What is a Unit Ramp Signal?

A signal is defined as a single-valued function of one or more independent variables which contain some information. Examples of signals are electric current and voltage, human speech, etc.

Unit Ramp Signal

A ramp function or ramp signal is a type of standard signal which starts at t = 0 and increases linearly with time. The unit ramp function has unit slop.

Continuous-Time Unit Ramp Signal

The continuous-time unit ramp signal is that function which starts at t = 0 and increases linearly with time. It is denoted by r(t). Mathematically, the continuous-time unit ramp signal is defined as follows −

$$\mathrm{r(t) \:=\: \left\{\begin{matrix} 1\; \; \:for\;\: t\:\geq\: 0\ 0\; \;\: for\: t\:\lt\: 0\ \end{matrix}\right.}$$

Also,

$$\mathrm{r(t) \:=\: t\: u(t)}$$

From the above equation, it is clear that the ramp signal is a signal whose magnitude varies linearly. The graphical representation of the continuous-time unit ramp signal is shown in Figure-1.

Discrete-Time Unit Ramp Sequence

The discrete time unit ramp signal is that function which starts from n = 0 and increases linearly. It is denoted by r(n). It is signal whose amplitude varies linearly with time n. mathematically, the discrete time unit ramp sequence is defined as −

$$\mathrm{r(n) \:=\: \left\{\begin{matrix} n\; \; \:for\;\: n\:\geq\: 0\ 0\; \;\: for\: n\:\lt\: 0\ \end{matrix}\right.}$$

Or,

$$\mathrm{r(n) \:=\: n\: u(n)}$$

The graphical representation of a discrete-time unit ramp sequence is shown in Figure-2.

Relationship between Unit Ramp and Unit Step Signals

The unit ramp signal can be obtained by integrating the unit step signal with respect to time. In other words, a unit step signal can be obtained by differentiating the unit ramp signal.

The unit step signal is given by,

$$\mathrm{u(t)\:=\:\left\{\begin{matrix} 1\; \;\: for\;\: t\:\geq\: 0\ 0\; \;\: for\: t\:\lt\: 0\ \end{matrix}\right.}$$

Therefore, the unit ramp function is,

$$\mathrm{r(t)\:=\:\int\: u(t)dt\:=\:\int dt\:=\:t;\; \; \; \:for\: t\:\geq\: 0 }$$

Also,

$$\mathrm{u(t)\:=\:\frac{\mathrm{d} }{\mathrm{d} t}r(t)}$$