int_{0}^{T}x\\\\left ( t \\\\right )e^{j\\\\pi \\\\left (\\\\cot \\\\alpha \\\\right ) t^{2}}e^{-j2\\\\pi \\\\left (\\\\csc \\\\alpha \\\\right ) ut}dt\\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: (5)\" class=\"mathcode\" src=\"2025-06-06sg2wu2it0lc.png\" \/><\/p>\n<p>\u5176\u4e2d<img decoding=\"async\" alt=\"A_{\\\\alpha }&#061;\\\\sqrt{\\\\frac{1-j\\\\cot\\\\alpha }{2\\\\pi}}\" class=\"mathcode\" src=\"2025-06-0641gujjengqj.png\" \/>&#xff0c;<img decoding=\"async\" alt=\"\\\\alpha&#061;p\\\\frac{\\\\pi}{2}\" src=\"2025-06-06nbwwqalll01.png\" \/>&#xff0c;<img decoding=\"async\" alt=\"p\" src=\"2025-06-060wu3viouidu.png\" \/>\u8868\u793a\u9636\u6b21&#xff0c;<img decoding=\"async\" alt=\"u\" src=\"2025-06-06e1xyfkfbyw0.png\" \/>\u8868\u793a\u5206\u6570\u9636\u5085\u91cc\u53f6\u57df&#xff0c;\u5f53\u9636\u6b21\u4e3a1\u65f6&#xff0c;\u4e0a\u5f0f\u9000\u5316\u4e3a\u5085\u91cc\u53f6\u53d8\u6362\u3002<\/p>\n<h2 id=\"%E4%B8%89%E3%80%81%E9%9D%9E%E7%BA%BF%E6%80%A7%E6%97%B6%E9%A2%91%E5%88%86%E6%9E%90\">\u4e09\u3001\u975e\u7ebf\u6027\u65f6\u9891\u5206\u6790<\/h2>\n<h3 id=\"3.1%E3%80%81%E4%BA%8C%E6%AC%A1%E5%9E%8B%E6%97%B6%E9%A2%91%E5%88%86%E6%9E%90\">3.1\u3001\u4e8c\u6b21\u578b\u65f6\u9891\u5206\u6790<\/h3>\n<p>Wigner-Ville\u5206\u5e03&#xff08;WVD&#xff09;<\/p>\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u6301\u7eed\u65f6\u95f4<img decoding=\"async\" alt=\"T\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a24ffe.png\" \/>\u7684\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u7684Wigner-Ville\u5206\u5e03\u4e3a<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"X_{\\\\text{WVD}}\\\\left ( t ,f \\\\right )&#061;\\\\int_{0}^{T}x\\\\left ( t &#043;\\\\tau \/2\\\\right )x^{*}\\\\left ( t -\\\\tau \/2\\\\right )e^{-j2\\\\pi f \\\\tau}d\\\\tau \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: (6)\" class=\"mathcode\" src=\"2025-06-06oxfplog5asd.png\" \/><\/p>\n<p>\u7531\u4e8e\u662f\u4e8c\u6b21\u578b\u65f6\u9891\u5206\u5e03&#xff0c;\u5f53\u5b58\u5728\u65f6\u9891\u6df7\u53e0\u60c5\u51b5\u65f6&#xff0c;\u5176\u65f6\u9891\u56fe\u5b58\u5728\u4ea4\u53c9\u9879\u3002<\/p>\n<p>\u5e73\u6ed1Wigner-Ville\u5206\u5e03&#xff08;SWVD&#xff09;<\/p>\n<p>\u5229\u7528\u5e73\u6ed1\u51fd\u6570<img decoding=\"async\" alt=\"\\\\phi \\\\left ( t,f \\\\right )\" class=\"mathcode\" src=\"2025-06-06c0hd0rbtwsh.png\" \/>\u5bf9Wigner-Ville\u5206\u5e03\u8fdb\u884c\u4e8c\u7ef4\u5377\u79ef\u5904\u7406\u5f97\u5230\u5e73\u6ed1Wigner-Ville\u5206\u5e03<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"X_{\\\\text{SWVD}}\\\\left ( t ,f \\\\right )&#061;\\\\phi \\\\left ( t ,f \\\\right )\\\\ast X_{\\\\text{WVD}}\\\\left ( t ,f \\\\right )\\\\\\\\ &#061;\\\\int_{-\\\\infty }^{\\\\infty}\\\\int_{-\\\\infty}^{\\\\infty}\\\\phi \\\\left ( x ,y \\\\right )X_{\\\\text{WVD}}\\\\left ( t -x,f -y\\\\right )dx dy \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: (7)\" class=\"mathcode\" src=\"2025-06-06uyifioipbwo.png\" \/><\/p>\n<p>\u5e73\u6ed1\u51fd\u6570\u53ef\u4e3a\u9ad8\u65af\u6838&#xff1a;<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"\\\\phi \\\\left ( t,f \\\\right )&#061;e^{-\\\\alpha t^{2}-\\\\beta f^{2}}\\\\, \\\\, \\\\, \\\\, \\\\, \\\\, \\\\, \\\\, \\\\alpha ,\\\\beta &gt;0\" class=\"mathcode\" src=\"2025-06-06bs1ptl1tqr4.png\" \/><\/p>\n<p>\u901a\u8fc7\u727a\u7272\u65f6\u9891\u5206\u8fa8\u7387\u6765\u6291\u5236\u4ea4\u53c9\u9879\u3002<\/p>\n<p>\u4f2aWigner-Ville\u5206\u5e03&#xff08;PWVD&#xff09;<\/p>\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u6301\u7eed\u65f6\u95f4<img