{"id":45484,"date":"2025-07-30T00:06:26","date_gmt":"2025-07-29T16:06:26","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/45484.html"},"modified":"2025-07-30T00:06:26","modified_gmt":"2025-07-29T16:06:26","slug":"%e8%ae%ba%e6%96%87%e7%bf%bb%e8%af%91%e4%b8%8e%e8%a7%a3%e8%af%bb-%e9%9b%b7%e8%be%be%e9%98%b5%e5%88%97%e4%bf%a1%e5%8f%b7%e5%a4%84%e7%90%86%e4%b8%ad%e7%9a%84%e6%9c%80%e5%a4%a7%e4%bc%bc","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/45484.html","title":{"rendered":"\u8bba\u6587\u7ffb\u8bd1\u4e0e\u89e3\u8bfb\u2014\u2014\u96f7\u8fbe\u9635\u5217\u4fe1\u53f7\u5904\u7406\u4e2d\u7684\u6700\u5927\u4f3c\u7136\u65b9\u6cd5"},"content":{"rendered":"

\u96f7\u8fbe\u9635\u5217\u4fe1\u53f7\u5904\u7406\u4e2d\u7684\u6700\u5927\u4f3c\u7136\u65b9\u6cd5<\/h2>\n

A. L. Swindlehurst and P. Stoica, \u201cMaximum likelihood methods in radar array signal processing,\u201d in Proceedings of the IEEE, vol. 86, no. 2, pp. 421-441, Feb. 1998, doi: 10.1109\/5.659495.<\/p>\n

\u672c\u6587\u63a2\u8ba8\u4e86\u5728\u96f7\u8fbe\u76ee\u6807\u4fe1\u53f7\u88ab\u4f20\u611f\u5668\u9635\u5217\u89c2\u6d4b\u65f6，\u5982\u4f55\u5728\u672a\u77e5\u4e8c\u9636\u7edf\u8ba1\u7279\u6027\u7684\u5e72\u6270\u73af\u5883\u4e2d，\u4f7f\u7528\u9c81\u68d2\u4e14\u8ba1\u7b97\u9ad8\u6548\u7684\u6700\u5927\u4f3c\u7136\u7b97\u6cd5\u6765\u4f30\u8ba1\u76ee\u6807\u53c2\u6570\u3002\u6587\u7ae0\u63d0\u51fa\u4e86\u4e24\u79cd\u6570\u636e\u6a21\u578b：\u4e00\u79cd\u4f7f\u7528\u76ee\u6807\u5230\u8fbe\u65b9\u5411\u548c\u4fe1\u53f7\u5e45\u5ea6\u4f5c\u4e3a\u53c2\u6570，\u53e6\u4e00\u79cd\u662f\u4f7f\u7528\u901a\u7528\u76ee\u6807"\u7a7a\u95f4\u7279\u5f81"\u7684\u7b80\u5316\u975e\u7ed3\u6784\u5316\u6a21\u578b\u3002\u901a\u8fc7\u5f15\u5165\u6269\u5c55\u4e0d\u53d8\u6027\u539f\u7406，\u5c55\u793a\u4e86\u5982\u4f55\u5c06\u4ece\u7b80\u5355\u6a21\u578b\u83b7\u5f97\u7684\u8f83\u4e0d\u7cbe\u786e\u7684\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u8fdb\u884c\u6539\u8fdb，\u4ee5\u6e10\u8fd1\u5730\u8fbe\u5230\u4f7f\u7528\u7ed3\u6784\u5316\u6a21\u578b\u6240\u80fd\u83b7\u5f97\u7684\u6027\u80fd\u3002<\/p>\n

\u5f15\u8a00<\/h3>\n

\u5728\u4e3b\u52a8\u96f7\u8fbe\u7cfb\u7edf\u4e2d，\u4e3b\u8981\u76ee\u6807\u662f\u5728\u566a\u58f0\u3001\u5730\u6742\u6ce2\u548c\u7535\u5b50\u5bf9\u6297\u63aa\u65bd（\u5982\u5e72\u6270）\u5b58\u5728\u7684\u60c5\u51b5\u4e0b\u68c0\u6d4b\u76ee\u6807\u5e76\u4f30\u8ba1\u5176\u53c2\u6570\u3002\u5bf9\u4e8e\u673a\u8f7d\u96f7\u8fbe\u5e94\u7528，\u611f\u5174\u8da3\u7684\u53c2\u6570\u5305\u62ec\u76ee\u6807\u4fe1\u53f7\u5e45\u5ea6（\u4e0e\u5176\u96f7\u8fbe\u622a\u9762\u79ef\u76f8\u5173）\u3001\u5230\u8fbe\u65b9\u5411（DOA）\u3001\u8ddd\u79bb\u548c\u591a\u666e\u52d2\u9891\u7387\u3002\u5373\u4f7f\u53ea\u6709\u5355\u4e2a\u76ee\u6807\u5b58\u5728，\u7531\u4e8e\u9700\u8981\u641c\u7d22\u7684\u53c2\u6570\u7a7a\u95f4\u4f53\u79ef\u5de8\u5927，\u8fd9\u4e2a\u68c0\u6d4b\u548c\u4f30\u8ba1\u95ee\u9898\u4e5f\u76f8\u5f53\u8270\u5de8\u3002\u56e0\u6b64，\u8ba1\u7b97\u7684\u7b80\u4fbf\u6027\u548c\u6548\u7387\u81f3\u5173\u91cd\u8981\u3002<\/p>\n

\u6807\u51c6\u89e3\u51b3\u65b9\u6848\u6d89\u53ca\u4f7f\u7528\u7ecf\u5178\u7684\u65f6\u7a7a\u6ee4\u6ce2\u5668，\u53ef\u4ee5\u662f\u6570\u636e\u65e0\u5173\u7684（\u5982\u5ef6\u8fdf\u6c42\u548c\u6ce2\u675f\u5f62\u6210\u5668）\u6216"\u81ea\u9002\u5e94"\u7684（\u5982\u4f7f\u7528\u7ebf\u6027\u7ea6\u675f\u3001\u6700\u5927\u4fe1\u53f7\u5e72\u6270\u566a\u58f0\u6bd4\u51c6\u5219\u7b49）\u3002\u6700\u5927\u4f3c\u7136（ML）\u65b9\u6cd5\u5728\u8fd9\u4e2a\u95ee\u9898\u4e0a\u6ca1\u6709\u88ab\u5e7f\u6cdb\u8003\u8651，\u4e3b\u8981\u662f\u56e0\u4e3a\u5b83\u4eec\u88ab\u8ba4\u4e3a\u8ba1\u7b97\u8fc7\u4e8e\u590d\u6742，\u800c\u4e14\u5728\u63a5\u53d7\u6240\u9700\u7684\u5efa\u6a21\u5047\u8bbe\u65b9\u9762\u5b58\u5728\u4e00\u4e9b\u963b\u529b\u3002\u6700\u8fd1\u7684\u5de5\u4f5c\u8868\u660e，\u5bf9\u4e8e\u9ad8\u65af\u5e72\u6270\u4e2d\u5177\u6709\u672a\u77e5\u7a7a\u95f4\u534f\u65b9\u5dee\u7684\u5355\u4e2a\u76ee\u6807\u6e90，ML\u89e3\u53ef\u4ee5\u901a\u8fc7\u5bf9\u76ee\u6807DOA\u548c\u591a\u666e\u52d2\u8fdb\u884c\u4e8c\u7ef4\u641c\u7d22，\u7136\u540e\u8fdb\u884c\u5e7f\u4e49\u4f3c\u7136\u6bd4\u68c0\u9a8c（GLRT）\u6765\u83b7\u5f97\u3002<\/p>\n

\u96f7\u8fbe\u6570\u636e\u6a21\u578b<\/h3>\n

\u7ed3\u6784\u5316\u9635\u5217\u6a21\u578b<\/h4>\n

\u5047\u8bbe\u4e00\u4e2amm<\/span><\/span>m<\/span><\/span><\/span><\/span><\/span>\u5143\u5929\u7ebf\u9635\u5217\u4f7f\u7528\u67d0\u7ec4\u53d1\u5c04\u6743\u91cd\u53d1\u5c04\u4e00\u7cfb\u5217\u8109\u51b2\u3002\u5728\u7ed9\u5b9a\u7684\u76f8\u5e72\u5904\u7406\u95f4\u9694（CPI）\u671f\u95f4，\u5b58\u5728\u4e8e\u7279\u5b9a\u8ddd\u79bb\u5355\u5143\u4e2d\u7684\u76ee\u6807\u53ef\u4ee5\u5efa\u6a21\u4e3a\u4ea7\u751f\u4ee5\u4e0b\u57fa\u5e26\u5411\u91cf\u4fe1\u53f7（\u7ecf\u8fc7\u8109\u51b2\u538b\u7f29\u548c\u89e3\u8c03\u540e）：<\/p>\n

target\u00a0signal=b0a(\u03b80)ej\u03c90t,t=1,\u22ef\u2009,N\\\\text{target signal} = b_0\\\\mathbf{a}(\\\\theta_0)e^{j\\\\omega_0 t}, \\\\quad t = 1, \\\\cdots, N<\/span><\/span>target\u00a0signal<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>e<\/span><\/span>j<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2db0b_0<\/span><\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u4fe1\u53f7\u7684\u590d\u632f\u5e45，\u03c90\\\\omega_0<\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u7531\u9635\u5217\u5e73\u53f0\u548c\u76ee\u6807\u4e4b\u95f4\u7684\u76f8\u5bf9\u8fd0\u52a8\u5f15\u8d77\u7684\u591a\u666e\u52d2\u9891\u79fb，a(\u03b8)\\\\mathbf{a}(\\\\theta)<\/span><\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span>\u662f\u9635\u5217\u5bf9\u4ece\u65b9\u4f4d\u89d2\u03b80\\\\theta_0<\/span><\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5230\u8fbe\u7684\u5e73\u9762\u6ce2\u7684\u54cd\u5e94\u3002\u5411\u91cfa(\u03b8)\\\\mathbf{a}(\\\\theta)<\/span><\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span>\u88ab\u5047\u5b9a\u4e3a\u7f29\u653e，\u4f7f\u5f97\u5176\u7b2c\u4e00\u4e2a\u5143\u7d20\u5bf9\u6240\u6709\u03b8\\\\theta<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>\u90fd\u4e3a1\u3002<\/p>\n

\u5047\u8bbe\u5728\u4efb\u4f55\u7ed9\u5b9a\u7684\u8ddd\u79bb\u5355\u5143\u4e2d\u6700\u591a\u5b58\u5728\u4e00\u4e2a\u76ee\u6807，\u9635\u5217\u5728\u65f6\u95f4tt<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u7684\u91c7\u6837\u5411\u91cf\u8f93\u51fa\u6216"\u5feb\u7167"\u53ef\u4ee5\u5199\u6210：<\/p>\n

x(t)=b0a(\u03b80)ej\u03c90t+e(t)\u2208Cm,t=1,\u22ef\u2009,N\\\\mathbf{x}(t) = b_0\\\\mathbf{a}(\\\\theta_0)e^{j\\\\omega_0 t} + \\\\mathbf{e}(t) \\\\in \\\\mathbb{C}^m, \\\\quad t = 1, \\\\cdots, N<\/span><\/span>x<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>e<\/span><\/span>j<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>C<\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u5982\u679c\u6ca1\u6709\u76ee\u6807\u5b58\u5728\u5219b0b_0<\/span><\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u53d6\u4e3a\u96f6，e(t)\\\\mathbf{e}(t)<\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u7531\u4e8e\u6742\u6ce2\u3001\u63a5\u6536\u673a\u566a\u58f0\u548c\u5e72\u6270\u5f15\u8d77\u7684\u5e72\u6270\u3002\u7531\u4e8ee(t)\\\\mathbf{e}(t)<\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u4f9d\u8d56\u4e8e\u5982\u6b64\u591a\u7684\u7269\u7406\u53d8\u91cf，\u4e3a\u5176\u63a8\u5bfc\u786e\u5b9a\u6027\u6a21\u578b\u662f\u6709\u95ee\u9898\u7684，\u901a\u5e38\u4e5f\u662f\u4e0d\u5b9e\u9645\u7684\u3002\u9996\u5148，\u6211\u4eec\u9009\u62e9\u5c06e(t)\\\\mathbf{e}(t)<\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5efa\u6a21\u4e3a\u5728CPI\u7684NN<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u4e2a\u6837\u672c\u4e0a\u7684\u5e73\u7a33\u3001\u65f6\u95f4\u767d\u6027\u9ad8\u65af\u968f\u673a\u8fc7\u7a0b，\u5177\u6709\u96f6\u5747\u503c\u548c\u672a\u77e5\u7a7a\u95f4\u534f\u65b9\u5deeQ\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u975e\u7ed3\u6784\u5316\u9635\u5217\u6a21\u578b<\/h4>\n

