信息处理 - LTI 系统互为逆 - 吾爱随笔录

所以我的问题是这样的：

假设我们有两个具有脉冲响应的 LTI 系统确定这些系统是否互为逆。
$h_{1} (t) = \frac{1}{2} δ (t - 1) and h_{2} (t) = 2 δ (t + 1) .$ $h_1(t) = \frac 12\delta(t-1)\quad\text{and}\quad h_2(t)=2\delta(t+1).$

这是我所做的，但我被卡住了。我很感激这方面的一些帮助

h_{1} (t) ⋆ h_{2} (t) \overset{?}{=} δ (t)

$h_1(t)\star h_2(t)\overset{?}{=}\delta(t)$

\begin{aligned} \int_{- \infty}^{\infty} h_{1} (τ) h_{2} (t - τ) d τ & = \int_{- \infty}^{\infty} \frac{1}{2} δ (τ - 1) 2 δ (t - τ + 1) d τ \\ = \int_{- \infty}^{\infty} δ (τ - 1) δ (t - τ + 1) d τ \end{aligned}

$\begin{align} \int_{-\infty}^\infty h_1(\tau)h_2(t-\tau)d\tau &=\int_{-\infty}^\infty \frac12\delta(\tau-1)2\delta(t-\tau+1)d\tau\\ &=\int_{-\infty}^\infty \delta(\tau-1)\delta(t-\tau+1)d\tau\\ \end{align}$

u = τ - 1 ⟹ \frac{d u}{d τ} = 1 ⟹ d u = d τ

$u = \tau - 1\implies \frac{du}{d\tau}=1\implies du = d\tau$

⟹ \int_{- \infty}^{\infty} h_{1} (τ) h_{2} (t - τ) d τ = \int_{- \infty}^{\infty} δ (u) δ (t - u) d τ

$\implies \int_{-\infty}^\infty h_1(\tau)h_2(t-\tau)d\tau = \int_{-\infty}^\infty \delta(u)\delta(t-u)d\tau$