decoding=\"async\" alt=\"T\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a24ffe.png\" \/>\u7684\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u7684\u4f2aWigner-Ville\u5206\u5e03\u4e3a<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"X_{\\\\text{PWVD}}\\\\left ( t ,f \\\\right )&#061;\\\\int_{0}^{T}x_{w}\\\\left ( t &#043;\\\\tau \/2\\\\right )x_{w}^{*}\\\\left ( t -\\\\tau \/2\\\\right )e^{-j2\\\\pi f \\\\tau}d\\\\tau \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: (8)\" class=\"mathcode\" src=\"2025-06-06jft4xke55ut.png\" \/><\/p>\n<p>\u5176\u4e2d<img decoding=\"async\" alt=\"x_{w}\\\\left ( t \\\\right )&#061;x\\\\left ( t \\\\right )w\\\\left ( t-\\\\tau \\\\right )\" class=\"mathcode\" src=\"2025-06-06hrhex2f212t.png\" \/>&#xff0c;\u662f\u5bf9\u4fe1\u53f7\u7684\u52a0\u7a97\u6ede\u540e\u3002\u51f8\u663e\u7ec6\u8282<\/p>\n<p>\u5e73\u6ed1\u4f2aWigner-Ville\u5206\u5e03&#xff08;SPWVD&#xff09;<\/p>\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u6301\u7eed\u65f6\u95f4<img decoding=\"async\" alt=\"T\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a24ffe.png\" \/>\u7684\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u7684\u4f2aWigner-Ville\u5206\u5e03\u4e3a<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"X_{\\\\text{SPWVD}}\\\\left ( t ,f \\\\right )&#061;\\\\int_{-\\\\infty }^{\\\\infty }h\\\\left ( \\\\tau \\\\right )\\\\int_{-\\\\infty }^{\\\\infty }g\\\\left ( u-\\\\tau \\\\right )x\\\\left ( t &#043;\\\\tau \/2\\\\right )x^{*}\\\\left ( t -\\\\tau \/2\\\\right )e^{-j2\\\\pi f \\\\tau}dud\\\\tau \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: (9)\" class=\"mathcode\" src=\"2025-06-06c3rhr1gtncb.png\" \/><\/p>\n<p>\u5176\u4e2d<img decoding=\"async\" alt=\"h\\\\left ( \\\\tau \\\\right )\" class=\"mathcode\" src=\"2025-06-06wwvpbei4a0s.png\" \/>\u548c<img decoding=\"async\" alt=\"g\\\\left ( \\\\tau \\\\right )\" class=\"mathcode\" src=\"2025-06-06ufma1egnlxk.png\" \/>\u90fd\u662f\u5e73\u6ed1\u7a97\u3002<\/p>\n<p>Cohen\u7c7b\u5206\u5e03<\/p>\n<p>\u00a0 \u6301\u7eed\u65f6\u95f4<img decoding=\"async\" alt=\"T\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a24ffe.png\" \/>\u7684\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u7684Cohen\u7c7b\u5206\u5e03\u4e3a<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"X_{\\\\text{CD}}\\\\left ( t ,f \\\\right )&#061;\\\\frac{1}{2\\\\pi}\\\\int \\\\int \\\\int x\\\\left ( u &#043;\\\\tau \/2\\\\right )x^{*}\\\\left ( u -\\\\tau \/2\\\\right )g\\\\left ( \\\\theta ,\\\\tau \\\\right )e^{-j2\\\\pi\\\\left (\\\\theta t&#043; f \\\\tau -u\\\\theta \\\\right )}dud\\\\tau d\\\\theta \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: (10)\" class=\"mathcode\" src=\"2025-06-06y4zoh3w3vub.png\" \/><\/p>\n<p><img decoding=\"async\" alt=\"t\" class=\"mathcode\" src=\"2025-06-06mwp1ep1l00l.png\" \/>\u4e3a\u65f6\u95f4&#xff0c;<img decoding=\"async\" alt=\"f\" class=\"mathcode\" src=\"2025-06-063xzptwdykq5.png\" \/>\u4e3a\u9891\u7387&#xff0c;<img decoding=\"async\" alt=\"\\\\tau\" class=\"mathcode\" src=\"2025-06-06xln3puotqr1.png\" \/>\u4e3a\u65f6\u79fb&#xff0c;<img decoding=\"async\" alt=\"\\\\theta\" class=\"mathcode\" src=\"2025-06-06qmooko1la1n.png\" \/>\u4e3a\u9891\u79fb&#xff0c;<img decoding=\"async\" alt=\"u\" class=\"mathcode\" src=\"2025-06-06e1xyfkfbyw0.png\" \/>\u4e3a\u79ef\u5206\u53d8\u91cf\u3002<img decoding=\"async\" alt=\"g\\\\left ( \\\\theta ,\\\\tau \\\\right )\" class=\"mathcode\" src=\"2025-06-06t05s3qwh3in.png\" \/>\u4e3a\u65f6\u9891\u5206\u5e03\u6838\u51fd\u6570\u3002<\/p>\n<p>\u00a0Choi-Williams\u5206\u5e03<\/p>\n<p>\u5f53<img decoding=\"async\" alt=\"g\\\\left ( \\\\theta ,\\\\tau \\\\right )&#061;e^{-\\\\frac{\\\\pi ^{2}\\\\theta ^{2}\\\\tau ^{2}}{2\\\\sigma^{2}} }\" class=\"mathcode\" src=\"2025-06-06ts5ctsm1q1i.png\" \/>&#xff0c;Cohen\u7c7b\u5206\u5e03\u4e3a\u00a0Choi-Williams\u5206\u5e03&#xff1a;<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"X_{\\\\text{CWD}}\\\\left ( t ,f \\\\right )&#061;\\\\int \\\\int \\\\sqrt{\\\\frac{1 }{4\\\\pi \\\\sigma\\\\tau ^{2}}}e^{-\\\\frac{ \\\\left (t-u \\\\right )^{2}}{4\\\\sigma\\\\tau ^{2}}}x\\\\left ( u &#043;\\\\tau \/2\\\\right )x^{*}\\\\left ( u -\\\\tau \/2\\\\right )e^{-j2\\\\pi f \\\\tau}dud\\\\tau \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: (11)\" class=\"mathcode\" src=\"2025-06-06lx142y2xabt.png\" \/><\/p>\n<p>\u6ce8&#xff1a;Affine\u7c7b\u65f6\u9891\u5206\u5e03\u662f\u5728Wigner-Ville\u5206\u5e03\u7684\u57fa\u7840\u4e0a\u589e\u52a0\u6838\u51fd\u6570\u6765\u8fdb\u884c\u4eff\u5c04\u5e73\u6ed1&#xff0c;\u4fdd\u6301\u4e86\u539f\u6709Wigner-Ville\u5206\u5e03\u7684\u65f6\u79fb\u548c\u4f38\u7f29\u79fb\u4e0d\u53d8\u6027\u3002<\/p>\n<h3 id=\"3.2%E3%80%81%E9%9D%9E%E7%BA%BF%E6%80%A7%E6%97%B6%E9%A2%91%E5%88%86%E6%9E%90\">3.2\u3001\u975e\u7ebf\u6027\u65f6\u9891\u5206\u6790<\/h3>\n<p>\u7ecf\u9a8c\u6a21\u6001\u5206\u89e3&#xff08;EMD&#xff09;<\/p>\n<p>\u00a0 \u00a0 \u00a0 \u00a0EMD\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u7684\u65f6\u9891\u5206\u6790\u65b9\u6cd5&#xff0c;\u9002\u7528\u4e8e\u975e\u7ebf \u6027\u3001\u975e\u5e73\u7a33\u4fe1\u53f7\u7684\u5904\u7406&#xff0c;\u5b83\u53ef\u4ee5\u4f9d\u636e\u4fe1\u53f7\u5185\u90e8\u7684\u65f6\u95f4\u5c3a\u5ea6\u7279\u5f81\u81ea\u9002\u5e94\u5206\u89e3\u4fe1\u53f7&#xff0c;\u5176 \u57fa\u51fd\u6570\u5373\u4e3a\u4fe1\u53f7\u672c\u8eab&#xff0c;\u662f\u7531\u6570\u636e\u9a71\u52a8\u7684\u4fe1\u53f7\u5206\u89e3&#xff0c;\u9002\u7528\u4e8e\u7a81\u53d8\u4fe1\u53f7\u7684\u5904\u7406\u5206\u6790\u3002\u5177\u4f53\u6b65\u9aa4\u5982\u4e0b&#xff1a;<\/p>\n<p>1&#xff09;\u5bf9\u539f\u59cb\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u5168\u90e8\u6781\u5927\u3001\u5c0f\u503c\u70b9\u5206\u522b\u91c7\u7528\u4e09\u6b21\u6837\u6761\u63d2\u503c\u8fdb\u884c\u8fde\u63a5&#xff0c;\u6c42\u51fa\u4e0a\u3001 \u4e0b\u5305\u7edc\u66f2\u7ebf<img decoding=\"async\" alt=\"e_{u}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06oab0q1aoij5.png\" \/>\u3001<img decoding=\"async\" alt=\"e_{d}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06tzu55l2z05y.png\" \/>&#xff1b;<\/p>\n<p>2&#xff09;\u6c42\u53d6\u4e0a\u3001\u4e0b\u5305\u7edc\u66f2\u7ebf\u5747\u503c<img decoding=\"async\" alt=\"m_{1}\\\\left ( t \\\\right )&#061;\\\\frac{e_{u}\\\\left ( t \\\\right )&#043;e_{d}\\\\left ( t \\\\right )}{2}\" class=\"mathcode\" src=\"2025-06-06mnldpnaco4l.png\" \/>&#xff1b;<\/p>\n<p>3&#xff09;\u8ba1\u7b97\u539f\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u4e0e\u5305\u7edc\u5747\u503c<img decoding=\"async\" alt=\"m_{1}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06puonekom01c.png\" \/>\u7684\u5dee\u503c&#xff0c;\u5f97\u5230\u4fe1\u53f7\u5206\u91cf<img decoding=\"async\" alt=\"d_{1}\\\\left ( t \\\\right )&#061;x\\\\left ( t \\\\right )-m_{1}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06at0cottdzud.png\" \/>&#xff1b;<\/p>\n<p>4&#xff09;\u5224\u65ad\u5206\u91cf<img decoding=\"async\" alt=\"d_{1}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06zjiko3mipmn.png\" \/>\u662f\u5426\u6ee1\u8db3IMF\u5fc5\u8981\u6761\u4ef6(<span style=\"color:#fe2c24\">IMF\u5fc5\u8981\u6761\u4ef6<\/span>:\u5728IMF\u5206\u91cf\u6574\u4f53\u4e2d&#xff0c;\u5c40\u90e8\u6781\u503c\u70b9\u4e0e\u8fc7\u96f6\u70b9\u4e2a\u6570\u5e94\u76f8\u7b49\u6216\u76f8\u5dee\u81f3\u591a\u4e3a1&#xff0c;\u6781 \u503c\u70b9\u5305\u62ec\u6781\u5927\u503c\u70b9\u4e0e\u6781\u5c0f\u503c\u70b9&#xff1b;\u5bf9\u4e8eIMF\u5206\u91cf\u4efb\u610f\u70b9&#xff0c;\u7531\u6781\u5927\u3001\u5c0f\u503c\u70b9\u786e\u5b9a\u7684\u4e0a\u3001\u4e0b\u5305\u7edc\u5747\u503c\u4e3a\u96f6\u3002)&#xff0c;\u82e5\u6ee1\u8db3\u5219\u4ee4<img