信号与系统——卷积的性质

对于信号与系统这门课来说，其卷积是这门课中非常重要的一个知识点。下面就来说一说卷积的性质。

  • 交换律

f(t) * g(t) = g(t) * f(t)    ，进而可以推出：

f_{1}(t) *f_{2}(t) = \int_{-\infty }^{\infty}f_{1}(\tau )f_{2}(t-\tau )d\tau = \int_{-\infty }^{\infty}f_{2}(\lambda )f_{1}(t-\lambda ) = f_{2}(t) * f_{1}(t)

 

  • 分配律

f(t) * [g(t) + k(t) ] = f(t) * g(t) + f(t) * k(t)

  •  结合律

f(t) * [g(t) * k(t) ] = g(t) * [ f(t) * k(t) ]

  •  微分与积分

\frac{\mathrm{d} }{\mathrm{d}t}[f_{1})(t) *f_{2})(t) ] = \frac{\mathrm{d} }{\mathrm{d} t}\int_{-\infty }^{\infty}f_{1}(\tau )f_{2}(t-\tau )d\tau = \int_{-\infty }^{\infty}f_{1}(\tau )\frac{\mathrm{d}f_{2}(t-\tau ) }{\mathrm{d} t}d\tau = f_{1}(t ) * \frac{\mathrm{d} f_{2}(t )}{\mathrm{d} t}

所以可以得到：\frac{\mathrm{d} }{\mathrm{d} x}[f_{2}(t) *f_{1}(t)] = f_{2}(t) * \frac{\mathrm{d} f_{1}(t)}{\mathrm{d} x}，

所以可证得交换律。

  • 一些重要的卷积公式

f(t) * \sigma (t) = \int_{-\infty }^{\infty}f(\tau )\sigma(t-\tau )d\tau = \int_{-\infty }^{\infty}f(\tau )\sigma(\tau -t) d\tau =f(t)

因为 \sigma(x) = \sigma(-x)，所以\sigma (t-\tau ) = \sigma (\tau -t)

f(t) * \sigma (t-t_{0}) = \int_{-\infty }^{\infty}f(\tau )\sigma(t-t_{0}-\tau )d\tau =f(t-t_{0})

f(t) * {\sigma}'(t) = {f}'(t)

依据   f(t) * u(t) = \int_{-\infty }^{t}f(\lambda )d\lambda    ,可推出  f(t) * \sigma ^{(k)}(t) = f^{(k)}(t)       与  f(t) * \sigma ^{(k)}(t-t_{0}) = f^{(k)}(t-t_{0})

posted @ 2021-04-13 13:34  Xa_L  阅读(1073)  评论(0)  编辑  收藏  举报