I've just learned about linear systems and impulse response functions. I know that in a linear system consisting of $h_1[k]$ and $h_2[k]$, the impulse response of the system is $h_1[k] \ast h_2[k]$.
In an exercise I was given the following system responses:
\begin{align*} h_1(0) &= a \\ h_1(1) &= b \\ h_2(0) &= c \\ h_2(1) &= d \\ h_1(n) = h_2(n) &= 0 \; \text{otherwise} \\ \end{align*}
The question was whether it would be possible to construct the parameters $a$, $b$, $c$, and $d$ in such a way that the linear system $h_1 \ast h_2$ would result in a moving average of the order 3 being applied to the input.
Well, I started with the convolution $h[k] = h_1[k] \ast h_2[k]$ for $k = 0, 1$:
\begin{equation} h[k] = [ac, bc + ad, bd] \end{equation}
I understand that for a moving average of order 3, the required convolution kernel would look like this:
\begin{equation} [1/3, 1/3, 1/3] \end{equation}
But that means I have to solve the following equation:
\begin{align*} ac &= 1/3 \\ bc + ad &= 1/3 \\ bd &= 1/3 \\ \end{align*}
And that's where I'm stuck. From my guts I'd say it's not possible to get to an answer here, because even if $a = 1$ and $b = 1$, $c$ and $d$ would have to be $1/3$ and $c + d$ would have to be $1/3$ as well. Which isn't solvable.