decoding=\"async\" alt=\"c_{1}\\\\left ( t \\\\right )&#061;d_{1}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06qqsvdkd4gdq.png\" \/>&#xff0c;\u8bb0\u4f5cIMF \u7684\u7b2c\u4e00\u4e2a\u5206\u91cf&#xff0c;\u82e5\u4e0d\u6ee1\u8db3\u5219\u5c06<img decoding=\"async\" alt=\"d_{1}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06zjiko3mipmn.png\" \/>\u5e26\u5165\u6b65\u9aa41&#xff09;\u4e2d\u4f5c\u4e3a\u521d\u59cb\u8f93\u5165\u5f00\u59cb\u518d\u4e00\u8f6e\u8ba1\u7b97&#xff0c; \u91cd\u590d1)-3&#xff09;\u7684\u8fc7\u7a0bk\u6b21&#xff0c;\u76f4\u81f3&#xff08;c&#xff09;\u7684\u8f93\u51fa\u5206\u91cf<img decoding=\"async\" alt=\"d_{1k}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06vf5tmgvasql.png\" \/>\u6ee1\u8db3IMF\u7ea6\u675f\u6761\u4ef6&#xff0c;\u5219\u8bb0IMF \u7b2c\u4e00\u5206\u91cf\u4e3a<img decoding=\"async\" alt=\"c_{1}\\\\left ( t \\\\right )&#061;d_{1k}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06xqwsie5ejlb.png\" \/>&#xff1b;<\/p>\n<p>5&#xff09;\u5c06\u539f\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u4e0eIMF\u5206\u91cf<img decoding=\"async\" alt=\"c_{1}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06qlucc4fynk0.png\" \/>\u4f5c\u5dee&#xff0c;\u8bb0\u4e3a<img decoding=\"async\" alt=\"r_{1}\\\\left ( t \\\\right )&#061;s\\\\left ( t \\\\right )-c_{1}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-0622kld3gwhlg.png\" \/>&#xff0c;\u5bf9<img decoding=\"async\" alt=\"r_{1}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06q3w2te5ubhb.png\" \/>\u8fdb\u884c\u6b65\u9aa4 1)-4&#xff09;\u7684\u8fc7\u7a0b&#xff0c;\u4e0d\u65ad\u5faa\u73af\u76f4\u5230\u51fa\u73b0\u6b8b\u4f59\u9879<img decoding=\"async\" alt=\"r_{n}\\\\left ( t \\\\right )&#061;r_{n-1}\\\\left ( t \\\\right )-c_{n}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06fu5xesvjilb.png\" \/>\u4e3a\u4e00\u5355\u8c03\u51fd\u6570\u6216\u5e38\u6570\u65f6&#xff0c;\u505c\u6b62\u5206\u89e3\u3002<\/p>\n<p>\u7b5b\u5206\u51fa\u7684\u7b2c\u4e00\u4e2a\u6a21\u6001\u5206\u91cf\u5305\u542b\u539f\u59cb\u4fe1\u53f7\u7684\u6700\u9ad8\u9891\u7387\u6210\u5206&#xff0c;\u4e4b\u540e\u5206\u89e3\u51fa\u7684IMF\u5206\u91cf\u9891\u7387\u4f9d\u6b21\u9012\u51cf\u3002\u539f\u59cb\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u7ecf\u7531EMD\u8fc7\u7a0b\u5f97\u5230\u7684IMF\u5206\u91cf\u548c\u6b8b\u4f59\u9879\u53ef\u8868\u793a\u4e3a&#xff1a;<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )&#061;\\\\sum_{i&#061;1}^{n}c_{i}\\\\left ( t \\\\right )&#043;r_{n}\\\\left ( t \\\\right )\\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: \\\\: (12)\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?x%5Cleft%20%28%20t%20%5Cright%20%29%3D%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dc_%7Bi%7D%5Cleft%20%28%20t%20%5Cright%20%29&amp;plus;r_%7Bn%7D%5Cleft%20%28%20t%20%5Cright%20%29%5C%3A%20%5C%3A%20%5C%3A%20%5C%3A%20%5C%3A%20%5C%3A%20%5C%3A%20%5C%3A%20%5C%3A%20%2812%29\" \/><\/p>\n<p>\u53d8\u5206\u6a21\u6001\u5206\u89e3&#xff08;VMD&#xff09;<\/p>\n<p>\u00a0 \u00a0 \u00a0 VMD\u53ef\u4ee5\u5c06\u4fe1\u53f7\u5206\u89e3\u4e3a\u5177\u6709\u56fa\u5b9a\u5e26\u5bbd\u548c\u4e2d\u5fc3\u9891\u7387\u7684\u6709\u9650\u56fa\u6709\u632f\u8361\u6a21\u6001&#xff0c;\u5e76\u8981\u6c42\u5404\u6a21\u6001\u4fe1\u53f7\u7684\u9891\u8c31\u80fd\u91cf\u4e3a\u7d27\u652f\u6491&#xff0c;\u5177\u4f53\u6b65\u9aa4\u5982\u4e0b&#xff1a;<\/p>\n<p>1&#xff09;\u8bbe\u5b9a\u539f\u59cb\u4fe1\u53f7<img decoding=\"async\" alt=\"x\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/06\/20250606102530-6842c21a40126.png\" \/>\u53ef\u4ee5\u88ab\u5206\u89e3\u4e3aK\u4e2a\u6a21\u6001&#xff08;\u53ef\u4ee5\u901a\u8fc7EMD\u83b7\u5f97&#xff09;&#xff0c;\u6bcf\u4e2a\u6a21\u6001\u51fd\u6570<img