# Accumulator with decay?

I am using an accumulator in the form of X + [Y-1], X plus the unit delay of the output. Is there an accumulator that follows this form, but will fall to 0 over time, similar to a differentiator?

$$y(n)=\alpha x(n)+(1-\alpha)y(n-1),\quad 0<\alpha<1\tag{1}$$
You could also just multiply $y(n-1)$ with a constant less than 1. The reason why in (1) $x(n)$ is multiplied by $\alpha$ is to guarantee a DC gain of 1.