{"id":589,"date":"2026-05-24T21:48:18","date_gmt":"2026-05-24T12:48:18","guid":{"rendered":"https:\/\/wuhanqing.cn\/wordpress\/?p=589"},"modified":"2026-05-25T11:48:00","modified_gmt":"2026-05-25T02:48:00","slug":"%e4%bb%8e%e7%82%b9%e7%a7%af%e5%88%b0%e5%86%85%e7%a7%af%e7%a9%ba%e9%97%b4%ef%bc%9a%e8%97%8f%e5%9c%a8%e7%ba%bf%e6%80%a7%e4%bb%a3%e6%95%b0%e3%80%81%e4%bf%a1%e5%8f%b7%e4%b8%8eai%e8%83%8c%e5%90%8e%e7%9a%84","status":"publish","type":"post","link":"https:\/\/wuhanqing.cn\/wordpress\/2026\/05\/24\/%e4%bb%8e%e7%82%b9%e7%a7%af%e5%88%b0%e5%86%85%e7%a7%af%e7%a9%ba%e9%97%b4%ef%bc%9a%e8%97%8f%e5%9c%a8%e7%ba%bf%e6%80%a7%e4%bb%a3%e6%95%b0%e3%80%81%e4%bf%a1%e5%8f%b7%e4%b8%8eai%e8%83%8c%e5%90%8e%e7%9a%84\/","title":{"rendered":"\u4ece\u70b9\u79ef\u5230\u5185\u79ef\u7a7a\u95f4\uff1a\u85cf\u5728\u7ebf\u6027\u4ee3\u6570\u3001\u4fe1\u53f7\u4e0eAI\u80cc\u540e\u7684\u540c\u4e00\u5957\u8bed\u8a00"},"content":{"rendered":"

# \u4ece\u70b9\u79ef\u5230\u5185\u79ef\u7a7a\u95f4\uff1a\u85cf\u5728\u7ebf\u6027\u4ee3\u6570\u3001\u4fe1\u53f7\u4e0eAI\u80cc\u540e\u7684\u540c\u4e00\u5957\u8bed\u8a00<\/p>\n

## \u6458\u8981<\/p>\n

\u5185\u79ef\uff08Inner Product\uff09\u662f\u7ebf\u6027\u4ee3\u6570\u3001\u6cdb\u51fd\u5206\u6790\u3001\u4fe1\u53f7\u5904\u7406\u3001\u673a\u5668\u5b66\u4e60\u4e0e\u91cf\u5b50\u529b\u5b66\u4e2d\u5171\u7528\u7684\u6838\u5fc3\u4ee3\u6570\u7ed3\u6784\u3002\u672c\u6587\u4ee5\"\u5185\u79ef\"\u4e3a\u552f\u4e00\u4e3b\u7ebf\uff0c\u4ece\u6709\u9650\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u4e2d\u7684\u70b9\u79ef\u51fa\u53d1\uff0c\u4f9d\u6b21\u5f15\u5165\u5185\u79ef\u7a7a\u95f4\u516c\u7406\u3001\u6b63\u4ea4\u5206\u89e3\u3001\u6700\u5c0f\u4e8c\u4e58\u6295\u5f71\u3001\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u3001\u5085\u91cc\u53f6\u7ea7\u6570\u4e0e\u53d8\u6362\u3001\u5377\u79ef\u3001\u79bb\u6563\u4f59\u5f26\u53d8\u6362\u3001\u5c0f\u6ce2\u5206\u6790\u3001\u81ea\u6ce8\u610f\u529b\u673a\u5236\u3001\u6838\u65b9\u6cd5\u4ee5\u53ca\u91cf\u5b50\u529b\u5b66\u4e2d\u7684\u6001\u77e2\u91cf\u6295\u5f71\uff0c\u63ed\u793a\u8fd9\u4e9b\u770b\u4f3c\u5206\u5c5e\u4e0d\u540c\u5b66\u79d1\u7684\u6982\u5ff5\u5728\u6570\u5b66\u7ed3\u6784\u4e0a\u7684\u7edf\u4e00\u6027\uff1a\u5b9a\u4e49\u5185\u79ef \u2192 \u5efa\u7acb\u6b63\u4ea4\u57fa \u2192 \u6295\u5f71\u5206\u89e3 \u2192 \u63d0\u53d6\u4fe1\u606f\u3002\u672c\u6587\u65e8\u5728\u4e3a\u8bfb\u8005\u63d0\u4f9b\u4e00\u5f20\u8d2f\u7a7f\u6570\u5b66\u3001\u5de5\u7a0b\u4e0e\u7269\u7406\u7684\u8ba4\u77e5\u56fe\u8c31\u3002<\/p>\n

## \u524d\u8a00 \u4e07\u7269\u7686\u6295\u5f71<\/p>\n

\u6570\u5b66\u4e0e\u5de5\u7a0b\u79d1\u5b66\u4e2d\u5b58\u5728\u7740\u4e00\u4e2a\u53cd\u590d\u51fa\u73b0\u7684\u6a21\u5f0f\uff1a\u5c06\u4e00\u4e2a\u590d\u6742\u5bf9\u8c61\u5206\u89e3\u4e3a\u82e5\u5e72\"\u57fa\u672c\u6210\u5206\"\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u800c\u5206\u89e3\u7684\u5de5\u5177\u6b63\u662f\u6295\u5f71\uff08projection\uff09\u3002\u6295\u5f71\u64cd\u4f5c\u7684\u672c\u8d28\u662f\u5185\u79ef\u2014\u2014\u4e00\u79cd\u5ea6\u91cf\"\u76f8\u4f3c\u6027\"\u7684\u4e8c\u5143\u8fd0\u7b97\u3002\u4ece\u5085\u91cc\u53f6\u5206\u6790\u4e2d\u5c06\u4fe1\u53f7\u5206\u89e3\u4e3a\u4e0d\u540c\u9891\u7387\u7684\u6b63\u5f26\u6ce2\uff0c\u5230\u6700\u5c0f\u4e8c\u4e58\u6cd5\u4e2d\u5bfb\u627e\u6570\u636e\u7684\u6700\u4f73\u62df\u5408\u76f4\u7ebf\uff0c\u518d\u5230\u91cf\u5b50\u529b\u5b66\u4e2d\u6d4b\u91cf\u5904\u4e8e\u53e0\u52a0\u6001\u7684\u7c92\u5b50\uff0c\u8fd9\u4e9b\u8fc7\u7a0b\u5171\u4eab\u540c\u4e00\u5957\u6570\u5b66\u8bed\u8a00\uff1a**\u5b9a\u4e49\u5185\u79ef \u2192 \u5efa\u7acb\u6b63\u4ea4\u57fa \u2192 \u6295\u5f71 \u2192 \u6b63\u4ea4\u5206\u89e3 \u2192 \u63d0\u53d6\u4fe1\u606f**\u3002<\/p>\n

\u672c\u6587\u7684\u76ee\u6807\u662f\u7cfb\u7edf\u6027\u5730\u9610\u660e\u8fd9\u4e00\u7edf\u4e00\u6846\u67b6\u3002\u6211\u4eec\u5c06\u4ece\u6700\u719f\u6089\u7684\u5411\u91cf\u70b9\u79ef\u51fa\u53d1\uff0c\u9010\u6b65\u62bd\u8c61\u81f3\u5185\u79ef\u7a7a\u95f4\u4e0e\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\uff0c\u5e76\u5c55\u793a\u8fd9\u4e00\u7ed3\u6784\u5982\u4f55\u5728\u5fae\u79ef\u5206\u3001\u4fe1\u53f7\u5904\u7406\u3001\u4eba\u5de5\u667a\u80fd\u4e0e\u91cf\u5b50\u529b\u5b66\u4e2d\u53cd\u590d\u51fa\u73b0\u3002\u8bfb\u8005\u65e0\u9700\u9884\u5148\u5177\u5907\u6cdb\u51fd\u5206\u6790\u80cc\u666f\uff0c\u53ea\u9700\u638c\u63e1\u57fa\u672c\u7684\u7ebf\u6027\u4ee3\u6570\u4e0e\u5fae\u79ef\u5206\u77e5\u8bc6\u3002<\/p>\n

---<\/p>\n

## \u7b2c\u4e00\u7ae0 \u5185\u79ef\u7684\u672c\u4f53 \u2014\u2014 \u5ea6\u91cf\u76f8\u4f3c\u6027\u7684\u57fa\u672c\u8fd0\u7b97<\/p>\n

### 1.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49<\/p>\n

\u5185\u79ef\uff08Inner Product\uff09\u7684\u6982\u5ff5\u8d77\u6e90\u4e8e\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55\u4e2d\u7684\u70b9\u79ef\uff08Dot Product\uff09\uff0c\u4f46\u5176\u6570\u5b66\u5185\u6db5\u5728\u6cdb\u51fd\u5206\u6790\u4e2d\u5f97\u5230\u4e86\u6781\u5927\u7684\u63a8\u5e7f\u3002\u672c\u8282\u4ece\u6709\u9650\u7ef4\u60c5\u5f62\u51fa\u53d1\uff0c\u9010\u6b65\u5efa\u7acb\u5185\u79ef\u7684\u4e25\u683c\u5b9a\u4e49\u3002<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 1.1 \u70b9\u79ef (Dot Product)
\n\u8bbe $\\mathbb{R}^n$ \u4e3a $n$ \u7ef4\u5b9e\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u3002\u5bf9\u4e8e\u4efb\u610f\u4e24\u4e2a\u5411\u91cf $\\mathbf{a} = (a_1, a_2, \\dots, a_n)$ \u4e0e $\\mathbf{b} = (b_1, b_2, \\dots, b_n)$\uff0c\u5176\u70b9\u79ef\u5b9a\u4e49\u4e3a\u5bf9\u5e94\u5206\u91cf\u4e58\u79ef\u4e4b\u548c$^{[1]}$\uff1a<\/p>\n

$$
\n\\langle \\mathbf{a}, \\mathbf{b} \\rangle = \\mathbf{a} \\cdot \\mathbf{b} = \\sum_{i=1}^{n} a_i b_i.
\n\\tag{1.1}
\n$$<\/p>\n

\u70b9\u79ef\u662f\u4e00\u4e2a\u5c06\u4e24\u4e2a\u5411\u91cf\u6620\u5c04\u4e3a\u4e00\u4e2a\u6807\u91cf\u7684\u4e8c\u5143\u8fd0\u7b97\u3002\u5176\u51e0\u4f55\u89e3\u91ca\u7531\u4f59\u5f26\u5b9a\u7406\u7ed9\u51fa\uff1a<\/p>\n

$$
\n\\mathbf{a} \\cdot \\mathbf{b} = \\|\\mathbf{a}\\| \\|\\mathbf{b}\\| \\cos\\theta,
\n\\tag{1.2}
\n$$<\/p>\n

\u5176\u4e2d $\\|\\mathbf{a}\\| = \\sqrt{\\langle \\mathbf{a}, \\mathbf{a} \\rangle}$ \u4e3a\u5411\u91cf\u7684\u6b27\u51e0\u91cc\u5f97\u8303\u6570\uff08$L_2$ \u8303\u6570\uff09\uff0c$\\theta$ \u4e3a\u4e24\u5411\u91cf\u4e4b\u95f4\u7684\u5939\u89d2\u3002
\n```<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 1.2 \u5185\u79ef\u7a7a\u95f4 (Inner Product Space)
\n\u8bbe $V$ \u4e3a\u57df $\\mathbb{F}$\uff08$\\mathbb{R}$ \u6216 $\\mathbb{C}$\uff09\u4e0a\u7684\u5411\u91cf\u7a7a\u95f4\u3002\u6620\u5c04 $\\langle \\cdot, \\cdot \\rangle: V \\times V \\to \\mathbb{F}$ \u79f0\u4e3a\u4e00\u4e2a\u5185\u79ef\uff0c\u82e5\u5176\u6ee1\u8db3\u4ee5\u4e0b\u4e09\u6761\u516c\u7406$^{[8][9]}$\uff1a<\/p>\n

1. **\u5171\u8f6d\u5bf9\u79f0\u6027\uff08Conjugate Symmetry\uff09**\uff1a$\\langle \\mathbf{u}, \\mathbf{v} \\rangle = \\overline{\\langle \\mathbf{v}, \\mathbf{u} \\rangle}$\uff0c\u5176\u4e2d\u4e0a\u5212\u7ebf\u8868\u793a\u590d\u5171\u8f6d\u3002\u5bf9\u4e8e\u5b9e\u5411\u91cf\u7a7a\u95f4\uff0c\u9000\u5316\u4e3a\u5bf9\u79f0\u6027 $\\langle \\mathbf{u}, \\mathbf{v} \\rangle = \\langle \\mathbf{v}, \\mathbf{u} \\rangle$\u3002
\n2. **\u5bf9\u7b2c\u4e00\u53d8\u5143\u7684\u7ebf\u6027\u6027\uff08Linearity in the First Argument\uff09**\uff1a$\\langle \\alpha\\mathbf{u} + \\beta\\mathbf{v}, \\mathbf{w} \\rangle = \\alpha\\langle \\mathbf{u}, \\mathbf{w} \\rangle + \\beta\\langle \\mathbf{v}, \\mathbf{w} \\rangle$\uff0c\u5bf9\u4efb\u610f $\\alpha, \\beta \\in \\mathbb{F}$ \u6210\u7acb\u3002
\n3. **\u6b63\u5b9a\u6027\uff08Positive Definiteness\uff09**\uff1a$\\langle \\mathbf{v}, \\mathbf{v} \\rangle \\geq 0$\uff0c\u4e14 $\\langle \\mathbf{v}, \\mathbf{v} \\rangle = 0$ \u5f53\u4e14\u4ec5\u5f53 $\\mathbf{v} = \\mathbf{0}$\u3002<\/p>\n

\u7531\u5185\u79ef\u53ef\u8bf1\u5bfc\u51fa\u8303\u6570 $\\|\\mathbf{v}\\| = \\sqrt{\\langle \\mathbf{v}, \\mathbf{v} \\rangle}$\uff0c\u8fdb\u800c\u8bf1\u5bfc\u51fa\u5ea6\u91cf $d(\\mathbf{u}, \\mathbf{v}) = \\|\\mathbf{u} - \\mathbf{v}\\|$\u3002\u56e0\u6b64\uff0c\u5185\u79ef\u7a7a\u95f4\u81ea\u7136\u662f\u4e00\u4e2a\u8d4b\u8303\u7a7a\u95f4\uff0c\u8fdb\u800c\u662f\u4e00\u4e2a\u5ea6\u91cf\u7a7a\u95f4\u3002
\n```<\/p>\n

```ad-theorem
\ntitle: \u5b9a\u7406 1.1 \u67ef\u897f-\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f (Cauchy\u2013Schwarz Inequality)
\n\u5bf9\u5185\u79ef\u7a7a\u95f4 $V$ \u4e2d\u7684\u4efb\u610f\u5411\u91cf $\\mathbf{u}, \\mathbf{v}$\uff0c\u6709<\/p>\n

$$
\n|\\langle \\mathbf{u}, \\mathbf{v} \\rangle| \\leq \\|\\mathbf{u}\\| \\|\\mathbf{v}\\|,
\n\\tag{1.3}
\n$$<\/p>\n

\u7b49\u53f7\u6210\u7acb\u5f53\u4e14\u4ec5\u5f53 $\\mathbf{u}$ \u4e0e $\\mathbf{v}$ \u7ebf\u6027\u76f8\u5173\u3002\u8be5\u4e0d\u7b49\u5f0f\u662f\u5185\u79ef\u7a7a\u95f4\u4e2d\u6700\u57fa\u672c\u7684\u4e0d\u7b49\u5f0f\uff0c\u5b83\u4fdd\u8bc1\u4e86\u5939\u89d2 $\\theta$ \u7684\u5b9a\u4e49 $\\cos\\theta = \\langle \\mathbf{u}, \\mathbf{v} \\rangle \/ (\\|\\mathbf{u}\\|\\|\\mathbf{v}\\|)$ \u7684\u53d6\u503c\u59cb\u7ec8\u843d\u5728 $[-1, 1]$ \u533a\u95f4\u5185\u3002
\n```<\/p>\n

### 1.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf<\/p>\n

\u5185\u79ef\u7684\u51e0\u4f55\u76f4\u89c2\u53ef\u4ee5\u7528\"\u6295\u5f71\"\u6765\u7406\u89e3\u3002\u7ed9\u5b9a\u5411\u91cf $\\mathbf{a}$ \u4e0e $\\mathbf{b}$\uff0c$\\mathbf{a}$ \u5728 $\\mathbf{b}$ \u65b9\u5411\u4e0a\u7684\u6807\u91cf\u6295\u5f71\uff08scalar projection\uff09\u4e3a<\/p>\n

$$
\n\\text{comp}_{\\mathbf{b}} \\mathbf{a} = \\|\\mathbf{a}\\| \\cos\\theta = \\frac{\\langle \\mathbf{a}, \\mathbf{b} \\rangle}{\\|\\mathbf{b}\\|}.
\n\\tag{1.4}
\n$$<\/p>\n

\u8be5\u503c\u5ea6\u91cf\u4e86 $\\mathbf{a}$ \u5728 $\\mathbf{b}$ \u65b9\u5411\u4e0a\u7684\"\u5206\u91cf\u5927\u5c0f\"\u3002\u5185\u79ef $\\langle \\mathbf{a}, \\mathbf{b} \\rangle$ \u5219\u53ef\u89c6\u4e3a\u8be5\u6295\u5f71\u957f\u5ea6\u4e58\u4ee5 $\\|\\mathbf{b}\\|$\uff0c\u5373\"\u6295\u5f71\u957f\u5ea6 \u00d7 \u57fa\u5ea7\u957f\u5ea6\"$^{[10]}$\u3002<\/p>\n

\u5f53 $\\langle \\mathbf{a}, \\mathbf{b} \\rangle = 0$ \u65f6\uff0c\u79f0 $\\mathbf{a}$ \u4e0e $\\mathbf{b}$ **\u6b63\u4ea4\uff08orthogonal\uff09**\uff0c\u6b64\u65f6 $\\mathbf{a}$ \u5728 $\\mathbf{b}$ \u65b9\u5411\u4e0a\u7684\u6295\u5f71\u4e3a\u96f6\u3002\u6b63\u4ea4\u610f\u5473\u7740\u4e24\u4e2a\u5411\u91cf\u5728\u65b9\u5411\u4e0a\u5b8c\u5168\u72ec\u7acb\uff0c\u4e92\u4e0d\u5305\u542b\u5bf9\u65b9\u7684\u4fe1\u606f\u3002<\/p>\n

\u82e5\u5c06\u5411\u91cf\u5f52\u4e00\u5316\uff08\u5373\u9664\u4ee5\u81ea\u8eab\u8303\u6570\uff09\uff0c\u5219\u5185\u79ef\u9000\u5316\u4e3a\u5939\u89d2\u4f59\u5f26\uff1a<\/p>\n

$$
\n\\cos\\theta = \\frac{\\langle \\mathbf{a}, \\mathbf{b} \\rangle}{\\|\\mathbf{a}\\| \\|\\mathbf{b}\\|}.
\n\\tag{1.5}
\n$$<\/p>\n

\u8be5\u91cf\u79f0\u4e3a**\u4f59\u5f26\u76f8\u4f3c\u5ea6\uff08Cosine Similarity\uff09**$^{[13]}$\uff0c\u5b83\u6d88\u9664\u4e86\u5411\u91cf\u5c3a\u5ea6\u7684\u5f71\u54cd\uff0c\u7eaf\u7cb9\u5ea6\u91cf\u65b9\u5411\u4e0a\u7684\u76f8\u4f3c\u6027\uff0c\u5728\u4fe1\u606f\u68c0\u7d22\u4e0e\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4e2d\u6709\u7740\u5e7f\u6cdb\u5e94\u7528\u3002<\/p>\n

### 1.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)<\/p>\n

```ad-example
\ntitle: \u4f8b\u9898 1.1 \u9ad8\u7ef4\u5411\u91cf\u7684\u70b9\u79ef\u4e0e\u6295\u5f71
\n\u7ed9\u5b9a\u56db\u7ef4\u7a7a\u95f4\u4e2d\u7684\u5411\u91cf $\\mathbf{a} = (1, 2, 3, 4)$ \u548c $\\mathbf{b} = (2, 0, -1, 3)$\uff0c\u8ba1\u7b97\uff1a(1) \u70b9\u79ef\uff1b(2) \u5404\u81ea\u7684\u8303\u6570\uff1b(3) $\\mathbf{a}$ \u5728 $\\mathbf{b}$ \u4e0a\u7684\u6807\u91cf\u6295\u5f71\uff1b(4) \u4f59\u5f26\u76f8\u4f3c\u5ea6\u3002<\/p>\n

**\u89e3** (1) \u7531\u5b9a\u4e49 (1.1)\uff1a<\/p>\n

$$
\n\\langle \\mathbf{a}, \\mathbf{b} \\rangle = 1\\times 2 + 2\\times 0 + 3\\times (-1) + 4\\times 3 = 2 + 0 - 3 + 12 = 11.
\n$$<\/p>\n

(2) \u8303\u6570\u8ba1\u7b97\uff1a<\/p>\n

$$
\n\\|\\mathbf{a}\\| = \\sqrt{1^2 + 2^2 + 3^2 + 4^2} = \\sqrt{30}, \\quad
\n\\|\\mathbf{b}\\| = \\sqrt{2^2 + 0^2 + (-1)^2 + 3^2} = \\sqrt{14}.
\n$$<\/p>\n

(3) \u6807\u91cf\u6295\u5f71\u7531 (1.4) \u7ed9\u51fa\uff1a<\/p>\n

$$
\n\\text{comp}_{\\mathbf{b}} \\mathbf{a} = \\frac{\\langle \\mathbf{a}, \\mathbf{b} \\rangle}{\\|\\mathbf{b}\\|} = \\frac{11}{\\sqrt{14}} \\approx 2.940.
\n$$<\/p>\n

(4) \u4f59\u5f26\u76f8\u4f3c\u5ea6\u7531 (1.5) \u7ed9\u51fa\uff1a<\/p>\n

$$
\n\\cos\\theta = \\frac{11}{\\sqrt{30 \\times 14}} = \\frac{11}{\\sqrt{420}} \\approx 0.5367.
\n$$<\/p>\n

\u8be5\u7ed3\u679c\u8868\u660e $\\mathbf{a}$ \u4e0e $\\mathbf{b}$ \u5728\u56db\u7ef4\u7a7a\u95f4\u4e2d\u5448\u7ea6 $57.5^\\circ$ \u7684\u5939\u89d2\uff0c\u5177\u6709\u4e2d\u7b49\u7a0b\u5ea6\u7684\u6b63\u76f8\u5173\u6027\u3002
\n```<\/p>\n

```ad-example
\ntitle: \u4f8b\u9898 1.2 \u8fde\u7eed\u51fd\u6570\u7684\u5185\u79ef\uff1a\u4ea4\u6d41\u7535\u8def\u7684\u5e73\u5747\u529f\u7387
\n\u5728\u4ea4\u6d41\u7535\u8def\u4e2d\uff0c\u7535\u538b $v(t) = V_m\\cos(\\omega t)$ \u4e0e\u7535\u6d41 $i(t) = I_m\\cos(\\omega t + \\phi)$ \u4e4b\u95f4\u5b58\u5728\u76f8\u4f4d\u5dee $\\phi$\u3002\u6c42\u4e00\u4e2a\u5b8c\u6574\u5468\u671f $T = 2\\pi\/\\omega$ \u5185\u7684\u5e73\u5747\u529f\u7387\u3002<\/p>\n

**\u89e3** \u5e73\u5747\u529f\u7387\u7684\u7269\u7406\u5b9a\u4e49\u4e3a\u7535\u538b\u4e0e\u7535\u6d41\u4e58\u79ef\u5728\u4e00\u4e2a\u5468\u671f\u5185\u7684\u5747\u503c\uff0c\u6570\u5b66\u4e0a\u5373\u4e8c\u8005\u5728 $L^2[0,T]$ \u7a7a\u95f4\u4e2d\u7684\u5185\u79ef\uff08\u9664\u4ee5\u5468\u671f\u957f\u5ea6\uff09\uff1a<\/p>\n

$$
\nP = \\frac{1}{T} \\int_0^T v(t) i(t) \\, dt.
\n\\tag{1.6}
\n$$<\/p>\n

\u4ee3\u5165\u8868\u8fbe\u5f0f\uff1a<\/p>\n

$$
\nP = \\frac{1}{T} \\int_0^T V_m I_m \\cos(\\omega t) \\cos(\\omega t + \\phi) \\, dt.
\n$$<\/p>\n

\u5229\u7528\u79ef\u5316\u548c\u5dee\u516c\u5f0f $\\cos\\alpha \\cos\\beta = \\frac{1}{2}[\\cos(\\alpha+\\beta) + \\cos(\\alpha-\\beta)]$\uff1a<\/p>\n

$$
\nP = \\frac{V_m I_m}{2T} \\int_0^T [\\cos(2\\omega t + \\phi) + \\cos(-\\phi)] \\, dt.
\n$$<\/p>\n

\u7531\u4e8e $\\cos(-\\phi) = \\cos\\phi$\uff0c\u4e14 $\\int_0^T \\cos(2\\omega t + \\phi) dt = 0$\uff08\u6b63\u5f26\u51fd\u6570\u5728\u5b8c\u6574\u5468\u671f\u4e0a\u7684\u79ef\u5206\u4e3a\u96f6\uff09\uff0c\u53ef\u5f97\uff1a<\/p>\n

$$
\nP = \\frac{V_m I_m}{2T} \\cdot T \\cos\\phi = \\frac{1}{2} V_m I_m \\cos\\phi.
\n\\tag{1.7}
\n$$<\/p>\n

\u5f0f (1.7) \u4e2d\u7684 $\\cos\\phi$ \u79f0\u4e3a**\u529f\u7387\u56e0\u6570\uff08power factor\uff09**\u3002\u5f53\u7535\u538b\u4e0e\u7535\u6d41\u540c\u76f8\uff08$\\phi = 0$\uff09\u65f6\uff0c$\\cos\\phi = 1$\uff0c\u5e73\u5747\u529f\u7387\u8fbe\u5230\u6700\u5927\u503c\uff1b\u5f53\u4e8c\u8005\u6b63\u4ea4\uff08$\\phi = \\pi\/2$\uff09\u65f6\uff0c$\\cos\\phi = 0$\uff0c\u5e73\u5747\u529f\u7387\u4e3a\u96f6\u3002\u8fd9\u4e00\u7ed3\u679c\u63ed\u793a\u4e86\u4ea4\u6d41\u7535\u8def\u4e2d\u6709\u529f\u529f\u7387\u4e0e\u65e0\u529f\u529f\u7387\u7684\u6570\u5b66\u672c\u8d28\uff1a\u53ea\u6709\u540c\u76f8\u5206\u91cf\uff08\u5373\u5185\u79ef\u975e\u96f6\u7684\u5206\u91cf\uff09\u624d\u4ea7\u751f\u51c0\u80fd\u91cf\u4f20\u9012\u3002
\n```<\/p>\n

### 1.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528<\/p>\n

\u5185\u79ef\u6700\u57fa\u672c\u7684\u5de5\u7a0b\u5e94\u7528\u4e4b\u4e00\u51fa\u73b0\u5728\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4e2d\u3002**\u8bcd\u5d4c\u5165\uff08Word Embedding\uff09**$^{[21]}$ \u6280\u672f\u5c06\u6bcf\u4e2a\u5355\u8bcd\u6620\u5c04\u4e3a $\\mathbb{R}^d$ \u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u7a20\u5bc6\u5411\u91cf\uff0c\u4f7f\u5f97\u8bed\u4e49\u76f8\u8fd1\u7684\u8bcd\u5728\u5411\u91cf\u7a7a\u95f4\u4e2d\u5f7c\u6b64\u9760\u8fd1\u3002\u4e24\u4e2a\u8bcd\u4e4b\u95f4\u7684\u8bed\u4e49\u76f8\u4f3c\u5ea6\u901a\u5e38\u7528\u4f59\u5f26\u76f8\u4f3c\u5ea6 (1.5) \u6765\u5ea6\u91cf\u3002<\/p>\n

\u56fe 1 \u5c55\u793a\u4e86\u4e94\u4e2a\u82f1\u6587\u5355\u8bcd\u7684 $5 \\times 5$ \u4f59\u5f26\u76f8\u4f3c\u5ea6\u70ed\u529b\u56fe\uff0c\u7531\u672c\u6587\u9644\u5e26\u7684 main.py<\/a> \u4ee3\u7801\u751f\u6210\u3002\u56fe\u4e2d\u53ef\u89c1\uff0c`king` \u4e0e `queen` \u7684\u4f59\u5f26\u76f8\u4f3c\u5ea6\u8f83\u9ad8\uff08\u63a5\u8fd1 $0.92$\uff09\uff0c\u800c `king` \u4e0e `apple` \u7684\u4f59\u5f26\u76f8\u4f3c\u5ea6\u8f83\u4f4e\uff08\u7ea6 $0.10$\uff09\uff0c\u8fd9\u4e00\u5b9a\u91cf\u7ed3\u679c\u4e0e\u4eba\u7c7b\u7684\u8bed\u4e49\u76f4\u89c9\u9ad8\u5ea6\u4e00\u81f4\u3002<\/p>\n

<\/p>\n

**\u56fe 1\uff1a\u8bcd\u5411\u91cf\u7684\u4f59\u5f26\u76f8\u4f3c\u5ea6\u70ed\u529b\u56fe\u3002** \u56fe\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c $(i,j)$ \u8868\u793a\u7b2c $i$ \u4e2a\u8bcd\u4e0e\u7b2c $j$ \u4e2a\u8bcd\u7684\u8bcd\u5411\u91cf\u4e4b\u95f4\u7684\u4f59\u5f26\u76f8\u4f3c\u5ea6\u3002\u989c\u8272\u8d8a\u6696\uff08\u63a5\u8fd1 1\uff09\u8868\u793a\u8bed\u4e49\u8d8a\u76f8\u8fd1\uff0c\u989c\u8272\u8d8a\u51b7\uff08\u63a5\u8fd1 0\uff09\u8868\u793a\u8bed\u4e49\u8d8a\u65e0\u5173\u3002\u8be5\u56fe\u7531 main.py<\/a> \u4e2d\u7684\u968f\u673a\u8bcd\u5411\u91cf\u6a21\u62df\u751f\u6210\u3002<\/p>\n

\u4f59\u5f26\u76f8\u4f3c\u5ea6\u5728\u63a8\u8350\u7cfb\u7edf\u3001\u4fe1\u606f\u68c0\u7d22\u3001\u6587\u672c\u5206\u7c7b\u7b49\u4efb\u52a1\u4e2d\u5747\u6709\u5e7f\u6cdb\u5e94\u7528\u3002\u5176\u6838\u5fc3\u601d\u60f3\u59cb\u7ec8\u5982\u4e00\uff1a\u5c06\u975e\u7ed3\u6784\u5316\u6570\u636e\u5d4c\u5165\u5411\u91cf\u7a7a\u95f4\uff0c\u7528\u5185\u79ef\u5ea6\u91cf\u76f8\u4f3c\u6027\uff0c\u518d\u57fa\u4e8e\u76f8\u4f3c\u6027\u8fdb\u884c\u68c0\u7d22\u6216\u6392\u5e8f\u3002<\/p>\n

---<\/p>\n

## \u7b2c\u4e8c\u7ae0 \u6b63\u4ea4\u4e0e\u6b63\u4ea4\u8865\u7a7a\u95f4 \u2014\u2014 \u72ec\u7acb\u6027\u7684\u4ee3\u6570\u523b\u753b<\/p>\n

### 2.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49<\/p>\n

\u5185\u79ef\u63d0\u4f9b\u4e86\u5ea6\u91cf\u5411\u91cf\u95f4\"\u76f8\u4f3c\u5ea6\"\u7684\u5de5\u5177\u3002\u5f53\u4e24\u4e2a\u5411\u91cf\u7684\u5185\u79ef\u4e3a\u96f6\u65f6\uff0c\u5b83\u4eec\u5728\u65b9\u5411\u4e0a\u5b8c\u5168\u72ec\u7acb\uff0c\u8fd9\u4e00\u6027\u8d28\u79f0\u4e3a**\u6b63\u4ea4\uff08orthogonality\uff09**\u3002<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 2.1 \u6b63\u4ea4
\n\u8bbe $V$ \u4e3a\u5185\u79ef\u7a7a\u95f4\u3002\u82e5 $\\langle \\mathbf{u}, \\mathbf{v} \\rangle = 0$\uff0c\u5219\u79f0\u5411\u91cf $\\mathbf{u}$ \u4e0e $\\mathbf{v}$ \u6b63\u4ea4\uff0c\u8bb0\u4f5c $\\mathbf{u} \\perp \\mathbf{v}$\u3002
\n```<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 2.2 \u6b63\u4ea4\u8865 (Orthogonal Complement)
\n\u8bbe $W$ \u4e3a\u5185\u79ef\u7a7a\u95f4 $V$ \u7684\u5b50\u7a7a\u95f4\u3002$W$ \u7684\u6b63\u4ea4\u8865\u5b9a\u4e49\u4e3a$^{[2]}$<\/p>\n

$$
\nW^\\perp = \\{ \\mathbf{v} \\in V \\mid \\langle \\mathbf{v}, \\mathbf{w} \\rangle = 0,\\ \\forall \\mathbf{w} \\in W \\}.
\n\\tag{2.1}
\n$$<\/p>\n

$W^\\perp$ \u662f $V$ \u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\uff0c\u4e14 $W \\cap W^\\perp = \\{\\mathbf{0}\\}$\u3002
\n```<\/p>\n

