Time shift and addition

Other than hearing a possible silence (for causal signals) before the signal onset, there is no audible effect of a pure delay.

What happens, however, when a signal $x(n)$ is added to a delayed version of itself $x(n-M)$?

We have created a digital filter, with a formula for going from input $x(n)$ to output $y(n)$ given by the difference equation

$$y(n) = x(n) + x(n-M).$$