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‎2020/05/12/DTFT/index.html

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<link rel="dns-prefetch" href="http://yoursite.com">
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<title>DTFT | Hexo</title>
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<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1">
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<meta name="description" content="DTFT \begin{aligned} X(f) &amp;&#x3D; \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt\\ x^*(t) &amp;&#x3D; \int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df\\ \end{aligned}When $\hat{x}(t) &#x3D; x(t) p(t) &#x3D; x(t) \cdot \sum \delta">
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<meta name="description" content="DTFT \begin{aligned} X(f) &amp;&#x3D; \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt\\ x^*(t) &amp;&#x3D; \int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df\\ \end{aligned}">
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<meta property="og:type" content="article">
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<meta property="og:title" content="DTFT">
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<meta property="og:url" content="http://yoursite.com/2020/05/12/DTFT/index.html">
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<meta property="og:site_name" content="Hexo">
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<meta property="og:description" content="DTFT \begin{aligned} X(f) &amp;&#x3D; \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt\\ x^*(t) &amp;&#x3D; \int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df\\ \end{aligned}When $\hat{x}(t) &#x3D; x(t) p(t) &#x3D; x(t) \cdot \sum \delta">
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<meta property="og:description" content="DTFT \begin{aligned} X(f) &amp;&#x3D; \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt\\ x^*(t) &amp;&#x3D; \int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df\\ \end{aligned}">
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<meta property="og:locale" content="en_US">
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<meta property="article:published_time" content="2020-05-12T23:16:42.809Z">
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<meta property="article:modified_time" content="2020-02-23T22:43:31.872Z">
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<meta property="article:published_time" content="2020-05-13T03:56:48.428Z">
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<meta property="article:modified_time" content="2020-05-13T03:56:48.428Z">
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<meta property="article:author" content="dassein">
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<meta property="article:tag" content="DSP">
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<meta property="article:tag" content="math">
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<a href="/2020/05/12/DTFT/" class="archive-article-date">
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<time datetime="2020-05-12T23:16:42.809Z" itemprop="datePublished"><i class="icon-calendar icon"></i>2020-05-12</time>
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<time datetime="2020-05-13T03:56:48.428Z" itemprop="datePublished"><i class="icon-calendar icon"></i>2020-05-12</time>
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\begin{aligned}
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X(f) &= \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt\\
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x^*(t) &= \int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df\\
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\end{aligned}</script><p>When $\hat{x}(t) = x(t) p(t) = x(t) \cdot \sum \delta(t-nT)= \sum x(nT) \delta(t-nT)$, then we have $\lambda &lt; 1$ that</p>
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\end{aligned}</script><a id="more"></a>
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<p>When $\hat{x}(t) = x(t) p(t) = x(t) \cdot \sum \delta(t-nT)= \sum x(nT) \delta(t-nT)$, then we have $\lambda &lt; 1$ that</p>
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<script type="math/tex; mode=display">
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\begin{aligned}
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\hat{X}(f) &= \int_{-\infty}^{\infty} \hat{x}(t) e^{-j2\pi ft} dt = \sum_n x(nT) \int_{-\infty}^{\infty} \delta(t-nT)e^{-j2\pi ft} dt\\
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&=\sum_n x(nT) e^{-j2 \pi n (f/f_s)}\\
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\hat{x}^*(t) &= \int_{-\infty}^{\infty} \hat{X}(f) e^{j2\pi ft} df\\
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x(nT) &= \int_{-\infty}^{\infty} \hat{x}^*(t) \frac{\sin(\pi(t-nT))}{\pi(t-nT)} dt\\
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&= \int_{-0.5}^{0.5} \hat{X}(f) e^{j2\pi (f/f_s)n} d(f/f_s)\\
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&= \int_{-0.5}^{0.5} \hat{X}(f) e^{j2\pi (f/f_s)n} d(f/f_s)\\
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\end{aligned}</script><p>Now define $\omega = 2\pi f/f_s$, then</p>
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<script type="math/tex; mode=display">
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X(e^{jw}) \equiv \hat{X}(f) = \sum_n x(nT) [e^{j\omega}]^{-n}\\
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&= \sum_{k} \Bigg[\frac{f_s}{N} \left\{ \sum_{n=0}^{N-1} x(nT) e^{-j2\pi (k/N)n} \right\} \Bigg] \int_{-\infty}^{\infty} e^{j2\pi ft} \delta(f-k\frac{f_s}{N}) df \\
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&= \sum_{k} \Bigg[\frac{f_s}{N} \left\{ \sum_{n=0}^{N-1} x(nT) e^{-j2\pi (k/N)n} \right\} \Bigg] e^{j2\pi (k/N)f_st} \\
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x(nT) &= \int_{-\infty}^{\infty} \tilde{x}^*(t) \frac{\sin(\pi(t-nT))}{\pi(t-nT)} dt\\
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&= \int_{-0.5}^{0.5} \tilde{X}(f) e^{j2\pi (f/f_s)n} d(f/f_s) = \int_{0}^{1} \tilde{X}(f) e^{j2\pi (f/f_s)n} d(f/f_s)\\
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&= \int_{-0.5}^{0.5} \tilde{X}(f) e^{j2\pi (f/f_s)n} d(f/f_s) = \int_{0}^{1} \tilde{X}(f) e^{j2\pi (f/f_s)n} d(f/f_s)\\
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&= \int_{0}^{1} e^{j2\pi (f/f_s)n} d(f/f_s) \sum_{k} \delta(f-k\frac{f_s}{N}) \Bigg[\frac{f_s}{N} \left\{ \sum_{n=0}^{N-1} x(nT) e^{-j2\pi (k/N)n} \right\} \Bigg]\\
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&= \sum_{k} \int_{0}^{1} e^{j2\pi (\frac{f}{f_s})n} \delta(\frac{f}{f_s}-\frac{k}{N}) d(\frac{f}{f_s})
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&= \sum_{k} \int_{0}^{1} e^{j2\pi (\frac{f}{f_s})n} \delta(\frac{f}{f_s}-\frac{k}{N}) d(\frac{f}{f_s})
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\Bigg[\frac{1}{N} \left\{ \sum_{n=0}^{N-1} x(nT) e^{-j2\pi (k/N)n} \right\} \Bigg]\\
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&= \sum_{k} \int_{0}^{1} e^{j2\pi (\frac{k}{N})n} \delta(\frac{f}{f_s}-\frac{k}{N}) d(\frac{f}{f_s})
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\Bigg[\frac{1}{N} \left\{ \sum_{n=0}^{N-1} x(nT) e^{-j2\pi (k/N)n} \right\} \Bigg]\\
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&= \sum_{k=0}^{N-1} e^{j2\pi (\frac{k}{N})n} \Bigg[\frac{1}{N} \left\{ \sum_{n=0}^{N-1} x(nT) e^{-j2\pi (k/N)n} \right\} \Bigg]\\
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&= \frac{1}{N} \sum_{k=0}^{N-1} \left\{ \sum_{n=0}^{N-1} x(nT) e^{-j2\pi (\frac{k}{N}) n} \right\} e^{j2\pi (\frac{k}{N})n}
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&= \sum_{k} \int_{0}^{1} e^{j2\pi (\frac{k}{N})n} \delta(\frac{f}{f_s}-\frac{k}{N}) d(\frac{f}{f_s})
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\Bigg[\frac{1}{N} \left\{ \sum_{n=0}^{N-1} x(nT) e^{-j2\pi (k/N)n} \right\} \Bigg]\\
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&= \sum_{k=0}^{N-1} e^{j2\pi (\frac{k}{N})n} \Bigg[\frac{1}{N} \left\{ \sum_{n=0}^{N-1} x(nT) e^{-j2\pi (k/N)n} \right\} \Bigg]\\
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&= \frac{1}{N} \sum_{k=0}^{N-1} \left\{ \sum_{n=0}^{N-1} x(nT) e^{-j2\pi (\frac{k}{N}) n} \right\} e^{j2\pi (\frac{k}{N})n}
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\end{aligned}</script>
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‎2020/05/12/Hexo 的主题中渲染 MathJax 数学公式/index.html

