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class="h pbm"> <h1 class="ph"> 频域分析随机过程的四个优势 </h1> <p class="xg2"> <span class="xg1">已有 11521 次阅读</span> <span class="xg1">2024-11-22 13:46</span> <span class="pipe">|</span><span class="xg1">个人分类:<a href="https://blog.sciencenet.cn/home.php?mod=space&uid=3418723&do=blog&classid=177005&view=me">随机过程</a></span><span class="pipe">|</span><span class="xg1">系统分类:<a href="https://blog.sciencenet.cn/home.php?mod=space&do=blog&view=all&uid=3418723&catid=1">科研笔记</a></span> </p> </div> <div id="blog_article" class="d cl" ><p style="line-height: 2em; text-indent: 2em;"><span style="text-wrap-mode: nowrap; font-size: 18px; font-family: 宋体, SimSun; color: #000000;">《</span><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;"><span style="text-wrap-mode: nowrap;"><span style="font-size: 18px; font-family: 宋体, SimSun; text-wrap-mode: nowrap; color: #000000;">随机过程》是研究随机运动现象及规律的<strong><span style="font-size: 18px; font-family: 宋体, SimSun; text-wrap-mode: nowrap; color: #000000;">应用数学理论</span></strong>，</span>但是</span>《随机过程》<span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">的<strong><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">概率分析方法</span></strong></span>在描述和解决实际问题时，会遇到如下2个难题：</span></p><p style="line-height: 2em; text-indent: 2em;"><strong><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">（1）</span></strong><strong><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">无法描述实际问题</span></strong></p><p style="line-height: 2em; text-indent: 2em;"><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">无法用数学解析式描述<span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">随机过程中的</span>任何一个样本函数，例如建立单个布朗粒子的运动学方程、惯性陀螺仪的随机游走误差模型和金融市场的股票价格模型等。</span></p><p style="line-height: 2em; text-indent: 2em;"><strong><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">（2）</span></strong><strong><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">无法解决实际问题</span></strong></p><p style="line-height: 2em; text-indent: 2em;"><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">不能提供从样本函数中提取有用信息，并衰减随机干扰的方法及工具，例如实时补偿惯性陀螺仪中的随机游走误差，解决飞机、舰船、火箭、卫星、宇宙飞船和导弹等航行体的精确导航和精确制导问题。</span></p><p style="line-height: 2em; text-indent: 2em;"><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">《随机信号分析》的频域分析方法可轻易地解决《随机过程》在时域遇到的上述两个问题。</span></p><p style="line-height: 2em; text-indent: 2em;"><strong><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">1、</span></strong><strong><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">可用确定性的数学解析式描述样本函数</span></strong></p><p style="line-height: 2em; text-indent: 2em;"><span style="font-size: 18px; font-family: 宋体, SimSun; color: #000000;">例如，布朗运动样本函数在频域的功率谱密度函数可用确定性的解析式表示为</span></p><p style="text-indent: 2em;"></p><p style="line-height: 2em; text-indent: 2em;"><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;"><img class="kfformula" 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data-latex="P(.omega )=.frac {{N}_{0}} {{.omega }^{2}}"/></span></p><p style="line-height: 2em;"><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">式中<em>N</em><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;"><sub><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">0</span></sub></span>为布朗运动瞬时速度的平均功率。</span></p><p style="line-height: 2em; text-indent: 2em;"><strong><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">2、</span></strong><strong><span style="text-wrap-mode: nowrap; font-size: 18px; font-family: 宋体, SimSun; color: #000000;">所有样本函数在频域具有相同的数学模型</span></strong></p><p style="line-height: 2em; text-indent: 2em;"><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">所有样本函数在时域的<span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">数学</span>描述<span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">虽然<span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">各不相</span></span>同，但它们在频域的功率谱密度函数是相同的。