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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd"> <html> <head> <title>page_100</title> <link rel="stylesheet" href="reset.css" type="text/css" media="all"> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> </head> <body> <table summary="top nav" border="0" width="100%"> <tr> <td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_99.html">< previous page</a></td> <td id="ebook_previous" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_100</strong></td> <td align="right" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_101.html">next page ></a></td> </tr> <tr> <td id="ebook_page" align="left" colspan="3" style="background: #ffffff; padding: 20px;"> <table border="0" width="100%" cellpadding="0"><tr><td align="center"> <table border="0" cellpadding="2" cellspacing="0" width="100%"><tr><td align="left"></td> <td align="right"></td> </tr></table></td></tr><tr><td align="left"><p></p><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="right"><font face="Times New Roman, Times, Serif" size="2" color="#FF0000">Page 100</font></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">To achieve this objective, assume that the vector <img src="59d309f86810e74b55acdfae11c7d758.gif" border="0" alt="XCIRCM.GIF" width="20" height="16" /> is known and</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="baf2b23f4d4826a459e73ac2f0be3b28.gif" border="0" alt="0100-01.GIF" width="157" height="19" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">Let the new measurement be defined as <i>y</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>+1</sub></font><font face="Times New Roman, Times, Serif" size="3"> = H</font><font face="Times New Roman, Times, Serif" size="1"><sub><i>m</i>+1</sub></font><font face="Times New Roman, Times, Serif" size="3">x + <i>n</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>+1</sub></font><font face="Times New Roman, Times, Serif" size="3">, where var(<i>n</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>+1</sub></font><font face="Times New Roman, Times, Serif" size="3"> <i>n</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>j</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="3">) = <i>R</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>+1,<i>j</i></sub></font><font face="Times New Roman, Times, Serif" size="3"></font><font face="Symbol" size="3">d</font><font face="Times New Roman, Times, Serif" size="1"><sub><i>m</i>+1,<i>j</i></sub></font><font face="Times New Roman, Times, Serif" size="3"> for all <i>j</i> (<i>R</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>+1</sub></font><font face="Times New Roman, Times, Serif" size="3"> will be used as a shorthand notation for <i>R</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>+1,<i>m</i>+1</sub></font><font face="Times New Roman, Times, Serif" size="3"> in the following). From Eq. (4.6),</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="87abfd75afef8bc0f78a8e7cc4a45e7b.gif" border="0" alt="0100-02.GIF" width="276" height="18" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">Therefore the information available for estimating x after measurement <i>m + </i>1 is</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="432ce87e936a8f2869999977f7f59744.gif" border="0" alt="0100-03.GIF" width="141" height="41" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">Constructing a hypothetical set of <i>n</i> + 1 measurements yields</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="01165ca7d7131f8bd26529018f506b43.gif" border="0" alt="0100-04.GIF" width="339" height="39" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">If the weighting matrix R</font><font face="Times New Roman, Times, Serif" size="2"><sup>-1</sup></font><font face="Times New Roman, Times, Serif" size="3"> is chosen as corresponding to <img src="e3d2c24047dec98a1999416e7fe8cf7c.gif" border="0" alt="C0100-02.GIF" width="98" height="20" />, Eq. (4.11) results in</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="e2c5eb033ec1e9e53c283d5d8c0f253a.gif" border="0" alt="0100-05.GIF" width="383" height="76" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">This formula shows the amount by which each new measurement increases the information matrix. Note that if all measurements are noisy (i.e., <i>R</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>j</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="3"></font><font face="Symbol" size="3">鹿</font><font face="Times New Roman, Times, Serif" size="3"> 0 for any <i>j</i>), then <img src="bb99a581f9d47192b0fb12dbc2b379e5.gif" border="0" alt="C0100-03.GIF" width="96" height="20" /> is always finite and positive; the information matrix is monotonically increasing, and the variance of the estimate (i.e, P</font><font face="Times New Roman, Times, Serif" size="1"><sub><i>m</i>+1</sub></font><font face="Times New Roman, Times, Serif" size="3">) monotonically decreases with each new measurement. From Eq. (4.19),</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="665ce31fd771f24d06f7f7dfc671cdf7.gif" border="0" alt="0100-06.GIF" width="342" height="24" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="48a77c82e2650464505fdb2177331981.gif" border="0" alt="0100-07.GIF" width="364" height="24" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">which will be needed in the following derivation.</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">The new estimate is calculated with Eq. (4.10) as</font><font face="Times New Roman, Times, Serif" size="3" color="#FFFF00"></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="15442ec9de1d188b8fc9a43bb002c8bb.gif" border="0" alt="0100-08.GIF" width="392" height="116" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr></table><p><font size="0"></font></p>聽 </td> </tr> <tr> <td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_99.html">< previous page</a></td> <td id="ebook_next" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_100</strong></td> <td align="right" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_101.html">next page ></a></td> </tr> </table> </body> </html>⌨️ 快捷键说明
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