```ad-theorem
\ntitle: \u5b9a\u7406 2.1 \u6b63\u4ea4\u5206\u89e3\u5b9a\u7406 (Orthogonal Decomposition Theorem)
\n\u8bbe $W$ \u4e3a\u5185\u79ef\u7a7a\u95f4 $V$ \u7684\u6709\u9650\u7ef4\u5b50\u7a7a\u95f4\u3002\u5219\u5bf9\u4efb\u610f $\\mathbf{x} \\in V$\uff0c\u5b58\u5728\u552f\u4e00\u7684\u5206\u89e3<\/p>\n

$$
\n\\mathbf{x} = \\mathbf{x}_W + \\mathbf{x}_{W^\\perp},
\n\\tag{2.2}
\n$$<\/p>\n

\u5176\u4e2d $\\mathbf{x}_W \\in W$\uff0c$\\mathbf{x}_{W^\\perp} \\in W^\\perp$\uff0c\u4e14 $\\langle \\mathbf{x}_W, \\mathbf{x}_{W^\\perp} \\rangle = 0$\u3002<\/p>\n

\u5411\u91cf $\\mathbf{x}_W$ \u79f0\u4e3a $\\mathbf{x}$ \u5728 $W$ \u4e0a\u7684**\u6b63\u4ea4\u6295\u5f71\uff08orthogonal projection\uff09**\uff0c\u8bb0\u4f5c $\\operatorname{proj}_W \\mathbf{x}$\u3002\u8be5\u5206\u89e3\u7684\u552f\u4e00\u6027\u7531\u5185\u79ef\u7684\u6b63\u5b9a\u6027\u4fdd\u8bc1\u3002
\n```<\/p>\n

```ad-theorem
\ntitle: \u5b9a\u7406 2.2 \u6295\u5f71\u7b97\u5b50
\n\u82e5 $\\{\\mathbf{w}_1, \\dots, \\mathbf{w}_k\\}$ \u662f $W$ \u7684\u4e00\u7ec4\u6807\u51c6\u6b63\u4ea4\u57fa\uff0c\u5219 $\\mathbf{x}$ \u5728 $W$ \u4e0a\u7684\u6b63\u4ea4\u6295\u5f71\u53ef\u663e\u5f0f\u8868\u793a\u4e3a<\/p>\n

$$
\n\\operatorname{proj}_W \\mathbf{x} = \\sum_{i=1}^k \\langle \\mathbf{x}, \\mathbf{w}_i \\rangle \\mathbf{w}_i.
\n\\tag{2.3}
\n$$<\/p>\n

\u5f0f (2.3) \u662f\u5185\u79ef\u7406\u8bba\u4e2d\u6700\u91cd\u8981\u7684\u516c\u5f0f\u4e4b\u4e00\uff1a\u5b83\u8868\u660e\u6295\u5f71\u7cfb\u6570\u6b63\u662f\u5185\u79ef\u672c\u8eab\u3002
\n```<\/p>\n

### 2.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf<\/p>\n

\u6b63\u4ea4\u7684\u51e0\u4f55\u610f\u4e49\u662f\"\u72ec\u7acb\u6027\"\u3002\u5728 $\\mathbb{R}^3$ \u4e2d\uff0c\u7ed9\u5b9a\u4e00\u5f20\u8fc7\u539f\u70b9\u7684\u5e73\u9762 $W$\uff0c\u5176\u6b63\u4ea4\u8865 $W^\\perp$ \u662f\u5782\u76f4\u4e8e\u8be5\u5e73\u9762\u7684\u76f4\u7ebf\u3002\u7a7a\u95f4\u4e2d\u4efb\u610f\u5411\u91cf $\\mathbf{x}$ \u53ef\u552f\u4e00\u5730\u5206\u89e3\u4e3a\u5e73\u9762\u5185\u7684\u5206\u91cf\u4e0e\u6cd5\u7ebf\u65b9\u5411\u7684\u5206\u91cf\uff0c\u4e8c\u8005\u76f8\u4e92\u5782\u76f4\u3001\u4e92\u4e0d\u5305\u542b\u5bf9\u65b9\u7684\u4fe1\u606f\u3002<\/p>\n

\u8fd9\u4e00\u6982\u5ff5\u53ef\u4ee5\u63a8\u5e7f\u81f3\u51fd\u6570\u7a7a\u95f4\uff1a\u82e5\u4e24\u4e2a\u51fd\u6570\u5728\u67d0\u4e2a\u533a\u95f4\u4e0a\u7684\u5185\u79ef\u4e3a\u96f6\uff0c\u5219\u79f0\u5b83\u4eec\u5728 $L^2$ \u610f\u4e49\u4e0b\u6b63\u4ea4\u3002\u4f8b\u5982\uff0c$\\sin(mx)$ \u4e0e $\\sin(nx)$\uff08$m \\neq n$\uff09\u5728 $[0, 2\\pi]$ \u4e0a\u6b63\u4ea4\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u4eec\u4f5c\u4e3a\"\u4fe1\u53f7\"\u4e92\u4e0d\u5e72\u6270\u2014\u2014\u8fd9\u6b63\u662f\u9891\u5206\u590d\u7528\u6280\u672f\u7684\u6570\u5b66\u57fa\u7840\u3002<\/p>\n

### 2.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)<\/p>\n

```ad-example
\ntitle: \u4f8b\u9898 2.1 \u6b63\u4ea4\u8865\u4e0e\u6b63\u4ea4\u5206\u89e3
\n\u5728 $\\mathbb{R}^3$ \u4e2d\uff0c\u7ed9\u5b9a\u5b50\u7a7a\u95f4 $W = \\operatorname{span}\\{\\mathbf{v}_1, \\mathbf{v}_2\\}$\uff0c\u5176\u4e2d<\/p>\n

$$
\n\\mathbf{v}_1 = \\begin{bmatrix} 1 \\\\ 1 \\\\ 0 \\end{bmatrix},\\quad
\n\\mathbf{v}_2 = \\begin{bmatrix} 1 \\\\ 0 \\\\ 1 \\end{bmatrix}.
\n$$<\/p>\n

\u5bf9\u5411\u91cf $\\mathbf{x} = \\begin{bmatrix} 2 \\\\ 3 \\\\ 4 \\end{bmatrix}$\uff0c\u6c42\u5176\u6b63\u4ea4\u5206\u89e3 $\\mathbf{x} = \\mathbf{x}_W + \\mathbf{x}_{W^\\perp}$\u3002<\/p>\n

**\u89e3** \u7b2c\u4e00\u6b65\uff1a\u6c42 $W^\\perp$ \u7684\u57fa\u3002\u8bbe $\\mathbf{n} = (n_1, n_2, n_3)^T \\in W^\\perp$\uff0c\u5219 $\\langle \\mathbf{n}, \\mathbf{v}_1 \\rangle = \\langle \\mathbf{n}, \\mathbf{v}_2 \\rangle = 0$\uff1a<\/p>\n

$$
\n\\begin{cases}
\nn_1 + n_2 = 0, \\\\
\nn_1 + n_3 = 0.
\n\\end{cases}
\n$$<\/p>\n

\u89e3\u5f97 $n_2 = n_3 = -n_1$\u3002\u53d6 $n_1 = 1$\uff0c\u5f97 $\\mathbf{n} = (1, -1, -1)^T$\uff0c\u6545 $W^\\perp = \\operatorname{span}\\{\\mathbf{n}\\}$\u3002<\/p>\n

\u7b2c\u4e8c\u6b65\uff1a\u6c42 $\\mathbf{x}$ \u5728 $W^\\perp$ \u4e0a\u7684\u6295\u5f71\u3002\u7531\u6295\u5f71\u516c\u5f0f (2.3)\uff1a<\/p>\n

$$
\n\\mathbf{x}_{W^\\perp} = \\operatorname{proj}_{\\mathbf{n}} \\mathbf{x} = \\frac{\\langle \\mathbf{x}, \\mathbf{n} \\rangle}{\\|\\mathbf{n}\\|^2} \\mathbf{n}.
\n$$<\/p>\n

\u8ba1\u7b97\u5185\u79ef\uff1a$\\langle \\mathbf{x}, \\mathbf{n} \\rangle = 2\\times 1 + 3\\times(-1) + 4\\times(-1) = -5$\u3002
\n\u8ba1\u7b97\u8303\u6570\u5e73\u65b9\uff1a$\\|\\mathbf{n}\\|^2 = 1^2 + (-1)^2 + (-1)^2 = 3$\u3002
\n\u56e0\u6b64\uff1a<\/p>\n

$$
\n\\mathbf{x}_{W^\\perp} = \\frac{-5}{3} \\begin{bmatrix} 1 \\\\ -1 \\\\ -1 \\end{bmatrix} = \\begin{bmatrix} -5\/3 \\\\ 5\/3 \\\\ 5\/3 \\end{bmatrix}.
\n$$<\/p>\n

\u7b2c\u4e09\u6b65\uff1a\u6c42 $\\mathbf{x}$ \u5728 $W$ \u4e0a\u7684\u6295\u5f71\u3002\u7531\u6b63\u4ea4\u5206\u89e3\u5b9a\u7406 (2.2)\uff1a<\/p>\n

$$
\n\\mathbf{x}_W = \\mathbf{x} - \\mathbf{x}_{W^\\perp} = \\begin{bmatrix} 2 \\\\ 3 \\\\ 4 \\end{bmatrix} - \\begin{bmatrix} -5\/3 \\\\ 5\/3 \\\\ 5\/3 \\end{bmatrix} = \\begin{bmatrix} 11\/3 \\\\ 4\/3 \\\\ 7\/3 \\end{bmatrix}.
\n$$<\/p>\n

\u7b2c\u56db\u6b65\uff1a\u9a8c\u8bc1 $\\mathbf{x}_W \\in W$\u3002\u6c42\u89e3 $\\alpha, \\beta$ \u4f7f\u5f97 $\\mathbf{x}_W = \\alpha\\mathbf{v}_1 + \\beta\\mathbf{v}_2$\uff1a<\/p>\n

$$
\n\\begin{bmatrix} \\alpha + \\beta \\\\ \\alpha \\\\ \\beta \\end{bmatrix} = \\begin{bmatrix} 11\/3 \\\\ 4\/3 \\\\ 7\/3 \\end{bmatrix} \\implies \\alpha = \\frac{4}{3},\\ \\beta = \\frac{7}{3}.
\n$$<\/p>\n

\u9a8c\u8bc1\u901a\u8fc7\u3002\u540c\u65f6\u53ef\u9a8c\u8bc1 $\\langle \\mathbf{x}_W, \\mathbf{x}_{W^\\perp} \\rangle = 0$\uff0c\u786e\u8ba4\u5206\u89e3\u7684\u6b63\u4ea4\u6027\u3002
\n```<\/p>\n

### 2.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528<\/p>\n

\u6b63\u4ea4\u8865\u7684\u6982\u5ff5\u5728\u901a\u4fe1\u5de5\u7a0b\u4e2d\u6709\u7740\u76f4\u63a5\u5e94\u7528\u3002**Gram-Schmidt \u6b63\u4ea4\u5316\u7b97\u6cd5**$^{[3]}$ \u6b63\u662f\u57fa\u4e8e\u6b63\u4ea4\u5206\u89e3\u7684\u601d\u60f3\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\uff0c\u901a\u8fc7\u9010\u6b21\u51cf\u53bb\u5df2\u5904\u7406\u65b9\u5411\u4e0a\u7684\u6295\u5f71\u5206\u91cf\uff0c\u6784\u9020\u51fa\u4e00\u7ec4\u6807\u51c6\u6b63\u4ea4\u57fa\u3002\u8be5\u7b97\u6cd5\u662f QR \u5206\u89e3\u7684\u7406\u8bba\u57fa\u7840\u3002<\/p>\n

\u5728 5G \u6beb\u7c73\u6ce2\u901a\u4fe1\u7684\u6ce2\u675f\u6210\u5f62\uff08beamforming\uff09\u8bbe\u8ba1\u4e2d\uff0c\u9700\u8981\u5c06\u53d1\u5c04\u4fe1\u53f7\u5411\u91cf\u7f6e\u4e8e\u5176\u4ed6\u7528\u6237\u4fe1\u53f7\u5b50\u7a7a\u95f4\u7684\u6b63\u4ea4\u8865\u4e2d\uff0c\u4ece\u800c\u5728\u7406\u8bba\u4e0a\u5b9e\u73b0\u96f6\u5e72\u6270\u2014\u2014\u53ea\u8981\u4fe1\u53f7\u5411\u91cf\u4e0e\u5e72\u6270\u5b50\u7a7a\u95f4\u6b63\u4ea4\uff0c\u65e0\u8bba\u53d1\u5c04\u529f\u7387\u591a\u5927\uff0c\u90fd\u4e0d\u4f1a\u5bf9\u76f8\u90bb\u7528\u6237\u4ea7\u751f\u5e72\u6270\u3002<\/p>\n

---<\/p>\n

## \u7b2c\u4e09\u7ae0 \u6700\u5c0f\u4e8c\u4e58\u6cd5\uff08\u7ebf\u6027\u4ee3\u6570\u89c6\u89d2\uff09 \u2014\u2014 \u4e0d\u53ef\u89e3\u65b9\u7a0b\u7684\u6700\u4f18\u8fd1\u4f3c<\/p>\n

### 3.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49<\/p>\n

\u5728\u5b9e\u9645\u5de5\u7a0b\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u7ecf\u5e38\u9047\u5230**\u8d85\u5b9a\u7cfb\u7edf\uff08overdetermined system\uff09**\uff1a\u65b9\u7a0b\u4e2a\u6570\u591a\u4e8e\u672a\u77e5\u6570\u4e2a\u6570\u7684\u7ebf\u6027\u7cfb\u7edf $A\\mathbf{x} = \\mathbf{b}$\uff0c\u5176\u4e2d $A \\in \\mathbb{R}^{m \\times n}$\uff0c$m > n$\u3002\u8fd9\u6837\u7684\u7cfb\u7edf\u901a\u5e38\u4e0d\u5b58\u5728\u7cbe\u786e\u89e3\uff0c\u56e0\u4e3a $\\mathbf{b}$ \u4e0d\u5728 $A$ \u7684\u5217\u7a7a\u95f4 $\\operatorname{Col}(A)$ \u4e2d\u3002<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 3.1 \u6700\u5c0f\u4e8c\u4e58\u95ee\u9898
\n\u5bf9\u4e8e $A \\in \\mathbb{R}^{m \\times n}$\uff08$m > n$\uff09\u548c $\\mathbf{b} \\in \\mathbb{R}^m$\uff0c\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u4e3a<\/p>\n

$$
\n\\min_{\\mathbf{x} \\in \\mathbb{R}^n} \\| A\\mathbf{x} - \\mathbf{b} \\|^2.
\n\\tag{3.1}
\n$$<\/p>\n

\u8be5\u95ee\u9898\u7684\u51e0\u4f55\u89e3\u91ca\u662f\uff1a\u5728 $A$ \u7684\u5217\u7a7a\u95f4\u4e2d\u5bfb\u627e\u4e0e $\\mathbf{b}$ \u8ddd\u79bb\u6700\u8fd1\u7684\u5411\u91cf $\\hat{\\mathbf{b}} = A\\hat{\\mathbf{x}}$\u3002
\n```<\/p>\n

```ad-theorem
\ntitle: \u5b9a\u7406 3.1 \u6b63\u89c4\u65b9\u7a0b (Normal Equation)
\n\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898 (3.1) \u7684\u89e3 $\\hat{\\mathbf{x}}$ \u6ee1\u8db3<\/p>\n

$$
\nA^T A \\hat{\\mathbf{x}} = A^T \\mathbf{b}.
\n\\tag{3.2}
\n$$<\/p>\n

**\u8bc1\u660e** \u8bbe $\\mathbf{r}(\\mathbf{x}) = \\mathbf{b} - A\\mathbf{x}$ \u4e3a\u6b8b\u5dee\u5411\u91cf\u3002\u7531\u6b63\u4ea4\u5206\u89e3\u5b9a\u7406\uff0c$\\hat{\\mathbf{b}} = A\\hat{\\mathbf{x}}$ \u662f $\\mathbf{b}$ \u5728 $\\operatorname{Col}(A)$ \u4e0a\u7684\u6b63\u4ea4\u6295\u5f71\u5f53\u4e14\u4ec5\u5f53\u6b8b\u5dee $\\mathbf{r}(\\hat{\\mathbf{x}})$ \u5782\u76f4\u4e8e $\\operatorname{Col}(A)$\uff0c\u5373<\/p>\n

$$
\n\\langle A\\mathbf{y}, \\mathbf{r}(\\hat{\\mathbf{x}}) \\rangle = 0, \\quad \\forall \\mathbf{y} \\in \\mathbb{R}^n.
\n$$<\/p>\n

\u7b49\u4ef7\u5730\uff0c$A^T \\mathbf{r}(\\hat{\\mathbf{x}}) = \\mathbf{0}$\uff0c\u5373 $A^T(\\mathbf{b} - A\\hat{\\mathbf{x}}) = \\mathbf{0}$\uff0c\u6574\u7406\u5373\u5f97 (3.2)\u3002$\\square$<\/p>\n

\u5f53 $A$ \u5217\u6ee1\u79e9\u65f6\uff0c$A^T A$ \u53ef\u9006\uff0c\u89e3\u53ef\u663e\u5f0f\u5199\u4e3a<\/p>\n

$$
\n\\hat{\\mathbf{x}} = (A^T A)^{-1} A^T \\mathbf{b}.
\n\\tag{3.3}
\n$$
\n```<\/p>\n

### 3.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf<\/p>\n

\u6700\u5c0f\u4e8c\u4e58\u6cd5\u7684\u51e0\u4f55\u672c\u8d28\u5982\u56fe 2 \u6240\u793a\u3002\u6570\u636e\u5411\u91cf $\\mathbf{b}$ \u4f4d\u4e8e\u9ad8\u7ef4\u7a7a\u95f4 $\\mathbb{R}^m$ \u4e2d\uff0c\u800c\u6a21\u578b\u7684\u53ef\u8fbe\u96c6 $\\operatorname{Col}(A)$ \u662f $\\mathbb{R}^m$ \u7684\u4e00\u4e2a $n$ \u7ef4\u5b50\u7a7a\u95f4\u3002\u7531\u4e8e $\\mathbf{b}$ \u901a\u5e38\u4e0d\u5728\u8be5\u5b50\u7a7a\u95f4\u5185\uff0c\u6211\u4eec\u65e0\u6cd5\u7cbe\u786e\u6c42\u89e3 $A\\mathbf{x} = \\mathbf{b}$\u3002\u6700\u4f18\u7b56\u7565\u662f\u5c06 $\\mathbf{b}$ \u6b63\u4ea4\u6295\u5f71\u5230 $\\operatorname{Col}(A)$ \u4e0a\uff0c\u5f97\u5230 $\\hat{\\mathbf{b}}$\uff0c\u518d\u53cd\u89e3\u7cfb\u6570 $\\hat{\\mathbf{x}}$\u3002<\/p>\n

\u6b8b\u5dee\u5411\u91cf $\\mathbf{e} = \\mathbf{b} - \\hat{\\mathbf{b}}$ \u5782\u76f4\u4e8e $\\operatorname{Col}(A)$\uff0c\u5373 $\\mathbf{e} \\perp \\operatorname{Col}(A)$\u3002\u8fd9\u4e00\u6b63\u4ea4\u6761\u4ef6 $A^T \\mathbf{e} = \\mathbf{0}$ \u6b63\u662f\u6b63\u89c4\u65b9\u7a0b (3.2) \u7684\u7b49\u4ef7\u8868\u8ff0\u3002<\/p>\n

### 3.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)<\/p>\n

```ad-example
\ntitle: \u4f8b\u9898 3.1 \u7ebf\u6027\u56de\u5f52\u7684\u6700\u5c0f\u4e8c\u4e58\u89e3
\n\u7ed9\u5b9a\u4e09\u4e2a\u6570\u636e\u70b9 $(1,2), (2,3), (3,5)$\uff0c\u6c42\u6700\u4f73\u62df\u5408\u76f4\u7ebf $y = kx + c$\u3002<\/p>\n

**\u89e3** \u5c06\u95ee\u9898\u5199\u4e3a\u77e9\u9635\u5f62\u5f0f $A\\mathbf{x} = \\mathbf{b}$\uff1a<\/p>\n

$$
\n\\begin{bmatrix} 1 & 1 \\\\ 2 & 1 \\\\ 3 & 1 \\end{bmatrix} \\begin{bmatrix} k \\\\ c \\end{bmatrix} = \\begin{bmatrix} 2 \\\\ 3 \\\\ 5 \\end{bmatrix}.
\n$$<\/p>\n

\u7531\u4e8e $3 > 2$\uff0c\u8be5\u7cfb\u7edf\u8d85\u5b9a\uff0c\u65e0\u7cbe\u786e\u89e3\u3002\u4f7f\u7528\u6b63\u89c4\u65b9\u7a0b (3.2)\u3002<\/p>\n

\u7b2c\u4e00\u6b65\uff1a\u8ba1\u7b97 $A^T A$\uff1a<\/p>\n

$$
\nA^T A = \\begin{bmatrix} 1 & 2 & 3 \\\\ 1 & 1 & 1 \\end{bmatrix} \\begin{bmatrix} 1 & 1 \\\\ 2 & 1 \\\\ 3 & 1 \\end{bmatrix} = \\begin{bmatrix} 14 & 6 \\\\ 6 & 3 \\end{bmatrix}.
\n$$<\/p>\n

\u7b2c\u4e8c\u6b65\uff1a\u8ba1\u7b97 $A^T \\mathbf{b}$\uff1a<\/p>\n

$$
\nA^T \\mathbf{b} = \\begin{bmatrix} 1 & 2 & 3 \\\\ 1 & 1 & 1 \\end{bmatrix} \\begin{bmatrix} 2 \\\\ 3 \\\\ 5 \\end{bmatrix} = \\begin{bmatrix} 23 \\\\ 10 \\end{bmatrix}.
\n$$<\/p>\n

\u7b2c\u4e09\u6b65\uff1a\u6c42\u89e3\u6b63\u89c4\u65b9\u7a0b\uff1a<\/p>\n

$$
\n\\begin{bmatrix} 14 & 6 \\\\ 6 & 3 \\end{bmatrix} \\begin{bmatrix} k \\\\ c \\end{bmatrix} = \\begin{bmatrix} 23 \\\\ 10 \\end{bmatrix}.
\n$$<\/p>\n

\u7531\u7b2c\u4e8c\u884c\u5f97 $6k + 3c = 10$\uff0c\u5373 $c = (10 - 6k)\/3$\u3002\u4ee3\u5165\u7b2c\u4e00\u884c\uff1a<\/p>\n

$$
\n14k + 6 \\cdot \\frac{10 - 6k}{3} = 23 \\implies 14k + 20 - 12k = 23 \\implies 2k = 3 \\implies k = \\frac{3}{2}.
\n$$<\/p>\n

\u56de\u4ee3\u5f97 $c = \\frac{10 - 9}{3} = \\frac{1}{3}$\u3002\u56e0\u6b64\u6700\u4f73\u62df\u5408\u76f4\u7ebf\u4e3a<\/p>\n

$$
\ny = \\frac{3}{2}x + \\frac{1}{3}.
\n\\tag{3.4}
\n$$<\/p>\n

\u7b2c\u56db\u6b65\uff1a\u9a8c\u8bc1\u6b63\u4ea4\u6027\u3002\u8ba1\u7b97\u62df\u5408\u503c $\\hat{\\mathbf{b}} = A\\hat{\\mathbf{x}}$ \u548c\u6b8b\u5dee $\\mathbf{e}$\uff1a<\/p>\n

$$
\n\\hat{\\mathbf{b}} = \\begin{bmatrix} 1\\times 1.5 + 1\/3 \\\\ 2\\times 1.5 + 1\/3 \\\\ 3\\times 1.5 + 1\/3 \\end{bmatrix} = \\begin{bmatrix} 11\/6 \\\\ 10\/3 \\\\ 29\/6 \\end{bmatrix},\\quad
\n\\mathbf{e} = \\mathbf{b} - \\hat{\\mathbf{b}} = \\begin{bmatrix} 1\/6 \\\\ -1\/3 \\\\ 1\/6 \\end{bmatrix}.
\n$$<\/p>\n

