# Impulse and Step Response

*Pieter P*

Here, we'll discuss 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 is defined as a discrete function that is one when is zero, and zero everywhere else: An alternative notation is . This value is one if and zero if :

#### Impulse response

The impulse response of a DTLTI system is defined as the output of the system when a Kronecker delta function is applied to its input: The letter will be used to refer to the impulse response of a system. As we'll see later, the impulse response can be used to define the system.

#### 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:
As you can see, all terms where are zero, so only the -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 is defined as a discrete function that is zero when is negative, and one if is zero or positive:

#### Step Response

Just like the impulse response, we can define the step response as the output of the system when the Heaviside step function is applied to the input: The step response is an important tool when investigating how a system responds to transients.

Unlike the impulse response, there is no specific 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: