卷积积分¶
\[ (f_1 * f_2)(t) := \int_{-\infty}^{+\infty} f_1(\tau)f_2(t-\tau) \mathrm{d}\tau \]
性质¶
- 交换律 \(f_1 * f_2 = f_2 * f_1\)
- 分配律 \(f_1 * (f_2 + f_3) = (f_1 * f_2) + (f_1 * f_3)\)
- 结合律 \(f_1 * (f_2 * f_3) = (f_1 * f_2) * f_3\)
- \(f * \delta = f\)
时移¶
微积分¶
\[ f^{(-1)}(t) := \int_{-\infty}^{t} f(\tau) \mathrm{d}\tau \]
若 \(f = f_1 * f_2\) 则
\[ f' = f_1' * f_2 = f_1 * f_2' \]
杜阿密尔积分¶
在 LTI 系统中，若激励为 \(f(t)\) 则零状态响应
\[ y_{zs}(t)=f(t)*h(t)=f'(t)*g(t) \]
图解法计算卷积¶
常用卷积¶
| \(f_1\) | \(f_2\) | \(f_1*f_2\) |
|---|---|---|
| \(f(t)\) | \(\varepsilon(t)\) | \(\displaystyle\int_{-\infty}^{t} f(\tau) \mathrm{d}\tau\) |
| \(\varepsilon(t)\) | \(\varepsilon(t)\) | \(t\varepsilon(t)\) |
| \(t\varepsilon(t)\) | \(\varepsilon(t)\) | \(\dfrac{1}{2}t^2\varepsilon(t)\) |
| \(e^{-\alpha t}\varepsilon(t)\) | \(\varepsilon(t)\) | \(\dfrac{1}{\alpha}(1-e^{-\alpha t})\varepsilon(t)\) |
| \(e^{-\alpha t}\varepsilon(t)\) | \(e^{-\alpha t}\varepsilon(t)\) | \(te^{-\alpha t}\varepsilon(t)\) |
| \(te^{-\alpha t}\varepsilon(t)\) | \(e^{-\alpha t}\varepsilon(t)\) | \(\dfrac{1}{2}t^2e^{-\alpha t}\varepsilon(t)\) |
两个门函数的卷积¶
常用的卷积，可以直接记下来。
