This is my system. We know that h1(t) = δ(t-1) and h2(t) = δ(t+1).
Now my problem is how to find y(t). I know that the exit of the adder will give us
y1{t} = x(t) - h2(t) correct?
That means that y(t) = h1 * y1{t} = δ(t-1)*x(t) - δ(t+1)*δ(t-1) = x(t-1) - δ(t).
However, I don't seem to understand how to draw the unit step response of y(t).
When setting up equations for a feedback loop, I usually start at the output and follow the path around once to see what happens to it.
If I start at $y(t)$, I go through the bottom block and get $y(t)*\delta(t+1)$, which yields $y(t+1)$.
Then I get to the summation block, and out pops $x(t)-y(t+1)$.
This signal then goes through the top block, becoming $\left[x(t)-y(t+1)\right]*\delta(t-1)=x(t-1)-y(t)$.
The signal that pops out of the top block is the output $y(t)$, so we can set up the equation:
$y(t)=x(t-1)-y(t)$.
Solving for y(t) gives: $y(t)=\frac{1}{2}x(t-1)$.