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:
$h_{3||4+2}=\sum_{k=-\infty}^{\infty}(δ[n-k] − (1/2)δ[n-k − 1])(−δ[n]−(1/2)^n σ[n])$
$h_{3||4+2}=−δ[n]−(1/2)^n σ[n]-\frac{1}{2}(−δ[n-1]−(1/2)^{n-1} σ[n-1])=−δ[n]-\frac{1}{2}^nδ[n]+\frac{1}{2}δ[n-1]$
$h_{3||4+2+1}=(\frac{1}{2})^nδ[n]-δ[n-1]$
But the solution is $a=1$ and $b=-1$. How is $a=1$?