\u9a8c\u8bc1 $A^T \\mathbf{e} = \\mathbf{0}$\uff1a<\/p>\n

$$
\nA^T \\mathbf{e} = \\begin{bmatrix} 1 & 2 & 3 \\\\ 1 & 1 & 1 \\end{bmatrix} \\begin{bmatrix} 1\/6 \\\\ -1\/3 \\\\ 1\/6 \\end{bmatrix} = \\begin{bmatrix} 1\/6 - 2\/3 + 3\/6 \\\\ 1\/6 - 1\/3 + 1\/6 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}.
\n$$<\/p>\n

\u6b63\u4ea4\u6761\u4ef6\u6ee1\u8db3\uff0c\u786e\u8ba4 $\\hat{\\mathbf{b}}$ \u662f $\\mathbf{b}$ \u5728 $\\operatorname{Col}(A)$ \u4e0a\u7684\u6b63\u4ea4\u6295\u5f71\u3002
\n```<\/p>\n

### 3.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528<\/p>\n

\u6700\u5c0f\u4e8c\u4e58\u6cd5\u662f\u7edf\u8ba1\u5b66\u4e2d**\u56de\u5f52\u5206\u6790\uff08Regression Analysis\uff09**$^{[5][25]}$ \u7684\u57fa\u7840\u3002\u56fe 2 \u5c55\u793a\u4e86\u4e0a\u8ff0\u4f8b\u9898\u7684\u51e0\u4f55\u76f4\u89c2\uff1a\u9ed1\u8272\u5706\u70b9\u4e3a\u539f\u59cb\u6570\u636e\u70b9\uff0c\u7ea2\u8272\u76f4\u7ebf\u4e3a\u6700\u5c0f\u4e8c\u4e58\u62df\u5408\u7ed3\u679c\uff0c\u7070\u8272\u865a\u7ebf\u8868\u793a\u6b8b\u5dee\uff08\u5373 $\\mathbf{b}$ \u5230 $\\operatorname{Col}(A)$ \u7684\u5782\u76f4\u8ddd\u79bb\uff09\u3002<\/p>\n

<\/p>\n

**\u56fe 2\uff1a\u6700\u5c0f\u4e8c\u4e58\u6cd5\u7684\u51e0\u4f55\u76f4\u89c2\u3002** \u9ed1\u8272\u5706\u70b9\u4e3a\u6570\u636e\u70b9\uff0c\u7ea2\u8272\u76f4\u7ebf\u4e3a\u62df\u5408\u7ed3\u679c\u3002\u6b8b\u5dee\u5411\u91cf $\\mathbf{e}$ \u5782\u76f4\u4e8e\u5217\u7a7a\u95f4 $\\operatorname{Col}(A)$\uff0c\u6b63\u4ea4\u6027\u6761\u4ef6 $A^T \\mathbf{e} = \\mathbf{0}$ \u7684\u6570\u503c\u9a8c\u8bc1\u7ed3\u679c\u4e3a $\\|A^T \\mathbf{e}\\|_2 \\approx 1.92 \\times 10^{-14}$\uff08\u7531 main.py<\/a>) \u8ba1\u7b97\uff09\uff0c\u5728\u6d6e\u70b9\u7cbe\u5ea6\u8303\u56f4\u5185\u4e3a\u96f6\u3002<\/p>\n

\u6700\u5c0f\u4e8c\u4e58\u6cd5\u5728\u5de5\u7a0b\u4e2d\u6709\u7740\u5e7f\u6cdb\u5e94\u7528\uff1a\u5361\u5c14\u66fc\u6ee4\u6ce2\u7684\u6d4b\u91cf\u66f4\u65b0\u6b65\u9aa4\u3001\u7cfb\u7edf\u8fa8\u8bc6\u4e2d\u7684\u53c2\u6570\u4f30\u8ba1\u3001\u673a\u5668\u5b66\u4e60\u4e2d\u7684\u7ebf\u6027\u56de\u5f52\u6a21\u578b\uff0c\u5176\u6838\u5fc3\u5747\u53ef\u5f52\u7ed3\u4e3a\u6c42\u89e3\u6b63\u89c4\u65b9\u7a0b (3.2)\u3002<\/p>\n

---<\/p>\n

## \u7b2c\u56db\u7ae0 \u4ece\u6709\u9650\u7ef4\u8d70\u5411\u65e0\u9650\u7ef4 \u2014\u2014 \u51fd\u6570\u4f5c\u4e3a\u5411\u91cf<\/p>\n

### 4.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49<\/p>\n

\u524d\u51e0\u7ae0\u8ba8\u8bba\u7684\u5185\u79ef\u5747\u5c40\u9650\u4e8e\u6709\u9650\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4 $\\mathbb{R}^n$\u3002\u7136\u800c\uff0c\u5185\u79ef\u7684\u6982\u5ff5\u53ef\u4ee5\u81ea\u7136\u5730\u63a8\u5e7f\u5230\u65e0\u9650\u7ef4\u51fd\u6570\u7a7a\u95f4\u3002\u8fd9\u4e00\u63a8\u5e7f\u662f\u6cdb\u51fd\u5206\u6790\u7684\u6838\u5fc3\u5185\u5bb9\uff0c\u4e5f\u662f\u8fde\u63a5\u7ebf\u6027\u4ee3\u6570\u4e0e\u4fe1\u53f7\u5904\u7406\u3001\u91cf\u5b50\u529b\u5b66\u7684\u6865\u6881\u3002<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 4.1 $L^2$ \u5185\u79ef
\n\u8bbe $f, g: [a, b] \\to \\mathbb{R}$ \u4e3a\u5e73\u65b9\u53ef\u79ef\u51fd\u6570\uff0c\u5373 $\\int_a^b [f(x)]^2 dx < \\infty$\u3002\u5b9a\u4e49\u5176\u5185\u79ef\u4e3a\n\n$$\n\\langle f, g \\rangle = \\int_a^b f(x) g(x) \\, dx.\n\\tag{4.1}\n$$\n\n\u8be5\u5185\u79ef\u8bf1\u5bfc\u7684\u8303\u6570\u4e3a\n\n$$\n\\|f\\| = \\sqrt{\\langle f, f \\rangle} = \\sqrt{\\int_a^b [f(x)]^2 \\, dx},\n\\tag{4.2}\n$$\n\n\u79f0\u4e3a $L^2$ \u8303\u6570\uff0c\u7269\u7406\u4e0a\u5e38\u89e3\u91ca\u4e3a\u4fe1\u53f7\u7684\"\u80fd\u91cf\"\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u4e49 4.2 \u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4 (Hilbert Space)\n\u5b8c\u5907\u7684\u5185\u79ef\u7a7a\u95f4\u79f0\u4e3a\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4$^{[6][8]}$\u3002\u5177\u4f53\u800c\u8a00\uff0c\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4 $\\mathcal{H}$ \u662f\u4e00\u4e2a\u5185\u79ef\u7a7a\u95f4\uff0c\u5176\u4e2d\u4efb\u610f\u67ef\u897f\u5e8f\u5217\u5747\u5728 $\\mathcal{H}$ \u4e2d\u6536\u655b\uff08\u5373\u7a7a\u95f4\u662f\u5b8c\u5907\u7684\uff09\u3002\n\n\u6709\u9650\u7ef4\u5185\u79ef\u7a7a\u95f4 $\\mathbb{R}^n$ \u662f\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u7684\u7279\u4f8b\u3002\u65e0\u9650\u7ef4\u7684\u4f8b\u5b50\u5305\u62ec $L^2[a,b]$\uff08\u5e73\u65b9\u53ef\u79ef\u51fd\u6570\u7a7a\u95f4\uff09\u548c $\\ell^2$\uff08\u5e73\u65b9\u53ef\u548c\u5e8f\u5217\u7a7a\u95f4\uff09\u3002\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u7684\u5b8c\u5907\u6027\u4fdd\u8bc1\u4e86\u5085\u91cc\u53f6\u7ea7\u6570\u7b49\u65e0\u7a77\u7ea7\u6570\u5c55\u5f00\u7684\u6536\u655b\u6027\u3002\n```\n\n```ad-theorem\ntitle: \u5b9a\u7406 4.1 $L^2$ \u7a7a\u95f4\u4e2d\u7684\u67ef\u897f-\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f\n\u5bf9 $L^2[a,b]$ \u4e2d\u7684\u4efb\u610f\u51fd\u6570 $f, g$\uff0c\u6709\n\n$$\n\\left| \\int_a^b f(x) g(x) \\, dx \\right| \\leq \\sqrt{\\int_a^b [f(x)]^2 \\, dx} \\cdot \\sqrt{\\int_a^b [g(x)]^2 \\, dx}.\n\\tag{4.3}\n$$\n```\n\n### 4.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf\n\n\u5c06\u51fd\u6570\u89c6\u4e3a\u5411\u91cf\u7684\u5173\u952e\u5728\u4e8e\u7406\u89e3\"\u9010\u70b9\u5bf9\u5e94\"\u7684\u601d\u60f3\u3002\u5728 $\\mathbb{R}^n$ \u4e2d\uff0c\u5411\u91cf $\\mathbf{v} = (v_1, \\dots, v_n)$ \u7684\u7b2c $i$ \u4e2a\u5206\u91cf $v_i$ \u5bf9\u5e94\u4e8e\u7b2c $i$ \u4e2a\u5750\u6807\u8f74\u4e0a\u7684\u53d6\u503c\u3002\u5728\u51fd\u6570\u7a7a\u95f4\u4e2d\uff0c\u6bcf\u4e2a $x \\in [a,b]$ \u5bf9\u5e94\u4e00\u4e2a\u72ec\u7acb\u7684\"\u5750\u6807\u8f74\"\uff0c\u51fd\u6570\u503c $f(x)$ \u5373\u4e3a\u8be5\u5750\u6807\u8f74\u4e0a\u7684\u5206\u91cf\u3002\u56e0\u6b64\uff0c\u51fd\u6570 $f$ \u672c\u8d28\u4e0a\u662f\u4e00\u4e2a\u5177\u6709\u4e0d\u53ef\u6570\u65e0\u7a77\u591a\u4e2a\u5206\u91cf\u7684\u5411\u91cf\u3002\n\n\u4e24\u4e2a\u51fd\u6570\u6b63\u4ea4\uff08$\\langle f, g \\rangle = 0$\uff09\u610f\u5473\u7740\u5b83\u4eec\u5728 $L^2$ \u610f\u4e49\u4e0b\"\u4e92\u4e0d\u5305\u542b\u5bf9\u65b9\u7684\u6210\u5206\"\u3002\u8fd9\u4e00\u6982\u5ff5\u5728\u4fe1\u53f7\u5904\u7406\u4e2d\u5177\u6709\u6df1\u523b\u7684\u7269\u7406\u542b\u4e49\uff1a\u6b63\u4ea4\u7684\u4fe1\u53f7\u53ef\u4ee5\u5728\u540c\u4e00\u4fe1\u9053\u4e2d\u4f20\u8f93\u800c\u4e92\u4e0d\u5e72\u6270\u3002\n\n### 4.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u9898 4.1 \u51fd\u6570\u7a7a\u95f4\u4e2d\u7684\u6b63\u4ea4\u6027\u4e0e\u8ddd\u79bb\u5ea6\u91cf\n\u5728\u533a\u95f4 $[-1, 1]$ \u4e0a\uff0c\u7ed9\u5b9a $f(x) = x$ \u548c $g(x) = x^2$\u3002\u5224\u65ad\u5b83\u4eec\u662f\u5426\u6b63\u4ea4\uff0c\u5e76\u8ba1\u7b97\u5404\u81ea\u7684\u8303\u6570\u53ca\u51fd\u6570\u95f4\u8ddd\u79bb\u3002\n\n**\u89e3** (1) \u8ba1\u7b97\u5185\u79ef\uff1a\n\n$$\n\\langle f, g \\rangle = \\int_{-1}^{1} x \\cdot x^2 \\, dx = \\int_{-1}^{1} x^3 \\, dx = \\left[ \\frac{x^4}{4} \\right]_{-1}^{1} = \\frac{1}{4} - \\frac{1}{4} = 0.\n$$\n\n\u56e0\u6b64 $\\langle f, g \\rangle = 0$\uff0c$f$ \u4e0e $g$ \u5728 $[-1,1]$ \u4e0a\u6b63\u4ea4\u3002\u539f\u56e0\u662f $x^3$ \u4e3a\u5947\u51fd\u6570\uff0c\u5728\u5bf9\u79f0\u533a\u95f4\u4e0a\u79ef\u5206\u4e3a\u96f6\u3002\n\n(2) \u8ba1\u7b97\u8303\u6570\uff1a\n\n$$\n\\|f\\| = \\sqrt{\\int_{-1}^{1} x^2 \\, dx} = \\sqrt{\\left[ \\frac{x^3}{3} \\right]_{-1}^{1}} = \\sqrt{\\frac{2}{3}} \\approx 0.8165,\n$$\n\n$$\n\\|g\\| = \\sqrt{\\int_{-1}^{1} x^4 \\, dx} = \\sqrt{\\left[ \\frac{x^5}{5} \\right]_{-1}^{1}} = \\sqrt{\\frac{2}{5}} \\approx 0.6325.\n$$\n\n(3) \u8ba1\u7b97\u51fd\u6570\u95f4\u8ddd\u79bb\uff1a\n\n$$\n\\|f - g\\|^2 = \\int_{-1}^{1} (x - x^2)^2 \\, dx = \\int_{-1}^{1} (x^2 - 2x^3 + x^4) \\, dx = \\frac{2}{3} + 0 + \\frac{2}{5} = \\frac{16}{15},\n$$\n\n\u6545 $d(f, g) = \\|f - g\\| = \\sqrt{16\/15} \\approx 1.0328$\u3002\n\n\u8be5\u4f8b\u9898\u8868\u660e\uff1a\u5947\u51fd\u6570\u4e0e\u5076\u51fd\u6570\u5728\u5bf9\u79f0\u533a\u95f4\u4e0a\u5929\u7136\u6b63\u4ea4\u3002\u8fd9\u4e00\u6027\u8d28\u5728\u5085\u91cc\u53f6\u5206\u6790\u4e2d\u81f3\u5173\u91cd\u8981\u2014\u2014\u5b83\u4fdd\u8bc1\u4e86\u6b63\u5f26\u57fa\u4e0e\u4f59\u5f26\u57fa\u4e4b\u95f4\u7684\u6b63\u4ea4\u6027\u3002\n```\n\n### 4.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528\n\n\u51fd\u6570\u5185\u79ef\u5728\u5de5\u7a0b\u4e2d\u6700\u76f4\u63a5\u7684\u5e94\u7528\u662f**\u5339\u914d\u6ee4\u6ce2\uff08Matched Filter\uff09**\u3002\u5728\u96f7\u8fbe\u548c\u901a\u4fe1\u7cfb\u7edf\u4e2d\uff0c\u63a5\u6536\u4fe1\u53f7 $r(t)$ \u4e0e\u53d1\u5c04\u6a21\u677f $s(t)$ \u7684\u5185\u79ef\n\n$$\n\\langle r, s \\rangle = \\int_{-\\infty}^{\\infty} r(t) s(t) \\, dt\n$$\n\n\u7528\u4e8e\u68c0\u6d4b\u76ee\u6807\u662f\u5426\u5b58\u5728\u3002\u5f53\u56de\u6ce2\u4e2d\u5b58\u5728\u76ee\u6807\u53cd\u5c04\u65f6\uff0c\u5185\u79ef\u503c\u663e\u8457\u589e\u5927\u3002\u8fd9\u672c\u8d28\u4e0a\u662f\u51fd\u6570\u7a7a\u95f4\u4e2d\u7684\"\u76f8\u4f3c\u5ea6\u68c0\u6d4b\"\u3002\n\n\u6b64\u5916\uff0c**\u6838\u65b9\u6cd5\uff08Kernel Methods\uff09**$^{[22]}$ \u7684\u6838\u5fc3\u601d\u60f3\u662f\u5c06\u6570\u636e\u70b9\u6620\u5c04\u5230\u518d\u751f\u6838\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\uff08RKHS\uff09\uff0c\u5728\u8be5\u65e0\u9650\u7ef4\u7a7a\u95f4\u4e2d\u8ba1\u7b97\u5185\u79ef\uff0c\u4ece\u800c\u9690\u5f0f\u5730\u5b9e\u73b0\u9ad8\u7ef4\u7279\u5f81\u53d8\u6362\u3002\u6211\u4eec\u5c06\u5728\u7b2c\u5341\u4e8c\u7ae0\u6df1\u5165\u63a2\u8ba8\u3002\n\n---\n\n## \u7b2c\u4e94\u7ae0 \u4e09\u89d2\u51fd\u6570\u6b63\u4ea4\u6027 \u2014\u2014 \u9891\u7387\u57df\u7684\u57fa\u51fd\u6570\n\n### 5.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49\n\n\u5728\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4 $L^2[-\\pi, \\pi]$ \u4e2d\uff0c\u4e09\u89d2\u51fd\u6570\u7cfb\u6784\u6210\u4e00\u7ec4\u91cd\u8981\u7684\u6b63\u4ea4\u57fa\u3002\u8003\u8651\u51fd\u6570\u96c6\u5408\n\n$$\n\\{1,\\ \\sin x,\\ \\cos x,\\ \\sin 2x,\\ \\cos 2x,\\ \\dots,\\ \\sin nx,\\ \\cos nx,\\ \\dots\\}.\n$$\n\n```ad-theorem\ntitle: \u5b9a\u7406 5.1 \u4e09\u89d2\u51fd\u6570\u7684\u6b63\u4ea4\u6027\n\u5728\u533a\u95f4 $[-\\pi, \\pi]$ \u4e0a\uff0c\u4e09\u89d2\u51fd\u6570\u7cfb\u6ee1\u8db3\u4ee5\u4e0b\u6b63\u4ea4\u5173\u7cfb$^{[4]}$\uff1a\n\n$$\n\\int_{-\\pi}^{\\pi} \\sin(mx) \\cos(nx) \\, dx = 0, \\quad \\forall m, n,\n\\tag{5.1}\n$$\n\n$$\n\\int_{-\\pi}^{\\pi} \\sin(mx) \\sin(nx) \\, dx = 0, \\quad m \\neq n,\n\\tag{5.2}\n$$\n\n$$\n\\int_{-\\pi}^{\\pi} \\cos(mx) \\cos(nx) \\, dx = 0, \\quad m \\neq n.\n\\tag{5.3}\n$$\n\n\u540c\u9891\u7387\u7684\u81ea\u5185\u79ef\u975e\u96f6\uff1a\n\n$$\n\\int_{-\\pi}^{\\pi} \\sin^2(nx) \\, dx = \\pi, \\quad\n\\int_{-\\pi}^{\\pi} \\cos^2(nx) \\, dx = \\pi.\n\\tag{5.4}\n$$\n\n**\u8bc1\u660e** \u8fd9\u4e9b\u5173\u7cfb\u53ef\u7531\u4e09\u89d2\u51fd\u6570\u7684\u79ef\u5316\u548c\u5dee\u516c\u5f0f\u76f4\u63a5\u5bfc\u51fa\u3002\u4f8b\u5982\uff0c\u5bf9 (5.2)\uff1a\n\n$$\n\\sin(mx)\\sin(nx) = \\frac{1}{2}[\\cos((m-n)x) - \\cos((m+n)x)].\n$$\n\n\u5f53 $m \\neq n$ \u65f6\uff0c$\\cos((m-n)x)$ \u548c $\\cos((m+n)x)$ \u5728 $[-\\pi, \\pi]$ \u4e0a\u7684\u79ef\u5206\u5747\u4e3a\u96f6\u3002$\\square$\n```\n\n### 5.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf\n\n\u4e09\u89d2\u51fd\u6570\u6b63\u4ea4\u6027\u7684\u51e0\u4f55\u610f\u4e49\u662f\uff1a\u4e0d\u540c\u9891\u7387\u7684\u6b63\u5f26\u6ce2\u548c\u4f59\u5f26\u6ce2\u5728 $L^2$ \u7a7a\u95f4\u4e2d\u76f8\u4e92\u5782\u76f4\u3002\u8fd9\u610f\u5473\u7740\u5b83\u4eec\u4f5c\u4e3a\"\u4fe1\u53f7\"\u4e92\u4e0d\u5e72\u6270\u2014\u2014\u8fd9\u6b63\u662f\u9891\u5206\u590d\u7528\u6280\u672f\u7684\u6570\u5b66\u57fa\u7840\u3002\n\n\u5728\u901a\u4fe1\u7cfb\u7edf\u4e2d\uff0c\u4e0d\u540c\u7528\u6237\u7684\u6570\u636e\u53ef\u4ee5\u8c03\u5236\u5230\u76f8\u4e92\u6b63\u4ea4\u7684\u8f7d\u6ce2\u4e0a\u540c\u65f6\u4f20\u8f93\uff0c\u63a5\u6536\u7aef\u901a\u8fc7\u5185\u79ef\u8fd0\u7b97\u5373\u53ef\u5206\u79bb\u5404\u8def\u4fe1\u53f7\uff0c\u5373\u4f7f\u5b83\u4eec\u5728\u65f6\u57df\u4e0a\u5b8c\u5168\u91cd\u53e0\u3002\u8fd9\u4e00\u539f\u7406\u5728\u73b0\u4ee3\u65e0\u7ebf\u901a\u4fe1\u7684**\u9891\u57df\uff08Frequency Domain\uff09**$^{[16]}$ \u5206\u6790\u4e2d\u5c45\u4e8e\u6838\u5fc3\u5730\u4f4d\u3002\n\n### 5.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u9898 5.1 \u4e09\u89d2\u51fd\u6570\u6b63\u4ea4\u6027\u7684\u624b\u5de5\u9a8c\u8bc1\n\u5728 $[-\\pi, \\pi]$ \u4e0a\u9a8c\u8bc1\u4ee5\u4e0b\u4e09\u7ec4\u5185\u79ef\u3002\n\n**\u60c5\u5f62 A\uff1a$\\langle \\sin(2x), \\cos(3x) \\rangle$**\n\n$$\n\\langle \\sin(2x), \\cos(3x) \\rangle = \\int_{-\\pi}^{\\pi} \\sin(2x)\\cos(3x) \\, dx.\n$$\n\n\u7531\u79ef\u5316\u548c\u5dee $\\sin\\alpha\\cos\\beta = \\frac{1}{2}[\\sin(\\alpha+\\beta) + \\sin(\\alpha-\\beta)]$\uff1a\n\n$$\n\\sin(2x)\\cos(3x) = \\frac{1}{2}[\\sin(5x) + \\sin(-x)] = \\frac{1}{2}[\\sin(5x) - \\sin(x)].\n$$\n\n\u7531\u4e8e $\\int_{-\\pi}^{\\pi} \\sin(kx) \\, dx = 0$ \u5bf9\u4efb\u610f\u6574\u6570 $k$ \u6210\u7acb\uff0c\u6545\n\n$$\n\\langle \\sin(2x), \\cos(3x) \\rangle = \\frac{1}{2} \\times 0 - \\frac{1}{2} \\times 0 = 0.\n$$\n\n**\u60c5\u5f62 B\uff1a$\\langle \\sin(2x), \\sin(3x) \\rangle$**\n\n\u7531 $\\sin\\alpha\\sin\\beta = \\frac{1}{2}[\\cos(\\alpha-\\beta) - \\cos(\\alpha+\\beta)]$\uff1a\n\n$$\n\\sin(2x)\\sin(3x) = \\frac{1}{2}[\\cos(-x) - \\cos(5x)] = \\frac{1}{2}[\\cos(x) - \\cos(5x)].\n$$\n\n\u7531\u4e8e $\\int_{-\\pi}^{\\pi} \\cos(kx) \\, dx = 0$ \u5bf9 $k \\neq 0$ \u6210\u7acb\uff0c\u6545\n\n$$\n\\langle \\sin(2x), \\sin(3x) \\rangle = \\frac{1}{2} \\times 0 - \\frac{1}{2} \\times 0 = 0.\n$$\n\n**\u60c5\u5f62 C\uff1a$\\langle \\sin(2x), \\sin(2x) \\rangle$\uff08\u81ea\u5185\u79ef\uff09**\n\n\u5229\u7528\u500d\u89d2\u516c\u5f0f $\\sin^2\\theta = (1 - \\cos 2\\theta)\/2$\uff1a\n\n$$\n\\langle \\sin(2x), \\sin(2x) \\rangle = \\int_{-\\pi}^{\\pi} \\frac{1 - \\cos(4x)}{2} \\, dx = \\frac{1}{2} \\cdot 2\\pi - 0 = \\pi.\n$$\n\n\u8be5\u7ed3\u679c\u8bf4\u660e $\\|\\sin(2x)\\| = \\sqrt{\\pi}$\uff0c\u8fd9\u6b63\u662f\u5085\u91cc\u53f6\u7ea7\u6570\u4e2d\u7cfb\u6570\u5206\u6bcd\u51fa\u73b0 $\\pi$ \u7684\u539f\u56e0\u3002\n```\n\n### 5.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528\n\n**\u6b63\u4ea4\u9891\u5206\u590d\u7528\uff08OFDM\uff09** \u662f\u73b0\u4ee3 4G\/5G \u65e0\u7ebf\u901a\u4fe1\u7684\u6838\u5fc3\u6280\u672f$^{[16]}$\u3002\u5b83\u5c06\u9ad8\u901f\u6570\u636e\u6d41\u5206\u5272\u4e3a\u591a\u4e2a\u4f4e\u901f\u5b50\u6d41\uff0c\u5206\u522b\u8c03\u5236\u5230\u76f8\u4e92\u6b63\u4ea4\u7684\u5b50\u8f7d\u6ce2\u4e0a\u5e76\u884c\u4f20\u8f93\u3002\u7531\u4e8e\u5b50\u8f7d\u6ce2\u95f4\u7684\u6b63\u4ea4\u6027\n\n$$\n\\int_0^T \\sin(2\\pi f_k t) \\cdot \\sin(2\\pi f_l t) \\, dt = 0, \\quad k \\neq l,\n$$\n\n\u63a5\u6536\u7aef\u53ef\u901a\u8fc7\u5185\u79ef\u8fd0\u7b97\u5b8c\u7f8e\u5206\u79bb\u5404\u5b50\u8f7d\u6ce2\u4fe1\u53f7\uff0c\u5373\u4f7f\u5b83\u4eec\u5728\u9891\u8c31\u4e0a\u4e25\u91cd\u91cd\u53e0\u3002\u8fd9\u6781\u5927\u5730\u63d0\u9ad8\u4e86\u9891\u8c31\u5229\u7528\u7387\u3002\n\n---\n\n## \u7b2c\u516d\u7ae0 \u5085\u91cc\u53f6\u7ea7\u6570\u4e0e\u5085\u91cc\u53f6\u53d8\u6362 \u2014\u2014 \u51fd\u6570\u5728\u4e09\u89d2\u57fa\u4e0a\u7684\u6295\u5f71\n\n### 6.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49\n\n\u4e09\u89d2\u51fd\u6570\u7cfb\u7684\u6b63\u4ea4\u6027\u4f7f\u5f97\u6211\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u5468\u671f\u51fd\u6570\u5206\u89e3\u4e3a\u4e0d\u540c\u9891\u7387\u4e09\u89d2\u51fd\u6570\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u8fd9\u4e00\u5206\u89e3\u79f0\u4e3a**\u5085\u91cc\u53f6\u7ea7\u6570\uff08Fourier Series\uff09**$^{[11]}$\u3002\n\n```ad-theorem\ntitle: \u5b9a\u7406 6.1 \u5085\u91cc\u53f6\u7ea7\u6570\n\u8bbe $f(t)$ \u662f\u4ee5 $2\\pi$ \u4e3a\u5468\u671f\u7684\u5e73\u65b9\u53ef\u79ef\u51fd\u6570\uff0c\u5219\u5176\u5085\u91cc\u53f6\u7ea7\u6570\u5c55\u5f00\u4e3a\n\n$$\nf(t) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} [a_n \\cos(nt) + b_n \\sin(nt)],\n\\tag{6.1}\n$$\n\n\u5176\u4e2d\u7cfb\u6570\u7531\u5185\u79ef\u7ed9\u51fa\uff1a\n\n$$\na_0 = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\, dt,\n\\tag{6.2}\n$$\n\n$$\na_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\cos(nt) \\, dt = \\frac{\\langle f, \\cos(nt) \\rangle}{\\|\\cos(nt)\\|^2},\n\\tag{6.3}\n$$\n\n$$\nb_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\sin(nt) \\, dt = \\frac{\\langle f, \\sin(nt) \\rangle}{\\|\\sin(nt)\\|^2}.\n\\tag{6.4}\n$$\n\n\u5f0f (6.3)-(6.4) \u63ed\u793a\u4e86\u5085\u91cc\u53f6\u7cfb\u6570\u7684\u672c\u8d28\uff1a\u5b83\u4eec\u5c31\u662f\u51fd\u6570 $f$ \u5728\u5404\u4e09\u89d2\u57fa\u4e0a\u7684\u6295\u5f71\u7cfb\u6570\uff08\u5185\u79ef\u9664\u4ee5\u57fa\u7684\u8303\u6570\u5e73\u65b9\uff09\uff0c\u4e0e\u6709\u9650\u7ef4\u5411\u91cf\u5728\u6b63\u4ea4\u57fa\u4e0a\u7684\u5750\u6807\u8ba1\u7b97\u5b8c\u5168\u4e00\u81f4\u3002\n\n\u5f53\u5468\u671f $T \\to \\infty$ \u65f6\uff0c\u5085\u91cc\u53f6\u7ea7\u6570\u8fc7\u6e21\u4e3a**\u5085\u91cc\u53f6\u53d8\u6362\uff08Fourier Transform\uff09**$^{[12]}$\uff1a\n\n$$\nX(f) = \\int_{-\\infty}^{\\infty} x(t) e^{-j2\\pi ft} \\, dt = \\langle x(t), e^{j2\\pi ft} \\rangle.\n\\tag{6.5}\n$$\n\n\u5085\u91cc\u53f6\u53d8\u6362\u5c06\u65f6\u57df\u51fd\u6570 $x(t)$ \u6295\u5f71\u5230\u590d\u6307\u6570\u57fa $e^{j2\\pi ft}$ \u4e0a\uff0c\u5f97\u5230\u9891\u57df\u8868\u793a $X(f)$\u3002\n```\n\n### 6.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf\n\n\u5085\u91cc\u53f6\u53d8\u6362\u7684\u51e0\u4f55\u672c\u8d28\u662f\"\u63a2\u9488\"\u601d\u60f3\uff1a\u7528\u4e0d\u540c\u9891\u7387\u7684\u590d\u6307\u6570\u632f\u8361\u4f5c\u4e3a\u63a2\u9488\uff0c\u4e0e\u5f85\u5206\u6790\u4fe1\u53f7\u505a\u5185\u79ef\u3002\u82e5\u4fe1\u53f7\u5305\u542b\u67d0\u9891\u7387\u6210\u5206\uff0c\u5219\u5185\u79ef\u503c\u8f83\u5927\uff08\u4ea7\u751f\u9891\u8c31\u5cf0\u503c\uff09\uff1b\u82e5\u4e0d\u5305\u542b\uff0c\u5219\u5185\u79ef\u503c\u63a5\u8fd1\u4e8e\u96f6\u3002\u9891\u8c31\u56fe\u4e0a\u7684\u6bcf\u4e00\u4e2a\u5cf0\uff0c\u5bf9\u5e94\u4fe1\u53f7\u5728\u8be5\u9891\u7387\u57fa\u4e0a\u7684\u6295\u5f71\u5f3a\u5ea6\u3002\n\n### 6.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u9898 6.1 \u5468\u671f\u65b9\u6ce2\u7684\u5085\u91cc\u53f6\u7ea7\u6570\u5c55\u5f00\n\u7ed9\u5b9a\u5468\u671f\u4e3a $2\\pi$ \u7684\u65b9\u6ce2\n\n$$\nf(t) = \\begin{cases}\n1, & 0 < t < \\pi, \\\\\n-1, & -\\pi < t < 0,\n\\end{cases}\n$$\n\n\u6c42\u5176\u5085\u91cc\u53f6\u7ea7\u6570\u7cfb\u6570\u3002\n\n**\u89e3** $f(t)$ \u4e3a\u5947\u51fd\u6570\uff0c\u6545 $a_0 = a_n = 0$\uff08\u4f59\u5f26\u7cfb\u6570\u5168\u4e3a\u96f6\uff09\u3002\u4ec5\u9700\u8ba1\u7b97 $b_n$\u3002\n\n$$\nb_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\sin(nt) \\, dt = \\frac{1}{\\pi} \\left( \\int_{-\\pi}^{0} (-\\sin(nt)) \\, dt + \\int_{0}^{\\pi} \\sin(nt) \\, dt \\right).\n$$\n\n\u8ba1\u7b97\u7b2c\u4e00\u9879\uff1a$\\int_{-\\pi}^{0} -\\sin(nt) \\, dt = \\left[ \\frac{\\cos(nt)}{n} \\right]_{-\\pi}^{0} = \\frac{1}{n} - \\frac{\\cos(-n\\pi)}{n} = \\frac{1 - (-1)^n}{n}$\u3002\n\n\u8ba1\u7b97\u7b2c\u4e8c\u9879\uff1a$\\int_{0}^{\\pi} \\sin(nt) \\, dt = \\left[ -\\frac{\\cos(nt)}{n} \\right]_{0}^{\\pi} = -\\frac{\\cos(n\\pi)}{n} + \\frac{1}{n} = \\frac{1 - (-1)^n}{n}$\u3002\n\n\u56e0\u6b64\uff1a\n\n$$\nb_n = \\frac{1}{\\pi} \\cdot \\frac{2[1 - (-1)^n]}{n} = \\begin{cases}\n\\dfrac{4}{n\\pi}, & n \\text{ \u4e3a\u5947\u6570}, \\\\[6pt]\n0, & n \\text{ \u4e3a\u5076\u6570}.\n\\end{cases}\n\\tag{6.6}\n$$\n\n\u6545\u65b9\u6ce2\u7684\u5085\u91cc\u53f6\u7ea7\u6570\u5c55\u5f00\u4e3a\n\n$$\nf(t) = \\frac{4}{\\pi} \\sum_{k=0}^{\\infty} \\frac{\\sin((2k+1)t)}{2k+1} = \\frac{4}{\\pi} \\left( \\sin t + \\frac{1}{3}\\sin 3t + \\frac{1}{5}\\sin 5t + \\cdots \\right).\n\\tag{6.7}\n$$\n\n\u6570\u503c\u9a8c\u8bc1\uff1a\u53d6 $t = \\pi\/2$\uff0c\u524d 3 \u9879\u8fd1\u4f3c\u4e3a\n\n$$\nf(\\pi\/2) \\approx \\frac{4}{\\pi} \\left( 1 - \\frac{1}{3} + \\frac{1}{5} \\right) = \\frac{52}{15\\pi} \\approx 1.103,\n$$\n\n\u5df2\u63a5\u8fd1\u771f\u5b9e\u503c $1$\u3002\u66f4\u591a\u9879\u5c06\u6536\u655b\u4e8e\u65b9\u6ce2\uff08\u5409\u5e03\u65af\u73b0\u8c61\u5728\u95f4\u65ad\u70b9\u5904\u4ea7\u751f\u7ea6 $9\\%$ \u7684\u8fc7\u51b2\uff09\u3002\n```\n\n### 6.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528\n\n\u56fe 3 \u5c55\u793a\u4e86\u5085\u91cc\u53f6\u53d8\u6362\u7684\u5178\u578b\u5e94\u7528\u3002\u4e00\u4e2a\u5305\u542b 50 Hz\u3001120 Hz \u548c 260 Hz \u4e09\u4e2a\u9891\u7387\u5206\u91cf\u7684\u5e26\u566a\u4fe1\u53f7 $x(t)$\uff0c\u5176\u65f6\u57df\u6ce2\u5f62\u770b\u4f3c\u6742\u4e71\u65e0\u7ae0\u3002\u7ecf\u5085\u91cc\u53f6\u53d8\u6362\u540e\uff0c\u9891\u8c31\u56fe\u5728\u5bf9\u5e94\u9891\u7387\u5904\u6e05\u6670\u5448\u73b0\u4e09\u4e2a\u5cf0\u503c\u2014\u2014\u8fd9\u6b63\u662f\u4fe1\u53f7\u5728\u5404\u9891\u7387\u57fa\u4e0a\u7684\u6295\u5f71\u5f3a\u5ea6\u3002\n\n<\/p>\n

**\u56fe 3\uff1a\u5085\u91cc\u53f6\u53d8\u6362\u7684\u9891\u57df\u6295\u5f71\u3002** \u4e0a\u56fe\u4e3a\u542b\u566a\u591a\u97f3\u4fe1\u53f7 $x(t) = 1.2\\sin(2\\pi\\cdot 50t) + 0.7\\sin(2\\pi\\cdot 120t) + 0.4\\sin(2\\pi\\cdot 260t) + \\eta(t)$ \u7684\u65f6\u57df\u6ce2\u5f62\uff1b\u4e0b\u56fe\u4e3a\u5e45\u5ea6\u9891\u8c31\uff0c\u5728 50\u3001120\u3001260 Hz \u5904\u51fa\u73b0\u663e\u8457\u5cf0\u503c\u3002\u8be5\u56fe\u7531 main.py<\/a> \u4e2d\u7684 `np.fft.rfft`\uff08\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\uff09\u751f\u6210\uff0c\u5176\u672c\u8d28\u662f\u8ba1\u7b97\u65f6\u57df\u91c7\u6837\u5411\u91cf\u4e0e\u590d\u6307\u6570\u57fa\u5411\u91cf\u7684\u5185\u79ef\u3002<\/p>\n

\u5085\u91cc\u53f6\u5206\u6790\u7684\u5e94\u7528\u904d\u53ca\u5de5\u7a0b\u5404\u9886\u57df\uff1aMP3 \u97f3\u9891\u538b\u7f29\u901a\u8fc7\u820d\u5f03\u4eba\u8033\u4e0d\u654f\u611f\u7684\u9ad8\u9891\u5206\u91cf\u6765\u51cf\u5c0f\u6570\u636e\u91cf\uff1bJPEG \u56fe\u50cf\u538b\u7f29\u4f7f\u7528\u79bb\u6563\u4f59\u5f26\u53d8\u6362\uff08DCT\uff09$^{[18]}$ \u5c06\u56fe\u50cf\u5757\u6295\u5f71\u5230\u9891\u7387\u57fa\u4e0a\uff1b\u5fc3\u7535\u56fe\uff08ECG\uff09\u4fe1\u53f7\u7684\u9891\u57df\u8bca\u65ad\u5219\u5229\u7528\u9891\u8c31\u7279\u5f81\u8bc6\u522b\u75c5\u7406\u6a21\u5f0f\u3002<\/p>\n

---<\/p>\n

## \u7b2c\u4e03\u7ae0 \u4ece\u9891\u57df\u5230\u590d\u9891\u57df \u2014\u2014 \u62c9\u666e\u62c9\u65af\u4e0e Z \u53d8\u6362<\/p>\n

### 7.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49<\/p>\n

\u5085\u91cc\u53f6\u53d8\u6362\u8981\u6c42\u4fe1\u53f7\u6ee1\u8db3\u7edd\u5bf9\u53ef\u79ef\u6761\u4ef6 $\\int_{-\\infty}^{\\infty} |f(t)|\\,dt < \\infty$\u3002\u5bf9\u4e8e\u6307\u6570\u53d1\u6563\u4fe1\u53f7\u5982 $f(t) = e^{2t}$\uff08$t \\geq 0$\uff09\uff0c\u5176\u80fd\u91cf\u968f $t$ \u589e\u957f\u800c\u53d1\u6563\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u7684\u5185\u79ef $\\langle f(t), e^{-j\\omega t} \\rangle$ \u4e0d\u6536\u655b\u3002\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u9700\u5c06\u63a2\u6d4b\u57fa\u5e95\u4ece\u7eaf\u865a\u6307\u6570 $e^{-j\\omega t}$ \u63a8\u5e7f\u4e3a\u5177\u6709\u5b9e\u90e8\u8870\u51cf\u56e0\u5b50\u7684\u590d\u6307\u6570 $e^{-st}$\uff0c\u5176\u4e2d $s = \\sigma + j\\omega$\u3002\n\n```ad-definition\ntitle: \u5b9a\u4e49 7.1 \u62c9\u666e\u62c9\u65af\u53d8\u6362\n\u8bbe $f(t)$ \u662f\u5b9a\u4e49\u5728 $[0, \\infty)$ \u4e0a\u7684\u51fd\u6570\uff0c\u5176**\u62c9\u666e\u62c9\u65af\u53d8\u6362**\u5b9a\u4e49\u4e3a $^{[14]}$\uff1a\n\n$$F(s) = \\mathcal{L}\\{f(t)\\} = \\int_0^{\\infty} f(t) e^{-st}\\,dt, \\quad s = \\sigma + j\\omega \\in \\mathbb{C} \\tag{7.1}$$\n\n\u5f53 $s$ \u7684\u5b9e\u90e8 $\\sigma$ \u8db3\u591f\u5927\u65f6\uff0c\u8870\u51cf\u56e0\u5b50 $e^{-\\sigma t}$ \u53ef\u538b\u5236 $f(t)$ \u7684\u53d1\u6563\u8d8b\u52bf\uff0c\u4f7f\u79ef\u5206\u6536\u655b\u3002\u4f7f (7.1) \u6536\u655b\u7684 $s$ \u503c\u96c6\u5408\u79f0\u4e3a**\u6536\u655b\u57df\uff08Region of Convergence, ROC\uff09**\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u4e49 7.2 Z \u53d8\u6362\n\u8bbe $x[n]$ \u662f\u5b9a\u4e49\u5728 $\\mathbb{Z}$ \u4e0a\u7684\u79bb\u6563\u5e8f\u5217\uff0c\u5176**Z \u53d8\u6362**\u5b9a\u4e49\u4e3a $^{[15]}$\uff1a\n\n$$X(z) = \\mathcal{Z}\\{x[n]\\} = \\sum_{n=-\\infty}^{\\infty} x[n] z^{-n}, \\quad z = re^{j\\omega} \\in \\mathbb{C} \\tag{7.2}$$\n\nZ \u53d8\u6362\u53ef\u89c6\u4e3a\u62c9\u666e\u62c9\u65af\u53d8\u6362\u5728\u79bb\u6563\u57df\u4e2d\u7684\u5bf9\u5e94\uff1a\u4ee4 $z = e^{sT}$\uff08$T$ \u4e3a\u91c7\u6837\u5468\u671f\uff09\uff0c\u5219 $z$ \u5e73\u9762\u4e0a\u7684\u5355\u4f4d\u5706 $|z| = 1$ \u5bf9\u5e94 $s$ \u5e73\u9762\u4e0a\u7684\u865a\u8f74 $s = j\\omega$\u3002\n\n\u4ece\u5185\u79ef\u89c6\u89d2\u770b\uff0c\u62c9\u666e\u62c9\u65af\u53d8\u6362\u548c Z \u53d8\u6362\u5747\u53ef\u7406\u89e3\u4e3a\u4fe1\u53f7\u4e0e\u590d\u6307\u6570\u57fa\u51fd\u6570\u7684\u5185\u79ef\uff1a\n\n$$\\mathcal{L}\\{f(t)\\} = \\langle f(t), e^{st} \\rangle, \\quad \\mathcal{Z}\\{x[n]\\} = \\langle x[n], z^n \\rangle$$\n\n\u5176\u4e2d\u57fa\u51fd\u6570 $e^{st}$ \u548c $z^n$ \u5305\u542b\u4e86\u5e45\u5ea6\u8870\u51cf\uff08\u901a\u8fc7 $\\sigma$ \u6216 $r$\uff09\u548c\u76f8\u4f4d\u65cb\u8f6c\uff08\u901a\u8fc7 $\\omega$\uff09\u4e24\u4e2a\u81ea\u7531\u5ea6\uff0c\u6bd4\u5085\u91cc\u53f6\u53d8\u6362\u7684\u57fa\u51fd\u6570\u66f4\u5177\u8868\u8fbe\u80fd\u529b\u3002\n```\n\n### 7.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf\n\n\u5085\u91cc\u53f6\u53d8\u6362\u7684\u57fa\u5e95 $e^{-j\\omega t}$ \u662f\u590d\u5e73\u9762\u5355\u4f4d\u5706\u4e0a\u7684\u5300\u901f\u65cb\u8f6c\u5411\u91cf\uff0c\u6a21\u957f\u6052\u4e3a 1\u3002\u5bf9\u4e8e\u53d1\u6563\u4fe1\u53f7 $e^{2t}$\uff0c\u88ab\u79ef\u51fd\u6570 $|e^{2t} \\cdot e^{-j\\omega t}| = e^{2t}$ \u968f $t$ \u589e\u957f\u800c\u53d1\u6563\uff0c\u79ef\u5206\u6c38\u4e0d\u6536\u655b\u3002\n\n\u62c9\u666e\u62c9\u65af\u53d8\u6362\u7684\u57fa\u5e95 $e^{-(\\sigma + j\\omega)t} = e^{-\\sigma t} e^{-j\\omega t}$ \u589e\u52a0\u4e86\u4e00\u4e2a\"\u8870\u51cf\u65cb\u94ae\" $\\sigma$\u3002\u5f53 $\\sigma > 2$ \u65f6\uff0c$e^{-\\sigma t}$ \u7684\u8870\u51cf\u901f\u7387\u8d85\u8fc7 $e^{2t}$ \u7684\u53d1\u6563\u901f\u7387\uff0c\u5185\u79ef\u79ef\u5206\u6536\u655b\u3002\u5728\u590d $s$ \u5e73\u9762\u4e0a\uff1a<\/p>\n

- **\u6536\u655b\u57df\uff08ROC\uff09**\uff1a\u4f7f\u53d8\u6362\u6536\u655b\u7684 $s$ \u503c\u533a\u57df\uff1b
\n- **\u6781\u70b9\uff08Pole\uff09**\uff1a\u4f7f $F(s)$ \u5206\u6bcd\u4e3a\u96f6\u3001\u53d8\u6362\u53d1\u6563\u5230\u65e0\u7a77\u7684\u70b9\uff1b
\n- **\u96f6\u70b9\uff08Zero\uff09**\uff1a\u4f7f $F(s)$ \u5206\u5b50\u4e3a\u96f6\u3001\u53d8\u6362\u4e3a\u96f6\u7684\u70b9\u3002<\/p>\n

\u6781\u70b9\u7684\u4f4d\u7f6e\u76f4\u63a5\u51b3\u5b9a\u4e86\u7cfb\u7edf\u7684\u7a33\u5b9a\u6027\uff1a\u6240\u6709\u6781\u70b9\u4f4d\u4e8e\u5de6\u534a\u5e73\u9762\uff08$\\text{Re}(s) < 0$\uff09\u65f6\u7cfb\u7edf\u7a33\u5b9a\uff1b\u4efb\u4e00\u6781\u70b9\u4f4d\u4e8e\u53f3\u534a\u5e73\u9762\u65f6\u7cfb\u7edf\u53d1\u6563\u3002\n\nZ \u53d8\u6362\u7684\u51e0\u4f55\u89e3\u91ca\u7c7b\u4f3c\uff1a$z = re^{j\\omega}$\uff0c$r$ \u63a7\u5236\u5e45\u5ea6\u7f29\u653e\uff0c$\\omega$ \u63a7\u5236\u76f8\u4f4d\u65cb\u8f6c\u3002\u6536\u655b\u57df\u4e3a $|z| > R$\uff08\u53f3\u8fb9\u5e8f\u5217\uff09\u6216 $|z| < R$\uff08\u5de6\u8fb9\u5e8f\u5217\uff09\u7684\u73af\u5f62\/\u5916\u90e8\u533a\u57df\u3002\u6781\u70b9\u4f4d\u4e8e\u5355\u4f4d\u5706\u5185\u65f6\u79bb\u6563\u7cfb\u7edf\u7a33\u5b9a\u3002\n\n### 7.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u9898 7.1 \u53d1\u6563\u51fd\u6570\u7684\u62c9\u666e\u62c9\u65af\u53d8\u6362\u2014\u2014\u6781\u70b9\u4e0e\u6536\u655b\u57df\u5206\u6790\n\n\u7ed9\u5b9a\u6307\u6570\u53d1\u6563\u51fd\u6570 $f(t) = e^{2t}$\uff08$t \\geq 0$\uff09\uff0c\u8ba1\u7b97\u5176\u62c9\u666e\u62c9\u65af\u53d8\u6362\u5e76\u5206\u6790\u6536\u655b\u57df\u4e0e\u6781\u70b9\u3002\n\n**\u89e3**\uff1a\u4ee3\u5165\u62c9\u666e\u62c9\u65af\u53d8\u6362\u5b9a\u4e49\u5f0f (7.1)\uff1a\n\n$$F(s) = \\int_0^{\\infty} e^{2t} \\cdot e^{-st}\\,dt = \\int_0^{\\infty} e^{-(s-2)t}\\,dt$$\n\n\u4ee4 $a = s - 2 = (\\sigma - 2) + j\\omega$\uff0c\u5219\uff1a\n\n$$F(s) = \\int_0^{\\infty} e^{-at}\\,dt = \\left[-\\frac{1}{a}e^{-at}\\right]_{t=0}^{t=\\infty}$$\n\n\u5f53 $t \\to \\infty$ \u65f6\uff0c$e^{-at} \\to 0$ \u8981\u6c42 $\\text{Re}(a) > 0$\uff0c\u5373 $\\text{Re}(s - 2) > 0$\uff0c\u4ea6\u5373 $\\sigma > 2$\u3002\u5728\u6b64\u6761\u4ef6\u4e0b\uff1a<\/p>\n

$$F(s) = 0 - \\left(-\\frac{1}{a}\\right) = \\frac{1}{a} = \\frac{1}{s - 2}$$<\/p>\n

\u56e0\u6b64\uff1a<\/p>\n

$$\\mathcal{L}\\{e^{2t}\\} = \\frac{1}{s - 2}, \\quad \\text{ROC: } \\text{Re}(s) > 2, \\quad \\text{Pole: } s = 2$$<\/p>\n

**\u5206\u6790**\uff1a\u5085\u91cc\u53f6\u53d8\u6362\u5bf9\u5e94 $\\sigma = 0$\uff0c\u800c $s = j\\omega$ \u7684\u5b9e\u90e8\u4e3a 0\uff0c\u5c0f\u4e8e 2\uff0c\u4e0d\u5728\u6536\u655b\u57df\u5185\u2014\u2014\u8fd9\u89e3\u91ca\u4e86\u4e3a\u4f55 $e^{2t}$ \u7684\u5085\u91cc\u53f6\u53d8\u6362\u4e0d\u5b58\u5728\u3002\u62c9\u666e\u62c9\u65af\u53d8\u6362\u901a\u8fc7\u5f15\u5165\u5b9e\u90e8\u81ea\u7531\u5ea6 $\\sigma$\uff0c\u5c06\u79ef\u5206\u8def\u5f84\u4ece\u865a\u8f74\u63a8\u5e7f\u5230\u590d\u5e73\u9762\u53f3\u534a\u5e73\u9762\uff0c\u4ece\u800c\u80fd\u591f\u5904\u7406\u53d1\u6563\u4fe1\u53f7\u3002
\n```<\/p>\n

```ad-example
\ntitle: \u4f8b\u9898 7.2 \u79bb\u6563\u5e8f\u5217\u7684 Z \u53d8\u6362\u2014\u2014\u6536\u655b\u57df\u4e0e\u7a33\u5b9a\u6027\u5206\u6790<\/p>\n

\u7ed9\u5b9a\u79bb\u6563\u5e8f\u5217 $x[n] = (0.5)^n u[n]$\uff0c\u5176\u4e2d $u[n]$ \u4e3a\u5355\u4f4d\u9636\u8dc3\u51fd\u6570\uff08$n < 0$ \u65f6\u4e3a 0\uff0c$n \\geq 0$ \u65f6\u4e3a 1\uff09\u3002\u8ba1\u7b97\u5176 Z \u53d8\u6362\u5e76\u5206\u6790\u6536\u655b\u57df\u4e0e\u7a33\u5b9a\u6027\u3002\n\n**\u89e3**\uff1a\u4ee3\u5165 Z \u53d8\u6362\u5b9a\u4e49\u5f0f (7.2)\uff1a\n\n$$X(z) = \\sum_{n=0}^{\\infty} (0.5)^n z^{-n} = \\sum_{n=0}^{\\infty} (0.5 z^{-1})^n$$\n\n\u6b64\u4e3a\u51e0\u4f55\u7ea7\u6570\u3002\u5f53 $|0.5 z^{-1}| < 1$ \u5373 $|z| > 0.5$ \u65f6\u7ea7\u6570\u6536\u655b\uff1a<\/p>\n

$$X(z) = \\frac{1}{1 - 0.5z^{-1}} = \\frac{z}{z - 0.5}, \\quad \\text{ROC: } |z| > 0.5$$<\/p>\n

\u6536\u655b\u57df\u662f\u4ee5\u539f\u70b9\u4e3a\u5706\u5fc3\u3001\u534a\u5f84\u4e3a 0.5 \u7684\u5706\u5916\u90e8\u533a\u57df\u3002\u5355\u4f4d\u5706 $|z| = 1$ \u5b8c\u5168\u4f4d\u4e8e\u6536\u655b\u57df\u5185\uff0c\u610f\u5473\u7740\u8be5\u5e8f\u5217\u7684\u79bb\u6563\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362\uff08DTFT\uff0c\u5bf9\u5e94 $z = e^{j\\omega}$\uff09\u5b58\u5728\u3002\u6781\u70b9\u4f4d\u4e8e $z = 0.5$\uff0c\u5728\u5355\u4f4d\u5706\u5185\u90e8\uff0c\u6545\u8be5\u7cfb\u7edf\u7a33\u5b9a\u3002
\n```<\/p>\n

### 7.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528<\/p>\n

\u62c9\u666e\u62c9\u65af\u53d8\u6362\u662f\u63a7\u5236\u7406\u8bba\u7684\u57fa\u77f3\u3002\u5728\u53cd\u9988\u63a7\u5236\u7cfb\u7edf\u4e2d\uff0c\u7cfb\u7edf\u7684\u4f20\u9012\u51fd\u6570 $H(s)$ \u7684\u6781\u70b9\u4f4d\u7f6e\u76f4\u63a5\u51b3\u5b9a\u7a33\u5b9a\u6027\uff1a<\/p>\n

- \u6240\u6709\u6781\u70b9\u4f4d\u4e8e\u5de6\u534a\u5e73\u9762\uff08$\\text{Re}(s) < 0$\uff09\uff1a\u7cfb\u7edf\u7a33\u5b9a\uff0c\u51b2\u6fc0\u54cd\u5e94\u6307\u6570\u8870\u51cf\uff1b\n- \u5b58\u5728\u6781\u70b9\u4f4d\u4e8e\u53f3\u534a\u5e73\u9762\uff08$\\text{Re}(s) > 0$\uff09\uff1a\u7cfb\u7edf\u53d1\u6563\uff0c\u51b2\u6fc0\u54cd\u5e94\u6307\u6570\u589e\u957f\uff1b
\n- \u6781\u70b9\u4f4d\u4e8e\u865a\u8f74\uff08$\\text{Re}(s) = 0$\uff09\uff1a\u7cfb\u7edf\u4e34\u754c\u7a33\u5b9a\uff0c\u51b2\u6fc0\u54cd\u5e94\u7b49\u5e45\u632f\u8361\u3002<\/p>\n

Z \u53d8\u6362\u662f\u6570\u5b57\u4fe1\u53f7\u5904\u7406\u7684\u6838\u5fc3\u3002\u6570\u5b57\u6ee4\u6ce2\u5668\u7684\u9891\u7387\u54cd\u5e94\u7531 $H(z)$ \u5728\u5355\u4f4d\u5706\u4e0a\u7684\u53d6\u503c\u51b3\u5b9a\uff0c\u7a33\u5b9a\u6027\u7531\u6781\u70b9\u662f\u5426\u5168\u90e8\u4f4d\u4e8e\u5355\u4f4d\u5706\u5185\u51b3\u5b9a\u3002IIR \u6ee4\u6ce2\u5668\u8bbe\u8ba1\u672c\u8d28\u4e0a\u662f\u5728 $z$ \u5e73\u9762\u4e0a\u914d\u7f6e\u6781\u70b9\u548c\u96f6\u70b9\uff0c\u4ee5\u903c\u8fd1\u76ee\u6807\u9891\u7387\u54cd\u5e94\u3002<\/p>\n

---<\/p>\n

## \u7b2c\u516b\u7ae0 \u5377\u79ef\u7684\u672c\u8d28 \u2014\u2014 \"\u6ed1\u52a8\u7684\u5185\u79ef\"<\/p>\n

### 8.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49<\/p>\n

\u5377\u79ef\uff08Convolution\uff09\u662f\u4fe1\u53f7\u5904\u7406\u3001\u63a7\u5236\u7406\u8bba\u548c\u6df1\u5ea6\u5b66\u4e60\u4e2d\u6700\u6838\u5fc3\u7684\u8fd0\u7b97\u4e4b\u4e00 $^{[17]}$\u3002\u4ece\u5185\u79ef\u89c6\u89d2\u770b\uff0c\u5377\u79ef\u7684\u672c\u8d28\u662f**\u6ed1\u52a8\u7a97\u53e3\u4e0a\u7684\u5185\u79ef\u5e8f\u5217**\u3002<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 8.1 \u5377\u79ef
\n\u8bbe $f, g: \\mathbb{R} \\to \\mathbb{R}$ \u4e3a\u4e24\u4e2a\u8fde\u7eed\u51fd\u6570\uff0c\u5176**\u5377\u79ef**\u5b9a\u4e49\u4e3a\uff1a<\/p>\n

$$(f * g)(t) = \\int_{-\\infty}^{\\infty} f(\\tau) g(t - \\tau)\\,d\\tau \\tag{8.1}$$<\/p>\n

\u5bf9\u4e8e\u79bb\u6563\u5e8f\u5217 $x, h: \\mathbb{Z} \\to \\mathbb{R}$\uff0c\u5176**\u79bb\u6563\u5377\u79ef**\u5b9a\u4e49\u4e3a\uff1a<\/p>\n

$$(x * h)[n] = \\sum_{k=-\\infty}^{\\infty} x[k]\\, h[n - k] \\tag{8.2}$$
\n```<\/p>\n