\u5728\u4e0a\u9762\u7684\u8ba8\u8bba\u4e2d，\u6211\u4eec\u5df2\u7ecf\u660e\u786e\u5730\u7528\u03b80\\\\theta_0<\/span><\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u6765\u8868\u793a\u9635\u5217\u54cd\u5e94\u3002\u5f53\u7136，\u8981\u4f30\u8ba1\u03b80\\\\theta_0<\/span><\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5411\u91cf\u503c\u51fd\u6570a(\u03b8)\\\\mathbf{a}(\\\\theta)<\/span><\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5fc5\u987b\u5bf9\u6240\u6709\u611f\u5174\u8da3\u7684\u03b8\\\\theta<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>\u90fd\u662f\u5df2\u77e5\u7684\u3002\u51fd\u6570a(\u03b8)\\\\mathbf{a}(\\\\theta)<\/span><\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span>\u53ef\u4ee5\u901a\u8fc7\u9635\u5217\u7684\u5b9e\u9a8c\u6216\u5206\u6790\u6821\u51c6\u83b7\u5f97，\u4f46\u8fd9\u79cd\u6821\u51c6\u7a0b\u5e8f\u6c38\u8fdc\u4e0d\u4f1a\u5b8c\u5168\u51c6\u786e，\u8fd9\u662f\u7531\u4e8e\u5929\u7ebf\u5143\u4ef6\u7684\u589e\u76ca\u548c\u76f8\u4f4d\u54cd\u5e94\u8bef\u5dee\u3001\u4e92\u8026\u3001\u91cf\u5316\u548c\u63d2\u503c\u8bef\u5dee，\u4ee5\u53ca\u7531\u4e8e\u5929\u6c14\u3001\u73af\u5883\u7b49\u53d8\u5316\u5bfc\u81f4\u7684\u63a5\u6536\u673a\u7535\u5b50\u7279\u6027\u6f02\u79fb\u3002\u6839\u636e\u5b9e\u9645\u9635\u5217\u54cd\u5e94\u4e0e\u5176\u6a21\u578b\u7684\u5dee\u5f02\u7a0b\u5ea6，\u68c0\u6d4b\u548c\u4f30\u8ba1\u6027\u80fd\u53ef\u80fd\u4f1a\u663e\u8457\u4e0b\u964d\u3002\u51fa\u4e8e\u8fd9\u4e2a\u539f\u56e0，\u6211\u4eec\u5728\u975e\u7ed3\u6784\u5316\u9635\u5217\u6a21\u578b\u4e0b\u5f00\u53d1\u4e86\u76ee\u6807\u68c0\u6d4b\u7684GLRT，\u8be5\u6a21\u578b\u672c\u8d28\u4e0a\u4e0d\u4f9d\u8d56\u4e8e\u9635\u5217\u6821\u51c6\u6570\u636e\u7684\u53ef\u7528\u6027\u3002\u8003\u8651\u4ee5\u4e0b\u6570\u636e\u6a21\u578b，\u5176\u4e2d\u4fe1\u53f7\u5e45\u5ea6\u548c\u9635\u5217\u54cd\u5e94\u5df2\u5408\u5e76\u4e3a\u5355\u4e2a\u975e\u7ed3\u6784\u5316\u5411\u91cf\u9879：<\/p>\n

x(t)=\u03b10ej\u03c90t+e(t),t=1,\u22ef\u2009,N\\\\mathbf{x}(t) = \\\\boldsymbol{\\\\alpha}_0 e^{j\\\\omega_0 t} + \\\\mathbf{e}(t), \\\\quad t = 1, \\\\cdots, N<\/span><\/span>x<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>j<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span>,<\/span><\/span><\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5411\u91cf\u03b10\\\\boldsymbol{\\\\alpha}_0<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u88ab\u79f0\u4e3a\u76ee\u6807\u7684"\u7a7a\u95f4\u7279\u5f81"，\u5728\u8fd9\u4e2a\u6a21\u578b\u4e2d\u53d6\u4ee3\u4e86b0a(\u03b80)b_0\\\\mathbf{a}(\\\\theta_0)<\/span><\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>，\u7b80\u5355\u5730\u88ab\u89c6\u4e3a\u4e00\u7ec4\u672a\u77e5\u7684\u590d\u5e38\u6570\u3002<\/p>\n

\u6269\u5c55\u4e0d\u53d8\u6027\u539f\u7406<\/h3>\n

\u5047\u8bbe\u4e3a\u4e86\u4f30\u8ba1\u4e0a\u8ff0\u96f7\u8fbe\u53c2\u6570，\u6211\u4eec\u6709\u4e00\u4e2a\u635f\u5931\u51fd\u6570（\u4f8b\u5982，ML\u51c6\u5219），\u53ef\u4ee5\u7528\u03b7\\\\boldsymbol{\\\\eta}<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u6216\u03b7~\\\\tilde{\\\\boldsymbol{\\\\eta}}<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u53c2\u6570\u5316\u3002\u5c06\u5f97\u5230\u7684\u51c6\u5219\u5206\u522b\u8868\u793a\u4e3aVN(\u03b7)V_N(\\\\boldsymbol{\\\\eta})<\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b7<\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u548cV~N(\u03b7~)\\\\tilde{V}_N(\\\\tilde{\\\\boldsymbol{\\\\eta}})<\/span><\/span><\/span>V<\/span><\/span><\/span>~<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u5982\u679c\u5b58\u5728\u4e00\u4e2a（\u53ef\u9006\u7684）\u6620\u5c04\u03b7=g(\u03b7~)\u2208D\u03b7,\u2200\u03b7~\u2208D\u03b7~\\\\boldsymbol{\\\\eta} = g(\\\\tilde{\\\\boldsymbol{\\\\eta}}) \\\\in D_\\\\eta, \\\\quad \\\\forall\\\\tilde{\\\\boldsymbol{\\\\eta}} \\\\in D_{\\\\tilde{\\\\eta}}<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>g<\/span>(<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>D<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u2200<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span><\/span><\/span>D<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u4f7f\u5f97\u51c6\u5219\u76f8\u7b49，\u7ecf\u5178\u7684\u4f30\u8ba1\u7406\u8bba\u4e0d\u53d8\u6027\u539f\u7406\u53ef\u4ee5\u88ab\u8c03\u7528\u6765\u8bc1\u660e\u6700\u5c0f\u5316VN(\u03b7)V_N(\\\\boldsymbol{\\\\eta})<\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b7<\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u548cV~N(\u03b7~)\\\\tilde{V}_N(\\\\tilde{\\\\boldsymbol{\\\\eta}})<\/span><\/span><\/span>V<\/span><\/span><\/span>~<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u7b49\u4ef7\u6027\u3002<\/p>\n

\u6269\u5c55\u4e0d\u53d8\u6027\u539f\u7406（EXIP）\u901a\u8fc7\u4ee5\u4e0b\u5b9a\u7406\u7ed9\u51fa：<\/p>\n

\u5b9a\u74061：\u5b9a\u4e49
\n\u03b7^=arg\u2061min\u2061\u03b7VN(\u03b7)\\\\hat{\\\\boldsymbol{\\\\eta}} = \\\\arg\\\\min_{\\\\boldsymbol{\\\\eta}} V_N(\\\\boldsymbol{\\\\eta})<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>arg<\/span><\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span><\/span><\/span>min<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b7<\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>
\n\u03b7~^=arg\u2061min\u2061\u03b7~V~N(\u03b7~)\\\\hat{\\\\tilde{\\\\boldsymbol{\\\\eta}}} = \\\\arg\\\\min_{\\\\tilde{\\\\boldsymbol{\\\\eta}}} \\\\tilde{V}_N(\\\\tilde{\\\\boldsymbol{\\\\eta}})<\/span><\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>arg<\/span><\/span><\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>min<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span><\/span>~<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u5047\u8bbe\u5b58\u5728\u6ee1\u8db3\u03b7~=f(\u03b7)\u2208D\u03b7~,\u2200\u03b7\u2208D\u03b7\\\\tilde{\\\\boldsymbol{\\\\eta}} = f(\\\\boldsymbol{\\\\eta}) \\\\in D_{\\\\tilde{\\\\eta}}, \\\\quad \\\\forall\\\\boldsymbol{\\\\eta} \\\\in D_\\\\eta<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>f<\/span>(<\/span>\u03b7<\/span><\/span><\/span>)<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>D<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u2200<\/span>\u03b7<\/span><\/span><\/span><\/span>\u2208<\/span><\/span><\/span><\/span>D<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u51fd\u6570ff<\/span><\/span>f<\/span><\/span><\/span><\/span><\/span>\u3002\u5047\u8bbeff<\/span><\/span>f<\/span><\/span><\/span><\/span><\/span>\u5728\u5176\u8f93\u5165\u53c2\u6570\u7684\u6574\u4e2a\u5b9a\u4e49\u57df\u4e0a\u662f\u4e00\u5bf9\u4e00\u7684\u4e14\u5b9a\u4e49\u826f\u597d\u3002\u5982\u679clim\u2061N\u2192\u221e\u03b7~^=lim\u2061N\u2192\u221ef(\u03b7^)\\\\lim_{N \\\\to \\\\infty} \\\\hat{\\\\tilde{\\\\boldsymbol{\\\\eta}}} = \\\\lim_{N \\\\to \\\\infty} f(\\\\hat{\\\\boldsymbol{\\\\eta}})<\/span><\/span>lim<\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>lim<\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>，\u5219<\/p>\n

\u03b7^=arg\u2061min\u2061\u03b7[\u03b7~^\u2212f(\u03b7)]TW[\u03b7~^\u2212f(\u03b7)]\\\\hat{\\\\boldsymbol{\\\\eta}} = \\\\arg\\\\min_{\\\\boldsymbol{\\\\eta}} [\\\\hat{\\\\tilde{\\\\boldsymbol{\\\\eta}}} – f(\\\\boldsymbol{\\\\eta})]^T \\\\mathbf{W} [\\\\hat{\\\\tilde{\\\\boldsymbol{\\\\eta}}} – f(\\\\boldsymbol{\\\\eta})]<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>arg<\/span><\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span><\/span><\/span>min<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>f<\/span>(<\/span>\u03b7<\/span><\/span><\/span>)<\/span>]<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>W<\/span>[<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>f<\/span>(<\/span>\u03b7<\/span><\/span><\/span>)]<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6e10\u8fd1\u5730（\u5bf9\u4e8e\u5927NN<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>）\u7b49\u4ef7\u4e8e\u4f30\u8ba1\u03b7^\\\\hat{\\\\boldsymbol{\\\\eta}}<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2d<\/p>\n

W=E{\u2202V~N(\u03b7~^)\u2202\u03b7~^\u2202\u03b7~^T}\u03b7~^=\u03b7~\\\\mathbf{W} = \\\\mathcal{E}\\\\left\\\\{\\\\frac{\\\\partial \\\\tilde{V}_N(\\\\hat{\\\\tilde{\\\\boldsymbol{\\\\eta}}})}{\\\\partial \\\\hat{\\\\tilde{\\\\boldsymbol{\\\\eta}}} \\\\partial \\\\hat{\\\\tilde{\\\\boldsymbol{\\\\eta}}}^T}\\\\right\\\\}_{\\\\hat{\\\\tilde{\\\\boldsymbol{\\\\eta}}} = \\\\tilde{\\\\boldsymbol{\\\\eta}}}<\/span><\/span>W<\/span><\/span>=<\/span><\/span><\/span><\/span>E<\/span><\/span>{<\/span><\/span><\/span><\/span>\u2202<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span><\/span>V<\/span><\/span><\/span>~<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6700\u5927\u4f3c\u7136\u4f30\u8ba1<\/h3>\n

\u7ed3\u6784\u5316\u6a21\u578b<\/h4>\n

\u5b8c\u5168\u53c2\u6570\u5316\u6a21\u578b\u7684NN<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u4e2a\u6570\u636e\u6837\u672c\u7684\u8d1f\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u5728\u4e58\u6cd5\u548c\u52a0\u6cd5\u5e38\u6570\u5185\u5bb9\u6613\u8868\u793a\u4e3a：<\/p>\n

VN(\u03b7)=log\u2061\u2223Q\u2223+Tr{Q\u22121C(b,\u03b8,\u03c9)}V_N(\\\\boldsymbol{\\\\eta}) = \\\\log |\\\\mathbf{Q}| + \\\\text{Tr}\\\\{\\\\mathbf{Q}^{-1}\\\\mathbf{C}(b, \\\\theta, \\\\omega)\\\\}<\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b7<\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>log<\/span><\/span><\/span>\u2223<\/span>Q<\/span>\u2223<\/span><\/span>+<\/span><\/span><\/span><\/span>Tr<\/span><\/span>{<\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span>(<\/span>b<\/span>,<\/span><\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)}<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d
\nC(b,\u03b8,\u03c9)=1N\u2211t=1N[x(t)\u2212ba(\u03b8)ej\u03c9t][x(t)\u2212ba(\u03b8)ej\u03c9t]\u2217\\\\mathbf{C}(b, \\\\theta, \\\\omega) = \\\\frac{1}{N}\\\\sum_{t=1}^{N}[\\\\mathbf{x}(t) – b\\\\mathbf{a}(\\\\theta)e^{j\\\\omega t}][\\\\mathbf{x}(t) – b\\\\mathbf{a}(\\\\theta)e^{j\\\\omega t}]^*<\/span><\/span>C<\/span>(<\/span>b<\/span>,<\/span><\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span>x<\/span>(<\/span>t<\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>b<\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span>e<\/span><\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span>[<\/span>x<\/span>(<\/span>t<\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>b<\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span>e<\/span><\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5173\u4e8eQ\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>\u548cbb<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>\u7684\u6700\u5c0f\u5316\u53ef\u4ee5\u663e\u5f0f\u6267\u884c\u3002\u4f7f\u7528\u6807\u51c6\u77e9\u9635\u5fae\u79ef\u5206\u7ed3\u679c，\u5173\u4e8eQ\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>\u7684\u51c6\u5219\u68af\u5ea6\u5bb9\u6613\u8868\u793a\u4e3a：<\/p>\n

