Memoryless

### Definition

(1)
\begin{align} y[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]=\sum_{k=-\infty}^{\infty}h[k]x[n-k] \nonumber \end{align}

$y[n]$ should depend only on the value of the input at the same time. $y[n]$ can't depend on $x[n-k]$ for any $k \neq 0$.

(2)
\begin{align} h[k]=0 \hspace{0.5cm}\forall k \neq 0 \nonumber \end{align}

If $h[k]=0 \hspace{0.2cm}\forall k \neq 0$ or $h[n]=0 \hspace{0.2cm}\forall n \neq 0$, then the LTI system is memoryless.

### Example

Given $h[n]$ as shown and $y[n]=2x[n]+x[n-1]$.

To make this memoryless $h[n]$ should be $0$ for $n=1$.