Unit Sample

The Kronecker Delta Function as a Discrete-Time Signal, denoted $\delta [n]$. Serves as a primitive building-block signal from which more complex discrete-time signals can be constructed.

$\delta [n]=\{\begin{array}{ll}1,& n=0\\ 0,& \text{otherwise}\end{array}$

Impulse Representation

Impulse

In Discrete-Time land, an Impulse $\delta [n]$ is a Signal that is the Unit Sample

$$\begin{array}{r}\delta [n]=\{\begin{array}{ll}1,& n=0\\ 0,& n\ne 0\end{array}\\ \\ \delta [n]=[1,\vec{0}]\end{array}$$

Note that for Convolutions, this unit impulse is like the Identity

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

Similar to the Step Function, you can scale or shift the delta function. Furthermore, you can compose these scaled and shifted delta functions to create ANY given signal.

For example, $x[n]$ is a signal that goes: 1, 2, 3.

$$x[n]=1\cdot \delta [n]+2\cdot \delta [n-1]+3\cdot \delta [n-2]$$Link to original

Sifting Property

Sifting Property

You can also use the delta function to “sift” out the output of a signal at a given $n_0$

$$x[n_0]=\sum _{n=-\infty }^{\infty }x[n]\delta [n-n_0]$$

This can be used for either constructing or deconstructing a signal.

Link to original

Relationship to Unit Step

$\delta [n]=u[n]-u[n-1]$

Categorical Structure

From Category Theory point of view, $\delta [n]$ is the Identity Morphism for Convolution of a signal: $x[n]\ast \delta [n]=x[n]$ for any signal $x[n]$.