decoding=\"async\" alt=\"c_{k}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06mm3twlo5mxm.png\" \/>\u90fd\u5305\u542b\u4e00 \u4e2d\u5fc3\u9891\u7387<img decoding=\"async\" alt=\"f_{k}\" class=\"mathcode\" src=\"2025-06-06gi15q3gp45f.png\" \/>\u53ca\u5b83\u7684\u6709\u6548\u5e26\u5bbd,\u5bf9<img decoding=\"async\" alt=\"c_{k}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06mm3twlo5mxm.png\" \/>\u8fdb\u884c\u5e0c\u5c14\u4f2f\u7279\u53d8\u6362&#xff0c;\u5f97\u5355\u8fb9\u8c31<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"c{}&#039;_{k}\\\\left ( t \\\\right )&#061;\\\\left [ \\\\delta \\\\left ( t \\\\right ) &#043;\\\\frac{j}{\\\\pi t}\\\\right ]\\\\ast c_{k}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06h2gkpen0j1m.png\" \/><\/p>\n<p>2&#xff09;\u5c06\u89e3\u6790\u4fe1\u53f7<img decoding=\"async\" alt=\"c{}&#039;_{k}\\\\left ( t \\\\right )\" class=\"mathcode\" src=\"2025-06-06zcsw0msw4io.png\" \/>\u9891\u8c31\u642c\u79fb\u5230\u57fa\u5e26&#xff1a;<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"c{}&#039;&#039;_{k}\\\\left ( t \\\\right )&#061;c{}&#039;_{k}\\\\left ( t \\\\right )e^{-j2\\\\pi f_{k}t}\" class=\"mathcode\" src=\"2025-06-06rgiml5sbf12.png\" \/><\/p>\n<p>3&#xff09;\u6784\u5efa\u4f7f\u6240\u6709\u6a21\u6001\u5e26\u5bbd\u4e4b\u548c\u6700\u5c0f\u7684\u7ea6\u675f\u6a21\u578b&#xff1a;<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"\\\\begin{matrix} \\\\underset{c_{k}\\\\left ( t \\\\right ),f_{k}}{\\\\text{min}}\\\\sum_{k&#061;1}^{K}\\\\left \\\\| \\\\partial _{t} \\\\left [ \\\\left ( \\\\delta \\\\left ( t \\\\right ) &#043;\\\\frac{j}{\\\\pi t}\\\\right )\\\\ast c_{k}\\\\left ( t \\\\right )\\\\right ]e^{-j2\\\\pi f_{k}t}\\\\right \\\\|_{2}^{2}\\\\\\\\ s.t.\\\\sum_{k&#061;1}^{K}c_{k}\\\\left ( t \\\\right )&#061;x\\\\left ( t \\\\right ) \\\\end{matrix}\" class=\"mathcode\" src=\"2025-06-06a4qsijxwhi2.png\" \/><img decoding=\"async\" alt=\"\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?\" \/><\/p>\n<p>4&#xff09;\u6784\u9020\u62c9\u683c\u6717\u65e5\u5c55\u5f00\u5f0f\u5982\u4e0b&#xff1a;<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"L\\\\left \\\\{ c_{k}\\\\left ( t \\\\right ) ,f_{k},\\\\lambda \\\\right \\\\}&#061;\\\\alpha \\\\sum_{k&#061;1}^{K}\\\\left \\\\| \\\\partial _{t} \\\\left [ \\\\left ( \\\\delta \\\\left ( t \\\\right ) &#043;\\\\frac{j}{\\\\pi t}\\\\right )\\\\ast c_{k}\\\\left ( t \\\\right )\\\\right ]e^{-j2\\\\pi f_{k}t}\\\\right \\\\|_{2}^{2}&#043;\\\\left \\\\|x\\\\left ( t \\\\right )-\\\\sum_{k&#061;1}^{K}c_{k}\\\\left ( t \\\\right ) \\\\right \\\\|_{2}^{2}&#043;\\\\left \\\\langle \\\\lambda \\\\left ( t \\\\right ) ,x\\\\left ( t \\\\right )-\\\\sum_{k&#061;1}^{K}c_{k}\\\\left ( t \\\\right )\\\\right \\\\rangle\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?L%5Cleft%20%5C%7B%20c_%7Bk%7D%5Cleft%20%28%20t%20%5Cright%20%29%20%2Cf_%7Bk%7D%2C%5Clambda%20%5Cright%20%5C%7D%3D%5Calpha%20%5Csum_%7Bk%3D1%7D%5E%7BK%7D%5Cleft%20%5C%7C%20%5Cpartial%20_%7Bt%7D%20%5Cleft%20%5B%20%5Cleft%20%28%20%5Cdelta%20%5Cleft%20%28%20t%20%5Cright%20%29%20&amp;plus;%5Cfrac%7Bj%7D%7B%5Cpi%20t%7D%5Cright%20%29%5Cast%20c_%7Bk%7D%5Cleft%20%28%20t%20%5Cright%20%29%5Cright%20%5De%5E%7B-j2%5Cpi%20f_%7Bk%7Dt%7D%5Cright%20%5C%7C_%7B2%7D%5E%7B2%7D&amp;plus;%5Cleft%20%5C%7Cx%5Cleft%20%28%20t%20%5Cright%20%29-%5Csum_%7Bk%3D1%7D%5E%7BK%7Dc_%7Bk%7D%5Cleft%20%28%20t%20%5Cright%20%29%20%5Cright%20%5C%7C_%7B2%7D%5E%7B2%7D&amp;plus;%5Cleft%20%5Clangle%20%5Clambda%20%5Cleft%20%28%20t%20%5Cright%20%29%20%2Cx%5Cleft%20%28%20t%20%5Cright%20%29-%5Csum_%7Bk%3D1%7D%5E%7BK%7Dc_%7Bk%7D%5Cleft%20%28%20t%20%5Cright%20%29%5Cright%20%5Crangle\" \/><\/p>\n<p>5&#xff09;\u6700\u4f18\u89e3\u901a\u8fc7\u4ea4\u66ff\u65b9\u5411\u4e58\u5b50\u6cd5\u8ba1\u7b97\u5f97\u5230&#xff0c;\u521d\u59cb\u5316<img