Given is the illustrated circuit diagram of a linear, time-invariant, time-discrete system:

[![enter image description here](https://i.stack.imgur.com/ECvdo.png)](https://i.stack.imgur.com/ECvdo.png)

How do I show that the total system has the impulse response $h[n] = aδ[n] + bδ [n - 1] + cδ [n - 2]$ and determine the constants a,b and c?

with the following impulse responses of the subsystems:

$h\_1[n] = δ[n] + (1/2)δ[n − 1]$

$h\_2[n] = δ[n] − (1/2)δ[n − 1]$

$h\_3[n] = −δ[n]$

$h4[n] = − (1/2)^n σ[n]$

So what I have done:

$h\_{3||4}=−δ[n]−(1/2)^n σ[n]$

and than:

$h\_{3||4+2}=\sum\_{k=-\infty}^{\infty}(δ[n-k] − (1/2)δ[n-k − 1])(−δ[n]−(1/2)^n σ[n])$

But problem is that I don't know how to calculate this and than to do it in parellel with $h\_1$ to get this form $h[n] = aδ[n] + bδ [n - 1] + cδ [n - 2]$ ?