\u2202VN(\u03b7)\u2202Q=Q\u22121\u2212Q\u22121C(b,\u03b8,\u03c9)Q\u22121\\\\frac{\\\\partial V_N(\\\\boldsymbol{\\\\eta})}{\\\\partial \\\\mathbf{Q}} = \\\\mathbf{Q}^{-1} – \\\\mathbf{Q}^{-1}\\\\mathbf{C}(b, \\\\theta, \\\\omega)\\\\mathbf{Q}^{-1}<\/span><\/span><\/span><\/span>\u2202<\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b7<\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span>(<\/span>b<\/span>,<\/span><\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ece\u4e2d\u53ef\u4ee5\u6e05\u695a\u5730\u770b\u51fa，Q\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>\u7684ML\u4f30\u8ba1\u7531\u4e0b\u5f0f\u7ed9\u51fa：
\nQ^s=C(b,\u03b8,\u03c9)\\\\hat{\\\\mathbf{Q}}_s = \\\\mathbf{C}(b, \\\\theta, \\\\omega)<\/span><\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>C<\/span>(<\/span>b<\/span>,<\/span><\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5f53\u5c06\u6b64\u4ee3\u5165\u51c6\u5219\u65f6，\u96c6\u4e2d\u51c6\u5219\u4e3a：
\nVN(b,\u03b8,\u03c9)=log\u2061\u2223C(b,\u03b8,\u03c9)\u2223+mV_N(b, \\\\theta, \\\\omega) = \\\\log |\\\\mathbf{C}(b, \\\\theta, \\\\omega)| + m<\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>b<\/span>,<\/span><\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>log<\/span><\/span><\/span>\u2223<\/span>C<\/span>(<\/span>b<\/span>,<\/span><\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span>\u2223<\/span><\/span>+<\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5173\u4e8ebb<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>\u96c6\u4e2d，\u5f97\u5230：
\nb^s=a\u2217(\u03b8^s)R^s\u22121y(\u03c9^s)a\u2217(\u03b8^s)R^s\u22121a(\u03b8^s)\\\\hat{b}_s = \\\\frac{\\\\mathbf{a}^*(\\\\hat{\\\\theta}_s)\\\\hat{\\\\mathbf{R}}_s^{-1}\\\\mathbf{y}(\\\\hat{\\\\omega}_s)}{\\\\mathbf{a}^*(\\\\hat{\\\\theta}_s)\\\\hat{\\\\mathbf{R}}_s^{-1}\\\\mathbf{a}(\\\\hat{\\\\theta}_s)}<\/span><\/span><\/span>b<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>(<\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u03b8^s,\u03c9^s=arg\u2061max\u2061\u03b8,\u03c9\u2223a\u2217(\u03b8)R^\u22121y(\u03c9)\u22232[1\u2212y\u2217(\u03c9)R^\u22121y(\u03c9)][a\u2217(\u03b8)R^\u22121a(\u03b8)]\\\\hat{\\\\theta}_s, \\\\hat{\\\\omega}_s = \\\\arg\\\\max_{\\\\theta,\\\\omega} \\\\frac{|\\\\mathbf{a}^*(\\\\theta)\\\\hat{\\\\mathbf{R}}^{-1}\\\\mathbf{y}(\\\\omega)|^2}{[1 – \\\\mathbf{y}^*(\\\\omega)\\\\hat{\\\\mathbf{R}}^{-1}\\\\mathbf{y}(\\\\omega)][\\\\mathbf{a}^*(\\\\theta)\\\\hat{\\\\mathbf{R}}^{-1}\\\\mathbf{a}(\\\\theta)]}<\/span><\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>arg<\/span><\/span><\/span><\/span>\u03b8<\/span>,<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>max<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span>1<\/span><\/span>\u2212<\/span><\/span>y<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span>)<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)]<\/span>[<\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span>)]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)<\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d
\ny(\u03c9)=1N\u2211t=1Nx(t)e\u2212j\u03c9t\\\\mathbf{y}(\\\\omega) = \\\\frac{1}{N}\\\\sum_{t=1}^{N}\\\\mathbf{x}(t)e^{-j\\\\omega t}<\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span>(<\/span>t<\/span>)<\/span>e<\/span><\/span>\u2212<\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\nR^s=R^\u2212y(\u03c9)y\u2217(\u03c9)\\\\hat{\\\\mathbf{R}}_s = \\\\hat{\\\\mathbf{R}} – \\\\mathbf{y}(\\\\omega)\\\\mathbf{y}^*(\\\\omega)<\/span><\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)<\/span>y<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>
\nR^=1N\u2211t=1Nx(t)x\u2217(t)\\\\hat{\\\\mathbf{R}} = \\\\frac{1}{N}\\\\sum_{t=1}^{N}\\\\mathbf{x}(t)\\\\mathbf{x}^*(t)<\/span><\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span>(<\/span>t<\/span>)<\/span>x<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u975e\u7ed3\u6784\u5316\u6a21\u578b<\/h4>\n

\u975e\u7ed3\u6784\u5316\u6a21\u578b\u7684ML\u89e3\u53ef\u4ee5\u7c7b\u4f3c\u5730\u63a8\u5bfc\u3002\u03b7~\\\\tilde{\\\\boldsymbol{\\\\eta}}<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u8d1f\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u5177\u6709\u4e0e\u7ed3\u6784\u5316\u60c5\u51b5\u76f8\u540c\u7684\u5f62\u5f0f，\u4f46\u7528\u03b1\\\\boldsymbol{\\\\alpha}<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u66ff\u6362ba(\u03b8)b\\\\mathbf{a}(\\\\theta)<\/span><\/span>b<\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u5173\u4e8e\u03b1\\\\boldsymbol{\\\\alpha}<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u6700\u5c0f\u5316\u7ed9\u51fa：<\/p>\n

\u03b1^u=y(\u03c9)\\\\hat{\\\\boldsymbol{\\\\alpha}}_u = \\\\mathbf{y}(\\\\omega)<\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u4e0b\u6807uu<\/span><\/span>u<\/span><\/span><\/span><\/span><\/span>\u7528\u4e8e\u660e\u786e\u8868\u793a\u4f7f\u7528\u975e\u7ed3\u6784\u5316\u6a21\u578b\u83b7\u5f97\u7684\u4f30\u8ba1\u3002\u5c06\u6b64\u4ee3\u5165\u51c6\u5219\u5e76\u7b80\u5316\u540e，\u03c9\\\\omega<\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>\u7684\u89e3\u7b80\u5355\u5730\u4e3a：<\/p>\n

\u03c9^u=arg\u2061max\u2061\u03c9y\u2217(\u03c9)R^\u22121y(\u03c9)\\\\hat{\\\\omega}_u = \\\\arg\\\\max_\\\\omega \\\\mathbf{y}^*(\\\\omega)\\\\hat{\\\\mathbf{R}}^{-1}\\\\mathbf{y}(\\\\omega)<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>arg<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>max<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span>)<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64，\u975e\u7ed3\u6784\u5316ML\u4f30\u8ba1\u03c9\\\\omega<\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>\u7531\u4e0b\u5f0f\u7ed9\u51fa：
\n\u03c9^u=arg\u2061max\u2061\u03c9y\u2217(\u03c9)R^\u22121y(\u03c9)\\\\hat{\\\\omega}_u = \\\\arg\\\\max_\\\\omega \\\\mathbf{y}^*(\\\\omega)\\\\hat{\\\\mathbf{R}}^{-1}\\\\mathbf{y}(\\\\omega)<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>arg<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>max<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span>)<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f7f\u7528EXIP\u8fdb\u884c\u53c2\u6570\u4f30\u8ba1<\/h3>\n

\u6839\u636e\u5b9a\u74064，\u7ed9\u5b9a\u4ece\u975e\u7ed3\u6784\u5316\u6a21\u578b\u83b7\u5f97\u7684Q^u\\\\hat{\\\\mathbf{Q}}_u<\/span><\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3001\u03b1^u\\\\hat{\\\\boldsymbol{\\\\alpha}}_u<\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548c\u03c9^u\\\\hat{\\\\omega}_u<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u96f7\u8fbe\u76ee\u6807\u53c2\u6570\u7684\u6e10\u8fd1（\u5728NN<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u4e2d）\u6709\u6548\u4f30\u8ba1\u53ef\u4ee5\u5982\u4e0b\u627e\u5230：<\/p>\n

Q^e=Q^u\\\\hat{\\\\mathbf{Q}}_e = \\\\hat{\\\\mathbf{Q}}_u<\/span><\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\n\u03c9^e=\u03c9^u\\\\hat{\\\\omega}_e = \\\\hat{\\\\omega}_u<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

b^e,\u03b8^e=arg\u2061min\u2061b,\u03b8[((\u03b1^u)r(\u03b1^u)i)\u2212A(\u03b8)(brbi)]T\u22c5WQ[((\u03b1^u)r(\u03b1^u)i)\u2212A(\u03b8)(brbi)]\\\\hat{b}_e, \\\\hat{\\\\theta}_e = \\\\arg\\\\min_{b,\\\\theta} \\\\left[\\\\begin{pmatrix} (\\\\hat{\\\\boldsymbol{\\\\alpha}}_u)_r \\\\\\\\ (\\\\hat{\\\\boldsymbol{\\\\alpha}}_u)_i \\\\end{pmatrix} – \\\\mathbf{A}(\\\\theta)\\\\begin{pmatrix} b_r \\\\\\\\ b_i \\\\end{pmatrix}\\\\right]^T \\\\cdot \\\\mathbf{W}_Q \\\\left[\\\\begin{pmatrix} (\\\\hat{\\\\boldsymbol{\\\\alpha}}_u)_r \\\\\\\\ (\\\\hat{\\\\boldsymbol{\\\\alpha}}_u)_i \\\\end{pmatrix} – \\\\mathbf{A}(\\\\theta)\\\\begin{pmatrix} b_r \\\\\\\\ b_i \\\\end{pmatrix}\\\\right]<\/span><\/span><\/span>b<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>arg<\/span><\/span><\/span><\/span>b<\/span>,<\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>min<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span><\/span>(<\/span><\/span><\/span>(<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>A<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>(<\/span><\/span><\/span>b<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>]<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>W<\/span><\/span>Q<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span><\/span>(<\/span><\/span><\/span>(<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>A<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>(<\/span><\/span><\/span>b<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d
\nWQ=2N[Re{Q^u\u22121}\u2212Im{Q^u\u22121}Im{Q^u\u22121}Re{Q^u\u22121}]\\\\mathbf{W}_Q = 2N\\\\begin{bmatrix} \\\\text{Re}\\\\{\\\\hat{\\\\mathbf{Q}}_u^{-1}\\\\} & -\\\\text{Im}\\\\{\\\\hat{\\\\mathbf{Q}}_u^{-1}\\\\} \\\\\\\\ \\\\text{Im}\\\\{\\\\hat{\\\\mathbf{Q}}_u^{-1}\\\\} & \\\\text{Re}\\\\{\\\\hat{\\\\mathbf{Q}}_u^{-1}\\\\} \\\\end{bmatrix}<\/span><\/span>W<\/span><\/span>Q<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>N<\/span><\/span>[<\/span><\/span><\/span>Re<\/span><\/span>{<\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span>Im<\/span><\/span>{<\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>Im<\/span><\/span>{<\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span>Re<\/span><\/span>{<\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

