This is a homework problem that I've worked on and I want to confirm some of my reasoning.
I'm given two sets of input-output pairs of a particular system, $S$, that we know is linear time-invariant (LTI). Given those two pairs, find the impulse response h[n] of the system:
$$x_1[n] = [1, 0, 1] * S \longrightarrow y_1[n] = [0,1,0,2,0,1]$$
$$x_2[n] = [0, 1, 1] * S \longrightarrow y_2[n] = [0,0,1,1,1,1]$$
I found a linear combination of both inputs to yield the delta:
$$\delta[n] = x_1[n] - x_2[n-1] = [1, 0, 0]$$
Due to the system being LTI, the output to the delta is:
$$y[n] = h[n] = y_1[n] - y_2[n-1] = [0,1,0,1,-1,0]$$
Since the input is 3 samples long and the output is 6 samples long, then my impulse response $h[n]$ has to be 4 samples long. So I discard the last two to finally yield:
$$h[n] = [0,1,0,1]$$
Convolving the input signals with the newly found $h[n]$ yields the given output signals. I am however uncomfortable with having to invoke the required length of $h[n]$ to yield its final form by deleting the last two samples.
Did I go wrong somewhere?