Given is the illustrated circuit diagram of a linear, time-invariant, time-discrete system:
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 $h[n] = aδ[n] + bδ [n - 1] + cδ [n - 2]$ ?