decoding=\"async\" alt=\"\\\\hat{x}\\\\left ( f \\\\right )&#061;FFT\\\\left \\\\{ x\\\\left ( t \\\\right ) \\\\right \\\\}\" class=\"mathcode\" src=\"2025-06-06djdxb5rvorv.png\" \/>&#xff0c;<img decoding=\"async\" alt=\"\\\\hat{\\\\lambda}^{0} \\\\left ( f \\\\right )\" class=\"mathcode\" src=\"2025-06-06xc4csaiheeu.png\" \/>,<img decoding=\"async\" alt=\"f_{k}\" class=\"mathcode\" src=\"2025-06-06gi15q3gp45f.png\" \/>,<img decoding=\"async\" alt=\"\\\\alpha\" class=\"mathcode\" src=\"2025-06-06btsvjz4sr2b.png\" \/>,<img decoding=\"async\" alt=\"\\\\tau\" class=\"mathcode\" src=\"2025-06-06xln3puotqr1.png\" \/>,<img decoding=\"async\" alt=\"\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?\" \/><img decoding=\"async\" alt=\"\\\\hat{c}_{k}^{0}\\\\left ( f \\\\right )&#061;FFT\\\\left \\\\{ {c}_{k}\\\\left ( t \\\\right ) \\\\right \\\\}\" class=\"mathcode\" src=\"2025-06-06uas1tmbvpyv.png\" \/>\u7b2c<img decoding=\"async\" alt=\"n\" class=\"mathcode\" src=\"2025-06-06f131vrfctrg.png\" \/>\u6b21\u66f4\u65b0\u7ed3\u679c\u8868\u793a\u5982\u4e0b&#xff1a;<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"\\\\begin{matrix} \\\\hat{c}_{k}^{n&#043;1}\\\\left ( f \\\\right )&#061;\\\\frac{\\\\hat{x}\\\\left ( f \\\\right )-\\\\sum_{i&#061;1,i\\\\neq k}^{K}\\\\hat{c}_{i}^{n}\\\\left ( f \\\\right )&#043;\\\\frac{\\\\hat{\\\\lambda}^{n} \\\\left ( f \\\\right ) }{2}}{1&#043;8\\\\pi^{2}\\\\alpha \\\\left ( f-f_{k}\\\\right )^{2}}\\\\\\\\ f_{k}^{n&#043;1}&#061;\\\\frac{\\\\int 2\\\\pi f\\\\left | \\\\hat{c}_{k}^{n&#043;1}\\\\left ( f \\\\right ) \\\\right |^{2}df}{\\\\int \\\\left | \\\\hat{c}_{k}^{n&#043;1}\\\\left ( f \\\\right ) \\\\right |^{2}df}\\\\\\\\ \\\\hat{\\\\lambda}^{n&#043;1} \\\\left ( f \\\\right )&#061;\\\\hat{\\\\lambda}^{n} \\\\left ( f \\\\right )&#043;\\\\tau \\\\left [ \\\\hat{x}\\\\left ( f \\\\right )-\\\\sum_{i&#061;1}^{K}\\\\hat{c}_{i}^{n&#043;1}\\\\left ( f \\\\right ) \\\\right ] \\\\end{matrix}\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%5Cbegin%7Bmatrix%7D%20%5Chat%7Bc%7D_%7Bk%7D%5E%7Bn&amp;plus;1%7D%5Cleft%20%28%20f%20%5Cright%20%29%3D%5Cfrac%7B%5Chat%7Bx%7D%5Cleft%20%28%20f%20%5Cright%20%29-%5Csum_%7Bi%3D1%2Ci%5Cneq%20k%7D%5E%7BK%7D%5Chat%7Bc%7D_%7Bi%7D%5E%7Bn%7D%5Cleft%20%28%20f%20%5Cright%20%29&amp;plus;%5Cfrac%7B%5Chat%7B%5Clambda%7D%5E%7Bn%7D%20%5Cleft%20%28%20f%20%5Cright%20%29%20%7D%7B2%7D%7D%7B1&amp;plus;8%5Cpi%5E%7B2%7D%5Calpha%20%5Cleft%20%28%20f-f_%7Bk%7D%5Cright%20%29%5E%7B2%7D%7D%5C%5C%20f_%7Bk%7D%5E%7Bn&amp;plus;1%7D%3D%5Cfrac%7B%5Cint%202%5Cpi%20f%5Cleft%20%7C%20%5Chat%7Bc%7D_%7Bk%7D%5E%7Bn&amp;plus;1%7D%5Cleft%20%28%20f%20%5Cright%20%29%20%5Cright%20%7C%5E%7B2%7Ddf%7D%7B%5Cint%20%5Cleft%20%7C%20%5Chat%7Bc%7D_%7Bk%7D%5E%7Bn&amp;plus;1%7D%5Cleft%20%28%20f%20%5Cright%20%29%20%5Cright%20%7C%5E%7B2%7Ddf%7D%5C%5C%20%5Chat%7B%5Clambda%7D%5E%7Bn&amp;plus;1%7D%20%5Cleft%20%28%20f%20%5Cright%20%29%3D%5Chat%7B%5Clambda%7D%5E%7Bn%7D%20%5Cleft%20%28%20f%20%5Cright%20%29&amp;plus;%5Ctau%20%5Cleft%20%5B%20%5Chat%7Bx%7D%5Cleft%20%28%20f%20%5Cright%20%29-%5Csum_%7Bi%3D1%7D%5E%7BK%7D%5Chat%7Bc%7D_%7Bi%7D%5E%7Bn&amp;plus;1%7D%5Cleft%20%28%20f%20%5Cright%20%29%20%5Cright%20%5D%20%5Cend%7Bmatrix%7D\" \/><\/p>\n<p>6&#xff09;\u5f53\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6&#xff0c;\u8fed\u4ee3\u505c\u6b62<\/p>\n<p style=\"text-align:center\"><img decoding=\"async\" alt=\"\\\\sum_{k&#061;1}^{K}\\\\frac{\\\\left \\\\| \\\\hat{u}_{n&#043;1}^{k}\\\\left ( f \\\\right ) -\\\\hat{u}_{n}^{k}\\\\left ( f \\\\right )\\\\right \\\\|_{2}^{2}}{\\\\left \\\\| \\\\hat{u}_{n}^{k}\\\\left ( f \\\\right )\\\\right \\\\|_{2}^{2}}&lt;\\\\varepsilon\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%5Csum_%7Bk%3D1%7D%5E%7BK%7D%5Cfrac%7B%5Cleft%20%5C%7C%20%5Chat%7Bu%7D_%7Bn&amp;plus;1%7D%5E%7Bk%7D%5Cleft%20%28%20f%20%5Cright%20%29%20-%5Chat%7Bu%7D_%7Bn%7D%5E%7Bk%7D%5Cleft%20%28%20f%20%5Cright%20%29%5Cright%20%5C%7C_%7B2%7D%5E%7B2%7D%7D%7B%5Cleft%20%5C%7C%20%5Chat%7Bu%7D_%7Bn%7D%5E%7Bk%7D%5Cleft%20%28%20f%20%5Cright%20%29%5Cright%20%5C%7C_%7B2%7D%5E%7B2%7D%7D%3C%5Cvarepsilon\" \/><\/p>\n<\/p>\n<h2 id=\"%E5%9B%9B%E3%80%81MATLAB%E6%97%B6%E9%A2%91%E5%88%86%E6%9E%90%E5%87%BD%E6%95%B0\">\u56db\u3001MATLAB\u65f6\u9891\u5206\u6790\u51fd\u6570<\/h2>\n<p>  Time-Domain Processing<br \/>\n    ifestar2 &#8211; Instantaneous frequency estimation using AR2 modelisation.<br \/>\n    instfreq &#8211; Instantaneous frequency estimation.<br \/>\n    loctime  &#8211; Time localization characteristics.<\/p>\n<p>   Frequency-Domain Processing<br \/>\n    fmt      &#8211; Fast Mellin transform<br \/>\n    ifmt     &#8211; Inverse fast Mellin transform.<br \/>\n    locfreq  &#8211; Frequency localization characteristics.<br \/>\n    sgrpdlay &#8211; Group delay estimation.<\/p>\n<p>   Linear Time-Frequency Processing<br \/>\n    tfrgabor &#8211; Gabor representation.<br \/>\n<span style=\"color:#fe2c24\">    <\/span><span style=\"color:#fe2c24\">tfrstft<\/span><span style=\"color:#fe2c24\">  &#8211; Short time Fourier transform.<\/span><\/p>\n<p>   Bilinear Time-Frequency Processing in the Cohen&#039;s Class<br \/>\n    tfrbj    &#8211; Born-Jordan distribution.<br \/>\n    tfrbud   &#8211; Butterworth distribution.<br \/>\n<span style=\"color:#fe2c24\">    <\/span><span style=\"color:#fe2c24\">tfrcw<\/span><span style=\"color:#fe2c24\">    &#8211; Choi-Williams distribution. <\/span><br \/>\n    tfrgrd   &#8211; Generalized rectangular distribution.<br \/>\n    tfrmh    &#8211; Margenau-Hill distribution.<br \/>\n    tfrmhs   &#8211; Margenau-Hill-Spectrogram distribution.<br \/>\n    tfrmmce  &#8211; MMCE combination of spectrograms.<br \/>\n    tfrpage  &#8211; Page distribution.<br \/>\n    tfrpmh   &#8211; Pseudo Margenau-Hill distribution.<br \/>\n    tfrppage &#8211; Pseudo Page distribution.<br \/>\n    tfrpwv   &#8211; Pseudo Wigner-Ville distribution.<br \/>\n    tfrri    &#8211; Rihaczek distribution.<br \/>\n    tfrridb  &#8211; Reduced interference distribution (Bessel window).<br \/>\n    tfrridh  &#8211; Reduced interference distribution (Hanning window).<br \/>\n    tfrridn  &#8211; Reduced interference distribution (binomial window).<br \/>\n    tfrridt  &#8211; Reduced interference distribution (triangular window).<br \/>\n    tfrsp    &#8211; Spectrogram.<br \/>\n<span style=\"color:#fe2c24\">    <\/span><span style=\"color:#fe2c24\">tfrspwv<\/span><span style=\"color:#fe2c24\">  &#8211; Smoothed Pseudo Wigner-Ville distribution.  <\/span><br \/>\n<span style=\"color:#fe2c24\">    <\/span><span style=\"color:#fe2c24\">tfrwv<\/span><span style=\"color:#fe2c24\">    &#8211; Wigner-Ville distribution.<\/span><br \/>\n    tfrzam   &#8211; Zhao-Atlas-Marks distribution.               <\/p>\n<p>   Bilinear Time-Frequency Processing in the Affine Class<br \/>\n    tfrbert  &#8211; Unitary Bertrand distribution.<br \/>\n    tfrdfla  &#8211; D-Flandrin distribution.<br \/>\n    tfrscalo &#8211; Scalogram, for Morlet or Mexican hat wavelet.<br \/>\n    tfrspaw  &#8211; Smoothed Pseudo Affine Wigner distributions.