A(\u03b8)=[Re{a(\u03b8)}\u2212Im{a(\u03b8)}Im{a(\u03b8)}Re{a(\u03b8)}]\\\\mathbf{A}(\\\\theta) = \\\\begin{bmatrix} \\\\text{Re}\\\\{\\\\mathbf{a}(\\\\theta)\\\\} & -\\\\text{Im}\\\\{\\\\mathbf{a}(\\\\theta)\\\\} \\\\\\\\ \\\\text{Im}\\\\{\\\\mathbf{a}(\\\\theta)\\\\} & \\\\text{Re}\\\\{\\\\mathbf{a}(\\\\theta)\\\\} \\\\end{bmatrix}<\/span><\/span>A<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>Re<\/span><\/span>{<\/span>a<\/span>(<\/span>\u03b8<\/span>)}<\/span><\/span><\/span><\/span>Im<\/span><\/span>{<\/span>a<\/span>(<\/span>\u03b8<\/span>)}<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>Im<\/span><\/span>{<\/span>a<\/span>(<\/span>\u03b8<\/span>)}<\/span><\/span><\/span><\/span>Re<\/span><\/span>{<\/span>a<\/span>(<\/span>\u03b8<\/span>)}<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u53ef\u4ee5\u663e\u5f0f\u6c42\u89e3b^e\\\\hat{b}_e<\/span><\/span><\/span>b<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5f97\u5230\u4ec5\u4f9d\u8d56\u4e8e\u03b8\\\\theta<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>\u7684\u51c6\u5219：<\/p>\n

b^e=a\u2217(\u03b8^e)Q^u\u22121\u03b1^ua\u2217(\u03b8^e)Q^u\u22121a(\u03b8^e)\\\\hat{b}_e = \\\\frac{\\\\mathbf{a}^*(\\\\hat{\\\\theta}_e)\\\\hat{\\\\mathbf{Q}}_u^{-1}\\\\hat{\\\\boldsymbol{\\\\alpha}}_u}{\\\\mathbf{a}^*(\\\\hat{\\\\theta}_e)\\\\hat{\\\\mathbf{Q}}_u^{-1}\\\\mathbf{a}(\\\\hat{\\\\theta}_e)}<\/span><\/span><\/span>b<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u03b8^e=arg\u2061max\u2061\u03b8\u2223a\u2217(\u03b8)Q^u\u22121\u03b1^u\u22232a\u2217(\u03b8)Q^u\u22121a(\u03b8)\\\\hat{\\\\theta}_e = \\\\arg\\\\max_\\\\theta \\\\frac{|\\\\mathbf{a}^*(\\\\theta)\\\\hat{\\\\mathbf{Q}}_u^{-1}\\\\hat{\\\\boldsymbol{\\\\alpha}}_u|^2}{\\\\mathbf{a}^*(\\\\theta)\\\\hat{\\\\mathbf{Q}}_u^{-1}\\\\mathbf{a}(\\\\theta)}<\/span><\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>arg<\/span><\/span><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>max<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u68c0\u6d4b<\/h3>\n

\u5728\u672c\u8282\u4e2d，\u6211\u4eec\u8ba8\u8bba\u5728\u786e\u5b9a\u7ed9\u5b9a\u6570\u636e\u96c6\u4e2d\u662f\u5426\u5b58\u5728\u76ee\u6807\u65f6\u51fa\u73b0\u7684\u4e8c\u5143\u5047\u8bbe\u68c0\u9a8c\u95ee\u9898\u3002\u7279\u522b\u662f，\u9700\u8981\u51b3\u5b9a\u4ee5\u4e0b\u4e24\u4e2a\u5047\u8bbe\u4e2d\u54ea\u4e00\u4e2a\u5bf9x(t),t=1,\u22ef\u2009,N\\\\mathbf{x}(t), t = 1, \\\\cdots, N<\/span><\/span>x<\/span>(<\/span>t<\/span>)<\/span>,<\/span><\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u6709\u6548：<\/p>\n

H0:x(t)=e(t)H_0: \\\\mathbf{x}(t) = \\\\mathbf{e}(t)<\/span><\/span>H<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>x<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>
\nH1:x(t)=b0a(\u03b80)ej\u03c90t+e(t)H_1: \\\\mathbf{x}(t) = b_0\\\\mathbf{a}(\\\\theta_0)e^{j\\\\omega_0 t} + \\\\mathbf{e}(t)<\/span><\/span>H<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>x<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>e<\/span><\/span>j<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7ed3\u6784\u5316\u6a21\u578b\u7684GLRT<\/h4>\n

\u5982\u679cpH0p_{H_0}<\/span><\/span>p<\/span><\/span>H<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548cpH1p_{H_1}<\/span><\/span>p<\/span><\/span>H<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u5bf9\u5e94\u4e8e\u4e24\u4e2a\u5047\u8bbe\u7684\u6570\u636e\u6982\u7387\u5bc6\u5ea6，\u7ed3\u6784\u5316ML\u6a21\u578b\u7684GLRT\u53ef\u4ee5\u8868\u8ff0\u4e3a：<\/p>\n

\u03bb=pH1(x(1),\u22ef\u2009,x(N)\u2223b^s,\u03b8^s,\u03c9^s,Q^s)pH0(x(1),\u22ef\u2009,x(N)\u2223Q^s)\u2277H1H0\u03bbT\\\\lambda = \\\\frac{p_{H_1}(\\\\mathbf{x}(1), \\\\cdots, \\\\mathbf{x}(N)|\\\\hat{b}_s, \\\\hat{\\\\theta}_s, \\\\hat{\\\\omega}_s, \\\\hat{\\\\mathbf{Q}}_s)}{p_{H_0}(\\\\mathbf{x}(1), \\\\cdots, \\\\mathbf{x}(N)|\\\\hat{\\\\mathbf{Q}}_s)} \\\\underset{H_0}{\\\\overset{H_1}{\\\\gtrless}} \\\\lambda_T<\/span><\/span>\u03bb<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>H<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>(<\/span>1<\/span>)<\/span>,<\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span>x<\/span>(<\/span>N<\/span>)<\/span>\u2223<\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>H<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>(<\/span>1<\/span>)<\/span>,<\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span>x<\/span>(<\/span>N<\/span>)<\/span>\u2223<\/span><\/span>b<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2277<\/span><\/span><\/span><\/span>H<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2db^s,\u03b8^s,\u03c9^s,Q^s\\\\hat{b}_s, \\\\hat{\\\\theta}_s, \\\\hat{\\\\omega}_s, \\\\hat{\\\\mathbf{Q}}_s<\/span><\/span><\/span>b<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u7ed3\u6784\u5316ML\u4f30\u8ba1，\u03bbT\\\\lambda_T<\/span><\/span>\u03bb<\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u67d0\u4e2a\u9608\u503c\u3002\u7531\u4e8eH0H_0<\/span><\/span>H<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548cH1H_1<\/span><\/span>H<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e0b\u7684\u6570\u636e\u90fd\u662f\u9ad8\u65af\u7684，\u901a\u5e38\u53d6\u4e24\u8fb9\u7684\u5bf9\u6570，\u6b64\u65f6GLRT\u5f62\u5f0f\u4e3a：<\/p>\n

\u03bb=2log\u2061\u03bb\u2032=2N(log\u2061\u2223R^\u2223\u2212log\u2061\u2223C(b^s,\u03b8^s,\u03c9^s)\u2223)\u2277H1H02log\u2061\u03bbT=\u03bbT\u2032\\\\lambda = 2\\\\log \\\\lambda' = 2N(\\\\log |\\\\hat{\\\\mathbf{R}}| – \\\\log |\\\\mathbf{C}(\\\\hat{b}_s, \\\\hat{\\\\theta}_s, \\\\hat{\\\\omega}_s)|) \\\\underset{H_0}{\\\\overset{H_1}{\\\\gtrless}} 2\\\\log \\\\lambda_T = \\\\lambda_T'<\/span><\/span>\u03bb<\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span>log<\/span><\/span><\/span>\u03bb<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>N<\/span>(<\/span>log<\/span><\/span><\/span>\u2223<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>log<\/span><\/span><\/span>\u2223<\/span>C<\/span>(<\/span><\/span>b<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u2223<\/span>)<\/span><\/span>H<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2277<\/span><\/span><\/span><\/span>H<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span>log<\/span><\/span><\/span>\u03bb<\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>T<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u975e\u7ed3\u6784\u5316\u6a21\u578b\u7684GLRT<\/h4>\n

\u5bf9\u4e8e\u975e\u7ed3\u6784\u5316\u6a21\u578b，\u4e24\u4e2a\u7ade\u4e89\u5047\u8bbe\u662f：
\nH0:x(t)=e(t)H_0: \\\\mathbf{x}(t) = \\\\mathbf{e}(t)<\/span><\/span>H<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>x<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>
\nH1u:x(t)=\u03b10ej\u03c90t+e(t)H_1^u: \\\\mathbf{x}(t) = \\\\boldsymbol{\\\\alpha}_0 e^{j\\\\omega_0 t} + \\\\mathbf{e}(t)<\/span><\/span>H<\/span><\/span>1<\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>x<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>j<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5f97\u5230\u7684GLRT\u5728\u5f62\u5f0f\u4e0a\u7c7b\u4f3c\u4e8e\u7ed3\u6784\u5316\u60c5\u51b5，\u7528\u03b1^u\\\\hat{\\\\boldsymbol{\\\\alpha}}_u<\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u66ff\u6362b^s,\u03b8^s\\\\hat{b}_s, \\\\hat{\\\\theta}_s<\/span><\/span><\/span>b<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>：<\/p>\n

\u03bbu=2N(log\u2061\u2223R^\u2223\u2212log\u2061\u2223C(\u03b1^u,\u03c9^u)\u2223)\u2277H1H0\u03bbTu\\\\lambda_u = 2N(\\\\log |\\\\hat{\\\\mathbf{R}}| – \\\\log |\\\\mathbf{C}(\\\\hat{\\\\boldsymbol{\\\\alpha}}_u, \\\\hat{\\\\omega}_u)|) \\\\underset{H_0}{\\\\overset{H_1}{\\\\gtrless}} \\\\lambda_{T_u}<\/span><\/span>\u03bb<\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>N<\/span>(<\/span>log<\/span><\/span><\/span>\u2223<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>log<\/span><\/span><\/span>\u2223<\/span>C<\/span>(<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u2223<\/span>)<\/span><\/span>H<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2277<\/span><\/span><\/span><\/span>H<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>T<\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ece\u524d\u9762\u7684\u63a8\u5bfc\u53ef\u77e5：
\nlog\u2061\u2223C(\u03b1^u,\u03c9^u)\u2223=log\u2061(1\u2212zR(\u03c9^u))+log\u2061\u2223R^\u2223\\\\log |\\\\mathbf{C}(\\\\hat{\\\\boldsymbol{\\\\alpha}}_u, \\\\hat{\\\\omega}_u)| = \\\\log(1 – z_R(\\\\hat{\\\\omega}_u)) + \\\\log |\\\\hat{\\\\mathbf{R}}|<\/span><\/span>log<\/span><\/span><\/span>\u2223<\/span>C<\/span>(<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u2223<\/span><\/span>=<\/span><\/span><\/span><\/span>log<\/span><\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>z<\/span><\/span>R<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>))<\/span><\/span>+<\/span><\/span><\/span><\/span>log<\/span><\/span><\/span>\u2223<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64\u95ee\u9898\u662f\u786e\u5b9a\u03bbTu\\\\lambda_{T_u}<\/span><\/span>\u03bb<\/span><\/span>T<\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4f7f\u5f97：
\n\u03bbu=\u22122Nlog\u2061(1\u2212zR(\u03c9^u))>\u03bbTu\\\\lambda_u = -2N\\\\log(1 – z_R(\\\\hat{\\\\omega}_u)) > \\\\lambda_{T_u}<\/span><\/span>\u03bb<\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span>N<\/span><\/span>log<\/span><\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>z<\/span><\/span>R<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>))<\/span><\/span>><\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>T<\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5728H0H_0<\/span><\/span>H<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e0b\u4ee5\u7ed9\u5b9a\u7684PFA PfP_f<\/span><\/span>P<\/span><\/span>f<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u53d1\u751f\u3002\u7531\u4e8e\u5bf9\u6570\u662f\u5355\u8c03\u7684，\u4e0d\u7b49\u5f0f\u7b49\u4ef7\u4e8e\u6761\u4ef6：
\n2NzR(\u03c9^u)>2N(1\u2212exp\u2061(\u2212\u03bbTu\/2N))2Nz_R(\\\\hat{\\\\omega}_u) > 2N(1 – \\\\exp(-\\\\lambda_{T_u}\/2N))<\/span><\/span>2<\/span>N<\/span>z<\/span><\/span>R<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>><\/span><\/span><\/span><\/span>2<\/span>N<\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>exp<\/span>(<\/span>\u2212<\/span>\u03bb<\/span><\/span>T<\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/2<\/span>N<\/span>))<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5bf9\u4e8e\u5927NN<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u53ef\u4ee5\u8fdb\u4e00\u6b65\u8fd1\u4f3c\u4e3a：
\n2NzR(\u03c9^u)>\u03bbTu2Nz_R(\\\\hat{\\\\omega}_u) > \\\\lambda_{T_u}<\/span><\/span>2<\/span>N<\/span>z<\/span><\/span>R<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>><\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>T<\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4eff\u771f\u7ed3\u679c<\/h3>\n

\u6e10\u8fd1\u6027\u9a8c\u8bc1<\/h4>\n

\"\"<\/p>\n

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\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

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\u9c81\u68d2\u68c0\u6d4b<\/h4>\n

