Unit step response of a FIR filter

Question

Given a FIR filter with impulse response:

$$ h(n) = \begin{cases}1, &0 \leq n < 5\\ -1, &10 \leq n < 15 \\ 0, &\text{otherwise}\end{cases} $$

What would be the right approach to calculate the filter discrete unit step response?

Thanks

Answer 1

input $x[n]=u[n]$

output $y[n]=x[n]\ast h[n]=\displaystyle\sum_{k=-\infty}^{k=\infty}x[k]h[n-k]$

$$ y(n) = \begin{cases}\displaystyle\sum_{k=0}^{k=n}1.1=n+1 \ , &0 \leq n < 5\\ \displaystyle\sum_{k=0}^{k=4}1.1=5\ , &5 \leq n < 10 \\ \left[5+\displaystyle\sum_{k=10}^{k=n}1.(-1)\right]= 14-n\ , &10 \leq n < 15 \\ 0, &\text{otherwise}\end{cases} $$