2.1a.1 Kronecker Delta

Kronecker Delta (Discrete-Time Unit Impulse)

The kronecker delta (i.e. the discrete-time unit impulse) is defined as
$\delta [n]=\{\begin{array}{llc}0& & \text{if }n=0\\ 1& & \text{if }n\ne 0\end{array}.$
Any arbitrary discrete-time signal $x(n)$ can be decomposed into a linear combination of discrete-time impulses (Kronecker deltas), such that
$x[n]={\sum }_{k=-\infty }^{\infty }x[k]\delta [n-k].$

Properties

(Step Representation) $\delta [n]=u[n]-u[n-1]$.

The RHS is $1$ only at $n=0$. Any value $n<0$ will lead to $0-0=0$, whereas any value $n>0$ will lead to $1-1=0$.