\"\"<\/p>\n

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\"\"<\/p>\n

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\u9c81\u68d2\u4f30\u8ba1<\/h4>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

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\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

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\u7ed3\u8bba<\/h3>\n

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\u603b\u4e4b，\u4f7f\u7528\u975e\u7ed3\u6784\u5316\u9635\u5217\u6a21\u578b\u4e0eEXIP\u76f8\u7ed3\u5408\u4e3a\u5c06ML\u65b9\u6cd5\u5e94\u7528\u4e8e\u96f7\u8fbe\u4fe1\u53f7\u5904\u7406\u95ee\u9898\u63d0\u4f9b\u4e86\u4e00\u4e2a\u975e\u5e38\u7b80\u5355\u7684\u7a0b\u5e8f\u3002\u6240\u63d0\u51fa\u7684\u4f30\u8ba1\u548c\u68c0\u6d4b\u7b97\u6cd5\u7684\u8ba1\u7b97\u7b80\u5355\u6027\u548c\u9c81\u68d2\u6027\u4f7f\u5b83\u4eec\u503c\u5f97\u5728\u5b9e\u9645\u96f7\u8fbe\u5e94\u7528\u4e2d\u8ba4\u771f\u8003\u8651\u3002<\/p>\n

\u9644\u5f55<\/h3>\n

\u9644\u5f55A：\u7ed3\u6784\u5316ML\u4f30\u8ba1\u5668\u7684\u6027\u8d28<\/h4>\n

A. \u4e00\u81f4\u6027<\/h5>\n

\u9996\u5148\u5efa\u7acb\u7ed3\u6784\u5316ML\u7b97\u6cd5\u7684\u4e00\u81f4\u6027\u3002\u9996\u5148，\u6211\u4eec\u6709：<\/p>\n

lim\u2061N\u2192\u221ey(\u03c9)=lim\u2061N\u2192\u221e1N\u2211t=1N[b0a(\u03b80)ej(\u03c90\u2212\u03c9)t+e(t)e\u2212j\u03c9t]\\\\lim_{N \\\\to \\\\infty} \\\\mathbf{y}(\\\\omega) = \\\\lim_{N \\\\to \\\\infty} \\\\frac{1}{N}\\\\sum_{t=1}^{N}\\\\left[b_0\\\\mathbf{a}(\\\\theta_0)e^{j(\\\\omega_0-\\\\omega)t} + \\\\mathbf{e}(t)e^{-j\\\\omega t}\\\\right]<\/span><\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span><\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>e<\/span><\/span>j<\/span>(<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>\u03c9<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span>e<\/span><\/span>\u2212<\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

=lim\u2061N\u2192\u221e1N\u2211t=1Nb0a(\u03b80)ej(\u03c90\u2212\u03c9)t+lim\u2061N\u2192\u221e1N\u2211t=1Ne(t)e\u2212j\u03c9t= \\\\lim_{N \\\\to \\\\infty} \\\\frac{1}{N}\\\\sum_{t=1}^{N}b_0\\\\mathbf{a}(\\\\theta_0)e^{j(\\\\omega_0-\\\\omega)t} + \\\\lim_{N \\\\to \\\\infty} \\\\frac{1}{N}\\\\sum_{t=1}^{N}\\\\mathbf{e}(t)e^{-j\\\\omega t}<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>e<\/span><\/span>j<\/span>(<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>\u03c9<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span>e<\/span><\/span>\u2212<\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5982\u679c\u6211\u4eec\u4ee4\u03b4(\u03c9)\\\\delta(\\\\omega)<\/span><\/span>\u03b4<\/span>(<\/span>\u03c9<\/span>)<\/span><\/span><\/span><\/span><\/span>\u8868\u793aKronecker delta\u51fd\u6570，\u7b2c\u4e00\u9879\u7684\u6781\u9650\u7b80\u5355\u5730\u4e3ab0a(\u03b80)\u03b4(\u03c90\u2212\u03c9)b_0\\\\mathbf{a}(\\\\theta_0)\\\\delta(\\\\omega_0 – \\\\omega)<\/span><\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u03b4<\/span>(<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c9<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u56e0\u4e3a：<\/p>\n

lim\u2061N\u2192\u221e1N\u2211t=1Nej(\u03c90\u2212\u03c9)t={1\u5982\u679c\u00a0\u03c9=\u03c900\u5426\u5219\\\\lim_{N \\\\to \\\\infty} \\\\frac{1}{N}\\\\sum_{t=1}^{N}e^{j(\\\\omega_0-\\\\omega)t} = \\\\begin{cases} 1 & \\\\text{\u5982\u679c } \\\\omega = \\\\omega_0 \\\\\\\\ 0 & \\\\text{\u5426\u5219} \\\\end{cases}<\/span><\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>j<\/span>(<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>\u03c9<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5982\u679c<\/span>\u00a0<\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5426\u5219<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6839\u636e\u5927\u6570\u5b9a\u5f8b，\u7b2c\u4e8c\u9879\u4ee5\u6982\u73871\u4e00\u81f4\u6536\u655b\u5230\u96f6，\u56e0\u4e3ae(t)\\\\mathbf{e}(t)<\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u662f\u96f6\u5747\u503c\u7684\u3002\u56e0\u6b64：<\/p>\n

lim\u2061N\u2192\u221ey(\u03c9)={b0a(\u03b80)\u5982\u679c\u00a0\u03c9=\u03c900\u5426\u5219w.p.1\\\\lim_{N \\\\to \\\\infty} \\\\mathbf{y}(\\\\omega) = \\\\begin{cases} b_0\\\\mathbf{a}(\\\\theta_0) & \\\\text{\u5982\u679c } \\\\omega = \\\\omega_0 \\\\\\\\ 0 & \\\\text{\u5426\u5219} \\\\end{cases} \\\\quad \\\\text{w.p.1}<\/span><\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span><\/span><\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5982\u679c<\/span>\u00a0<\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5426\u5219<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>w.p.1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ee4R^=lim\u2061N\u2192\u221eR^\\\\hat{\\\\mathbf{R}} = \\\\lim_{N \\\\to \\\\infty} \\\\hat{\\\\mathbf{R}}<\/span><\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>lim<\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5219\u5728(16)(16)<\/span><\/span>(<\/span>16<\/span>)<\/span><\/span><\/span><\/span><\/span>\u4e2d\u7684ML\u51c6\u5219\u6e10\u8fd1\u5730\u7531\u4e0b\u5f0f\u7ed9\u51fa：<\/p>\n

VN(\u03b8,\u03c9)=\u2223b0\u22232\u2223a\u2217(\u03b8)R\u22121a(\u03b80)\u22232[1\u2212\u2223b0\u22232a\u2217(\u03b80)R\u22121a(\u03b80)]a\u2217(\u03b8)R\u22121a(\u03b8)V_N(\\\\theta, \\\\omega) = \\\\frac{|b_0|^2|\\\\mathbf{a}^*(\\\\theta)\\\\mathbf{R}^{-1}\\\\mathbf{a}(\\\\theta_0)|^2}{[1-|b_0|^2\\\\mathbf{a}^*(\\\\theta_0)\\\\mathbf{R}^{-1}\\\\mathbf{a}(\\\\theta_0)]\\\\mathbf{a}^*(\\\\theta)\\\\mathbf{R}^{-1}\\\\mathbf{a}(\\\\theta)}<\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>[<\/span>1<\/span><\/span>\u2212<\/span><\/span>\u2223<\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)]<\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>)<\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>)<\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6982\u7387\u4e3a1\u3002\u5728\u03c9\u2260\u03c90\\\\omega \\\\neq \\\\omega_0<\/span><\/span>\u03c9<\/span><\/span><\/span>\ue020<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u65f6，\u6781\u9650ML\u51c6\u5219\u4e3a\u96f6\u3002\u67ef\u897f-\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f\u53ef\u4ee5\u7528\u6765\u83b7\u5f97\u6781\u9650ML\u51c6\u5219\u7684\u4ee5\u4e0b\u4e0a\u754c：<\/p>\n

lim\u2061N\u2192\u221eVN(\u03b8,\u03c90)=\u2223b0\u22232\u2223a\u2217(\u03b8)R\u22121a(\u03b80)\u22232ca\u2217(\u03b8)R\u22121a(\u03b8)\u2264\u2223b0\u22232a\u2217(\u03b80)R\u22121a(\u03b80)c\\\\lim_{N \\\\to \\\\infty} V_N(\\\\theta, \\\\omega_0) = \\\\frac{|b_0|^2|\\\\mathbf{a}^*(\\\\theta)\\\\mathbf{R}^{-1}\\\\mathbf{a}(\\\\theta_0)|^2}{c\\\\mathbf{a}^*(\\\\theta)\\\\mathbf{R}^{-1}\\\\mathbf{a}(\\\\theta)} \\\\leq \\\\frac{|b_0|^2\\\\mathbf{a}^*(\\\\theta_0)\\\\mathbf{R}^{-1}\\\\mathbf{a}(\\\\theta_0)}{c}<\/span><\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>c<\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>)<\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>)<\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span><\/span><\/span>c<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2dc=1\u2212\u2223b0\u22232a\u2217(\u03b80)R\u22121a(\u03b80)c = 1 – |b_0|^2\\\\mathbf{a}^*(\\\\theta_0)\\\\mathbf{R}^{-1}\\\\mathbf{a}(\\\\theta_0)<\/span><\/span>c<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u2223<\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u5f53\u4e14\u4ec5\u5f53R\u22121\/2a(\u03b8)=\u03baR\u22121\/2a(\u03b80)\\\\mathbf{R}^{-1\/2}\\\\mathbf{a}(\\\\theta) = \\\\kappa\\\\mathbf{R}^{-1\/2}\\\\mathbf{a}(\\\\theta_0)<\/span><\/span>R<\/span><\/span>\u2212<\/span>1\/2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03ba<\/span>R<\/span><\/span>\u2212<\/span>1\/2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u65f6\u7b49\u5f0f\u6210\u7acb，\u8fd9\u8fdb\u800c\u53ea\u6709\u5f53a(\u03b8)=\u03baa(\u03b80)\\\\mathbf{a}(\\\\theta) = \\\\kappa\\\\mathbf{a}(\\\\theta_0)<\/span><\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03ba<\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u65f6\u624d\u53ef\u80fd，\u56e0\u4e3a\u6211\u4eec\u9690\u542b\u5730\u5047\u8bbe\u4e86\u4e00\u4e2a"\u65e0\u6b67\u4e49"\u7684\u9635\u5217\u3002<\/p>\n

\u867d\u7136\u6781\u9650ML\u51c6\u5219\u5728\u771f\u5b9e\u53c2\u6570\u503c\u5904\u5177\u6709\u5168\u5c40\u6700\u5927\u503c，\u4f46\u7531\u4e8e\u51c6\u5219\u5728\u03c9\\\\omega<\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>\u4e2d\u7684\u4e0d\u8fde\u7eed\u6027，\u5efa\u7acb\u4e00\u81f4\u6027\u662f\u590d\u6742\u7684\u3002\u4f7f\u7528\u7c7b\u4f3c\u4e8e\u6587\u732e\u4e2d\u5f15\u74061\u548c\u5b9a\u74061\u7684\u8bba\u8bc1\u53ef\u4ee5\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u3002\u5df2\u7ecf\u8bc1\u660e，\u53ea\u8981VN(\u03b8,\u03c9)V_N(\\\\theta, \\\\omega)<\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span><\/span><\/span><\/span><\/span>\u53ef\u4ee5\u5199\u6210\u4e24\u4e2a\u5206\u91cfVN(1)(\u03b8,\u03c9)V_N^{(1)}(\\\\theta, \\\\omega)<\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span><\/span><\/span><\/span><\/span>\u548cVN(2)(\u03b8,\u03c9)V_N^{(2)}(\\\\theta, \\\\omega)<\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>(<\/span>2<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u548c，\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6，\u5c31\u53ef\u4ee5\u5b9e\u73b0\u4e00\u81f4\u6027：<\/p>\n

  • VN(1)(\u03b8,\u03c9)\u21920V_N^{(1)}(\\\\theta, \\\\omega) \\\\to 0<\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u5728\u03c9\\\\omega<\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>\u4e2d\u4e00\u81f4；<\/li>\n
  • \u5bf9\u4e8e\u4efb\u4f55\u03f5>0\\\\epsilon > 0<\/span><\/span>\u03f5<\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u5b58\u5728\u03b4>0\\\\delta > 0<\/span><\/span>\u03b4<\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u548cNMN_M<\/span><\/span>N<\/span><\/span>M<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4f7f\u5f97\u5bf9\u6240\u6709\u2223(\u03b8,\u03c9)\u2212(\u03b80,\u03c90)\u2223>\u03f5|(\\\\theta, \\\\omega) – (\\\\theta_0, \\\\omega_0)| > \\\\epsilon<\/span><\/span>\u2223<\/span>(<\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u2223<\/span><\/span>><\/span><\/span><\/span><\/span>\u03f5<\/span><\/span><\/span><\/span><\/span>\u548cN>NMN > N_M<\/span><\/span>N<\/span><\/span>><\/span><\/span><\/span><\/span>N<\/span><\/span>M<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u6709VN(2)(\u03b8,\u03c9)<VN(2)(\u03b80,\u03c90)\u2212\u03b4V_N^{(2)}(\\\\theta, \\\\omega) < V_N^{(2)}(\\\\theta_0, \\\\omega_0) – \\\\delta<\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>(<\/span>2<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>)<\/span><\/span><<\/span><\/span><\/span><\/span>V<\/span><\/span>N<\/span><\/span><\/span><\/span>(<\/span>2<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span><\/span><\/span><\/span>\u3002<\/li>\n
    B. Cram\u00e9r-Rao\u754c<\/h5>\n

