https://icci.ink/categories/%E5%82%85%E9%87%8C%E5%8F%B6/Recent content in 傅里叶 on ICCIHugozh-cnFri, 18 Apr 2025 00:00:00 +0000https://icci.ink/study/theory/math-%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2%E7%9A%84%E6%80%A7%E8%B4%A8/Fri, 18 Apr 2025 00:00:00 +0000https://icci.ink/study/theory/math-%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2%E7%9A%84%E6%80%A7%E8%B4%A8/<p>傅里叶变换的性质</p> <h2 id="傅里叶变换的性质">傅里叶变换的性质<a class="anchor" href="#%e5%82%85%e9%87%8c%e5%8f%b6%e5%8f%98%e6%8d%a2%e7%9a%84%e6%80%a7%e8%b4%a8">#</a></h2> <table> <thead> <tr> <th>名称</th> <th>时域 ( f(t) )</th> <th>频域 ( F(j\omega) )</th> </tr> </thead> <tbody> <tr> <td>定义</td> <td>( f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(j\omega) e^{j\omega t} d\omega )</td> <td>( F(j\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt ) <br> ( F(j\omega) =</td> </tr> <tr> <td>线性</td> <td>( af_1(t) + bf_2(t) )</td> <td>( aF_1(j\omega) + bF_2(j\omega) )</td> </tr> <tr> <td>奇偶性</td> <td>( f(t) ) 为实函数 <br> ( f(t) = f(-t) ) <br> ( f(t) = -f(-t) )</td> <td>(</td> </tr> <tr> <td>反转</td> <td>( f(-t) )</td> <td>( F(-j\omega) )</td> </tr> <tr> <td>对称性</td> <td>( F(jt) )</td> <td>( 2\pi f(-\omega) )</td> </tr> <tr> <td>尺度变换</td> <td>( f(at), \quad a \neq 0 )</td> <td>( \frac{1}{</td> </tr> <tr> <td>时移特性</td> <td>( f(t \pm t_0) )</td> <td>( e^{\mp j\omega t_0} F(j\omega) )</td> </tr> <tr> <td>频移特性</td> <td>( f(t) e^{j\omega_0 t} )</td> <td>( F[j(\omega \mp \omega_0)] )</td> </tr> <tr> <td>卷积定理</td> <td>( f_1(t) * f_2(t) )</td> <td>( F_1(j\omega) F_2(j\omega) ) <br> ( f_1(t) f_2(t) ) <br> ( \frac{1}{2\pi} F_1(j\omega) * F_2(j\omega) )</td> </tr> <tr> <td>时域微分</td> <td>( f^{(n)}(t) )</td> <td>( (j\omega)^n F(j\omega) )</td> </tr> <tr> <td>时域积分</td> <td>( f^{(-1)}(t) )</td> <td>( \pi F(0) \delta(\omega) + \frac{1}{j\omega} F(j\omega) )</td> </tr> <tr> <td>频域微分</td> <td>(-jt)^n f(t)</td> <td>F^(n)(jω)</td> </tr> <tr> <td>频域积分</td> <td>π f(0)δ(t) + frac(1}{-jt) f(t)</td> <td>F^(-1)(jω)</td> </tr> <tr> <td>相关定理</td> <td>R12(τ) = int(-∞ to ∞) f1(t) f2(t-τ) dt</td> <td>ℱ[R12(τ)] = F1(jω) * F2(jω)</td> </tr> </tbody> </table> <p>请注意，表格中的公式仍然使用了 LaTeX 语法来表示数学表达式。如果您需要在特定的Markdown环境中使用这个表格，可能需要根据该环境的语法规则进行适当的调整。</p>