```ad-theorem
\ntitle: \u547d\u9898 8.1 \u5377\u79ef\u7684\u5185\u79ef\u89e3\u91ca
\n\u5728\u56fa\u5b9a\u65f6\u523b $t$\uff0c\u5377\u79ef\u8fd0\u7b97 $(f * g)(t)$ \u7b49\u4ef7\u4e8e\u51fd\u6570 $f(\\tau)$ \u4e0e\u7ffb\u8f6c\u5e73\u79fb\u540e\u7684 $g(\\tau)$ \u4e4b\u95f4\u7684\u5185\u79ef\uff1a<\/p>\n

$$(f * g)(t) = \\langle f(\\tau), g(t - \\tau) \\rangle = \\int f(\\tau) g(t - \\tau)\\,d\\tau \\tag{8.3}$$<\/p>\n

\u5176\u4e2d\u7ffb\u8f6c\u64cd\u4f5c $g(\\tau) \\to g(-\\tau)$ \u786e\u4fdd\u7cfb\u7edf\u6ee1\u8db3\u56e0\u679c\u6027\u2014\u2014\u5f53\u524d\u8f93\u51fa\u4ec5\u4f9d\u8d56\u4e8e\u5f53\u524d\u53ca\u8fc7\u53bb\u7684\u8f93\u5165\u3002
\n```<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 8.2 \u4e92\u76f8\u5173
\n\u4e0e\u5377\u79ef\u5bc6\u5207\u76f8\u5173\u7684\u8fd0\u7b97\u662f**\u4e92\u76f8\u5173\uff08Cross-Correlation\uff09**\uff1a<\/p>\n

$$(f \\star g)(t) = \\int_{-\\infty}^{\\infty} f(\\tau) g(\\tau + t)\\,d\\tau \\tag{8.4}$$<\/p>\n

\u4e92\u76f8\u5173\u4e0d\u542b\u7ffb\u8f6c\u64cd\u4f5c\uff0c\u76f4\u63a5\u8ba1\u7b97\u4fe1\u53f7\u5728\u4e0d\u540c\u504f\u79fb\u4e0b\u7684\u5185\u79ef\uff0c\u5e38\u7528\u4e8e\u6a21\u677f\u5339\u914d\u548c\u76f8\u4f3c\u5ea6\u68c0\u6d4b\u3002
\n```<\/p>\n

### 8.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf<\/p>\n

\u5377\u79ef\u7684\u51e0\u4f55\u8fc7\u7a0b\u53ef\u5206\u89e3\u4e3a\u56db\u4e2a\u6b65\u9aa4\uff1a<\/p>\n

1. **\u7ffb\u8f6c**\uff1a\u5c06\u6838\u51fd\u6570 $g(\\tau)$ \u7ffb\u8f6c\u4e3a $g(-\\tau)$\uff0c\u4f7f\u8fd0\u7b97\u6ee1\u8db3\u56e0\u679c\u6027\uff1b
\n2. **\u5e73\u79fb**\uff1a\u5c06\u7ffb\u8f6c\u540e\u7684\u6838\u5e73\u79fb $t$\uff0c\u5f97\u5230 $g(t - \\tau)$\uff1b
\n3. **\u76f8\u4e58**\uff1a\u5c06 $f(\\tau)$ \u4e0e $g(t - \\tau)$ \u9010\u70b9\u76f8\u4e58\uff1b
\n4. **\u79ef\u5206**\uff1a\u5bf9\u4e58\u79ef\u6c42\u548c\uff08\u79ef\u5206\uff09\uff0c\u5f97\u5230\u8be5\u65f6\u523b\u7684\u5185\u79ef\u503c\u3002<\/p>\n

\u968f\u7740 $t$ \u7684\u53d8\u5316\uff0c\u6838\u51fd\u6570\u6cbf\u65f6\u95f4\u8f74\u6ed1\u52a8\uff0c\u5728\u6bcf\u4e2a\u4f4d\u7f6e\u8ba1\u7b97\u4fe1\u53f7\u4e0e\u6838\u7684\u5185\u79ef\u3002\u5377\u79ef\u7ed3\u679c $y(t)$ \u5373\u4e3a\u5185\u79ef\u503c\u968f\u6ed1\u52a8\u4f4d\u7f6e\u7684\u53d8\u5316\u66f2\u7ebf\u3002\u5185\u79ef\u503c\u5927\u7684\u4f4d\u7f6e\uff0c\u8868\u793a\u4fe1\u53f7\u5c40\u90e8\u4e0e\u6838\u7684\u6ce2\u5f62\u6700\u4e3a\u76f8\u4f3c\u2014\u2014\u8fd9\u6b63\u662f**\u5339\u914d\u6ee4\u6ce2\uff08Matched Filter\uff09** \u7684\u539f\u7406\u3002<\/p>\n

\u5728\u56fe\u50cf\u5904\u7406\u4e2d\uff0c\u4e8c\u7ef4\u5377\u79ef\u6838\uff08Kernel\uff09\u5728\u56fe\u50cf\u4e0a\u6ed1\u52a8\uff0c\u6bcf\u4e2a\u4f4d\u7f6e\u8ba1\u7b97 $k \\times k$ \u90bb\u57df\u4e0e\u6838\u7684\u4e8c\u7ef4\u5185\u79ef\uff0c\u8f93\u51fa\u4e00\u5f20\"\u54cd\u5e94\u56fe\"\uff08Feature Map\uff09\u3002\u54cd\u5e94\u503c\u9ad8\u7684\u533a\u57df\u8868\u793a\u8be5\u5c40\u90e8\u56fe\u50cf\u5757\u4e0e\u5377\u79ef\u6838\u7684\u6a21\u5f0f\u6700\u4e3a\u5339\u914d\u3002<\/p>\n

### 8.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)<\/p>\n

```ad-example
\ntitle: \u4f8b\u9898 8.1 \u79bb\u6563\u5e8f\u5217\u7684\u6ed1\u52a8\u5185\u79ef\u5377\u79ef\u2014\u2014\u9010\u70b9\u624b\u7b97<\/p>\n

\u7ed9\u5b9a\u8f93\u5165\u5e8f\u5217 $x[n] = [1, 2, 3]$\uff08$n = 0, 1, 2$\uff09\u548c\u5377\u79ef\u6838 $h[n] = [0.5, 1, 0.5]$\uff08$n = 0, 1, 2$\uff09\u3002\u8ba1\u7b97\u5377\u79ef $y[n] = (x * h)[n]$\u3002<\/p>\n

**\u89e3**\uff1a\u6839\u636e\u79bb\u6563\u5377\u79ef\u516c\u5f0f (8.2)\uff0c\u9010\u70b9\u8ba1\u7b97\uff1a<\/p>\n

$n = 0$\uff1a
\n$$y[0] = \\sum_{k} x[k]h[0-k] = x[0]h[0] = 1 \\times 0.5 = 0.5$$<\/p>\n

$n = 1$\uff1a
\n$$y[1] = x[0]h[1] + x[1]h[0] = 1 \\times 1 + 2 \\times 0.5 = 2$$<\/p>\n

$n = 2$\uff1a
\n$$y[2] = x[0]h[2] + x[1]h[1] + x[2]h[0] = 1 \\times 0.5 + 2 \\times 1 + 3 \\times 0.5 = 4$$<\/p>\n

$n = 3$\uff1a
\n$$y[3] = x[1]h[2] + x[2]h[1] = 2 \\times 0.5 + 3 \\times 1 = 4$$<\/p>\n

$n = 4$\uff1a
\n$$y[4] = x[2]h[2] = 3 \\times 0.5 = 1.5$$<\/p>\n

\u56e0\u6b64 $y[n] = [0.5, 2, 4, 4, 1.5]$\u3002\u5728 $n = 2, 3$ \u5904\u5377\u79ef\u503c\u6700\u5927\uff08\u4e3a 4\uff09\uff0c\u6b64\u65f6\u8f93\u5165\u5e8f\u5217 $[1, 2, 3]$ \u4e0e\u7ffb\u8f6c\u6838 $[0.5, 1, 0.5]$ \u7684\u91cd\u53e0\u533a\u57df\u6700\u5927\uff0c\u5185\u79ef\u503c\u8fbe\u5230\u5cf0\u503c\u3002
\n```<\/p>\n

```ad-example
\ntitle: \u4f8b\u9898 8.2 Sobel \u8fb9\u7f18\u68c0\u6d4b\u2014\u2014\u4e8c\u7ef4\u5377\u79ef\u4f5c\u4e3a\u5185\u79ef\u6a21\u677f<\/p>\n

Sobel \u7b97\u5b50\u7531\u4e24\u4e2a $3 \\times 3$ \u7684\u5377\u79ef\u6838\u7ec4\u6210\uff0c\u5206\u522b\u68c0\u6d4b\u6c34\u5e73\u548c\u5782\u76f4\u65b9\u5411\u7684\u8fb9\u7f18\uff1a<\/p>\n

$$S_x = \\begin{bmatrix} 1 & 0 & -1 \\\\ 2 & 0 & -2 \\\\ 1 & 0 & -1 \\end{bmatrix}, \\quad S_y = \\begin{bmatrix} 1 & 2 & 1 \\\\ 0 & 0 & 0 \\\\ -1 & -2 & -1 \\end{bmatrix}$$<\/p>\n

\u7ed9\u5b9a $3 \\times 3$ \u5c40\u90e8\u56fe\u50cf\u5757\uff08\u7070\u5ea6\u503c\uff09\uff1a<\/p>\n

$$I = \\begin{bmatrix} 10 & 20 & 30 \\\\ 10 & 20 & 30 \\\\ 10 & 20 & 30 \\end{bmatrix}$$<\/p>\n

\u8be5\u56fe\u50cf\u5757\u5448\u73b0\u6c34\u5e73\u65b9\u5411\u7684\u4eae\u5ea6\u6e10\u53d8\uff08\u4ece\u5de6\u5230\u53f3\u53d8\u4eae\uff09\uff0c\u5782\u76f4\u65b9\u5411\u4eae\u5ea6\u5747\u5300\u3002<\/p>\n

**\u89e3**\uff1a\u8ba1\u7b97 Sobel X \u7b97\u5b50\u4e0e\u56fe\u50cf\u5757\u7684\u4e8c\u7ef4\u5185\u79ef\uff1a<\/p>\n

$$G_x = \\sum_{i=1}^{3} \\sum_{j=1}^{3} S_x(i,j) \\cdot I(i,j)$$<\/p>\n

$$= (1 \\times 10) + (0 \\times 20) + (-1 \\times 30) + (2 \\times 10) + (0 \\times 20) + (-2 \\times 30) + (1 \\times 10) + (0 \\times 20) + (-1 \\times 30)$$<\/p>\n

$$= 10 + 0 - 30 + 20 + 0 - 60 + 10 + 0 - 30 = -80$$<\/p>\n

\u8ba1\u7b97 Sobel Y \u7b97\u5b50\u7684\u4e8c\u7ef4\u5185\u79ef\uff1a<\/p>\n

$$G_y = (1 \\times 10) + (2 \\times 20) + (1 \\times 30) + (0 \\times 10) + (0 \\times 20) + (0 \\times 30) + (-1 \\times 10) + (-2 \\times 20) + (-1 \\times 30)$$<\/p>\n

$$= 10 + 40 + 30 + 0 + 0 + 0 - 10 - 40 - 30 = 0$$<\/p>\n

\u8fb9\u7f18\u5f3a\u5ea6\u4e3a\uff1a<\/p>\n

$$\\|\\nabla I\\| = \\sqrt{G_x^2 + G_y^2} = \\sqrt{(-80)^2 + 0^2} = 80$$<\/p>\n

**\u5206\u6790**\uff1a$|G_x| = 80$ \u8f83\u5927\uff0c\u8868\u660e\u6c34\u5e73\u65b9\u5411\u5b58\u5728\u663e\u8457\u4eae\u5ea6\u53d8\u5316\uff08\u5782\u76f4\u8fb9\u7f18\uff09\uff1b$G_y = 0$ \u8868\u660e\u5782\u76f4\u65b9\u5411\u4eae\u5ea6\u5747\u5300\u3002Sobel \u8fb9\u7f18\u68c0\u6d4b\u7684\u672c\u8d28\u662f\u7528\u4e24\u4e2a\u6b63\u4ea4\u7684\u5377\u79ef\u6838\uff08\u5185\u79ef\u6a21\u677f\uff09\u5728\u56fe\u50cf\u4e0a\u6ed1\u52a8\uff0c\u8ba1\u7b97\u6bcf\u4e2a\u50cf\u7d20\u90bb\u57df\u4e0e\u6a21\u677f\u7684\u4e8c\u7ef4\u5185\u79ef\uff0c\u5185\u79ef\u5e45\u503c\u5927\u7684\u4f4d\u7f6e\u5373\u4e3a\u8fb9\u7f18\u6240\u5728\u3002
\n```<\/p>\n

### 8.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528<\/p>\n

<\/p>\n

> **\u56fe 4\uff1a\u6ed1\u52a8\u5185\u79ef\u4e0e\u5339\u914d\u6ee4\u6ce2 (Matched Filter)**\u3002\u56fe\u4e2d\u84dd\u8272\u66f2\u7ebf\u4e3a\u542b\u566a\u968f\u673a\u5e8f\u5217 $x[n]$\uff0c\u7ea2\u8272\u66f2\u7ebf\u4e3a\u5377\u79ef\u54cd\u5e94\u3002\u6a21\u677f\u8109\u51b2 $h[n] = [0, 0.35, 1.0, 0.35, 0]$ \u6cbf\u65f6\u95f4\u8f74\u6ed1\u52a8\uff0c\u5728\u6bcf\u4e2a\u4f4d\u7f6e\u8ba1\u7b97 $\\sum x[k]h[n-k]$\u3002\u6a59\u8272\u6807\u8bb0\u5904\uff08$n \\approx 110, 265, 340$\uff09\u5377\u79ef\u503c\u8fbe\u5230\u5cf0\u503c\uff0c\u8868\u660e\u8fd9\u4e9b\u4f4d\u7f6e\u7684\u4fe1\u53f7\u5c40\u90e8\u6ce2\u5f62\u4e0e\u6a21\u677f\u6700\u4e3a\u5339\u914d\u3002\u73b0\u4ee3\u96f7\u8fbe\u4fe1\u53f7\u6355\u83b7\u7684\u6838\u5fc3\u539f\u7406\u5373\u6e90\u4e8e\u6b64\u6ed1\u52a8\u6295\u5f71\u673a\u5236\u3002<\/p>\n

<\/p>\n

> **\u56fe 5\uff1a\u4e8c\u7ef4\u5377\u79ef\u63d0\u53d6\u8fb9\u7f18\u7279\u5f81 (Sobel Edge Detection)**\u3002Sobel \u7b97\u5b50\u662f\u4e00\u5bf9\u6b63\u4ea4\u7684 $3 \\times 3$ \u5fae\u5206\u6a21\u677f\uff0c\u5206\u522b\u6cbf $x$ \u548c $y$ \u65b9\u5411\u68c0\u6d4b\u4eae\u5ea6\u68af\u5ea6\u3002\u5f53\u6a21\u677f\u5728\u7070\u5ea6\u56fe\u50cf\u4e0a\u6ed1\u52a8\u65f6\uff0c\u5728\u5e73\u5766\u533a\u57df\u6b63\u8d1f\u6295\u5f71\u76f8\u4e92\u62b5\u6d88\uff08\u5185\u79ef\u63a5\u8fd1\u96f6\uff09\uff0c\u800c\u5728\u8fb9\u7f18\u5904\u50cf\u7d20\u9636\u8dc3\u5bfc\u81f4\u5185\u79ef\u5e45\u503c\u663e\u8457\u589e\u5927\u3002\u901a\u8fc7 $\\|\\nabla I\\| = \\sqrt{G_x^2 + G_y^2}$ \u5408\u5e76\u4e24\u6b63\u4ea4\u5206\u91cf\uff0c\u5373\u53ef\u63d0\u53d6\u51fa\u7269\u7406\u4e16\u754c\u7684\u8fb9\u7f18\u4fe1\u606f\u3002\u8fd9\u662f\u8ba1\u7b97\u673a\u89c6\u89c9\u4e2d\u7279\u5f81\u63d0\u53d6\u7684\u5e95\u5c42\u57fa\u7840\u3002<\/p>\n

---<\/p>\n

## \u7b2c\u4e5d\u7ae0 \u79bb\u6563\u4f59\u5f26\u53d8\u6362\u4e0e JPEG \u538b\u7f29<\/p>\n

### 9.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49<\/p>\n

\u79bb\u6563\u4f59\u5f26\u53d8\u6362\uff08Discrete Cosine Transform, DCT\uff09\u662f JPEG \u56fe\u50cf\u538b\u7f29\u6807\u51c6\u7684\u6838\u5fc3\u7b97\u6cd5 $^{[18][19]}$\u3002\u4ece\u5185\u79ef\u89c6\u89d2\u770b\uff0cDCT \u5c06\u56fe\u50cf\u5757\u5411\u4e00\u7ec4\u79bb\u6563\u4f59\u5f26\u57fa\u51fd\u6570\u505a\u6b63\u4ea4\u6295\u5f71\uff0c\u5c06\u7a7a\u95f4\u57df\u7684\u50cf\u7d20\u503c\u53d8\u6362\u4e3a\u9891\u57df\u7cfb\u6570\u3002<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 9.1 \u4e8c\u7ef4 DCT
\n\u8bbe $f(x, y)$ \u4e3a $N \\times N$ \u7684\u56fe\u50cf\u5757\uff08$x, y = 0, 1, \\dots, N-1$\uff09\uff0c\u5176\u4e8c\u7ef4 DCT \u5b9a\u4e49\u4e3a\uff1a<\/p>\n

$$F(u, v) = \\frac{2}{N} C(u) C(v) \\sum_{x=0}^{N-1} \\sum_{y=0}^{N-1} f(x, y) \\cos\\left[\\frac{(2x+1)u\\pi}{2N}\\right] \\cos\\left[\\frac{(2y+1)v\\pi}{2N}\\right] \\tag{9.1}$$<\/p>\n

\u5176\u4e2d $u, v = 0, 1, \\dots, N-1$ \u4e3a\u9891\u7387\u7d22\u5f15\uff0c\u5f52\u4e00\u5316\u7cfb\u6570\u4e3a\uff1a<\/p>\n

$$C(k) = \\begin{cases} 1\/\\sqrt{2}, & k = 0 \\\\ 1, & k \\neq 0 \\end{cases}$$
\n```<\/p>\n

```ad-theorem
\ntitle: \u547d\u9898 9.1 DCT \u4f5c\u4e3a\u6b63\u4ea4\u6295\u5f71
\n\u5b9a\u4e49 $N \\times N$ \u4e2a DCT \u57fa\u51fd\u6570\uff1a<\/p>\n

$$B_{u,v}(x, y) = \\frac{2}{N} C(u) C(v) \\cos\\left[\\frac{(2x+1)u\\pi}{2N}\\right] \\cos\\left[\\frac{(2y+1)v\\pi}{2N}\\right]$$<\/p>\n

\u5219 $\\{B_{u,v}\\}$ \u6784\u6210 $\\mathbb{R}^{N \\times N}$ \u4e0a\u7684\u4e00\u7ec4\u5b8c\u5907\u6b63\u4ea4\u57fa\uff0c\u6ee1\u8db3\uff1a<\/p>\n

$$\\langle B_{u,v}, B_{u',v'} \\rangle = \\sum_{x=0}^{N-1} \\sum_{y=0}^{N-1} B_{u,v}(x, y) B_{u',v'}(x, y) = \\delta_{u,u'} \\delta_{v,v'}$$<\/p>\n

DCT \u7cfb\u6570 $F(u, v)$ \u6b63\u662f\u56fe\u50cf\u5757 $f$ \u5728\u57fa\u51fd\u6570 $B_{u,v}$ \u4e0a\u7684\u6295\u5f71\uff1a<\/p>\n

$$F(u, v) = \\langle f, B_{u,v} \\rangle = \\sum_{x=0}^{N-1} \\sum_{y=0}^{N-1} f(x, y) B_{u,v}(x, y) \\tag{9.2}$$
\n```<\/p>\n

```ad-theorem
\ntitle: \u547d\u9898 9.2 \u80fd\u91cf\u96c6\u4e2d\u6027
\n\u5bf9\u4e8e\u81ea\u7136\u56fe\u50cf\uff0cDCT \u7cfb\u6570\u7684\u80fd\u91cf\u4e3b\u8981\u96c6\u4e2d\u5728\u4f4e\u9891\u533a\u57df\uff08$u, v$ \u8f83\u5c0f\uff09\uff0c\u9ad8\u9891\u7cfb\u6570\uff08$u, v$ \u8f83\u5927\uff09\u5e45\u503c\u8d8b\u8fd1\u4e8e\u96f6\u3002JPEG \u538b\u7f29\u5229\u7528\u8fd9\u4e00\u7279\u6027\uff0c\u901a\u8fc7\u91cf\u5316\u820d\u5f03\u5fae\u5c0f\u7684\u9ad8\u9891\u7cfb\u6570\uff0c\u5728\u4fdd\u6301\u89c6\u89c9\u8d28\u91cf\u7684\u524d\u63d0\u4e0b\u5b9e\u73b0\u5927\u5e45\u538b\u7f29\u3002
\n```<\/p>\n

### 9.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf<\/p>\n

\u4e00\u4e2a $8 \\times 8$ \u7684\u56fe\u50cf\u5757\u53ef\u89c6\u4e3a 64 \u7ef4\u7a7a\u95f4\u4e2d\u7684\u5411\u91cf\u3002DCT \u57fa\u51fd\u6570\u6784\u6210\u8be5 64 \u7ef4\u7a7a\u95f4\u4e2d\u7684\u4e00\u7ec4\u5b8c\u5907\u6b63\u4ea4\u57fa\uff1a<\/p>\n

- **$B_{0,0}$\uff08DC \u57fa\uff09**\uff1a\u5e38\u6570\u51fd\u6570\uff0c\u5bf9\u5e94\u56fe\u50cf\u5757\u7684\u5e73\u5747\u4eae\u5ea6\uff1b
\n- **\u4f4e\u9891\u57fa**\uff08$u, v$ \u8f83\u5c0f\uff09\uff1a\u5e73\u6ed1\u6e10\u53d8\u6a21\u5f0f\uff0c\u5bf9\u5e94\u56fe\u50cf\u7684\u5927\u5c3a\u5ea6\u7ed3\u6784\uff1b
\n- **\u9ad8\u9891\u57fa**\uff08$u, v$ \u8f83\u5927\uff09\uff1a\u5bc6\u96c6\u632f\u8361\u6a21\u5f0f\uff0c\u5bf9\u5e94\u56fe\u50cf\u7684\u7ec6\u8282\u7eb9\u7406\u548c\u566a\u58f0\u3002<\/p>\n

\u5c06\u56fe\u50cf\u5757\u5411\u91cf\u5411\u8fd9 64 \u4e2a\u57fa\u65b9\u5411\u6295\u5f71\uff0c\u5f97\u5230 64 \u4e2a DCT \u7cfb\u6570\u3002\u5bf9\u4e8e\u81ea\u7136\u56fe\u50cf\uff0c\u6295\u5f71\u80fd\u91cf\u9ad8\u5ea6\u96c6\u4e2d\u5728\u4f4e\u9891\u7cfb\u6570\uff08\u5de6\u4e0a\u89d2\uff09\uff0c\u9ad8\u9891\u7cfb\u6570\uff08\u53f3\u4e0b\u89d2\uff09\u63a5\u8fd1\u4e8e\u96f6\u3002JPEG \u538b\u7f29\u901a\u8fc7\u91cf\u5316\u5c06\u5fae\u5c0f\u7684\u9ad8\u9891\u7cfb\u6570\u7f6e\u96f6\uff0c\u4ec5\u4fdd\u7559\u5c11\u6570\u4f4e\u9891\u7cfb\u6570\u5373\u53ef\u8fd1\u4f3c\u91cd\u5efa\u539f\u56fe\u50cf\u5757\u3002<\/p>\n

