Unit Step Function

The unit step function \(u\left(t\right)\) is 0 for negative of \(t\) and 1 for positive values of \(t\).

\(\boxed{u\left(t\right) = \begin{cases}0\,, & t < 0 \\ 1\,, & t > 0\end{cases}}\)

The unit step function is undefined at \(t = 0\), where it changes abruptly from 0 to 1. It is dimensionless.

If the abrupt change occurs at \(t = t_0\) (where \(t_0 > 0\)) instead of \(t = 0\), the unit step function becomes

\(\displaystyle u\left(t - t_0\right) = \begin{cases}0\,, & t < t_0 \\ 1\,, & t > t_0\end{cases}\)

meaning that \(u\left(t\right)\) is delayed by \(t_0\) seconds.

If the change is at \(t = -t_0\), the unit step function becomes

\(\displaystyle u\left(t + t_0\right) = \begin{cases}0\,, & t < -t_0 \\ 1\,, & t > -t_0\end{cases}\)

meaning that \(u\left(t\right)\) is advanced by \(t_0\) seconds.

The step function is used to represent an abrupt change in voltage or current, like the changes that occur in the circuits of control systems and digital computers.

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Unit Ramp Function

Integrating the unit step function \(u\left(t\right)\) results in the unit ramp function \(r\left(t\right)\).
Unit Impulse Function

The derivative of the unit step function \(u\left(t\right)\) is the unit impulse function \(\delta\left(t\right)\).
Time Response of Second-Order System

Using the unit-step signal \(u(t)\) as an input \(r(t)\) to the second-order system,
Step Response of an RL Circuit

When the DC source of an RL circuit is suddenly applied, the voltage or current source can be modeled as a step function, and the response is known as a step response.
Step Response of an RC Circuit

When the DC source of an RC circuit is suddenly applied, the voltage or current source can be modeled as a step function, and the response is known as a step response.
Step Response of a Circuit

The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source.
Singularity Functions

The three most widely used singularity functions in circuit analysis are the unit step, the unit impulse, and the unit ramp functions.
Laplace Transform

unit step
Fourier Transform

unit step
Discrete Singularity Functions

The discrete impulse is just a unit sample at \(n = 0\). It is completely free of the kind of ambiguities associated with analog impulse \(\delta(t)\) at \(t = 0\). The discrete unit step \(u[n]\) also has a well defined, unique value of \(u[0] = 1\) (unlike its analog counterpart). The signal \(x[n] = An\,u[n] = A\,r[n]\) describes a discrete ramp whose slope \(A\) is given by \(x[k] - x[k - 1]\), the difference between adjacent sample values.

The discrete versions of the unit impulse (or sample) \(\delta[n]\), unit step \(u[n]\), and unit ramp \(r[n]\) are defined as
Continuous Convolution

The convolution of a signal \(x(t)\) with an unit step \(u(t)\) is the running integral of the signal \(x(t)\).

While evaluating the convolution integral, keep in mind that \(x(\lambda)\) and \(h(t - \lambda)\) are functions of \(\lambda\) (and not \(t\)), but \(t\) is a constant with respect to \(\lambda\). Simplification of the integration limits due to the step functions in the convolution kernel occurs frequently in problem solving.

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