    \u7531\u4e8e\u7ed3\u6784\u5316ML\u4f30\u8ba1\u662f\u4e00\u81f4\u7684，\u5b83\u4eec\u5c06\u662f\u7edf\u8ba1\u6709\u6548\u7684，\u5176\u6e10\u8fd1\u65b9\u5dee\u5c06\u7531CRB\u7ed9\u51fa\u3002\u5728\u7ed3\u6784\u5316\u6a21\u578b\u4e0b，\u89c2\u6d4bx(1),\u22ef\u2009,x(N)\\\\mathbf{x}(1), \\\\cdots, \\\\mathbf{x}(N)<\/span><\/span>x<\/span>(<\/span>1<\/span>)<\/span>,<\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span>x<\/span>(<\/span>N<\/span>)<\/span><\/span><\/span><\/span><\/span>\u662f\u72ec\u7acb\u7684\u5706\u5f62\u9ad8\u65af\u968f\u673a\u53d8\u91cf，\u5747\u503c\u4e3a：<\/p>\n

    E{x(t)}=b0a(\u03b80)ej\u03c90t\\\\mathcal{E}\\\\{\\\\mathbf{x}(t)\\\\} = b_0\\\\mathbf{a}(\\\\theta_0)e^{j\\\\omega_0 t}<\/span><\/span>E<\/span>{<\/span>x<\/span>(<\/span>t<\/span>)}<\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>e<\/span><\/span>j<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u548c\u534f\u65b9\u5deeQ\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2d\u4e3a\u7b80\u6d01\u8d77\u89c1\u6211\u4eec\u7701\u7565\u4e86\u4e0b\u680700<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b，NN<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u4e2a\u89c2\u6d4b\u7684FIM\u7684\u5143\u7d20(i,j)(i,j)<\/span><\/span>(<\/span>i<\/span>,<\/span><\/span>j<\/span>)<\/span><\/span><\/span><\/span><\/span>\u53ef\u4ee5\u8868\u793a\u4e3a：<\/p>\n

    [JN]ij=N\u22c5tr{Q\u22121\u2202Q\u2202\u03b7iQ\u22121\u2202Q\u2202\u03b7j}+2Re{\u2211t=1N\u2202\u03bc\u2217(t)\u2202\u03b7iQ\u22121\u2202\u03bc(t)\u2202\u03b7j}[\\\\mathbf{J}_N]_{ij} = N \\\\cdot \\\\text{tr}\\\\left\\\\{\\\\mathbf{Q}^{-1}\\\\frac{\\\\partial \\\\mathbf{Q}}{\\\\partial \\\\eta_i}\\\\mathbf{Q}^{-1}\\\\frac{\\\\partial \\\\mathbf{Q}}{\\\\partial \\\\eta_j}\\\\right\\\\} + 2\\\\text{Re}\\\\left\\\\{\\\\sum_{t=1}^{N}\\\\frac{\\\\partial \\\\mu^*(t)}{\\\\partial \\\\eta_i}\\\\mathbf{Q}^{-1}\\\\frac{\\\\partial \\\\mu(t)}{\\\\partial \\\\eta_j}\\\\right\\\\}<\/span><\/span>[<\/span>J<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>N<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>tr<\/span><\/span><\/span>{<\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03b7<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>Q<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03b7<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>Q<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span>Re<\/span><\/span><\/span>{<\/span><\/span><\/span>t<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03b7<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03bc<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03b7<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03bc<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u7531\u4e8eQ\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>\u548c{b,\u03b8,\u03c9}\\\\{b, \\\\theta, \\\\omega\\\\}<\/span><\/span>{<\/span>b<\/span>,<\/span><\/span>\u03b8<\/span>,<\/span><\/span>\u03c9<\/span>}<\/span><\/span><\/span><\/span><\/span>\u4f9d\u8d56\u4e8e\u03b7\\\\boldsymbol{\\\\eta}<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u4e0d\u540c\u5143\u7d20，\u5f88\u660e\u663eJN\\\\mathbf{J}_N<\/span><\/span>J<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c06\u76f8\u5bf9\u4e8e\u4fe1\u53f7\u548c\u566a\u58f0\u53c2\u6570\u662f\u5757\u5bf9\u89d2\u7684\u3002\u7279\u522b\u5730，(103)(103)<\/span><\/span>(<\/span>103<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u7b2c\u4e00\u9879\u53ea\u5bf9\u566a\u58f0\u5757\u7ed9\u51fa\u975e\u96f6\u7ed3\u679c，\u800c\u7b2c\u4e8c\u9879\u53ea\u5bf9\u4fe1\u53f7\u5757\u975e\u96f6\u3002\u7531\u4e8e\u6211\u4eec\u53ea\u5173\u5fc3\u4fe1\u53f7\u53c2\u6570\u7684CRB，\u6211\u4eec\u53ea\u9700\u8981\u8003\u8651\u7b2c\u4e8c\u9879：<\/p>\n

    JN(s)=2Re{\u2211t,s=1N\u2202\u03bc\u2217(t)\u2202\u03b7sQ\u22121\u2202\u03bc(s)\u2202\u03b7sT}ej(\u03c90\u2212\u03c9)(s\u2212t)\\\\mathbf{J}_N^{(s)} = 2\\\\text{Re}\\\\left\\\\{\\\\sum_{t,s=1}^{N}\\\\frac{\\\\partial \\\\mu^*(t)}{\\\\partial \\\\boldsymbol{\\\\eta}_s}\\\\mathbf{Q}^{-1}\\\\frac{\\\\partial \\\\mu(s)}{\\\\partial \\\\boldsymbol{\\\\eta}_s^T}\\\\right\\\\}e^{j(\\\\omega_0-\\\\omega)(s-t)}<\/span><\/span>J<\/span><\/span>N<\/span><\/span><\/span><\/span>(<\/span>s<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>Re<\/span><\/span><\/span>{<\/span><\/span><\/span>t<\/span>,<\/span>s<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03b7<\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03bc<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03b7<\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03bc<\/span>(<\/span>s<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span>e<\/span><\/span>j<\/span>(<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>\u03c9<\/span>)<\/span>(<\/span>s<\/span>\u2212<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u6240\u9700\u7684\u504f\u5bfc\u6570\u5982\u4e0b：<\/p>\n

    \u2202\u03bc(t)\u2202Re{b}=a(\u03b8)ej\u03c9t\\\\frac{\\\\partial \\\\mu(t)}{\\\\partial \\\\text{Re}\\\\{b\\\\}} = \\\\mathbf{a}(\\\\theta)e^{j\\\\omega t}<\/span><\/span><\/span><\/span>\u2202<\/span>Re<\/span><\/span>{<\/span>b<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03bc<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span>e<\/span><\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u2202\u03bc(t)\u2202Im{b}=ja(\u03b8)ej\u03c9t\\\\frac{\\\\partial \\\\mu(t)}{\\\\partial \\\\text{Im}\\\\{b\\\\}} = j\\\\mathbf{a}(\\\\theta)e^{j\\\\omega t}<\/span><\/span><\/span><\/span>\u2202<\/span>Im<\/span><\/span>{<\/span>b<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03bc<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>j<\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span>e<\/span><\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u2202\u03bc(t)\u2202\u03b8=jba\u02d9(\u03b8)ej\u03c9t\\\\frac{\\\\partial \\\\mu(t)}{\\\\partial \\\\theta} = jb\\\\dot{\\\\mathbf{a}}(\\\\theta)e^{j\\\\omega t}<\/span><\/span><\/span><\/span>\u2202<\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03bc<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>jb<\/span><\/span>a<\/span><\/span><\/span>\u02d9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>)<\/span>e<\/span><\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u2202\u03bc(t)\u2202\u03c9=jtba(\u03b8)ej\u03c9t\\\\frac{\\\\partial \\\\mu(t)}{\\\\partial \\\\omega} = jtb\\\\mathbf{a}(\\\\theta)e^{j\\\\omega t}<\/span><\/span><\/span><\/span>\u2202<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03bc<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>j<\/span>t<\/span>b<\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span>e<\/span><\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u5176\u4e2da\u02d9(\u03b8)=\u2202a(\u03b8)\u2202\u03b8\\\\dot{\\\\mathbf{a}}(\\\\theta) = \\\\frac{\\\\partial \\\\mathbf{a}(\\\\theta)}{\\\\partial \\\\theta}<\/span><\/span><\/span>a<\/span><\/span><\/span>\u02d9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>a<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

    \u5c06\u8fd9\u4e9b\u4ee3\u5165(104)(104)<\/span><\/span>(<\/span>104<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5f97\u5230CRB\u8868\u8fbe\u5f0f\u3002\u4e3a\u4e86\u627e\u5230ML\u591a\u666e\u52d2\u548cDOA\u4f30\u8ba1\u7684\u65b9\u5dee，\u6211\u4eec\u4f7f\u7528\u5206\u5757\u5bf9\u79f0\u77e9\u9635\u9006\u7684\u4ee5\u4e0b\u516c\u5f0f：<\/p>\n

    [ABTBC]\u22121=[(A\u2212BTC\u22121B)\u22121\u2212A\u22121BTD\u2212DTBA\u22121D]\\\\begin{bmatrix} \\\\mathbf{A} & \\\\mathbf{B}^T \\\\\\\\ \\\\mathbf{B} & \\\\mathbf{C} \\\\end{bmatrix}^{-1} = \\\\begin{bmatrix} (\\\\mathbf{A} – \\\\mathbf{B}^T\\\\mathbf{C}^{-1}\\\\mathbf{B})^{-1} & -\\\\mathbf{A}^{-1}\\\\mathbf{B}^T\\\\mathbf{D} \\\\\\\\ -\\\\mathbf{D}^T\\\\mathbf{B}\\\\mathbf{A}^{-1} & \\\\mathbf{D} \\\\end{bmatrix}<\/span><\/span>[<\/span><\/span><\/span>A<\/span><\/span><\/span><\/span>B<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>(<\/span>A<\/span><\/span>\u2212<\/span><\/span>B<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span>)<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>D<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span>A<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>A<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>D<\/span><\/span><\/span><\/span>D<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u5176\u4e2dD=(C\u2212BA\u22121BT)\u22121\\\\mathbf{D} = (\\\\mathbf{C} – \\\\mathbf{B}\\\\mathbf{A}^{-1}\\\\mathbf{B}^T)^{-1}<\/span><\/span>D<\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>C<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>B<\/span>A<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662fJN\u22121\\\\mathbf{J}_N^{-1}<\/span><\/span>J<\/span><\/span>N<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u4e0b2\u00d722 \\\\times 2<\/span><\/span>2<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>\u5757\u3002\u4e3a\u65b9\u4fbf\u8d77\u89c1，\u5b9a\u4e49：<\/p>\n

    Qr=Re{Q}\\\\mathbf{Q}_r = \\\\text{Re}\\\\{\\\\mathbf{Q}\\\\}<\/span><\/span>Q<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Re<\/span><\/span>{<\/span>Q<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span>
    \nQi=Im{Q}\\\\mathbf{Q}_i = \\\\text{Im}\\\\{\\\\mathbf{Q}\\\\}<\/span><\/span>Q<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Im<\/span><\/span>{<\/span>Q<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span>
    \n\u03b1r=Re{b0a(\u03b80)}\\\\boldsymbol{\\\\alpha}_r = \\\\text{Re}\\\\{b_0\\\\mathbf{a}(\\\\theta_0)\\\\}<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Re<\/span><\/span>{<\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)}<\/span><\/span><\/span><\/span><\/span><\/span>
    \n\u03b1i=Im{b0a(\u03b80)}\\\\boldsymbol{\\\\alpha}_i = \\\\text{Im}\\\\{b_0\\\\mathbf{a}(\\\\theta_0)\\\\}<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Im<\/span><\/span>{<\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)}<\/span><\/span><\/span><\/span><\/span><\/span>
    \n\u03b4N=[\u03b3N\u2212\u03b2N2]\u03b10\u2217Q\u22121\u03b10\\\\delta_N = [\\\\gamma_N – \\\\beta_N^2]\\\\boldsymbol{\\\\alpha}_0^*\\\\mathbf{Q}^{-1}\\\\boldsymbol{\\\\alpha}_0<\/span><\/span>\u03b4<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span>\u03b3<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>N<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u03b2N\\\\beta_N<\/span><\/span>\u03b2<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3001\u03b3N\\\\gamma_N<\/span><\/span>\u03b3<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7531(21)(21)<\/span><\/span>(<\/span>21<\/span>)<\/span><\/span><\/span><\/span><\/span>\u548c(22)(22)<\/span><\/span>(<\/span>22<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5b9a\u4e49\u3002<\/p>\n

