Find convolution of u[n]-u[n-2] and u[n]-u[n-2]

Question

Question: Find convolution of $u[n]-u[n-2]$ and $u[n]-u[n-2]$

I have found that

$u[n]\cdot u[n]=n$,

$u[n]\cdot u[n-2]=n-2$,

$u[n-2]\cdot u[n-2]=n-4$

Use linear property, my answer is:

$n-2(n-2)+n-4=0$

But my teacher said my answer is wrong! Can anyone help me? Thanks

Answer 1

It is easy! We know that $u[n]-u[n-2]=\delta[n]+\delta[n-1]$ and delta property that $\delta[n-m_1]*\delta[n-m_2]=\delta[n-m_1-m_2]$. So:

$$(u[n]-u[n-2])*(u[n]-u[n-2])=(\delta[n]+\delta[n-1])*(\delta[n]+\delta[n-1])=\delta[n]+2\delta[n-1]+\delta[n-2]$$