</span></p><p style="line-height: 2em; text-indent: 2em;"><strong><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">3、</span></strong><strong><span style="text-wrap-mode: nowrap; font-size: 18px; font-family: 宋体, SimSun; color: #000000;">能分离有用信息和随机干扰</span></strong></p><p style="line-height: 2em; text-indent: 2em;"><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">样本函数中的有用信息和随机干扰在时域是叠加的，在频域是分离的（位于不同的频段）。</span></p><p style="line-height: 2em; text-indent: 2em;"><strong><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">4、</span></strong><strong><span style="text-wrap-mode: nowrap; font-size: 18px; font-family: 宋体, SimSun; color: #000000;">为滤波器处理随机信号提供准确的设计依据</span></strong></p><p style="line-height: 2em; text-indent: 2em;"><span style="white-space-collapse: preserve; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">根据样本函数的功率谱密度函数提供的信息，可准确设计滤波器的频率响应曲线，确保样本函数中的随机干扰被有效衰减，有用信号无失真通过。</span></p><p style="line-height: 2em; text-indent: 2em;"><span style="font-family: 宋体, SimSun; font-size: 18px; color: #000000;">总之，频域分析可将时域的随机性问题转化为频域的确定性问题，然后利用已有的数学分析工具研究随机过程，看到时域无法看到的随机过程特征及规律，因此，频域被专业学者们称为上帝的视角。</span></p><p style="line-height: 2em; text-indent: 2em;"><span style="font-family: 宋体, SimSun; font-size: 18px; color: #000000;">北大数院院长陈大岳教授在其负责编写的《应用随机过程》前言中写道：“<span style="font-size: 18px; white-space-collapse: preserve; font-family: 宋体, SimSun; color: #000000;">作者并无实际应用经验，但</span>我们深信，随机过程理论可以用来描述、理解、解决实际问题”。一个理论能否“用来描述、理解、解决实际问题”，只能通过实践的检验，这是科学基本常识，而不是<span style="font-family: 宋体, SimSun; font-size: 18px; color: #000000;">对没有实际应用经验的数学家们的</span>盲目“深信”。</span></p><p>      </p><p>             </p><p style="line-height: 2em;"><strong><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">参考</span></strong><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">：</span></p><p style="line-height: 2em; text-indent: 2em;"><span style="text-wrap-mode: nowrap; font-size: 18px; font-family: arial, helvetica, sans-serif; color: #000000;">[1] </span><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">工程师的随机过程分析工具降维碾压数学家</span></p><p style="line-height: 2em; text-indent: 2em;"><span style="text-wrap-mode: nowrap; color: #000000; font-family: 宋体, SimSun; font-size: 18px;"><a href="https://blog.sciencenet.cn/blog-3418723-1459016.html" target="_blank" target="_blank">https://blog.sciencenet.cn/blog-3418723-1459016.html</a>   </span></p><p style="line-height: 2em; text-indent: 2em;"><span style="text-wrap-mode: nowrap; font-size: 18px; font-family: arial, helvetica, sans-serif; color: #000000;">[2] </span><span style="text-wrap-mode: nowrap; font-family: 宋体, SimSun; font-size: 18px; color: #000000;">随机运动理论中的“地心说”与“日心说”</span></p><p style="line-height: 2em; text-indent: 2em;"><span style="text-wrap-mode: nowrap; color: #000000; font-family: 宋体, SimSun; font-size: 18px;"><a href="https://blog.sciencenet.cn/blog-3418723-1459406.html" target="_blank" target="_blank">https://blog.sciencenet.cn/blog-3418723-1459406.html</a>   </span></p><p>    </p><p>    </p> <br /><br /> <label style="font-size:13px; color:#850f0f">转载本文请联系原作者获取授权，同时请注明本文来自高宏科学网博客。<br />链接地址：</label><a href="https://blog.sciencenet.cn/blog-3418723-1461172.html" target="_blank" style="font-size:13px; color:#850f0f">https://blog.sciencenet.cn/blog-3418723-1461172.html </a> <br /><br />上一篇：<a href='blog-3418723-1461056.html' target='_black'>《随机过程》理论描述和解决实际问题时遇到的两个难题</a><br />下一篇：<a href='blog-3418723-1461460.html' target='_black'>为什么《随机过程》无法用解析式描述随机现象？</a>  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