Step response

The step response is the output signal of a linear, time-invariant system ( LTI ) system, where the step function is applied at the input. It is then called a transition function, if the height of the jump input is 1 ( unit step function ) and was divided by the height of the input jump.

Mathematical Description

The step response can be calculated with the impulse response as a convolution of the step function:

The step response is thus the time integral of the impulse response.

In the discrete:

Since the transfer function is the Laplace of the impulse response, it can also be determined by Laplace transform of the time derivative of the step response:

Conversely, it follows that:

In practice, jump signals can generate much more accurate than Dirac impulses (which are the input signal for the impulse response ). Through above relationship is also omits the step response of the transfer function of the system easy to determine. Thus, the step response is an important parameter of the system behavior and to describe systems of high relevance.

Example

The step function is suitable for a system as a test signal. If the input of an electronic circuit is a step function with height 2 V is applied, then you can at the output of the transmission link also notice a change in the voltage. The time course of this voltage is called step response, so it is the response of the system to the applied step function. On the picture you can see how the output signal slowly toward the value at input. If the step response looks nothing like the picture, it can be concluded on a system with a memory. The memory is a capacitor in this case. It is loaded by the 2 V at the input through the resistor, to the input voltage is reached. The system behaves like a PT1 element.