LTI systems inverse of each other So my question looks like this:


  Suppose that we have two LTI systems with impulse responses 
  $$h_1(t) = \frac 12\delta(t-1)\quad\text{and}\quad h_2(t)=2\delta(t+1).$$
  Determine whether these systems are inverses of one another.


Here's what I've done but I'm stuck. I'd appreciate some help with this

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

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

$$\implies \int_{-\infty}^\infty h_1(\tau)h_2(t-\tau)d\tau = \int_{-\infty}^\infty \delta(u)\delta(t-u)d\tau$$
So you have
$$
\int_{-\infty}^{+\infty} \delta(u) \delta(t-u) du
$$

Try using the sifting property of $\delta(t)$:

$$
\int_{-\infty}^{+\infty} f(u) \delta(T-u) du = f(T)
$$