### 9.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)<\/p>\n

```ad-example
\ntitle: \u4f8b\u9898 9.1 $2 \\times 2$ \u56fe\u50cf\u5757\u7684 DCT \u6295\u5f71\u7cfb\u6570\u624b\u7b97<\/p>\n

\u4e3a\u5c55\u793a DCT \u7684\u6295\u5f71\u672c\u8d28\uff0c\u8003\u8651 $N = 2$ \u7684\u5fae\u578b\u56fe\u50cf\u5757\u3002$2 \\times 2$ DCT \u57fa\u77e9\u9635\u4e3a\uff1a<\/p>\n

$$T = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix}$$<\/p>\n

$T$ \u662f\u6b63\u4ea4\u77e9\u9635\uff0c\u6ee1\u8db3 $T^T T = I$\u3002\u7ed9\u5b9a\u7070\u5ea6\u56fe\u50cf\u5757\uff1a<\/p>\n

$$I = \\begin{bmatrix} 100 & 80 \\\\ 60 & 40 \\end{bmatrix}$$<\/p>\n

\u4e8c\u7ef4 DCT \u53ef\u901a\u8fc7\u77e9\u9635\u4e58\u6cd5\u5b9e\u73b0\uff1a$F = T \\cdot I \\cdot T^T$\u3002<\/p>\n

**\u89e3**\uff1a<\/p>\n

**\u6b65\u9aa4 1**\uff1a\u8ba1\u7b97 $T \\cdot I$\u3002<\/p>\n

$$T \\cdot I = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix} \\begin{bmatrix} 100 & 80 \\\\ 60 & 40 \\end{bmatrix} = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 160 & 120 \\\\ 40 & 40 \\end{bmatrix}$$<\/p>\n

**\u6b65\u9aa4 2**\uff1a\u8ba1\u7b97 $(T \\cdot I) \\cdot T^T$\u3002<\/p>\n

$$F = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 160 & 120 \\\\ 40 & 40 \\end{bmatrix} \\cdot \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix} = \\frac{1}{2} \\begin{bmatrix} 160 & 120 \\\\ 40 & 40 \\end{bmatrix} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix}$$<\/p>\n

$$= \\frac{1}{2} \\begin{bmatrix} 280 & 40 \\\\ 80 & 0 \\end{bmatrix} = \\begin{bmatrix} 140 & 20 \\\\ 40 & 0 \\end{bmatrix}$$<\/p>\n

**\u6b65\u9aa4 3**\uff1a\u89e3\u8bfb DCT \u7cfb\u6570\u3002<\/p>\n

- $F(0,0) = 140$\uff1aDC \u7cfb\u6570\uff0c\u5bf9\u5e94\u56fe\u50cf\u5757\u5e73\u5747\u4eae\u5ea6\u3002$(100+80+60+40)\/4 = 70$\uff0c\u4e58\u4ee5 $N = 2$ \u5f97 140\u3002
\n- $F(0,1) = 20$\uff1a\u6c34\u5e73\u65b9\u5411\u9ad8\u9891\u5206\u91cf\uff0c\u53cd\u6620\u5de6\u53f3\u50cf\u7d20\u5dee\u5f02\u3002
\n- $F(1,0) = 40$\uff1a\u5782\u76f4\u65b9\u5411\u9ad8\u9891\u5206\u91cf\uff0c\u53cd\u6620\u4e0a\u4e0b\u50cf\u7d20\u5dee\u5f02\u3002
\n- $F(1,1) = 0$\uff1a\u5bf9\u89d2\u65b9\u5411\u9ad8\u9891\u5206\u91cf\uff0c\u4e3a\u96f6\u8bf4\u660e\u65e0\u5bf9\u89d2\u7eb9\u7406\u3002<\/p>\n

**\u5173\u952e\u89c2\u5bdf**\uff1a$F(1,1) = 0$\uff0c\u5373\u5bf9\u89d2\u65b9\u5411\u9ad8\u9891\u57fa\u4e0a\u7684\u6295\u5f71\u4e3a\u96f6\u2014\u2014\u8be5\u5206\u91cf\u53ef\u5b8c\u5168\u820d\u5f03\u800c\u4e0d\u635f\u5931\u4fe1\u606f\u3002\u8fd9\u6b63\u662f JPEG \u538b\u7f29\u7684\u6838\u5fc3\u539f\u7406\uff1a\u81ea\u7136\u56fe\u50cf\u7684\u5927\u90e8\u5206\u9ad8\u9891 DCT \u7cfb\u6570\u63a5\u8fd1\u4e8e\u96f6\uff0c\u91cf\u5316\u540e\u53d8\u4e3a\u96f6\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u5e45\u538b\u7f29\u3002
\n```<\/p>\n

### 9.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528<\/p>\n

JPEG \u538b\u7f29\u6d41\u7a0b\u5982\u4e0b\uff1a<\/p>\n

1. **\u5206\u5757**\uff1a\u5c06\u56fe\u50cf\u5206\u5272\u4e3a $8 \\times 8$ \u7684\u5757\uff1b
\n2. **DCT \u53d8\u6362**\uff1a\u5bf9\u6bcf\u4e2a\u5757\u6267\u884c\u4e8c\u7ef4 DCT\uff0c\u5f97\u5230 64 \u4e2a\u9891\u57df\u7cfb\u6570\uff1b
\n3. **\u91cf\u5316**\uff1a\u7528\u91cf\u5316\u77e9\u9635\u9664\u4ee5 DCT \u7cfb\u6570\uff08\u9ad8\u9891\u91cf\u5316\u6b65\u957f\u66f4\u5927\uff09\uff0c\u5c06\u5fae\u5c0f\u7cfb\u6570\u7f6e\u96f6\uff1b
\n4. **\u71b5\u7f16\u7801**\uff1a\u5bf9\u91cf\u5316\u540e\u7684\u7cfb\u6570\u8fdb\u884c Huffman \u6216\u7b97\u672f\u7f16\u7801\u3002<\/p>\n

\u5728\u89e3\u7801\u7aef\uff0c\u901a\u8fc7\u9006 DCT \u53d8\u6362\u91cd\u5efa\u56fe\u50cf\u5757\u3002\u7531\u4e8e\u820d\u5f03\u4e86\u4eba\u773c\u4e0d\u654f\u611f\u7684\u9ad8\u9891\u5206\u91cf\uff0cJPEG \u53ef\u5728\u4fdd\u6301\u89c6\u89c9\u8d28\u91cf\u7684\u524d\u63d0\u4e0b\u5c06\u56fe\u50cf\u538b\u7f29\u81f3\u539f\u59cb\u5927\u5c0f\u7684 $1\/10$ \u751a\u81f3\u66f4\u5c0f\u3002<\/p>\n

DCT \u8fd8\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u89c6\u9891\u538b\u7f29\uff08MPEG\u3001H.264\/AVC\u3001HEVC\uff09\u3001\u97f3\u9891\u538b\u7f29\uff08MP3 \u4e2d\u7684 MDCT \u53d8\u4f53\uff09\u4ee5\u53ca\u4fe1\u53f7\u5904\u7406\u4e2d\u7684\u53bb\u76f8\u5173\u548c\u7279\u5f81\u63d0\u53d6\u3002<\/p>\n

---<\/p>\n

## \u7b2c\u5341\u7ae0 \u5c0f\u6ce2\u53d8\u6362 \u2014\u2014 \u591a\u5206\u8fa8\u7387\u5185\u79ef<\/p>\n

### 10.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49<\/p>\n

\u5085\u91cc\u53f6\u53d8\u6362\u5c06\u4fe1\u53f7\u6295\u5f71\u5230\u65e0\u9650\u5ef6\u4f38\u7684\u6b63\u5f26\u6ce2\u57fa\u5e95\u4e0a\uff0c\u83b7\u5f97\u4e86\u5168\u5c40\u9891\u7387\u4fe1\u606f\uff0c\u4f46\u4e27\u5931\u4e86\u65f6\u95f4\u5b9a\u4f4d\u80fd\u529b\u2014\u2014\u65e0\u6cd5\u4ece\u9891\u8c31\u4e2d\u5f97\u77e5\u67d0\u4e00\u9891\u7387\u6210\u5206\u5728\u4f55\u65f6\u51fa\u73b0\u3002\u5bf9\u4e8e\u97f3\u4e50\u3001\u5730\u9707\u6ce2\u3001\u5fc3\u7535\u4fe1\u53f7\u7b49\u975e\u5e73\u7a33\u4fe1\u53f7\uff0c\u8fd9\u4e00\"\u65f6\u95f4\u76f2\u533a\"\u662f\u6839\u672c\u6027\u7684\u7f3a\u9677\u3002<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 10.1 \u77ed\u65f6\u5085\u91cc\u53f6\u53d8\u6362
\n\u4e3a\u5f25\u8865\u65f6\u95f4\u5b9a\u4f4d\u7684\u7f3a\u5931\uff0c\u77ed\u65f6\u5085\u91cc\u53f6\u53d8\u6362\uff08STFT\uff09\u5f15\u5165\u7a97\u51fd\u6570 $w(t)$\uff1a<\/p>\n

$$\\text{STFT}\\{f(t)\\}(\\tau, \\omega) = \\int_{-\\infty}^{\\infty} f(t) w(t - \\tau) e^{-j\\omega t}\\,dt$$<\/p>\n

\u4f46 STFT \u7684\u7a97\u957f\u56fa\u5b9a\u540e\uff0c\u65f6\u95f4\u5206\u8fa8\u7387 $\\Delta t$ \u4e0e\u9891\u7387\u5206\u8fa8\u7387 $\\Delta f$ \u53d7\u6d77\u68ee\u5821\u6d4b\u4e0d\u51c6\u539f\u7406\u7ea6\u675f $^{[16]}$\uff1a<\/p>\n

$$\\Delta t \\cdot \\Delta f \\geq \\frac{1}{4\\pi} \\tag{10.1}$$
\n```<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 10.2 \u5c0f\u6ce2\u53d8\u6362
\n\u5c0f\u6ce2\u53d8\u6362\u91c7\u7528\u4e00\u7ec4\u53ef\u4f38\u7f29\u3001\u53ef\u5e73\u79fb\u7684\u57fa\u51fd\u6570 $\\psi_{a,b}(t)$\uff0c\u4ece\u6839\u672c\u4e0a\u89e3\u51b3\u4e86\u65f6\u9891\u5206\u8fa8\u7387\u7684\u77db\u76fe $^{[17]}$\u3002\u8bbe $\\psi(t)$ \u4e3a\u6bcd\u5c0f\u6ce2\uff08Mother Wavelet\uff09\uff0c\u6ee1\u8db3 $\\int \\psi(t)\\,dt = 0$\uff08\u96f6\u5747\u503c\u6761\u4ef6\uff09\uff0c\u5219\u5c0f\u6ce2\u57fa\u51fd\u6570\u65cf\u5b9a\u4e49\u4e3a\uff1a<\/p>\n

$$\\psi_{a,b}(t) = \\frac{1}{\\sqrt{|a|}} \\psi\\left(\\frac{t - b}{a}\\right), \\quad a \\neq 0, \\; b \\in \\mathbb{R} \\tag{10.2}$$<\/p>\n

\u5176\u4e2d $a$ \u4e3a\u5c3a\u5ea6\u53c2\u6570\uff08\u63a7\u5236\u4f38\u7f29\uff0c\u5bf9\u5e94\u9891\u7387\uff09\uff0c$b$ \u4e3a\u5e73\u79fb\u53c2\u6570\uff08\u63a7\u5236\u4f4d\u7f6e\uff0c\u5bf9\u5e94\u65f6\u95f4\uff09\u3002\u5c0f\u6ce2\u57fa\u51fd\u6570\u5728\u65f6\u57df\u4e0a\u5177\u6709**\u7d27\u652f\u6491**\uff08Compact Support\uff09\u6027\u8d28\u2014\u2014\u53ea\u5728\u6709\u9650\u533a\u95f4\u5185\u975e\u96f6\u2014\u2014\u56e0\u6b64\u5929\u7136\u5177\u5907\u65f6\u95f4\u5b9a\u4f4d\u80fd\u529b\u3002
\n```<\/p>\n

```ad-definition
\ntitle: \u5b9a\u4e49 10.3 \u8fde\u7eed\u5c0f\u6ce2\u53d8\u6362
\n\u4fe1\u53f7 $f(t)$ \u7684\u8fde\u7eed\u5c0f\u6ce2\u53d8\u6362\uff08CWT\uff09\u5b9a\u4e49\u4e3a $f$ \u4e0e\u5c0f\u6ce2\u57fa\u51fd\u6570\u7684\u5185\u79ef\uff1a<\/p>\n

$$W_f(a, b) = \\langle f, \\psi_{a,b} \\rangle = \\int_{-\\infty}^{\\infty} f(t) \\cdot \\frac{1}{\\sqrt{|a|}} \\psi^*\\left(\\frac{t - b}{a}\\right) dt \\tag{10.3}$$
\n```<\/p>\n

```ad-theorem
\ntitle: \u547d\u9898 10.1 \u591a\u5206\u8fa8\u7387\u5206\u6790
\n\u5c0f\u6ce2\u53d8\u6362\u7684\u65f6\u9891\u5206\u8fa8\u7387\u968f\u5c3a\u5ea6 $a$ \u81ea\u9002\u5e94\u53d8\u5316\uff1a<\/p>\n

- **\u5c0f\u5c3a\u5ea6 $a$**\uff08\u9ad8\u9891\uff09\uff1a\u5c0f\u6ce2\u88ab\u538b\u7f29\uff0c\u65f6\u95f4\u5206\u8fa8\u7387\u9ad8\u3001\u9891\u7387\u5206\u8fa8\u7387\u4f4e\uff0c\u9002\u5408\u5206\u6790\u77ac\u6001\u4fe1\u53f7\uff1b
\n- **\u5927\u5c3a\u5ea6 $a$**\uff08\u4f4e\u9891\uff09\uff1a\u5c0f\u6ce2\u88ab\u62c9\u4f38\uff0c\u9891\u7387\u5206\u8fa8\u7387\u9ad8\u3001\u65f6\u95f4\u5206\u8fa8\u7387\u4f4e\uff0c\u9002\u5408\u5206\u6790\u957f\u671f\u8d8b\u52bf\u3002<\/p>\n

\u8fd9\u79cd**\u591a\u5206\u8fa8\u7387\u5206\u6790\uff08Multi-Resolution Analysis, MRA\uff09** \u7279\u6027\u662f\u5c0f\u6ce2\u53d8\u6362\u533a\u522b\u4e8e\u5085\u91cc\u53f6\u53d8\u6362\u548c STFT \u7684\u6838\u5fc3\u4f18\u52bf\u3002
\n```<\/p>\n

### 10.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf<\/p>\n

\u5c0f\u6ce2\u53d8\u6362\u7684\u51e0\u4f55\u8fc7\u7a0b\u53ef\u7406\u89e3\u4e3a\u4f7f\u7528\u4e00\u7ec4\u4e0d\u540c\u5c3a\u5bf8\u7684\"\u63a2\u9488\"\u6cbf\u65f6\u95f4\u8f74\u6ed1\u52a8\uff1a<\/p>\n

- **\u5927\u63a2\u9488\uff08\u5927\u5c3a\u5ea6 $a$\uff09**\uff1a\u8986\u76d6\u5bbd\u65f6\u95f4\u8303\u56f4\uff0c\u611f\u77e5\u4fe1\u53f7\u957f\u671f\u8d8b\u52bf\uff08\u4f4e\u9891\uff09\uff0c\u4f46\u65e0\u6cd5\u7cbe\u786e\u5b9a\u4f4d\u53d8\u5316\u65f6\u523b\uff1b
\n- **\u5c0f\u63a2\u9488\uff08\u5c0f\u5c3a\u5ea6 $a$\uff09**\uff1a\u8986\u76d6\u7a84\u65f6\u95f4\u8303\u56f4\uff0c\u7cbe\u786e\u5b9a\u4f4d\u4fe1\u53f7\u7a81\u53d8\u70b9\uff08\u9ad8\u9891\uff09\uff0c\u4f46\u770b\u4e0d\u5230\u6574\u4f53\u8d8b\u52bf\u3002<\/p>\n

\u5728\u6bcf\u4e2a\u4f4d\u7f6e $b$\uff0c\u8ba1\u7b97\u4fe1\u53f7 $f(t)$ \u4e0e\u63a2\u9488 $\\psi_{a,b}(t)$ \u7684\u5185\u79ef $W_f(a, b)$\u3002\u7ed3\u679c\u6784\u6210\u4e00\u5f20**\u5c3a\u5ea6\u56fe\uff08Scalogram\uff09**\uff0c\u6a2a\u8f74\u4e3a\u65f6\u95f4 $b$\uff0c\u7eb5\u8f74\u4e3a\u5c3a\u5ea6 $a$\uff08\u6216\u7b49\u6548\u9891\u7387\uff09\uff0c\u989c\u8272\u6df1\u6d45\u8868\u793a\u5185\u79ef\u5f3a\u5ea6\u3002<\/p>\n

\u4e0e\u5085\u91cc\u53f6\u53d8\u6362\u7684\u5bf9\u6bd4\uff1a\u5085\u91cc\u53f6\u53d8\u6362\u7528\u65e0\u9650\u957f\u7684\u6b63\u5f26\u6ce2\"\u5339\u914d\"\u6574\u4e2a\u4fe1\u53f7\uff0c\u5f97\u5230\u5168\u5c40\u9891\u8c31\uff1b\u5c0f\u6ce2\u53d8\u6362\u7528\u6709\u9650\u957f\u7684\u5c0f\u6ce2\"\u626b\u63cf\"\u4fe1\u53f7\uff0c\u5728\u6bcf\u4e00\u5904\u8bb0\u5f55\u5c40\u90e8\u5339\u914d\u5ea6\uff0c\u540c\u65f6\u4fdd\u7559\u65f6\u95f4\u548c\u9891\u7387\u4fe1\u606f\u3002<\/p>\n

### 10.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)<\/p>\n

```ad-example
\ntitle: \u4f8b\u9898 10.1 Haar \u5c0f\u6ce2\u5206\u89e3\u2014\u2014\u624b\u5de5\u8ba1\u7b97\u4e00\u7ea7\u4e0e\u4e8c\u7ea7\u5c0f\u6ce2\u53d8\u6362<\/p>\n

Haar \u5c0f\u6ce2\u662f\u6700\u7b80\u5355\u7684\u6b63\u4ea4\u5c0f\u6ce2\uff0c\u5176\u5c3a\u5ea6\u51fd\u6570 $\\phi(t)$ \u548c\u5c0f\u6ce2\u51fd\u6570 $\\psi(t)$ \u5b9a\u4e49\u4e3a\uff1a<\/p>\n