<br \/>\n    tfrunter &#8211; Unterberger distribution, active or passive form.<\/p>\n<p>   Reassigned Time-Frequency Processing<br \/>\n    tfrrgab  &#8211; Reassigned Gabor spectrogram.<br \/>\n    tfrrmsc  &#8211; Reassigned Morlet Scalogram time-frequency distribution.<br \/>\n    tfrrpmh  &#8211; Reassigned Pseudo Margenau-Hill distribution.<br \/>\n    tfrrppag &#8211; Reassigned Pseudo Page distribution.<br \/>\n    tfrrpwv  &#8211; Reassigned Pseudo Wigner-Ville distribution.<br \/>\n    tfrrsp   &#8211; Reassigned Spectrogram.<br \/>\n    tfrrspwv &#8211; Reassigned Smoothed Pseudo WV distribution.<\/p>\n<p>   Ambiguity Functions<br \/>\n    ambifunb &#8211; Narrow band ambiguity function.<br \/>\n    ambifuwb &#8211; Wide band ambiguity function.<\/p>\n<p>   Post-Processing or Help to the Interpretation<br \/>\n    friedman &#8211; Instantaneous frequency density.<br \/>\n    holder   &#8211; Estimate the Holder exponent through an affine TFR.<br \/>\n    htl      &#8211; Hough transform for detection of lines in images.<br \/>\n    margtfr  &#8211; Marginals and energy of a time-frequency representation.<br \/>\n    midscomp &#8211; Mid-point construction used in the interference diagram.<br \/>\n    momftfr  &#8211; Frequency moments of a time-frequency representation.<br \/>\n    momttfr  &#8211; Time moments of a time-frequency representation.<br \/>\n    plotsid  &#8211; Schematic interference diagram of FM signals.<br \/>\n    renyi    &#8211; Measure Renyi information.<br \/>\n    ridges   &#8211; Extraction of ridges.<br \/>\n    tfrideal &#8211; Ideal TFR for given frequency laws.<\/p>\n<p>   Visualization &amp; backup<br \/>\n    plotifl  &#8211; Plot normalized instantaneous frequency laws.<br \/>\n    tfrqview &#8211; Quick visualization of time-frequency representations.<br \/>\n    tfrview  &#8211; Visualization of time-frequency representations.<br \/>\n    tfrparam &#8211; Return the paramaters needed to display (or save) a TFR.<br \/>\n    tfrsave  &#8211; Save the parameters of a time-frequency representation.<\/p>\n<p>  Other<br \/>\n    disprog  &#8211; Display progression of a loop.<br \/>\n    divider  &#8211; Find dividers of integer such that product equals integer.<br \/>\n    dwindow  &#8211; Derive a window.<br \/>\n    integ    &#8211; Approximate integral.<br \/>\n    integ2d  &#8211; Approximate 2-D integral.<br \/>\n    izak     &#8211; Inverse Zak transform.<br \/>\n    istfr1   &#8211; returns true if a distribution is Cohen&#039;s class type 1 (spectrogram)<br \/>\n    istfr2   &#8211; returns true if a distribution is Cohen&#039;s class type 2 (Wigner-Ville)<br \/>\n    istfraff &#8211; returns true is a distribution is an affine class member<br \/>\n    kaytth   &#8211; Kay-Tretter filter computation.<br \/>\n    modulo   &#8211; Congruence of a vector.<br \/>\n    movcw4at &#8211; Four atoms rotating, analyzed by the Choi-Williams distribution.<br \/>\n    movpwdph &#8211; Influence of a phase-shift on the interferences of the pWVD.<br \/>\n    movpwjph &#8211; Influence of a jump of phase on the interferences of the pWVD.<br \/>\n    movsc2wv &#8211; Movie illustrating the passage from the scalogram to the WVD.<br \/>\n    movsp2wv &#8211; Movie illustrating the passage from the spectrogram to the WVD.<br \/>\n    movwv2at &#8211; Oscillating structure of the interferences of the WVD.<br \/>\n    odd      &#8211; Round towards nearest odd value.<br \/>\n    sigmerge &#8211; Add two signals with given energy ratio in dB.<br \/>\n    zak     &#8211; Zak transform.<\/p>\n<p>\u90e8\u5206\u6587\u732e<\/p>\n<p>[1]\u00a0Y. Chen and K. 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