https://icci.ink/categories/z%E5%8F%98%E6%8D%A2/Recent content in Z变换 on ICCIHugozh-cnFri, 18 Apr 2025 00:00:00 +0000https://icci.ink/study/theory/math-z%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-z%E5%8F%98%E6%8D%A2%E7%9A%84%E6%80%A7%E8%B4%A8/<p>Z变换的性质</p> <h2 id="z变换的性质">Z变换的性质<a class="anchor" href="#z%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>k 域</th> <th>z 域</th> </tr> </thead> <tbody> <tr> <td>定义</td> <td>(f(k)=\frac{1}{2\pi j}\oint_{C}F(z)z^{k-1}dz)</td> <td>(F(z)=\sum_{k=-\infty}^{\infty}f(k)z^{-k}, \alpha&lt;</td> </tr> <tr> <td>线性</td> <td>(a_1f_1(k)+a_2f_2(k))</td> <td>(a_1F_1(z)+a_2F_2(z), \max(\alpha_1,\alpha_2)&lt;</td> </tr> <tr> <td>移位</td> <td>双边变换</td> <td>(f(k+m))</td> </tr> <tr> <td></td> <td>单边变换</td> <td>(f(k-m), m&gt;0)</td> </tr> <tr> <td></td> <td></td> <td>(f(k+m), m&gt;0)</td> </tr> <tr> <td>z 域尺度变换</td> <td>(a^kf(k), a\neq0)</td> <td>(F\left(\frac{z}{a}\right), \alpha</td> </tr> <tr> <td>k 域卷积</td> <td>(f_1(k)*f_2(k))</td> <td>(F_1(z)F_2(z), \max(\alpha_1,\alpha_2)&lt;</td> </tr> <tr> <td>z 域微分</td> <td>(k^mf(k), m&gt;0)</td> <td>([-z\frac{d}{dz}]^mF(z), \alpha&lt;</td> </tr> <tr> <td>z 域积分</td> <td>(\frac{f(k)}{k+m}, k+m&gt;0)</td> <td>(z^n\int_{z}^{\infty}\frac{F(\eta)}{\eta^{n+1}}d\eta, \alpha&lt;</td> </tr> <tr> <td>k域反转</td> <td>f(-k)</td> <td></td> </tr> <tr> <td>部分和</td> <td>(\sum_{i=-\infty}^{k} f(i))</td> <td></td> </tr> <tr> <td>初值定理</td> <td>因果序列</td> <td>( f(0) = \lim_{z \to \infty} F(z) )</td> </tr> <tr> <td></td> <td></td> <td>( f(m) = \lim_{z \to \infty} z^m [F(z) - \sum_{k=0}^{m-1} f(k) z^{-k}] ), (</td> </tr> <tr> <td>终值定理</td> <td></td> <td>( f(\infty) = \lim_{z \to 1} (z-1) F(z) ), (\lim_{k \to \infty} f(k)) 收敛, (</td> </tr> </tbody> </table> <p>注: α、β为正实常数, 分别称为收敛域的内、外半径。</p>