$$\\phi(t) = \\begin{cases} 1, & 0 \\leq t < 1 \\\\ 0, & \\text{otherwise} \\end{cases}, \\quad \\psi(t) = \\begin{cases} 1, & 0 \\leq t < 0.5 \\\\ -1, & 0.5 \\leq t < 1 \\\\ 0, & \\text{otherwise} \\end{cases}$$\n\n\u7ed9\u5b9a\u957f\u5ea6\u4e3a 8 \u7684\u79bb\u6563\u4fe1\u53f7\uff1a\n\n$$x = [4, 6, 10, 12, 8, 6, 5, 5]$$\n\n\u624b\u5de5\u6267\u884c Haar \u5c0f\u6ce2\u5206\u89e3\u3002\n\n**\u89e3**\uff1a\n\n**\u6b65\u9aa4 1\uff1a\u4e00\u7ea7\u5206\u89e3\u2014\u2014\u8ba1\u7b97\u8fd1\u4f3c\u7cfb\u6570\u3002** \u8fd1\u4f3c\u7cfb\u6570\u901a\u8fc7\u5c3a\u5ea6\u51fd\u6570\u7684\u5185\u79ef\u83b7\u5f97\uff0c\u5373\u76f8\u90bb\u4e24\u70b9\u7684\u5e73\u5747\u503c\uff1a\n\n$$a_1 = \\frac{4+6}{2} = 5, \\quad a_2 = \\frac{10+12}{2} = 11, \\quad a_3 = \\frac{8+6}{2} = 7, \\quad a_4 = \\frac{5+5}{2} = 5$$\n\n\u8fd1\u4f3c\u7cfb\u6570\u5411\u91cf\uff1a$A^{(1)} = [5, 11, 7, 5]$\n\n**\u6b65\u9aa4 2\uff1a\u4e00\u7ea7\u5206\u89e3\u2014\u2014\u8ba1\u7b97\u7ec6\u8282\u7cfb\u6570\u3002** \u7ec6\u8282\u7cfb\u6570\u901a\u8fc7\u5c0f\u6ce2\u51fd\u6570\u7684\u5185\u79ef\u83b7\u5f97\uff0c\u5373\u76f8\u90bb\u4e24\u70b9\u5dee\u503c\u7684\u4e00\u534a\uff1a\n\n$$d_1 = \\frac{4-6}{2} = -1, \\quad d_2 = \\frac{10-12}{2} = -1, \\quad d_3 = \\frac{8-6}{2} = 1, \\quad d_4 = \\frac{5-5}{2} = 0$$\n\n\u7ec6\u8282\u7cfb\u6570\u5411\u91cf\uff1a$D^{(1)} = [-1, -1, 1, 0]$\n\n**\u6b65\u9aa4 3\uff1a\u9a8c\u8bc1\u91cd\u6784\u3002** \u4ece $A^{(1)}$ \u548c $D^{(1)}$ \u53ef\u5b8c\u7f8e\u6062\u590d\u539f\u59cb\u4fe1\u53f7\uff1a\n\n$$x_1 = a_1 + d_1 = 5 + (-1) = 4, \\quad x_2 = a_1 - d_1 = 5 - (-1) = 6$$\n$$x_3 = a_2 + d_2 = 11 + (-1) = 10, \\quad x_4 = a_2 - d_2 = 11 - (-1) = 12$$\n$$x_5 = a_3 + d_3 = 7 + 1 = 8, \\quad x_6 = a_3 - d_3 = 7 - 1 = 6$$\n$$x_7 = a_4 + d_4 = 5 + 0 = 5, \\quad x_8 = a_4 - d_4 = 5 - 0 = 5$$\n\n\u91cd\u6784\u5b8c\u5168\u6b63\u786e\u3002\n\n**\u6b65\u9aa4 4\uff1a\u4e8c\u7ea7\u5206\u89e3\u3002** \u5bf9\u8fd1\u4f3c\u7cfb\u6570 $A^{(1)} = [5, 11, 7, 5]$ \u7ee7\u7eed\u505a Haar \u5c0f\u6ce2\u53d8\u6362\uff1a\n\n$$a_1^{(2)} = \\frac{5+11}{2} = 8, \\quad a_2^{(2)} = \\frac{7+5}{2} = 6$$\n$$d_1^{(2)} = \\frac{5-11}{2} = -3, \\quad d_2^{(2)} = \\frac{7-5}{2} = 1$$\n\n\u4e8c\u7ea7\u8fd1\u4f3c\uff1a$A^{(2)} = [8, 6]$\uff0c\u4e8c\u7ea7\u7ec6\u8282\uff1a$D^{(2)} = [-3, 1]$\n\n**\u5173\u952e\u89c2\u5bdf**\uff1a\u539f\u59cb\u4fe1\u53f7\u9700 8 \u4e2a\u6570\u503c\u5b58\u50a8\u3002\u4e00\u7ea7\u5206\u89e3\u540e $A^{(1)}$\uff084 \u503c\uff09+ $D^{(1)}$\uff084 \u503c\uff09= 8 \u503c\uff0c\u672a\u538b\u7f29\u3002\u4f46\u82e5\u5c06\u7edd\u5bf9\u503c\u5c0f\u7684\u7ec6\u8282\u7cfb\u6570\uff08\u5982 $d_4 = 0$\uff09\u7f6e\u96f6\uff0c\u5219\u53ea\u9700\u5b58\u50a8 7 \u4e2a\u6709\u6548\u503c\u2014\u2014\u8fd9\u5c31\u662f\u5c0f\u6ce2\u538b\u7f29\u7684\u539f\u7406\u3002JPEG2000 \u6b63\u662f\u57fa\u4e8e\u5c0f\u6ce2\u53d8\u6362\uff08CDF 9\/7 \u5c0f\u6ce2\uff09\u5b9e\u73b0\u4e86\u6bd4 JPEG\uff08DCT\uff09\u66f4\u4f18\u7684\u538b\u7f29\u6027\u80fd\uff0c\u4e14\u65e0\u5757\u6548\u5e94\u3002\n```\n\n### 10.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528\n\n\u5c0f\u6ce2\u5206\u6790\u5728\u4fe1\u53f7\u5904\u7406\u9886\u57df\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff1a\n\n- **JPEG2000 \u56fe\u50cf\u538b\u7f29**\uff1a\u91c7\u7528 CDF 9\/7 \u5c0f\u6ce2\u8fdb\u884c\u591a\u7ea7\u5206\u89e3\uff0c\u6bd4 JPEG \u7684 DCT \u65b9\u6cd5\u538b\u7f29\u7387\u66f4\u9ad8\u4e14\u65e0\u5757\u6548\u5e94\uff1b\n- **\u5fc3\u7535\u56fe\uff08ECG\uff09\u5206\u6790**\uff1a\u5c0f\u6ce2\u53d8\u6362\u53ef\u7cbe\u786e\u5b9a\u4f4d QRS \u6ce2\u7fa4\uff0c\u7528\u4e8e\u5fc3\u5f8b\u5931\u5e38\u68c0\u6d4b\uff1b\n- **\u5730\u9707\u4fe1\u53f7\u5904\u7406**\uff1a\u5c0f\u6ce2\u65f6\u9891\u8c31\u53ef\u540c\u65f6\u63ed\u793a\u5730\u9707\u6ce2\u5230\u8fbe\u65f6\u95f4\u548c\u9891\u7387\u6210\u5206\uff1b\n- **\u6df1\u5ea6\u5b66\u4e60\u4e2d\u7684\u5c0f\u6ce2\u7f51\u7edc**\uff1a\u5c06\u5c0f\u6ce2\u53d8\u6362\u4f5c\u4e3a\u795e\u7ecf\u7f51\u7edc\u7684\u524d\u7f6e\u7279\u5f81\u63d0\u53d6\u5c42\uff0c\u7528\u4e8e\u5904\u7406\u975e\u5e73\u7a33\u4fe1\u53f7\u3002\n\n---\n\n## \u7b2c\u5341\u4e00\u7ae0 \u81ea\u6ce8\u610f\u529b\u673a\u5236 \u2014\u2014 AI \u7684\u5185\u79ef\u5f15\u64ce\n\n### 11.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49\n\n\u73b0\u4ee3\u4eba\u5de5\u667a\u80fd\uff0c\u5c24\u5176\u662f\u5927\u8bed\u8a00\u6a21\u578b\uff08LLM\uff09\u5982 GPT\u3001BERT \u7b49\uff0c\u5176\u5e95\u5c42\u8ba1\u7b97\u51e0\u4e4e\u5168\u90e8\u7531\u5185\u79ef\uff08\u70b9\u79ef\uff09\u6784\u6210\u3002Transformer \u67b6\u6784\u7684\u6838\u5fc3\u2014\u2014**\u81ea\u6ce8\u610f\u529b\u673a\u5236\uff08Self-Attention\uff09**\u2014\u2014\u672c\u8d28\u4e0a\u662f\u4e00\u7ec4\u5927\u89c4\u6a21\u7684\u3001\u5e76\u884c\u7684\u3001\u53ef\u5b66\u4e60\u7684\u5411\u91cf\u5185\u79ef\u8fd0\u7b97 $^{[18]}$\u3002\n\n```ad-definition\ntitle: \u5b9a\u4e49 11.1 \u7f29\u653e\u70b9\u79ef\u6ce8\u610f\u529b\n\u7ed9\u5b9a\u8f93\u5165\u5e8f\u5217\uff0c\u6bcf\u4e2a\u4f4d\u7f6e\u7684 token \u88ab\u7ebf\u6027\u6295\u5f71\u4e3a\u4e09\u4e2a\u5411\u91cf\uff1a\u67e5\u8be2\u5411\u91cf $Q$\u3001\u952e\u5411\u91cf $K$\u3001\u503c\u5411\u91cf $V$\u3002\u81ea\u6ce8\u610f\u529b\u8f93\u51fa\u5b9a\u4e49\u4e3a\uff1a\n\n$$\\text{Attention}(Q, K, V) = \\text{softmax}\\left(\\frac{QK^T}{\\sqrt{d_k}}\\right) V \\tag{11.1}$$\n\n\u5176\u4e2d $Q \\in \\mathbb{R}^{n \\times d_k}$\uff0c$K \\in \\mathbb{R}^{n \\times d_k}$\uff0c$V \\in \\mathbb{R}^{n \\times d_v}$\uff0c$n$ \u4e3a\u5e8f\u5217\u957f\u5ea6\uff0c$d_k$ \u4e3a\u67e5\u8be2\/\u952e\u7684\u7ef4\u5ea6\u3002\n```\n\n```ad-theorem\ntitle: \u547d\u9898 11.1 \u6ce8\u610f\u529b\u6743\u91cd\u4f5c\u4e3a\u5f52\u4e00\u5316\u5185\u79ef\n\u77e9\u9635 $QK^T$ \u4e2d\u7684\u7b2c $(i, j)$ \u5143\u7d20\u6b63\u662f\u7b2c $i$ \u4e2a\u67e5\u8be2\u5411\u91cf\u4e0e\u7b2c $j$ \u4e2a\u952e\u5411\u91cf\u7684\u5185\u79ef\uff1a\n\n$$(QK^T)_{ij} = \\langle Q_i, K_j \\rangle = Q_i \\cdot K_j = \\sum_{k=1}^{d_k} Q_{i,k} \\cdot K_{j,k} \\tag{11.2}$$\n\n\u8be5\u5185\u79ef\u503c\u8d8a\u5927\uff0c\u8868\u793a\u7b2c $i$ \u4e2a token \u4e0e\u7b2c $j$ \u4e2a token \u7684\u76f8\u5173\u6027\u8d8a\u9ad8\u3002\u7f29\u653e\u56e0\u5b50 $1\/\\sqrt{d_k}$ \u9632\u6b62\u5185\u79ef\u503c\u968f\u7ef4\u5ea6\u589e\u957f\u8fc7\u5927\u5bfc\u81f4 softmax \u68af\u5ea6\u6d88\u5931\u3002\u7ecf softmax \u5f52\u4e00\u5316\u540e\uff0c\u5185\u79ef\u503c\u8f6c\u5316\u4e3a\u6982\u7387\u6743\u91cd\uff0c\u7528\u4e8e\u5bf9\u503c\u5411\u91cf $V$ \u8fdb\u884c\u52a0\u6743\u6c42\u548c\u3002\n\n**\u591a\u5934\u6ce8\u610f\u529b**\u5c06\u4e0a\u8ff0\u8fc7\u7a0b\u5e76\u884c\u6267\u884c $h$ \u6b21\uff08$h$ \u4e3a\u6ce8\u610f\u529b\u5934\u6570\uff09\uff0c\u6bcf\u4e2a\u5934\u5b66\u4e60\u4e0d\u540c\u7684\u6295\u5f71\u5b50\u7a7a\u95f4\uff1a\n\n$$\\text{MultiHead}(Q, K, V) = \\text{Concat}(\\text{head}_1, \\dots, \\text{head}_h) W^O \\tag{11.3}$$\n\n\u5176\u4e2d $\\text{head}_i = \\text{Attention}(Q W_i^Q, K W_i^K, V W_i^V)$\u3002\n```\n\n### 11.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf\n\n\u81ea\u6ce8\u610f\u529b\u673a\u5236\u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u6267\u884c\u4e86\u4e00\u4e2a\u7cbe\u5999\u7684\"\u6295\u5f71-\u68c0\u7d22\"\u64cd\u4f5c\uff1a\n\n1. **\u67e5\u8be2\u5411\u91cf $Q_i$**\uff1a\u7f16\u7801\"\u8c01\u548c\u6211\u76f8\u5173\uff1f\"\u7684\u67e5\u8be2\u610f\u56fe\uff1b\n2. **\u952e\u5411\u91cf $K_j$**\uff1a\u7f16\u7801\"\u6211\u662f\u8c01\uff0c\u6211\u6709\u4ec0\u4e48\u7279\u5f81\uff1f\"\u7684\u6807\u8bc6\u4fe1\u606f\uff1b\n3. **\u5185\u79ef $\\langle Q_i, K_j \\rangle$**\uff1a\u8861\u91cf\u67e5\u8be2\u4e0e\u952e\u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u76f8\u4f3c\u5ea6\uff08\u5411\u91cf\u5939\u89d2\u4f59\u5f26\u7684\u7f29\u653e\u7248\u672c\uff09\uff1b\n4. **Softmax \u5f52\u4e00\u5316**\uff1a\u5c06\u76f8\u4f3c\u5ea6\u8f6c\u6362\u4e3a\u6982\u7387\u5206\u5e03\uff0c\u4f7f\u6a21\u578b\u805a\u7126\u4e8e\u6700\u76f8\u5173\u7684 token\uff1b\n5. **\u52a0\u6743\u6c42\u548c**\uff1a\u6839\u636e\u6ce8\u610f\u529b\u6743\u91cd\u4ece\u503c\u5411\u91cf\u4e2d\u63d0\u53d6\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002\n\n\u6574\u4e2a Transformer \u6a21\u578b\u53ef\u89c6\u4e3a\u4e00\u4e2a\u5de8\u5927\u7684**\u53ef\u5fae\u5185\u79ef\u5f15\u64ce**\uff1a\u6bcf\u4e00\u5c42\u90fd\u5728\u6267\u884c\u5185\u79ef\u8fd0\u7b97\uff0c\u901a\u8fc7\u53cd\u5411\u4f20\u64ad\u4e0d\u65ad\u8c03\u6574 $Q$\u3001$K$\u3001$V$ \u7684\u6295\u5f71\u77e9\u9635\uff0c\u4f7f\u5185\u79ef\u7ed3\u679c\u80fd\u591f\u51c6\u786e\u6355\u6349\u6570\u636e\u4e2d\u7684\u957f\u8ddd\u79bb\u4f9d\u8d56\u5173\u7cfb\u3002\n\n### 11.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u9898 11.1 \u624b\u5de5\u8ba1\u7b97 2 \u4e2a token \u7684\u81ea\u6ce8\u610f\u529b\n\n\u8003\u8651\u6781\u7b80\u5e8f\u5217\uff0c\u4ec5\u542b\u4e24\u4e2a token\uff1a\"\u6211\"\u548c\"\u7231\"\u3002\u7ecf\u5d4c\u5165\u548c\u7ebf\u6027\u6295\u5f71\u540e\uff08\u8bbe $d_k = 3$\uff09\uff1a\n\n$$Q = \\begin{bmatrix} 1 & 0 & 1 \\\\ 0 & 1 & 1 \\end{bmatrix}, \\quad\nK = \\begin{bmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{bmatrix}, \\quad\nV = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}$$\n\n\u7b2c\u4e00\u884c\u5bf9\u5e94\"\u6211\"\uff0c\u7b2c\u4e8c\u884c\u5bf9\u5e94\"\u7231\"\u3002\n\n**\u89e3**\uff1a\n\n**\u6b65\u9aa4 1\uff1a\u8ba1\u7b97 $QK^T$\uff08\u6240\u6709\u5185\u79ef\u5bf9\uff09\u3002**\n\n$$QK^T = \\begin{bmatrix} 1 & 0 & 1 \\\\ 0 & 1 & 1 \\end{bmatrix}\n\\begin{bmatrix} 1 & 0 \\\\ 1 & 1 \\\\ 0 & 1 \\end{bmatrix}$$\n\n\u9010\u5143\u7d20\u8ba1\u7b97\uff1a\n- $(QK^T)_{11} = \\langle Q_1, K_1 \\rangle = 1 \\times 1 + 0 \\times 1 + 1 \\times 0 = 1$\n- $(QK^T)_{12} = \\langle Q_1, K_2 \\rangle = 1 \\times 0 + 0 \\times 1 + 1 \\times 1 = 1$\n- $(QK^T)_{21} = \\langle Q_2, K_1 \\rangle = 0 \\times 1 + 1 \\times 1 + 1 \\times 0 = 1$\n- $(QK^T)_{22} = \\langle Q_2, K_2 \\rangle = 0 \\times 0 + 1 \\times 1 + 1 \\times 1 = 2$\n\n$$QK^T = \\begin{bmatrix} 1 & 1 \\\\ 1 & 2 \\end{bmatrix}$$\n\n**\u6b65\u9aa4 2\uff1a\u7f29\u653e\uff08\u9664\u4ee5 $\\sqrt{d_k} = \\sqrt{3} \\approx 1.732$\uff09\u3002**\n\n$$\\frac{QK^T}{\\sqrt{3}} = \\begin{bmatrix} 0.577 & 0.577 \\\\ 0.577 & 1.155 \\end{bmatrix}$$\n\n**\u6b65\u9aa4 3\uff1aSoftmax \u5f52\u4e00\u5316\uff08\u6309\u884c\uff09\u3002**\n\n\u7b2c\u4e00\u884c $[0.577, 0.577]$\uff1a\n$$e^{0.577} \\approx 1.781, \\quad \\text{sum} = 3.562$$\n$$\\text{softmax}_{11} = \\frac{1.781}{3.562} = 0.5, \\quad \\text{softmax}_{12} = \\frac{1.781}{3.562} = 0.5$$\n\n\u7b2c\u4e8c\u884c $[0.577, 1.155]$\uff1a\n$$e^{0.577} \\approx 1.781, \\quad e^{1.155} \\approx 3.174, \\quad \\text{sum} = 4.955$$\n$$\\text{softmax}_{21} = \\frac{1.781}{4.955} = 0.359, \\quad \\text{softmax}_{22} = \\frac{3.174}{4.955} = 0.641$$\n\n\u6ce8\u610f\u529b\u6743\u91cd\u77e9\u9635\uff1a\n\n$$\\text{Weights} = \\begin{bmatrix} 0.5 & 0.5 \\\\ 0.359 & 0.641 \\end{bmatrix}$$\n\n**\u6b65\u9aa4 4\uff1a\u52a0\u6743\u6c42\u548c\u5f97\u5230\u8f93\u51fa\u3002**\n\n$$\\text{Output} = \\text{Weights} \\cdot V = \\begin{bmatrix} 0.5 & 0.5 \\\\ 0.359 & 0.641 \\end{bmatrix}\n\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}$$\n\n- \"\u6211\"\u7684\u65b0\u8868\u793a\uff1a$0.5 \\times [1, 0] + 0.5 \\times [0, 1] = [0.5, 0.5]$\n- \"\u7231\"\u7684\u65b0\u8868\u793a\uff1a$0.359 \\times [1, 0] + 0.641 \\times [0, 1] = [0.359, 0.641]$\n\n**\u5173\u952e\u89c2\u5bdf**\uff1a\n- \"\u6211\"\u7684\u6ce8\u610f\u529b\u5747\u5300\u5206\u5e03\u5728\u4e24\u4e2a token \u4e0a\uff08\u5404 0.5\uff09\uff0c\u56e0\u5176\u4e0e\u4e24\u8005\u7684\u5185\u79ef\u76f8\u540c\uff1b\n- \"\u7231\"\u66f4\u5173\u6ce8\u81ea\u5df1\uff080.641\uff09\u800c\u975e\"\u6211\"\uff080.359\uff09\uff0c\u56e0\u5176\u4e0e\u81ea\u5df1\u7684\u5185\u79ef\uff082\uff09\u5927\u4e8e\u4e0e\"\u6211\"\u7684\u5185\u79ef\uff081\uff09\uff1b\n- \u8f93\u51fa\u5411\u91cf\u662f\u503c\u5411\u91cf\u7684\u52a0\u6743\u7ec4\u5408\uff0c\u6743\u91cd\u5b8c\u5168\u7531\u5185\u79ef\u51b3\u5b9a\u2014\u2014\u8fd9\u5c31\u662f\"\u901a\u8fc7\u5185\u79ef\u5b9e\u73b0\u4e0a\u4e0b\u6587\u611f\u77e5\u8868\u793a\"\u7684\u6838\u5fc3\u673a\u5236\u3002\n```\n\n### 11.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528\n\n\u81ea\u6ce8\u610f\u529b\u673a\u5236\u7684\u8ba1\u7b97\u91cf\u968f\u5e8f\u5217\u957f\u5ea6 $n$ \u5448 $O(n^2)$ \u589e\u957f\u3002\u5bf9\u4e8e GPT-4 \u7b49\u5927\u6a21\u578b\uff08\u4e0a\u4e0b\u6587\u957f\u5ea6\u53ef\u8fbe 128K\uff09\uff0c\u5355\u6b21\u524d\u5411\u4f20\u64ad\u9700\u6267\u884c\u6570\u5341\u4e07\u4ebf\u6b21\u5185\u79ef\u8fd0\u7b97\u3002\u4e3a\u52a0\u901f\u8ba1\u7b97\uff0c\u4e1a\u754c\u5f00\u53d1\u4e86\u591a\u79cd\u4f18\u5316\u6280\u672f\uff1a\n\n- **Flash Attention**\uff1a\u901a\u8fc7\u5206\u5757\u8ba1\u7b97\u548c\u5185\u5b58\u4f18\u5316\uff0c\u51cf\u5c11\u663e\u5b58\u8bfb\u5199\uff0c\u5c06\u6ce8\u610f\u529b\u8ba1\u7b97\u52a0\u901f 2\u20134 \u500d\uff1b\n- **\u7a00\u758f\u6ce8\u610f\u529b**\uff1a\u4ec5\u8ba1\u7b97\u90e8\u5206 token \u5bf9\u4e4b\u95f4\u7684\u5185\u79ef\uff08\u5982\u5c40\u90e8\u7a97\u53e3 + \u5168\u5c40 token\uff09\uff0c\u5c06\u590d\u6742\u5ea6\u964d\u81f3 $O(n \\log n)$\uff1b\n- **\u591a\u67e5\u8be2\u6ce8\u610f\u529b\uff08MQA\uff09**\uff1a\u591a\u4e2a\u67e5\u8be2\u5934\u5171\u4eab\u540c\u4e00\u7ec4\u952e\u503c\u5bf9\uff0c\u51cf\u5c11 KV \u7f13\u5b58\u5927\u5c0f\uff1b\n- **\u7ebf\u6027\u6ce8\u610f\u529b**\uff1a\u7528\u6838\u65b9\u6cd5\u8fd1\u4f3c softmax \u6ce8\u610f\u529b\uff0c\u5c06\u590d\u6742\u5ea6\u964d\u81f3 $O(n)$\u3002\n\n\u8fd9\u4e9b\u4f18\u5316\u672c\u8d28\u4e0a\u90fd\u662f\u5728\"\u51cf\u5c11\u5185\u79ef\u8ba1\u7b97\u6b21\u6570\"\u548c\"\u4fdd\u6301\u6a21\u578b\u8868\u8fbe\u80fd\u529b\"\u4e4b\u95f4\u5bfb\u627e\u6700\u4f18\u5e73\u8861\u3002\n\n---\n\n## \u7b2c\u5341\u4e8c\u7ae0 \u6838\u65b9\u6cd5 \u2014\u2014 \u9690\u5f0f\u9ad8\u7ef4\u5185\u79ef\n\n### 12.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49\n\n\u5728\u4f4e\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u6570\u636e\u5f80\u5f80\u662f\u7ebf\u6027\u4e0d\u53ef\u5206\u7684\u2014\u2014\u4f8b\u5982\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u540c\u5fc3\u5706\u6570\u636e\u65e0\u6cd5\u7528\u4e00\u6761\u76f4\u7ebf\u5206\u5f00\u3002\u4f20\u7edf\u505a\u6cd5\u662f\u624b\u52a8\u6784\u9020\u9ad8\u7ef4\u7279\u5f81\uff08\u5982 $x_1^2 + x_2^2$\uff09\uff0c\u4f46\u7279\u5f81\u5de5\u7a0b\u6210\u672c\u6781\u9ad8\u3002**\u6838\u65b9\u6cd5\uff08Kernel Method\uff09** \u7684\u6838\u5fc3\u601d\u60f3\u662f\uff1a\u4e0d\u663e\u5f0f\u8ba1\u7b97\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u5750\u6807\uff0c\u800c\u662f\u76f4\u63a5\u8ba1\u7b97\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef $^{[22]}$\u3002\u8fd9\u4e00\u6280\u5de7\u79f0\u4e3a**\u6838\u6280\u5de7\uff08Kernel Trick\uff09**\u3002\n\n```ad-definition\ntitle: \u5b9a\u4e49 12.1 \u6838\u51fd\u6570\n\u8bbe $\\phi: \\mathcal{X} \\to \\mathcal{H}$ \u662f\u4ece\u8f93\u5165\u7a7a\u95f4\u5230\u9ad8\u7ef4\uff08\u53ef\u80fd\u65e0\u7a77\u7ef4\uff09\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u7684\u975e\u7ebf\u6027\u6620\u5c04\u3002\u6838\u51fd\u6570 $k: \\mathcal{X} \\times \\mathcal{X} \\to \\mathbb{R}$ \u5b9a\u4e49\u4e3a\uff1a\n\n$$k(x, y) = \\langle \\phi(x), \\phi(y) \\rangle_{\\mathcal{H}} \\tag{12.1}$$\n\n\u6838\u51fd\u6570\u7684\u7cbe\u5999\u4e4b\u5904\u5728\u4e8e\uff1a\u6211\u4eec\u65e0\u9700\u77e5\u9053 $\\phi$ \u7684\u5177\u4f53\u5f62\u5f0f\uff0c\u53ea\u8981 $k(x, y)$ \u6ee1\u8db3 **Mercer \u6761\u4ef6**\uff08\u5bf9\u79f0\u4e14\u534a\u6b63\u5b9a\uff09\uff0c\u5b83\u5c31\u5bf9\u5e94\u67d0\u4e2a\u518d\u751f\u6838\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\uff08RKHS\uff09\u4e2d\u7684\u5185\u79ef\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u4e49 12.2 \u5e38\u89c1\u6838\u51fd\u6570\n\u5e38\u7528\u7684\u6838\u51fd\u6570\u5305\u62ec\uff1a\n\n- **\u7ebf\u6027\u6838**\uff1a$k(x, y) = x^T y$\uff08\u5373\u539f\u59cb\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef\uff09\uff1b\n- **\u591a\u9879\u5f0f\u6838**\uff1a$k(x, y) = (x^T y + c)^d$\uff08\u5bf9\u5e94 $d$ \u9636\u591a\u9879\u5f0f\u7279\u5f81\u7a7a\u95f4\uff09\uff1b\n- **\u9ad8\u65af\u5f84\u5411\u57fa\u6838\uff08RBF\uff09**\uff1a$k(x, y) = \\exp\\left(-\\frac{\\|x - y\\|^2}{2\\sigma^2}\\right)$\uff08\u5bf9\u5e94\u65e0\u7a77\u7ef4\u7279\u5f81\u7a7a\u95f4\uff09\uff1b\n- **Sigmoid \u6838**\uff1a$k(x, y) = \\tanh(\\alpha x^T y + c)$\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u4e49 12.3 \u652f\u6301\u5411\u91cf\u673a\n\u652f\u6301\u5411\u91cf\u673a\uff08SVM\uff09\u662f\u6838\u65b9\u6cd5\u6700\u7ecf\u5178\u7684\u5e94\u7528 $^{[23]}$\u3002SVM \u5728\u7279\u5f81\u7a7a\u95f4\u4e2d\u5bfb\u627e\u6700\u5927\u95f4\u9694\u8d85\u5e73\u9762\uff0c\u5176\u51b3\u7b56\u51fd\u6570\u4ec5\u4f9d\u8d56\u4e8e\u652f\u6301\u5411\u91cf\u4e0e\u5f85\u5206\u7c7b\u6837\u672c\u7684\u5185\u79ef\uff1a\n\n$$f(x) = \\text{sign}\\left(\\sum_{i=1}^{m} \\alpha_i y_i \\langle \\phi(x_i), \\phi(x) \\rangle + b\\right) = \\text{sign}\\left(\\sum_{i=1}^{m} \\alpha_i y_i k(x_i, x) + b\\right) \\tag{12.2}$$\n\n\u5176\u4e2d $x_i$ \u4e3a\u652f\u6301\u5411\u91cf\uff0c$y_i \\in \\{-1, +1\\}$ \u4e3a\u6807\u7b7e\uff0c$\\alpha_i$ \u4e3a\u5bf9\u5076\u53d8\u91cf\u3002\n```\n\n### 12.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf\n\n\u6838\u6280\u5de7\u7684\u51e0\u4f55\u76f4\u89c2\u53ef\u7528\"\u6298\u53e0-\u5c55\u5f00\"\u6765\u7406\u89e3\uff1a\n\n1. **\u8f93\u5165\u7a7a\u95f4**\uff1a\u6570\u636e\u70b9\u6742\u4e71\u5206\u5e03\u5728\u4f4e\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u7ebf\u6027\u5206\u7c7b\u5668\u65e0\u80fd\u4e3a\u529b\uff1b\n2. **\u9690\u5f0f\u6620\u5c04 $\\phi$**\uff1a\u5c06\u6570\u636e\u70b9\"\u5c55\u5f00\"\u5230\u9ad8\u7ef4\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e2d\uff0c\u539f\u672c\u7ea0\u7f20\u7684\u6570\u636e\u70b9\u88ab\"\u62c9\u76f4\"\uff1b\n3. **\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef**\uff1aSVM \u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u5bfb\u627e\u6700\u5927\u95f4\u9694\u8d85\u5e73\u9762\u2014\u2014\u7b49\u4ef7\u4e8e\u8f93\u5165\u7a7a\u95f4\u4e2d\u7684\u975e\u7ebf\u6027\u51b3\u7b56\u8fb9\u754c\uff1b\n4. **\u6838\u51fd\u6570 $k(x, y)$**\uff1a\u76f4\u63a5\u8fd4\u56de\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef\u503c\uff0c\u4eff\u4f5b\u6570\u636e\u5df2\u88ab\u6620\u5c04\u5230\u9ad8\u7ef4\u7a7a\u95f4\uff0c\u4f46\u8ba1\u7b97\u91cf\u4ecd\u4e0e\u4f4e\u7ef4\u7a7a\u95f4\u76f8\u540c\u3002\n\n**\u5173\u952e\u6d1e\u5bdf**\uff1aRBF \u6838 $\\exp(-\\gamma\\|x - y\\|^2)$ \u7684\u6cf0\u52d2\u5c55\u5f00\u5305\u542b\u6240\u6709\u9636\u7684\u591a\u9879\u5f0f\u7279\u5f81\uff0c\u56e0\u6b64 RBF \u6838 SVM \u7406\u8bba\u4e0a\u53ef\u903c\u8fd1\u4efb\u610f\u590d\u6742\u7684\u51b3\u7b56\u8fb9\u754c\u3002\n\n### 12.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u9898 12.1 \u4e8c\u7ef4\u5f02\u6216\uff08XOR\uff09\u95ee\u9898\u7684\u6838\u6280\u5de7\u2014\u2014\u624b\u5de5\u63a8\u5bfc\n\nXOR \u6570\u636e\u96c6\uff1a$x_1 = (-1, -1)$ \u6807\u7b7e $-1$\uff0c$x_2 = (1, 1)$ \u6807\u7b7e $-1$\uff0c$x_3 = (-1, 1)$ \u6807\u7b7e $+1$\uff0c$x_4 = (1, -1)$ \u6807\u7b7e $+1$\u3002\u5728\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\uff0cXOR \u6570\u636e\u7ebf\u6027\u4e0d\u53ef\u5206\u3002\n\n**\u89e3**\uff1a\n\n**\u6b65\u9aa4 1\uff1a\u9009\u62e9\u6838\u51fd\u6570\u5e76\u627e\u51fa\u9690\u5f0f\u6620\u5c04\u3002** \u53d6\u591a\u9879\u5f0f\u6838 $k(x, y) = (x^T y)^2$\u3002\u5c55\u5f00\uff1a\n\n$$(x^T y)^2 = (x_1 y_1 + x_2 y_2)^2 = x_1^2 y_1^2 + 2x_1 x_2 y_1 y_2 + x_2^2 y_2^2$$\n\n$$= \\langle (x_1^2, \\sqrt{2}x_1 x_2, x_2^2), (y_1^2, \\sqrt{2}y_1 y_2, y_2^2) \\rangle$$\n\n\u56e0\u6b64\u9690\u5f0f\u6620\u5c04\u4e3a $\\phi(x) = (x_1^2, \\sqrt{2}x_1 x_2, x_2^2)$\uff0c\u5c06\u4e8c\u7ef4\u6570\u636e\u6620\u5c04\u5230\u4e09\u7ef4\u7a7a\u95f4\u3002\n\n**\u6b65\u9aa4 2\uff1a\u8ba1\u7b97\u6570\u636e\u70b9\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u5750\u6807\u3002**\n\n$$\\phi(x_1) = \\phi(-1, -1) = (1, \\sqrt{2}, 1), \\quad \\phi(x_2) = \\phi(1, 1) = (1, \\sqrt{2}, 1)$$\n$$\\phi(x_3) = \\phi(-1, 1) = (1, -\\sqrt{2}, 1), \\quad \\phi(x_4) = \\phi(1, -1) = (1, -\\sqrt{2}, 1)$$\n\n**\u6b65\u9aa4 3\uff1a\u9a8c\u8bc1\u7ebf\u6027\u53ef\u5206\u6027\u3002** \u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\uff0c$x_1, x_2$\uff08\u6807\u7b7e $-1$\uff09\u5747\u4f4d\u4e8e $(1, \\sqrt{2}, 1)$\uff0c$x_3, x_4$\uff08\u6807\u7b7e $+1$\uff09\u5747\u4f4d\u4e8e $(1, -\\sqrt{2}, 1)$\u3002\u4e24\u7c7b\u70b9\u53ef\u88ab\u5e73\u9762 $z_2 = 0$\uff08\u5373 $\\sqrt{2}x_1 x_2 = 0$\uff09\u5b8c\u7f8e\u5206\u5f00\uff01\n\n**\u6b65\u9aa4 4\uff1a\u9a8c\u8bc1\u6838\u6280\u5de7\u3002** \u8ba1\u7b97 $k(x_1, x_3) = (x_1^T x_3)^2$\uff1a\n\n$$x_1^T x_3 = (-1)(-1) + (-1)(1) = 0, \\quad k(x_1, x_3) = 0^2 = 0$$\n\n\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\uff1a$\\langle \\phi(x_1), \\phi(x_3) \\rangle = 1 \\times 1 + \\sqrt{2} \\times (-\\sqrt{2}) + 1 \\times 1 = 0$\n\n\u4e24\u8005\u76f8\u7b49\uff0c\u9a8c\u8bc1\u4e86\u6838\u6280\u5de7\u7684\u6b63\u786e\u6027\u3002\n\n**\u6b65\u9aa4 5\uff1aSVM \u51b3\u7b56\u3002** \u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u6700\u5927\u95f4\u9694\u8d85\u5e73\u9762\u4e3a $z_2 = 0$\uff0c\u6cd5\u5411\u91cf $w = (0, 1, 0)$\uff0c\u504f\u7f6e $b = 0$\u3002\u652f\u6301\u5411\u91cf\u4e3a\u5168\u90e8\u56db\u4e2a\u70b9\uff0c$\\alpha_i = 1$\u3002\n\n\u5bf9\u4e8e\u6d4b\u8bd5\u70b9 $x = (0.5, -0.5)$\uff1a\n\n$$k(x_1, x) = ((-1)(0.5) + (-1)(-0.5))^2 = 0, \\quad k(x_2, x) = ((1)(0.5) + (1)(-0.5))^2 = 0$$\n$$k(x_3, x) = ((-1)(0.5) + (1)(-0.5))^2 = 1, \\quad k(x_4, x) = ((1)(0.5) + (-1)(-0.5))^2 = 1$$\n\n$$f(x) = \\text{sign}(-0 - 0 + 1 + 1) = \\text{sign}(2) = +1$$\n\n\u9884\u6d4b\u4e3a $+1$\uff0c\u6b63\u786e\u3002\n\n**\u5173\u952e\u89c2\u5bdf**\uff1a\u6211\u4eec\u4ece\u672a\u663e\u5f0f\u8ba1\u7b97 $\\phi(x)$\uff0c\u800c\u662f\u901a\u8fc7\u6838\u51fd\u6570 $k(x, y) = (x^T y)^2$ \u76f4\u63a5\u5f97\u5230\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef\u2014\u2014\u7528\u4f4e\u7ef4\u8ba1\u7b97\u91cf\u5b9e\u73b0\u9ad8\u7ef4\u5206\u7c7b\u80fd\u529b\u3002\n```\n\n### 12.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528\n\n\u6838\u65b9\u6cd5\u7684\u5e94\u7528\u8fdc\u4e0d\u6b62 SVM\uff1a\n\n- **\u6838\u4e3b\u6210\u5206\u5206\u6790\uff08Kernel PCA\uff09**\uff1a\u5728\u6838\u6620\u5c04\u540e\u7684\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u505a PCA\uff0c\u7528\u4e8e\u975e\u7ebf\u6027\u964d\u7ef4\uff1b\n- **\u6838\u5cad\u56de\u5f52\uff08Kernel Ridge Regression\uff09**\uff1a\u5c06\u7ebf\u6027\u5cad\u56de\u5f52\u63a8\u5e7f\u5230\u975e\u7ebf\u6027\u56de\u5f52\uff1b\n- **\u6838\u5747\u503c\u5339\u914d\uff08Kernel Mean Matching\uff09**\uff1a\u7528\u4e8e\u9886\u57df\u81ea\u9002\u5e94\u548c\u8fc1\u79fb\u5b66\u4e60\uff1b\n- **\u9ad8\u65af\u8fc7\u7a0b\uff08Gaussian Process\uff09**\uff1a\u4ee5\u6838\u51fd\u6570\u4f5c\u4e3a\u534f\u65b9\u5dee\u51fd\u6570\uff0c\u7528\u4e8e\u8d1d\u53f6\u65af\u4f18\u5316\u548c\u56de\u5f52\uff1b\n- **\u795e\u7ecf\u6b63\u5207\u6838\uff08NTK\uff09**\uff1a\u8fde\u63a5\u65e0\u9650\u5bbd\u795e\u7ecf\u7f51\u7edc\u4e0e\u6838\u65b9\u6cd5\uff0c\u4e3a\u6df1\u5ea6\u5b66\u4e60\u63d0\u4f9b\u7406\u8bba\u5206\u6790\u5de5\u5177\u3002\n\n---\n\n## \u7b2c\u5341\u4e09\u7ae0 \u91cf\u5b50\u529b\u5b66\u4e2d\u7684\u5185\u79ef \u2014\u2014 \u6982\u7387\u5373\u6295\u5f71\n\n### 13.1 \u7406\u8bba\u4e0e\u4e25\u683c\u5b9a\u4e49\n\n\u91cf\u5b50\u529b\u5b66\u5c06\u5185\u79ef\u7684\u6982\u5ff5\u63a8\u5411\u4e86\u7269\u7406\u4e16\u754c\u7684\u7ec8\u6781\u5c42\u9762\u3002\u5728\u91cf\u5b50\u529b\u5b66\u4e2d\uff0c\u7cfb\u7edf\u7684\u72b6\u6001\u7531\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4 $\\mathcal{H}$ \u4e2d\u7684**\u6001\u77e2\u91cf** $|\\psi\\rangle$ \u63cf\u8ff0\uff08\u72c4\u62c9\u514b\u7b26\u53f7\uff09$^{[26]}$\u3002\u6b64\u5904\u7684\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u901a\u5e38\u662f\u65e0\u7a77\u7ef4\u7684\u590d\u5185\u79ef\u7a7a\u95f4\u3002\n\n```ad-definition\ntitle: \u5b9a\u4e49 13.1 \u6001\u77e2\u91cf\u4e0e\u5185\u79ef\n\u6001\u77e2\u91cf $|\\psi\\rangle \\in \\mathcal{H}$ \u5305\u542b\u91cf\u5b50\u7cfb\u7edf\u7684\u5168\u90e8\u4fe1\u606f\u3002\u4e24\u4e2a\u6001\u7684\u5185\u79ef $\\langle \\phi | \\psi \\rangle$ \u662f\u4e00\u4e2a\u590d\u6570\uff0c\u5176\u6a21\u5e73\u65b9\u7ed9\u51fa\u6d4b\u91cf\u6982\u7387\u3002\n\n**\u516c\u7406 13.1\uff08\u73bb\u6069\u89c4\u5219\uff09** \u5f53\u7cfb\u7edf\u5904\u4e8e\u6001 $|\\psi\\rangle$ \u65f6\uff0c\u6d4b\u91cf\u53ef\u89c2\u6d4b\u91cf $\\hat{A}$ \u5f97\u5230\u672c\u5f81\u503c $\\lambda_n$ \u7684\u6982\u7387\u4e3a $^{[21]}$\uff1a\n\n$$P(\\lambda_n) = |\\langle a_n | \\psi \\rangle|^2 \\tag{13.1}$$\n\n\u5176\u4e2d $|a_n\\rangle$ \u4e3a $\\hat{A}$ \u5bf9\u5e94\u4e8e $\\lambda_n$ \u7684\u672c\u5f81\u6001\u3002\u6d4b\u91cf\u540e\uff0c\u7cfb\u7edf\u6001\u574d\u7f29\u5230 $|a_n\\rangle$\u3002\u73bb\u6069\u89c4\u5219\u7684\u672c\u8d28\u662f\uff1a**\u6982\u7387\u7b49\u4e8e\u6001\u77e2\u91cf\u5728\u6d4b\u91cf\u57fa\u4e0a\u7684\u6295\u5f71\u6a21\u5e73\u65b9**\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u4e49 13.2 \u53ef\u89c2\u6d4b\u91cf\u4e0e\u81ea\u4f34\u7b97\u5b50\n\u53ef\u89c2\u6d4b\u91cf\u5bf9\u5e94\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e0a\u7684\u81ea\u4f34\u7b97\u5b50\uff08Hermitian Operator\uff09$\\hat{A}$\uff0c\u6ee1\u8db3 $\\hat{A}^\\dagger = \\hat{A}$\u3002\u81ea\u4f34\u7b97\u5b50\u7684\u672c\u5f81\u503c\u4e3a\u5b9e\u6570\uff0c\u672c\u5f81\u6001\u6784\u6210\u5b8c\u5907\u6b63\u4ea4\u57fa\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u4e49 13.3 \u859b\u5b9a\u8c14\u65b9\u7a0b\n\u6001\u77e2\u91cf\u7684\u65f6\u95f4\u6f14\u5316\u7531\u859b\u5b9a\u8c14\u65b9\u7a0b\u63cf\u8ff0\uff1a\n\n$$i\\hbar \\frac{d}{dt} |\\psi(t)\\rangle = \\hat{H} |\\psi(t)\\rangle \\tag{13.2}$$\n\n\u5176\u4e2d $\\hat{H}$ \u4e3a\u54c8\u5bc6\u987f\u7b97\u7b26\uff08\u80fd\u91cf\u7b97\u5b50\uff09\u3002\u8be5\u65b9\u7a0b\u672c\u8d28\u4e0a\u662f\u5728\u65e0\u7a77\u7ef4\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e2d\u7684\u9149\u6f14\u5316\u65b9\u7a0b\u2014\u2014\u4fdd\u5185\u79ef\u7684\u65cb\u8f6c\u3002\n```\n\n### 13.2 \u51e0\u4f55\u4e0e\u7a7a\u95f4\u56fe\u50cf\n\n\u91cf\u5b50\u529b\u5b66\u7684\u51e0\u4f55\u56fe\u50cf\u4e0e\u7ecf\u5178\u5185\u79ef\u7a7a\u95f4\u6709\u7740\u6df1\u523b\u7684\u8054\u7cfb\uff1a\n\n1. **\u6001\u77e2\u91cf\u662f\u5355\u4f4d\u5411\u91cf**\uff1a\u7269\u7406\u4e0a\u8981\u6c42 $|\\psi\\rangle$ \u5f52\u4e00\u5316\uff0c\u5373 $\\langle \\psi | \\psi \\rangle = 1$\u3002\u6240\u6709\u53ef\u80fd\u7684\u6001\u77e2\u91cf\u6784\u6210\u590d\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e2d\u7684\u5355\u4f4d\u7403\u9762\u3002\n\n2. **\u6d4b\u91cf\u662f\u6b63\u4ea4\u6295\u5f71**\uff1a\u6d4b\u91cf\u64cd\u4f5c\u5c06\u6001\u77e2\u91cf $|\\psi\\rangle$ \u6295\u5f71\u5230\u672c\u5f81\u5b50\u7a7a\u95f4\u4e0a\u3002\u6295\u5f71\u957f\u5ea6 $|\\langle a_n | \\psi \\rangle|$ \u51b3\u5b9a\u6982\u7387\u5e45\uff0c\u5176\u5e73\u65b9\u4e3a\u6d4b\u91cf\u6982\u7387\u3002\n\n3. **\u6b63\u4ea4\u6001\u4e92\u65a5**\uff1a\u82e5 $\\langle \\phi | \\psi \\rangle = 0$\uff0c\u5219\u4e24\u4e2a\u6001\u6b63\u4ea4\uff08\u4e92\u65a5\uff09\u2014\u2014\u7cfb\u7edf\u5904\u4e8e $|\\psi\\rangle$ \u65f6\uff0c\u6d4b\u91cf\u5230 $|\\phi\\rangle$ \u7684\u6982\u7387\u4e3a\u96f6\u3002\n\n4. **\u7ea0\u7f20\u6001\u4e0d\u53ef\u5206**\uff1a\u5bf9\u4e8e\u590d\u5408\u7cfb\u7edf\uff0c\u82e5 $|\\psi\\rangle_{AB} \\neq |\\phi\\rangle_A \\otimes |\\chi\\rangle_B$\uff0c\u5219\u4e24\u4e2a\u5b50\u7cfb\u7edf\u5904\u4e8e\u7ea0\u7f20\u6001\u3002\u7ea0\u7f20\u6001\u7684\u6570\u5b66\u672c\u8d28\u662f\uff1a\u4e24\u4e2a\u5b50\u7cfb\u7edf\u7684\u5185\u79ef\u7ed3\u6784\u65e0\u6cd5\u5206\u89e3\u4e3a\u76f4\u79ef\u5f62\u5f0f\u3002\n\n### 13.3 \u786c\u6838\u4f8b\u9898\u8be6\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u9898 13.1 \u81ea\u65cb $1\/2$ \u7cfb\u7edf\u7684\u6d4b\u91cf\u6982\u7387\u2014\u2014\u5185\u79ef\u8ba1\u7b97\n\n\u8003\u8651\u7535\u5b50\u81ea\u65cb\uff0c\u5176\u6001\u53ef\u8868\u793a\u4e3a\u4e8c\u7ef4\u590d\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e2d\u7684\u5411\u91cf\u3002\u81ea\u65cb $z$ \u65b9\u5411\u672c\u5f81\u6001\uff1a\n\n$$| \\uparrow_z \\rangle = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}, \\quad | \\downarrow_z \\rangle = \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}$$\n\n\u81ea\u65cb $x$ \u65b9\u5411\u672c\u5f81\u6001\uff1a\n\n$$| \\uparrow_x \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix}, \\quad | \\downarrow_x \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix}$$\n\n\u7535\u5b50\u5904\u4e8e\u6001 $|\\psi\\rangle = \\frac{1}{\\sqrt{2}}| \\uparrow_z \\rangle + \\frac{1}{\\sqrt{2}}| \\downarrow_z \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix}$\u3002\n\n**\u89e3**\uff1a\n\n**\u6b65\u9aa4 1\uff1a\u9a8c\u8bc1\u5f52\u4e00\u5316\u3002**\n\n$$\\langle \\psi | \\psi \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & 1 \\end{pmatrix} \\cdot \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} = \\frac{1}{2}(1 + 1) = 1$$\n\n\u5f52\u4e00\u5316\u6210\u7acb\u3002\n\n**\u6b65\u9aa4 2\uff1a\u6d4b\u91cf $S_z$ \u7684\u6982\u7387\u3002**\n\n$$P(\\uparrow_z) = |\\langle \\uparrow_z | \\psi \\rangle|^2 = \\left| \\begin{pmatrix} 1 & 0 \\end{pmatrix} \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{\\sqrt{2}} \\right|^2 = \\frac{1}{2}$$\n\n$$P(\\downarrow_z) = |\\langle \\downarrow_z | \\psi \\rangle|^2 = \\left| \\begin{pmatrix} 0 & 1 \\end{pmatrix} \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{\\sqrt{2}} \\right|^2 = \\frac{1}{2}$$\n\n\u5404 50%\uff0c\u7b26\u5408\u9884\u671f\u3002\n\n**\u6b65\u9aa4 3\uff1a\u6d4b\u91cf $S_x$ \u7684\u6982\u7387\u3002**\n\n$$P(\\uparrow_x) = |\\langle \\uparrow_x | \\psi \\rangle|^2 = \\left| \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & 1 \\end{pmatrix} \\cdot \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{2}(1 + 1) \\right|^2 = 1$$\n\n$$P(\\downarrow_x) = |\\langle \\downarrow_x | \\psi \\rangle|^2 = \\left| \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & -1 \\end{pmatrix} \\cdot \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{2}(1 - 1) \\right|^2 = 0$$\n\n**\u5173\u952e\u89c2\u5bdf**\uff1a$|\\psi\\rangle = | \\uparrow_x \\rangle$\uff0c\u56e0\u6b64\u6d4b\u91cf $S_x$ \u65f6 100% \u5f97\u5230 $+\\hbar\/2$\u3002\u8fd9\u9a8c\u8bc1\u4e86\u5185\u79ef\u7684\u51e0\u4f55\u610f\u4e49\uff1a\u6001\u77e2\u91cf\u5b8c\u5168\u5bf9\u9f50\u65f6\uff08\u5185\u79ef\u6a21\u4e3a 1\uff09\uff0c\u6982\u7387\u4e3a 100%\uff1b\u6b63\u4ea4\u65f6\uff08\u5185\u79ef\u4e3a 0\uff09\uff0c\u6982\u7387\u4e3a 0\u3002\n\n**\u6b65\u9aa4 4\uff1a\u6d4b\u91cf\u540e\u7684\u6001\u574d\u7f29\u3002** \u5047\u8bbe\u6d4b\u91cf $S_z$ \u5f97\u5230 $+\\hbar\/2$\uff0c\u6001\u77e2\u91cf\u574d\u7f29\uff1a\n\n$$|\\psi\\rangle = \\frac{1}{\\sqrt{2}}| \\uparrow_z \\rangle + \\frac{1}{\\sqrt{2}}| \\downarrow_z \\rangle \\xrightarrow{\\text{\u6d4b\u91cf } S_z = +\\hbar\/2} |\\psi'\\rangle = | \\uparrow_z \\rangle$$\n\n\u6b64\u65f6\u518d\u6b21\u6d4b\u91cf $S_z$ \u5c06 100% \u5f97\u5230 $+\\hbar\/2$\uff0c\u4f46\u6d4b\u91cf $S_x$ \u53c8\u56de\u5230 50\/50 \u6982\u7387\u3002\u8fd9\u5c31\u662f\"\u6d4b\u91cf\u6539\u53d8\u72b6\u6001\"\u7684\u672c\u8d28\u2014\u2014\u6b63\u4ea4\u6295\u5f71\u64cd\u4f5c\u3002\n```\n\n### 13.4 \u5de5\u7a0b\u4e0e\u524d\u6cbf\u5e94\u7528\n\n\u91cf\u5b50\u5185\u79ef\u7684\u6982\u5ff5\u6b63\u5728\u50ac\u751f\u9769\u547d\u6027\u6280\u672f\uff1a\n\n- **\u91cf\u5b50\u8ba1\u7b97**\uff1a\u91cf\u5b50\u95e8\u64cd\u4f5c\u672c\u8d28\u4e0a\u662f\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e2d\u7684\u9149\u53d8\u6362\uff08\u4fdd\u5185\u79ef\u65cb\u8f6c\uff09\u3002Shor \u7b97\u6cd5\u548c Grover \u7b97\u6cd5\u5229\u7528\u91cf\u5b50\u6001\u7684\u53e0\u52a0\u548c\u5e72\u6d89\uff08\u5185\u79ef\u7684\u76f8\u4f4d\uff09\u5b9e\u73b0\u6307\u6570\u7ea7\u52a0\u901f\uff1b\n- **\u91cf\u5b50\u5bc6\u7801\u5b66**\uff1aBB84 \u534f\u8bae\u5229\u7528\u6d4b\u91cf\u57fa\u7684\u6b63\u4ea4\u6027\u68c0\u6d4b\u7a83\u542c\u2014\u2014\u7a83\u542c\u8005\u7684\u6d4b\u91cf\u4f1a\u574d\u7f29\u6001\u77e2\u91cf\uff0c\u6539\u53d8\u5185\u79ef\u7ed3\u679c\uff0c\u4ece\u800c\u88ab\u5408\u6cd5\u901a\u4fe1\u65b9\u53d1\u73b0\uff1b\n- **\u91cf\u5b50\u9690\u5f62\u4f20\u6001**\uff1a\u5229\u7528 Bell \u6001\uff08\u6700\u5927\u7ea0\u7f20\u6001\uff09\u7684\u5185\u79ef\u7ed3\u6784\u5b9e\u73b0\u91cf\u5b50\u4fe1\u606f\u7684\u8fdc\u7a0b\u4f20\u8f93\uff1b\n- **\u91cf\u5b50\u673a\u5668\u5b66\u4e60**\uff1a\u91cf\u5b50\u6838\u65b9\u6cd5\u5229\u7528\u91cf\u5b50\u6001\u5185\u79ef\u5728\u9ad8\u7ef4\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e2d\u9ad8\u6548\u8ba1\u7b97\u6838\u51fd\u6570\uff0c\u6709\u671b\u5b9e\u73b0\u91cf\u5b50\u4f18\u52bf\u3002\n\n---\n\n## \u7ec8\u7ae0 \u5927\u7edf\u4e00\u77e5\u8bc6\u56fe\u8c31\u4e0e\u54f2\u5b66\u5347\u534e\n\n### \u4e07\u7269\u7686\u6295\u5f71\u2014\u2014\u4e00\u5f20\u8d2f\u7a7f\u6240\u6709\u5b66\u79d1\u7684\u5185\u79ef\u56fe\u8c31\n\n\u56de\u987e\u5168\u6587\u6784\u5efa\u7684\u77e5\u8bc6\u4f53\u7cfb\uff0c\u4ece\u4e8c\u7ef4\u5411\u91cf\u70b9\u79ef\u5230\u65e0\u7a77\u7ef4\u590d\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u7684\u6001\u77e2\u91cf\u5185\u79ef\uff0c\u5185\u79ef\u6982\u5ff5\u8d2f\u7a7f\u4e86\u6570\u5b66\u3001\u7269\u7406\u3001\u5de5\u7a0b\u548c\u8ba1\u7b97\u673a\u79d1\u5b66\u7684\u6bcf\u4e00\u4e2a\u89d2\u843d\u3002\n\n**\u6838\u5fc3\u4e3b\u7ebf**\uff1a\u5185\u79ef $\\langle \\cdot, \\cdot \\rangle$ \u662f\u4e00\u4e2a**\u76f8\u4f3c\u5ea6\u5ea6\u91cf**\u3002\u65e0\u8bba\u5bf9\u8c61\u662f\u5411\u91cf\u3001\u51fd\u6570\u3001\u4fe1\u53f7\u3001\u56fe\u50cf\u8fd8\u662f\u91cf\u5b50\u6001\uff0c\u5185\u79ef\u90fd\u5728\u56de\u7b54\u540c\u4e00\u4e2a\u95ee\u9898\u2014\u2014\"\u8fd9\u4e24\u4e2a\u5bf9\u8c61\u6709\u591a\u76f8\u4f3c\uff1f\"\n\n**\u5927\u7edf\u4e00\u77e5\u8bc6\u56fe\u8c31**\uff1a\n\n| \u9886\u57df | \u5185\u79ef\u7684\u5177\u4f53\u5f62\u5f0f | \u51e0\u4f55\u89e3\u91ca | \u6838\u5fc3\u5e94\u7528 |\n|------|--------------|---------|---------|\n| \u7ebf\u6027\u4ee3\u6570 | $\\langle x, y \\rangle = x^T y$ | \u6295\u5f71\u957f\u5ea6 | \u6b63\u4ea4\u5206\u89e3\u3001\u6700\u5c0f\u4e8c\u4e58 |\n| \u51fd\u6570\u5206\u6790 | $\\langle f, g \\rangle = \\int fg$ | \u6ce2\u5f62\u76f8\u4f3c\u5ea6 | \u5085\u91cc\u53f6\u7ea7\u6570\u3001\u5c0f\u6ce2\u53d8\u6362 |\n| \u4fe1\u53f7\u5904\u7406 | $\\langle x, h \\rangle = \\sum x[n]h[n]$ | \u5339\u914d\u6ee4\u6ce2 | \u5377\u79ef\u3001\u76f8\u5173\u68c0\u6d4b |\n| \u6982\u7387\u7edf\u8ba1 | $\\text{Cov}(X,Y) = E[(X-\\mu_X)(Y-\\mu_Y)]$ | \u76f8\u5173\u6027\u65b9\u5411 | PCA\u3001\u56de\u5f52\u5206\u6790 |\n| \u673a\u5668\u5b66\u4e60 | $\\langle Q_i, K_j \\rangle$ | \u6ce8\u610f\u529b\u6743\u91cd | Transformer\u3001\u81ea\u6ce8\u610f\u529b |\n| \u56fe\u50cf\u5904\u7406 | $\\langle I, K \\rangle$ | \u7279\u5f81\u54cd\u5e94 | \u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u3001\u8fb9\u7f18\u68c0\u6d4b |\n| \u91cf\u5b50\u529b\u5b66 | $\\langle \\phi \\mid \\psi \\rangle$ | \u6982\u7387\u5e45 | \u6d4b\u91cf\u3001\u91cf\u5b50\u8ba1\u7b97 |\n| \u63a7\u5236\u7406\u8bba | $\\langle f, e^{-st} \\rangle$ | \u590d\u9891\u57df\u6295\u5f71 | \u62c9\u666e\u62c9\u65af\u53d8\u6362\u3001\u7a33\u5b9a\u6027\u5206\u6790 |\n\n### \u54f2\u5b66\u5347\u534e\u2014\u2014\u6295\u5f71\u5373\u8ba4\u77e5\n\n\u4ece\u54f2\u5b66\u5c42\u9762\u770b\uff0c\"\u4e07\u7269\u7686\u6295\u5f71\"\u4e0d\u4ec5\u662f\u4e00\u4e2a\u6570\u5b66\u8bba\u65ad\uff0c\u66f4\u662f\u4e00\u79cd\u8ba4\u77e5\u4e16\u754c\u7684\u65b9\u5f0f $^{[22]}$\uff1a\n\n1. **\u8ba4\u77e5\u5373\u6295\u5f71**\uff1a\u4eba\u7c7b\u8ba4\u8bc6\u4e16\u754c\u7684\u8fc7\u7a0b\uff0c\u672c\u8d28\u4e0a\u662f\u5c06\u5916\u90e8\u4e16\u754c\u7684\u590d\u6742\u4fe1\u606f\u6295\u5f71\u5230\u6709\u9650\u7684\u8ba4\u77e5\u57fa\u51fd\u6570\u4e0a\u3002\u6211\u4eec\u770b\u5230\u7684\u4e0d\u662f\"\u771f\u5b9e\u4e16\u754c\u672c\u8eab\"\uff0c\u800c\u662f\u771f\u5b9e\u4e16\u754c\u5728\u8ba4\u77e5\u57fa\u4e0a\u7684\u6295\u5f71\u7cfb\u6570\u3002\n\n2. **\u6b63\u4ea4\u5373\u72ec\u7acb**\uff1a\u5f53\u4e24\u4e2a\u6982\u5ff5\u6b63\u4ea4\u65f6\uff0c\u610f\u5473\u7740\u5b83\u4eec\u4e92\u4e0d\u5e72\u6270\u3001\u4e92\u4e0d\u91cd\u53e0\u3002\u6b63\u4ea4\u5206\u89e3\u662f\u7b80\u5316\u590d\u6742\u95ee\u9898\u7684\u7ec8\u6781\u6b66\u5668\u2014\u2014\u5c06\u590d\u6742\u7cfb\u7edf\u5206\u89e3\u4e3a\u4e92\u4e0d\u76f8\u5173\u7684\u72ec\u7acb\u6a21\u5757\u3002\n\n3. **\u6295\u5f71\u5373\u51b3\u7b56**\uff1a\u6700\u5c0f\u4e8c\u4e58\u6cd5\u8868\u660e\uff0c\u5f53\u7cbe\u786e\u89e3\u4e0d\u5b58\u5728\u65f6\uff0c\u6c42\u6295\u5f71\u662f\u6700\u4f18\u9009\u62e9\u3002\u5f53\u5b8c\u7f8e\u65b9\u6848\u4e0d\u53ef\u5f97\u65f6\uff0c\u5728\u53ef\u884c\u57df\u4e0a\u505a\u6b63\u4ea4\u6295\u5f71\uff0c\u5373\u4e3a\u6700\u4f18\u51b3\u7b56\u3002\n\n4. **\u57fa\u7684\u9009\u62e9\u51b3\u5b9a\u4e00\u5207**\uff1a\u5085\u91cc\u53f6\u9009\u62e9\u6b63\u5f26\u6ce2\u4e3a\u57fa\uff0c\u5c0f\u6ce2\u9009\u62e9\u7d27\u652f\u6491\u51fd\u6570\u4e3a\u57fa\uff0cTransformer \u9009\u62e9\u53ef\u5b66\u4e60\u7684\u6ce8\u610f\u529b\u57fa\u2014\u2014\u9009\u62e9\u4ec0\u4e48\u6837\u7684\u57fa\uff0c\u51b3\u5b9a\u4e86\u80fd\u770b\u5230\u4ec0\u4e48\u6837\u7684\u4e16\u754c\u3002\n\n### \u7ec8\u5c40\u601d\u8003\n\n\u5185\u79ef\u4e0d\u4ec5\u662f\u4e00\u4e2a\u6570\u5b66\u8fd0\u7b97\uff0c\u66f4\u662f\u8fde\u63a5\u5fae\u89c2\u4e0e\u5b8f\u89c2\u3001\u8fde\u7eed\u4e0e\u79bb\u6563\u3001\u786e\u5b9a\u6027\u4e0e\u6982\u7387\u6027\u7684**\u5143\u8bed\u8a00**\u3002\u4ece\u52fe\u80a1\u5b9a\u7406\u5230\u91cf\u5b50\u7ea0\u7f20\uff0c\u4ece\u6700\u5c0f\u4e8c\u4e58\u5230\u5927\u8bed\u8a00\u6a21\u578b\uff0c\u5185\u79ef\u4ee5\u5176\u7b80\u6d01\u800c\u6df1\u523b\u7684\u5f62\u5f0f\uff0c\u7edf\u4e00\u4e86\u4eba\u7c7b\u77e5\u8bc6\u5927\u53a6\u7684\u5404\u4e2a\u89d2\u843d\u3002\n\n---\n\n## \u9644\u5f55 \u672c\u6587\u56fe\u8868\u751f\u6210\u4ee3\u7801\n\n\u672c\u6587\u6240\u6709\u4e94\u5f20\u56fe\u8868\uff08\u4f59\u5f26\u76f8\u4f3c\u5ea6\u70ed\u529b\u56fe\u3001\u6700\u5c0f\u4e8c\u4e58\u6295\u5f71\u3001\u5085\u91cc\u53f6\u5206\u89e3\u3001\u5377\u79ef\u5339\u914d\u6ee4\u6ce2\u3001Sobel \u8fb9\u7f18\u68c0\u6d4b\uff09\u5747\u7531 main.py<\/a> \u7edf\u4e00\u751f\u6210\u3002\u8be5\u811a\u672c\u57fa\u4e8e Python \u7684\u79d1\u5b66\u8ba1\u7b97\u751f\u6001\uff08NumPy\u3001SciPy\u3001Matplotlib\uff09\uff0c\u56f4\u7ed5\"\u5185\u79ef\"\u8fd9\u4e00\u6838\u5fc3\u4e3b\u9898\uff0c\u5c06\u6587\u4e2d\u62bd\u8c61\u7684\u6570\u5b66\u6982\u5ff5\u8f6c\u5316\u4e3a\u76f4\u89c2\u7684\u53ef\u89c6\u5316\u56fe\u5f62\u3002<\/p>\n

