A discrete-time unit impulse, also known as the Kronecker delta function, is a discrete-time signal that has a value of 1 at time n=0, and 0 everywhere else. It is denoted as: $\delta(n) = \begin{cases} 0, & n \neq 0.\\ 1, & n = 0. \end{cases} $ ![[impulse.svg|500]] Any discrete-time signal can be expressed as a sum of scaled and time-shifted unit impulses: $ x(n) = \sum_{k = -\infty}^{+\infty} x(k) \delta(n-k) $ It can be used, for example, to express a [[discrete-time unit-step]] function: $ u(n) = \sum_{k = 0}^{+\infty} \delta(n-k) $