    \u4f30\u8ba1\u65b9\u5dee\u56e0\u6b64\u7531\u4e0b\u5f0f\u7ed9\u51fa：<\/p>\n

    E{(\u03b8^s\u2212\u03b80)2}=a\u2217Q\u22121a2N\u2223b\u22232[(a\u2217Q\u22121a)(a\u02d9\u2217Q\u22121a\u02d9)\u2212\u2223a\u2217Q\u22121a\u02d9\u22232]\\\\mathcal{E}\\\\{(\\\\hat{\\\\theta}_s – \\\\theta_0)^2\\\\} = \\\\frac{\\\\mathbf{a}^*\\\\mathbf{Q}^{-1}\\\\mathbf{a}}{2N|b|^2[(\\\\mathbf{a}^*\\\\mathbf{Q}^{-1}\\\\mathbf{a})(\\\\dot{\\\\mathbf{a}}^*\\\\mathbf{Q}^{-1}\\\\dot{\\\\mathbf{a}}) – |\\\\mathbf{a}^*\\\\mathbf{Q}^{-1}\\\\dot{\\\\mathbf{a}}|^2]}<\/span><\/span>E<\/span>{(<\/span><\/span>\u03b8<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>N<\/span>\u2223<\/span>b<\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[(<\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>)<\/span>(<\/span><\/span>a<\/span><\/span><\/span>\u02d9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span>\u02d9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span><\/span>\u2223<\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span>\u02d9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    E{(\u03c9^s\u2212\u03c90)2}=12N\u2223b\u22232(\u03b3N\u2212\u03b2N2)a\u2217Q\u22121a\u22486N3\u2223b\u22232a\u2217Q\u22121a\\\\mathcal{E}\\\\{(\\\\hat{\\\\omega}_s – \\\\omega_0)^2\\\\} = \\\\frac{1}{2N|b|^2(\\\\gamma_N – \\\\beta_N^2)\\\\mathbf{a}^*\\\\mathbf{Q}^{-1}\\\\mathbf{a}} \\\\approx \\\\frac{6}{N^3|b|^2\\\\mathbf{a}^*\\\\mathbf{Q}^{-1}\\\\mathbf{a}}<\/span><\/span>E<\/span>{(<\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>N<\/span>\u2223<\/span>b<\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b3<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span><\/span>N<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>b<\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u9644\u5f55B：\u975e\u7ed3\u6784\u5316ML\u4f30\u8ba1\u5668\u7684\u6027\u8d28<\/h4>\n

    A. \u4e00\u81f4\u6027<\/h5>\n

    \u975e\u7ed3\u6784\u5316ML\u7b97\u6cd5\u7684\u4e00\u81f4\u6027\u4ee5\u7c7b\u4f3c\u4e8e\u7ed3\u6784\u5316\u60c5\u51b5\u7684\u65b9\u5f0f\u5efa\u7acb\u3002\u6309\u7167(96)(96)<\/span><\/span>(<\/span>96<\/span>)<\/span><\/span><\/span><\/span><\/span>–(98)(98)<\/span><\/span>(<\/span>98<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u6211\u4eec\u6709：<\/p>\n

    lim\u2061N\u2192\u221ey(\u03c9)={\u03b10\u5982\u679c\u00a0\u03c9=\u03c900\u5426\u5219\\\\lim_{N \\\\to \\\\infty} \\\\mathbf{y}(\\\\omega) = \\\\begin{cases} \\\\boldsymbol{\\\\alpha}_0 & \\\\text{\u5982\u679c } \\\\omega = \\\\omega_0 \\\\\\\\ 0 & \\\\text{\u5426\u5219} \\\\end{cases}<\/span><\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5982\u679c<\/span>\u00a0<\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5426\u5219<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u56e0\u6b64：<\/p>\n

    lim\u2061N\u2192\u221ey\u2217(\u03c9)R^\u22121y(\u03c9)={\u03b10\u2217R\u22121\u03b10\u5982\u679c\u00a0\u03c9=\u03c900\u5426\u5219\\\\lim_{N \\\\to \\\\infty} \\\\mathbf{y}^*(\\\\omega)\\\\hat{\\\\mathbf{R}}^{-1}\\\\mathbf{y}(\\\\omega) = \\\\begin{cases} \\\\boldsymbol{\\\\alpha}_0^*\\\\mathbf{R}^{-1}\\\\boldsymbol{\\\\alpha}_0 & \\\\text{\u5982\u679c } \\\\omega = \\\\omega_0 \\\\\\\\ 0 & \\\\text{\u5426\u5219} \\\\end{cases}<\/span><\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span>)<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5982\u679c<\/span>\u00a0<\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5426\u5219<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u5c06\u51c6\u5219\u5206\u6210\u4e24\u9879：<\/p>\n

    y\u2217(\u03c9)R^\u22121y(\u03c9)=1N2\u2211t,s=1N\u03b10\u2217R\u22121\u03b10e\u2212j(\u03c90\u2212\u03c9)(t\u2212s)+1N2\u2211t,s=1N[\u03b10\u2217R\u22121e(t)e\u2212j\u03c9s+e\u2217(s)R\u22121\u03b10ej\u03c9t+e\u2217(s)R\u22121e(t)]e\u2212j\u03c9(t\u2212s)\\\\mathbf{y}^*(\\\\omega)\\\\hat{\\\\mathbf{R}}^{-1}\\\\mathbf{y}(\\\\omega) = \\\\frac{1}{N^2}\\\\sum_{t,s=1}^{N}\\\\boldsymbol{\\\\alpha}_0^*\\\\mathbf{R}^{-1}\\\\boldsymbol{\\\\alpha}_0 e^{-j(\\\\omega_0-\\\\omega)(t-s)} + \\\\frac{1}{N^2}\\\\sum_{t,s=1}^{N}[\\\\boldsymbol{\\\\alpha}_0^*\\\\mathbf{R}^{-1}\\\\mathbf{e}(t)e^{-j\\\\omega s} + \\\\mathbf{e}^*(s)\\\\mathbf{R}^{-1}\\\\boldsymbol{\\\\alpha}_0 e^{j\\\\omega t} + \\\\mathbf{e}^*(s)\\\\mathbf{R}^{-1}\\\\mathbf{e}(t)]e^{-j\\\\omega(t-s)}<\/span><\/span>y<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span>)<\/span><\/span>R<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>(<\/span>\u03c9<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>,<\/span>s<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>j<\/span>(<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>\u03c9<\/span>)<\/span>(<\/span>t<\/span>\u2212<\/span>s<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>,<\/span>s<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span>(<\/span>t<\/span>)<\/span>e<\/span><\/span>\u2212<\/span>j\u03c9<\/span>s<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>s<\/span>)<\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>s<\/span>)<\/span>R<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span>(<\/span>t<\/span>)]<\/span>e<\/span><\/span>\u2212<\/span>j\u03c9<\/span>(<\/span>t<\/span>\u2212<\/span>s<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u6211\u4eec\u518d\u6b21\u6709\u4e24\u4e2a\u6ee1\u8db3\u4e00\u81f4\u6027\u6761\u4ef6\u7684\u51fd\u6570\u3002\u56e0\u6b64，lim\u2061N\u2192\u221e\u03c9^u=\u03c90\\\\lim_{N \\\\to \\\\infty} \\\\hat{\\\\omega}_u = \\\\omega_0<\/span><\/span>lim<\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u6b64\u5916：<\/p>\n

    lim\u2061N\u2192\u221e\u03b1^u=lim\u2061N\u2192\u221ey(\u03c9^u)=\u03b10\\\\lim_{N \\\\to \\\\infty} \\\\hat{\\\\boldsymbol{\\\\alpha}}_u = \\\\lim_{N \\\\to \\\\infty} \\\\mathbf{y}(\\\\hat{\\\\omega}_u) = \\\\boldsymbol{\\\\alpha}_0<\/span><\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>N<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>(<\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    Q^u\\\\hat{\\\\mathbf{Q}}_u<\/span><\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u4e00\u81f4\u6027\u901a\u8fc7\u5c06\u03b1^u,\u03c9^u\\\\hat{\\\\boldsymbol{\\\\alpha}}_u, \\\\hat{\\\\omega}_u<\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165(25)(25)<\/span><\/span>(<\/span>25<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5f97\u5230\u3002<\/p>\n

    B. Cram\u00e9r-Rao\u754c<\/h5>\n

    \u975e\u7ed3\u6784\u5316\u6a21\u578b\u4e0b\u7684\u6570\u636ex(1),\u22ef\u2009,x(N)\\\\mathbf{x}(1), \\\\cdots, \\\\mathbf{x}(N)<\/span><\/span>x<\/span>(<\/span>1<\/span>)<\/span>,<\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span>x<\/span>(<\/span>N<\/span>)<\/span><\/span><\/span><\/span><\/span>\u662f\u72ec\u7acb\u7684\u5706\u5f62\u9ad8\u65af\u968f\u673a\u53d8\u91cf，\u5747\u503c\u4e3a\u03bcu(t)=\u03b10ej\u03c90t\\\\mu_u(t) = \\\\boldsymbol{\\\\alpha}_0e^{j\\\\omega_0 t}<\/span><\/span>\u03bc<\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>j<\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548c\u534f\u65b9\u5deeQ\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>\u3002\u4e0e\u7ed3\u6784\u5316\u60c5\u51b5\u4e00\u6837，\u975e\u7ed3\u6784\u5316CRB\u76f8\u5bf9\u4e8e\u4fe1\u53f7{\u03b1r,\u03b1i,\u03c9}\\\\{\\\\boldsymbol{\\\\alpha}_r, \\\\boldsymbol{\\\\alpha}_i, \\\\omega\\\\}<\/span><\/span>{<\/span>\u03b1<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03c9<\/span>}<\/span><\/span><\/span><\/span><\/span>\u548c\u566a\u58f0(Q)(\\\\mathbf{Q})<\/span><\/span>(<\/span>Q<\/span>)<\/span><\/span><\/span><\/span><\/span>\u53c2\u6570\u662f\u5757\u5bf9\u89d2\u7684\u3002FIM\u7684\u8ba1\u7b97\u4e0e(104)(104)<\/span><\/span>(<\/span>104<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5b8c\u5168\u76f8\u540c，\u7528\u03b7~\\\\tilde{\\\\boldsymbol{\\\\eta}}<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u66ff\u6362\u03b7\\\\boldsymbol{\\\\eta}<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u6240\u9700\u7684\u504f\u5bfc\u6570\u7531\u4e0b\u5f0f\u7ed9\u51fa：<\/p>\n

    \u2202\u03bc(t)\u2202\u03b1r=Iej\u03c9t\\\\frac{\\\\partial \\\\mu(t)}{\\\\partial \\\\boldsymbol{\\\\alpha}_r} = \\\\mathbf{I}e^{j\\\\omega t}<\/span><\/span><\/span><\/span>\u2202<\/span>\u03b1<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03bc<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span>e<\/span><\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u2202\u03bc(t)\u2202\u03b1i=jIej\u03c9t\\\\frac{\\\\partial \\\\mu(t)}{\\\\partial \\\\boldsymbol{\\\\alpha}_i} = j\\\\mathbf{I}e^{j\\\\omega t}<\/span><\/span><\/span><\/span>\u2202<\/span>\u03b1<\/span><\/span><\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03bc<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>j<\/span>I<\/span>e<\/span><\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u2202\u03bc(t)\u2202\u03c9=jt\u03b1ej\u03c9t\\\\frac{\\\\partial \\\\mu(t)}{\\\\partial \\\\omega} = jt\\\\boldsymbol{\\\\alpha}e^{j\\\\omega t}<\/span><\/span><\/span><\/span>\u2202<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03bc<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>j<\/span>t<\/span>\u03b1<\/span><\/span><\/span>e<\/span><\/span>j\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u5c06(117)(117)<\/span><\/span>(<\/span>117<\/span>)<\/span><\/span><\/span><\/span><\/span>–(119)(119)<\/span><\/span>(<\/span>119<\/span>)<\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165(104)(104)<\/span><\/span>(<\/span>104<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5f97\u5230FIM\u8868\u8fbe\u5f0f\u3002\u53ef\u4ee5\u901a\u8fc7\u6c42\u9006(120)(120)<\/span><\/span>(<\/span>120<\/span>)<\/span><\/span><\/span><\/span><\/span>\u83b7\u5f97CRB\u7684\u663e\u5f0f\u8868\u8fbe\u5f0f\u3002\u4e3a\u6b64，\u6211\u4eec\u4f7f\u7528(112)(112)<\/span><\/span>(<\/span>112<\/span>)<\/span><\/span><\/span><\/span><\/span>\u548c\u77e9\u9635\u6c42\u9006\u5f15\u7406\u3002<\/p>\n

