Discrete System in Time Domain

We previously defined convolution for discrete signals as,

$$x[n] \ast y[n] = \sum_{m=-\infty}^{+\infty} x[m] y[n - m]$$

We previous learned that,

$$x(t) = x(t) \ast \delta(t)$$

Similarly, it's obvious,

$$x[n] = x[n] \ast \delta[n]$$

For LTI system, we still have response of unit impulse,

$$h[n] = \mathcal{S}(\delta[n])$$

And, because of linearity,

$$y[n] = h[n] \ast x[n]$$