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When a signal gives the constant acceleration distinction of actual input signal, such a signal is known as **parabolic signal** or **parabolic function**. It is also known as unit **acceleration signal**. The unit parabolic signal starts at t = 0.

The continuous-time unit parabolic signal is a unit parabolic signal which is defined for every instant of positive time. It is denoted by p(t). Mathematically, p(t) is given as,

$$\mathrm{p(t)=\left\{\begin{matrix} \frac{t^{2}}{2}\; for\: t\geq 0\\ 0\; for\; t< 0\\ \end{matrix}\right.}$$

Also,

$$\mathrm{p(t)=\frac{t^{2}}{2}\;u(t)}$$

The graphical representation of the continuous-time parabolic signal p(t) is shown in Figure-1.

The discrete-time unit parabolic sequence is a unit parabolic signal which is defined only at discrete instants of positive time n. It is denoted by p(n). **Mathematically**, p(n) is given as,

$$\mathrm{p(n)=\left\{\begin{matrix} \frac{n^{2}}{2}\; for\: n\geq 0\\ 0\; for\; n< 0\\ \end{matrix}\right.}$$

$$\mathrm{p(n)=\frac{n^{2}}{2}\;u(n)}$$

The graphical representation of discrete-time unit parabolic signal p(n) is shown in Figure-2.

The unit parabolic signal can be obtained by integrating the unit ramp signal with respect to time. **In other words**, the unit ramp function is the time derivative of the unit parabolic function, i.e.,

$$\mathrm{p(t)=\int r(t)dt=\int t\: dt=\frac{t^{2}}{2}\; \; for\: t\geq 0}$$

Or

$$\mathrm{r\left ( t \right )=\frac{\mathrm{d} }{\mathrm{d} t}p(t)}$$

The unit parabolic signal can be obtained by double integrating the unit step signal. **In other words**, the unit step signal is a double derivative of parabolic signal, i.e.,

$$\mathrm{p(t)=\iint\: u(t)dt\: dt=\iint\: 1\: dt\: dt=\int t\: dt=\frac{t^{2}}{2}\; \; for\: t\geq 0 }$$

or

$$\mathrm{u(t)=\frac{\mathrm{d} ^{2}}{\mathrm{d} t^{2}}p(t)}$$

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