Let \(h_1[n]=\delta[n]+\delta[n-1]\) and \(h_2[n]=\delta[n-2]\) be connected in cascade. If \(x[n]=\delta[n]\), what is the output \(y[n]\) of the overall system?
In cascade the total response is \(h[n]=h_1[n]*h_2[n]\), which equals \(\delta[n-2]+\delta[n-3]\). With an impulse input, the output is that same response. Mixing up the shifts gives an incorrect output.
A linear time-invariant (LTI) system is a system that satisfies superposition and whose behavior does not change when the input is shifted. It is determined by its impulse response \(h[n]\).
\[y[n]=\sum_{k=-\infty}^{\infty} x[k]\,h[n-k]\]
More practice problems: linear time-invariant systems