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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 −
$$\mathrm{r(t)=\left\{\begin{matrix} 1\; \; for\; t\geq 0\ 0\; \; for\: t< 0\ \end{matrix}\right.}$$
Also,
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 −
$$\mathrm{r(n)=\left\{\begin{matrix} n\; \; for\; n\geq 0\ 0\; \; for\: n< 0\ \end{matrix}\right.}$$
Or,
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,
$$\mathrm{u(t)=\left\{\begin{matrix} 1\; \; for\; t\geq 0\ 0\; \; for\: t< 0\ \end{matrix}\right.}$$
Therefore, the unit ramp function is,
$$\mathrm{r(t)=\int\: u(t)dt=\int dt=t;\; \; \; for\: t\geq 0 }$$
Also,
$$\mathrm{u(t)=\frac{\mathrm{d} }{\mathrm{d} t}r(t)}$$
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