This page discusses two important signals that are often used in signal processing, the delta function or unit impulse, and the unit step function.

## The Kronecker Delta Function

The Kronecker delta function or unit impulse $\delta \left[n\right]$ is defined as a discrete function that is one when $n$ is zero, and zero everywhere else: $\begin{array}{}\text{(1)}& \delta :\mathbb{Z}\to \mathbb{R}:n↦\delta \left[n\right]\triangleq \left\{\begin{array}{ll}1& n=0\\ 0& n\ne 0\end{array}\end{array}$ An alternative notation is ${\delta }_{n,k}$. This function is one if $n=k$ and zero if $n\ne k$: ${\delta }_{n,k}\triangleq \delta \left[n-k\right]$

### Impulse response

The impulse response $h\left[n\right]$ of a DTLTI system $T$ is defined as the output of the system when a Kronecker delta function is applied to its input: $h\left[n\right]\triangleq T\left(\delta \left[n\right]\right)$ The letter $h$ will be used to refer to the impulse response of a system. As we'll see later, the impulse response can be used to fully define the system, it captures all of its properties.

### Properties of the Kronecker Delta Function

The most important property of the Kronecker delta is its ability to select a single term from an (infinite) sum: $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}x\left[n\right]\cdot \delta \left[k-n\right]\\ =\phantom{\rule{thickmathspace}{0ex}}& x\left[0\right]\cdot \delta \left[k-0\right]+x\left[1\right]\cdot \delta \left[k-1\right]+\dots +x\left[k\right]\cdot \delta \left[k-k\right]+\dots \\ =\phantom{\rule{thickmathspace}{0ex}}& x\left[k\right]\end{array}$ As you can see, all terms where $n\ne k$ are zero, so only the $k$-th term remains.
This is sometimes referred to as the sifting property of the delta function.

## The Heaviside Step Function

The (discrete) Heaviside step function or unit step function $u\left[n\right]$ (sometimes $H\left[n\right]$) is defined as a discrete function that is zero when $n$ is negative, and one if $n$ is zero or positive: $\begin{array}{}\text{(2)}& u:\mathbb{Z}\to \mathbb{R}:n↦u\left[n\right]\triangleq \left\{\begin{array}{ll}0& n<0\\ 1& n\ge 0\end{array}\end{array}$

### Step Response

Similar the impulse response, the step response is defined as the output of the system when the Heaviside step function is applied to the input: ${y}_{\text{step}}\left[n\right]\triangleq T\left(u\left[n\right]\right)$ The step response is an important tool when investigating how a system responds to transients.

Unlike the impulse response, there is no universal symbol or letter for the step response.

### Properties of the Heaviside Step Function

The step function can also be written as the cumulative sum of the delta function: $u\left[n\right]=\sum _{k=-\mathrm{\infty }}^{n}\delta \left[k\right]$