    \u9644\u5f55C：\u5b9a\u74064\u7684\u8bc1\u660e<\/h4>\n

    \u901a\u8fc7\u5c06\u4e00\u822cEXIP\u7ed3\u679c\u5e94\u7528\u4e8e\u7ed3\u6784\u5316\u548c\u975e\u7ed3\u6784\u5316ML\u96f7\u8fbe\u76ee\u6807\u53c2\u6570\u4f30\u8ba1\u5668\u7684\u51c6\u5219\u6765\u8bc1\u660e\u8be5\u5b9a\u7406\u3002\u7279\u522b\u5730，\u5c06\u5b9a\u4e49\u4e00\u4e2a\u6ee1\u8db3(7)(7)<\/span><\/span>(<\/span>7<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u51fd\u6570ff<\/span><\/span>f<\/span><\/span><\/span><\/span><\/span>，\u8be5\u51fd\u6570\u5173\u8054SML\u548cUML\u53c2\u6570，\u5e76\u5c06\u663e\u793a(9)(9)<\/span><\/span>(<\/span>9<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684WLS\u6700\u5c0f\u5316\u7b80\u5316\u4e3a(47)(47)<\/span><\/span>(<\/span>47<\/span>)<\/span><\/span><\/span><\/span><\/span>–(50)(50)<\/span><\/span>(<\/span>50<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3001(53)(53)<\/span><\/span>(<\/span>53<\/span>)<\/span><\/span><\/span><\/span><\/span>\u548c(54)(54)<\/span><\/span>(<\/span>54<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u56e0\u6b64，\u5b9a\u74061\u8868\u660e\u5206\u522b\u7531(47)(47)<\/span><\/span>(<\/span>47<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3001(48)(48)<\/span><\/span>(<\/span>48<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3001(53)(53)<\/span><\/span>(<\/span>53<\/span>)<\/span><\/span><\/span><\/span><\/span>\u548c(54)(54)<\/span><\/span>(<\/span>54<\/span>)<\/span><\/span><\/span><\/span><\/span>\u83b7\u5f97\u7684bb<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>\u548c\u03b8\\\\theta<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>\u7684\u4f30\u8ba1\u6e10\u8fd1\u5730\u4e0e\u7ed3\u6784\u5316ML\u7b97\u6cd5\u83b7\u5f97\u7684\u4f30\u8ba1\u4e00\u6837\u51c6\u786e，\u56e0\u6b64\u5728\u7edf\u8ba1\u4e0a\u662f\u6709\u6548\u7684\u3002<\/p>\n

    \u5173\u8054D\u03b7D_\\\\eta<\/span><\/span>D<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548cD\u03b7~D_{\\\\tilde{\\\\eta}}<\/span><\/span>D<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5143\u7d20\u7684\u51fd\u6570ff<\/span><\/span>f<\/span><\/span><\/span><\/span><\/span>\u7531\u4e0b\u5f0f\u9690\u5f0f\u5b9a\u4e49：<\/p>\n

    Qs=Q\\\\mathbf{Q}_s = \\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span>
    \n\u03c9s=\u03c9\\\\omega_s = \\\\omega<\/span><\/span>\u03c9<\/span><\/span>s<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span>
    \nRe{b0a(\u03b80)}=Re{\u03b1}=A(\u03b8)(brbi)\\\\text{Re}\\\\{b_0\\\\mathbf{a}(\\\\theta_0)\\\\} = \\\\text{Re}\\\\{\\\\boldsymbol{\\\\alpha}\\\\} = \\\\mathbf{A}(\\\\theta)\\\\begin{pmatrix} b_r \\\\\\\\ b_i \\\\end{pmatrix}<\/span><\/span>Re<\/span><\/span>{<\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)}<\/span><\/span>=<\/span><\/span><\/span><\/span>Re<\/span><\/span>{<\/span>\u03b1<\/span><\/span><\/span>}<\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>(<\/span><\/span><\/span>b<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
    \nIm{b0a(\u03b80)}=Im{\u03b1}\\\\text{Im}\\\\{b_0\\\\mathbf{a}(\\\\theta_0)\\\\} = \\\\text{Im}\\\\{\\\\boldsymbol{\\\\alpha}\\\\}<\/span><\/span>Im<\/span><\/span>{<\/span>b<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>(<\/span>\u03b8<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)}<\/span><\/span>=<\/span><\/span><\/span><\/span>Im<\/span><\/span>{<\/span>\u03b1<\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u5176\u4e2d：<\/p>\n

    A(\u03b8)=[Re{a(\u03b8)}\u2212Im{a(\u03b8)}Im{a(\u03b8)}Re{a(\u03b8)}]\\\\mathbf{A}(\\\\theta) = \\\\begin{bmatrix} \\\\text{Re}\\\\{\\\\mathbf{a}(\\\\theta)\\\\} & -\\\\text{Im}\\\\{\\\\mathbf{a}(\\\\theta)\\\\} \\\\\\\\ \\\\text{Im}\\\\{\\\\mathbf{a}(\\\\theta)\\\\} & \\\\text{Re}\\\\{\\\\mathbf{a}(\\\\theta)\\\\} \\\\end{bmatrix}<\/span><\/span>A<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>Re<\/span><\/span>{<\/span>a<\/span>(<\/span>\u03b8<\/span>)}<\/span><\/span><\/span><\/span>Im<\/span><\/span>{<\/span>a<\/span>(<\/span>\u03b8<\/span>)}<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>Im<\/span><\/span>{<\/span>a<\/span>(<\/span>\u03b8<\/span>)}<\/span><\/span><\/span><\/span>Re<\/span><\/span>{<\/span>a<\/span>(<\/span>\u03b8<\/span>)}<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u8fd9\u4e9b\u65b9\u7a0b\u8fde\u540c\u5728\u9644\u5f55A\u548cB\u4e2d\u5efa\u7acb\u7684\u4e24\u4e2a\u7b97\u6cd5\u7684\u4e00\u81f4\u6027，\u6ee1\u8db3\u5b9a\u74061\u7684\u6761\u4ef6(7)(7)<\/span><\/span>(<\/span>7<\/span>)<\/span><\/span><\/span><\/span><\/span>\u548c(8)(8)<\/span><\/span>(<\/span>8<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

    \u7531\u4e8e\u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\u4f7f\u7528\u4e86ML\u51c6\u5219，(10)(10)<\/span><\/span>(<\/span>10<\/span>)<\/span><\/span><\/span><\/span><\/span>\u4e2d\u7684\u6700\u4f18\u52a0\u6743\u88ab\u89c6\u4e3a\u7b49\u4e8eFIM，\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u662f\u9488\u5bf9\u975e\u7ed3\u6784\u5316\u6a21\u578b\u8bc4\u4f30\u7684\u3002\u5728\u9644\u5f55B\u4e2d\u663e\u793a，\u975e\u7ed3\u6784\u5316\u6a21\u578b\u7684FIM\u76f8\u5bf9\u4e8e\u4fe1\u53f7{\u03b1r,\u03b1i,\u03c9}\\\\{\\\\boldsymbol{\\\\alpha}_r, \\\\boldsymbol{\\\\alpha}_i, \\\\omega\\\\}<\/span><\/span>{<\/span>\u03b1<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03c9<\/span>}<\/span><\/span><\/span><\/span><\/span>\u548c\u566a\u58f0\u53c2\u6570\u662f\u5757\u5bf9\u89d2\u7684，\u56e0\u6b64(9)(9)<\/span><\/span>(<\/span>9<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u51c6\u5219\u5c06\u7531\u4e24\u9879\u7ec4\u6210，\u4e00\u9879\u4ec5\u6d89\u53ca\u03c9\\\\omega<\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>，\u53e6\u4e00\u9879\u4ec5\u6d89\u53caQ\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>\u7684\u5143\u7d20\u3002\u56e0\u6b64，\u8fd9\u4e24\u9879\u53ef\u4ee5\u5206\u522b\u6700\u5c0f\u5316\u3002\u7531\u4e8eQ\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>\u7684\u53c2\u6570\u5316\u5728\u7ed3\u6784\u5316\u548c\u975e\u7ed3\u6784\u5316\u6a21\u578b\u4e0b\u76f8\u540c，\u6d89\u53caQ\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>\u7684\u9879\u53ef\u4ee5\u901a\u8fc7\u7b80\u5355\u5730\u8bbe\u7f6eQ^e=Q^u\\\\hat{\\\\mathbf{Q}}_e = \\\\hat{\\\\mathbf{Q}}_u<\/span><\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4f7f\u5176\u4e3a\u96f6\u3002\u8fd9\u5efa\u7acb\u4e86(47)(47)<\/span><\/span>(<\/span>47<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

    \u6d89\u53ca\u03b7~\\\\tilde{\\\\boldsymbol{\\\\eta}}<\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u5269\u4f59\u9879\u901a\u8fc7\u5c06(129)(129)<\/span><\/span>(<\/span>129<\/span>)<\/span><\/span><\/span><\/span><\/span>\u548c(130)(130)<\/span><\/span>(<\/span>130<\/span>)<\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165(9)(9)<\/span><\/span>(<\/span>9<\/span>)<\/span><\/span><\/span><\/span><\/span>\u627e\u5230：<\/p>\n

    J=[(\u03b1^ur\u03b1^ui\u03c9^u)\u2212(A(\u03b8)00A(\u03b8)00)(brbi\u03c9)]TWQ~[(\u03b1^ur\u03b1^ui\u03c9^u)\u2212(A(\u03b8)00A(\u03b8)00)(brbi\u03c9)]J = \\\\left[\\\\begin{pmatrix} \\\\hat{\\\\boldsymbol{\\\\alpha}}_{ur} \\\\\\\\ \\\\hat{\\\\boldsymbol{\\\\alpha}}_{ui} \\\\\\\\ \\\\hat{\\\\omega}_u \\\\end{pmatrix} – \\\\begin{pmatrix} \\\\mathbf{A}(\\\\theta) & 0 \\\\\\\\ 0 & \\\\mathbf{A}(\\\\theta) \\\\\\\\ 0 & 0 \\\\end{pmatrix}\\\\begin{pmatrix} b_r \\\\\\\\ b_i \\\\\\\\ \\\\omega \\\\end{pmatrix}\\\\right]^T \\\\mathbf{W}_{\\\\tilde{Q}} \\\\left[\\\\begin{pmatrix} \\\\hat{\\\\boldsymbol{\\\\alpha}}_{ur} \\\\\\\\ \\\\hat{\\\\boldsymbol{\\\\alpha}}_{ui} \\\\\\\\ \\\\hat{\\\\omega}_u \\\\end{pmatrix} – \\\\begin{pmatrix} \\\\mathbf{A}(\\\\theta) & 0 \\\\\\\\ 0 & \\\\mathbf{A}(\\\\theta) \\\\\\\\ 0 & 0 \\\\end{pmatrix}\\\\begin{pmatrix} b_r \\\\\\\\ b_i \\\\\\\\ \\\\omega \\\\end{pmatrix}\\\\right]<\/span><\/span>J<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>A<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>W<\/span><\/span><\/span>Q<\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>A<\/span>(<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u5176\u4e2d\u52a0\u6743\u7531\u5728\u03b7~^\\\\hat{\\\\tilde{\\\\boldsymbol{\\\\eta}}}<\/span><\/span><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>~<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5904\u8bc4\u4f30\u7684FIM\u7ed9\u51fa\u3002\u5982\u679c\u6211\u4eec\u5173\u4e8e\u03c9\\\\omega<\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>\u6700\u5c0f\u5316(131)(131)<\/span><\/span>(<\/span>131<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u7b80\u5355\u7684\u8ba1\u7b97\u5f97\u51fa\u03c9^e=\u03c9^u\\\\hat{\\\\omega}_e = \\\\hat{\\\\omega}_u<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u8fd9\u4e5f\u9a8c\u8bc1\u4e86(48)(48)<\/span><\/span>(<\/span>48<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u03c9^e=\u03c9^u\\\\hat{\\\\omega}_e = \\\\hat{\\\\omega}_u<\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u4e8b\u5b9e\u4e5f\u8bc1\u660e\u4e86(131)(131)<\/span><\/span>(<\/span>131<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7b49\u4e8e(49)(49)<\/span><\/span>(<\/span>49<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u65b9\u7a0b(50)(50)<\/span><\/span>(<\/span>50<\/span>)<\/span><\/span><\/span><\/span><\/span>\u901a\u8fc7\u5c06(53)(53)<\/span><\/span>(<\/span>53<\/span>)<\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165(50)(50)<\/span><\/span>(<\/span>50<\/span>)<\/span><\/span><\/span><\/span><\/span>\u83b7\u5f97\u3002\u56e0\u6b64，\u5b9a\u74064\u4e2d\u7684\u6240\u6709\u9648\u8ff0\u90fd\u5f97\u5230\u9a8c\u8bc1，\u8bc1\u660e\u5b8c\u6210\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

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