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<title>Hexo 的主题中渲染 MathJax 数学公式 | Hexo</title>
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<meta name="description" content="Hexo 的主题中渲染 MathJax 数学公式https:&#x2F;&#x2F;blog.csdn.net&#x2F;wgshun616&#x2F;article&#x2F;details&#x2F;81019687 原因Hexo 默认使用 hexo-renderer-marked 引擎渲染网页，该引擎会把一些特殊的 markdown 符号转换为相应的 html 标签，比如在 markdown 语法中，下划线_代表斜体，会被渲染引擎处理为&lt;em&amp;">
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<meta name="description" content="Hexo 的主题中渲染 MathJax 数学公式https:&#x2F;&#x2F;blog.csdn.net&#x2F;wgshun616&#x2F;article&#x2F;details&#x2F;81019687">
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<meta property="og:url" content="http://yoursite.com/2020/05/12/Hexo%20%E7%9A%84%E4%B8%BB%E9%A2%98%E4%B8%AD%E6%B8%B2%E6%9F%93%20MathJax%20%E6%95%B0%E5%AD%A6%E5%85%AC%E5%BC%8F/index.html">
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<meta property="og:description" content="Hexo 的主题中渲染 MathJax 数学公式https:&#x2F;&#x2F;blog.csdn.net&#x2F;wgshun616&#x2F;article&#x2F;details&#x2F;81019687 原因Hexo 默认使用 hexo-renderer-marked 引擎渲染网页，该引擎会把一些特殊的 markdown 符号转换为相应的 html 标签，比如在 markdown 语法中，下划线_代表斜体，会被渲染引擎处理为&lt;em&amp;">
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<h2 id="Hexo-的主题中渲染-MathJax-数学公式"><a href="#Hexo-的主题中渲染-MathJax-数学公式" class="headerlink" title="Hexo 的主题中渲染 MathJax 数学公式"></a>Hexo 的主题中渲染 MathJax 数学公式</h2><p><a href="https://blog.csdn.net/wgshun616/article/details/81019687" target="_blank" rel="noopener">https://blog.csdn.net/wgshun616/article/details/81019687</a></p>
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<h3 id="原因"><a href="#原因" class="headerlink" title="原因"></a>原因</h3><p>Hexo 默认使用 hexo-renderer-marked 引擎渲染网页，该引擎会把一些特殊的 markdown 符号转换为相应的 html 标签，比如在 markdown 语法中，下划线<code>_</code>代表斜体，会被渲染引擎处理为<code>&lt;em&gt;</code>标签。</p>
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<p>因为类 Latex 格式书写的数学公式下划线<code>_</code>表示下标，有特殊的含义，如果被强制转换为<code>&lt;em&gt;</code>标签，那么 MathJax 引擎在渲染数学公式的时候就会出错。</p>
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<p>类似的语义冲突的符号还包括<code>*</code>, <code>{</code>, <code>}</code>, <code>\\</code>等。</p>
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</ul>

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