\u811a\u672c\u7684\u6838\u5fc3\u8bbe\u8ba1\u601d\u8def\u5982\u4e0b\uff1a<\/p>\n

1. **\u4f59\u5f26\u76f8\u4f3c\u5ea6**\uff1a\u901a\u8fc7 `cosine_similarity()` \u51fd\u6570\u8ba1\u7b97\u8bcd\u5d4c\u5165\u5411\u91cf\u95f4\u7684\u5f52\u4e00\u5316\u5185\u79ef\uff0c\u751f\u6210 $5 \\times 5$ \u70ed\u529b\u56fe\u77e9\u9635\u3002\u8be5\u51fd\u6570\u5b9e\u73b0\u516c\u5f0f (1.5) \u4e2d\u7684\u4f59\u5f26\u76f8\u4f3c\u5ea6\u5b9a\u4e49\u3002
\n2. **\u6700\u5c0f\u4e8c\u4e58\u6cd5**\uff1a\u5229\u7528 `np.linalg.lstsq` \u6c42\u89e3\u6b63\u89c4\u65b9\u7a0b $A^T A \\hat{x} = A^T b$\uff08\u5b9a\u7406 3.1\uff09\uff0c\u672c\u8d28\u4e0a\u662f\u5c06\u89c2\u6d4b\u5411\u91cf\u5411\u6a21\u578b\u7a7a\u95f4\u505a\u6b63\u4ea4\u6295\u5f71\u3002
\n3. **\u5085\u91cc\u53f6\u5206\u89e3**\uff1a\u901a\u8fc7 FFT \u5c06\u65f6\u57df\u4fe1\u53f7\u6295\u5f71\u5230\u9891\u7387\u57fa\u4e0a\uff08\u5b9a\u7406 6.1\uff09\uff0c\u9891\u8c31\u4e2d\u7684\u6bcf\u4e2a\u5cf0\u503c\u5bf9\u5e94\u4e00\u4e2a\u9891\u7387\u5206\u91cf\u7684\u5185\u79ef\u7cfb\u6570\u3002
\n4. **\u5377\u79ef\u4e0e\u5339\u914d\u6ee4\u6ce2**\uff1a\u5c06\u5377\u79ef\u89c6\u4e3a\u6ed1\u52a8\u7684\u5185\u79ef\u8fd0\u7b97\uff08\u5b9a\u4e49 8.1\uff09\uff0c\u7528\u6a21\u677f\u4e0e\u4fe1\u53f7\u9010\u70b9\u505a\u5185\u79ef\u6765\u68c0\u6d4b\u8109\u51b2\u4f4d\u7f6e\u3002
\n5. **Sobel \u8fb9\u7f18\u68c0\u6d4b**\uff1a\u5c06\u4e8c\u7ef4\u5377\u79ef\u6838\u4e0e\u56fe\u50cf\u505a\u5185\u79ef\uff08\u4f8b\u9898 8.2\uff09\uff0c\u8ba1\u7b97\u6bcf\u4e2a\u50cf\u7d20\u5904\u7684\u68af\u5ea6\u5e45\u503c\u3002<\/p>\n

\u4ee5\u4e0b\u662f\u811a\u672c\u4e2d\u751f\u6210\u4f59\u5f26\u76f8\u4f3c\u5ea6\u70ed\u529b\u56fe\u7684\u6838\u5fc3\u4ee3\u7801\u7247\u6bb5\uff1a<\/p>\n

def cosine_similarity(vec_a: np.ndarray, vec_b: np.ndarray) -> float:\r\n    dot_product = float(np.dot(vec_a, vec_b))\r\n    norm_a = np.linalg.norm(vec_a)\r\n    norm_b = np.linalg.norm(vec_b)\r\n    return dot_product \/ (norm_a * norm_b)\r\n\r\ndef build_semantic_demo() -> tuple[list[str], dict[str, np.ndarray], np.ndarray]:\r\n    tokens = [\"king\", \"queen\", \"man\", \"woman\", \"apple\"]\r\n    embeddings = {\r\n        \"king\": np.array([0.92, 0.10, 0.78, 0.25, 0.60]),\r\n        \"queen\": np.array([0.90, 0.12, 0.80, 0.30, 0.63]),\r\n        \"man\": np.array([0.88, 0.18, 0.40, 0.22, 0.35]),\r\n        \"woman\": np.array([0.86, 0.22, 0.42, 0.28, 0.38]),\r\n        \"apple\": np.array([0.05, 0.95, 0.08, 0.87, 0.10]),\r\n    }\r\n    matrix = np.array(\r\n        [[cosine_similarity(embeddings[left], embeddings[right]) for right in tokens] for left in tokens]\r\n    )\r\n    return tokens, embeddings, matrix\r\n<\/code><\/pre>\n
\u5b8c\u6574\u4ee3\u7801\u8bf7\u53c2\u7167 main.py<\/a><\/p>\n
## \u53c2\u8003\u6587\u732e<\/p>\n
[1] Wikipedia contributors. (2026, April 28). Dot product. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:42, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Dot_product&oldid=1351567929<\/a>.<\/p>\n
[2] Wikipedia contributors. (2025, November 3). Orthogonal complement. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:43, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Orthogonal_complement&oldid=1320174088<\/a>.<\/p>\n
[3] Wikipedia contributors. (2025, July 7). Orthogonalization. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:44, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Orthogonalization&oldid=1299273509<\/a>.<\/p>\n
[4] Wikipedia contributors. (2025, September 1). Orthogonal functions. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:46, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Orthogonal_functions&oldid=1308940353<\/a>.<\/p>\n
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