Kronecker Delta Function

The Kronecker Delta Function is given by ${\delta }_{ij}=\{\begin{array}{ll}0& i\ne j\\ 1& i=j\end{array}$

From MATH213:

Definition 1: Kronecker delta

The Kronecker delta function $\delta [t]$ is defined as the function from $\mathbb{Z}$ to $\{0,1\}$ given by

$\delta [t]=\{\begin{array}{ll}1& t=0\\ 0& else\end{array}$

Theorem 2

For all functions $f:\mathbb{Z}\to \mathbb{C}$, $f[t]={\sum }_{\tau =-\infty }^{\infty }f[\tau ]\delta [\tau -t]$

Theorem 2 shows how to decompose every function $f$ into a linear combination of impulse functions (i.e. $\{\delta [\tau -t]\mid \tau \in \mathbb{Z}\}$ is a basis for all DT functions.)

From CS370

They use a slightly different notation

${\delta }_{k,l}=\{\begin{array}{ll}0& k\ne l\\ 1& k=l\end{array}$

Related

• Unit Impulse Function