Laplace Transform of Ramp Function and Parabolic Function

Laplace Transform

The Laplace transform is a mathematical tool which is used to convert the differential equation in time domain into the algebraic equations in the frequency domain or s-domain.

Mathematically, if $\mathrm{\mathit{x\left ( t \right )}}$ is a time domain function, then its Laplace transform is defined as −

$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{=}X\left ( s \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\; dt\; \; \; \cdot \cdot \cdot \left ( \mathrm{1} \right )}}$$

Equation (1) gives the bilateral Laplace transform of the function $\mathrm{\mathit{x\left ( t \right )}}$ . But for the causal signals, the unilateral Laplace transform is applied, which is defined as,

$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{=}X\left ( s \right )\mathrm{=}\int_{\mathrm{0} }^{\infty }x\left ( t \right )e^{-st}\; dt\; \; \; \cdot \cdot \cdot \left ( \mathrm{2} \right )}}$$

Laplace Transform of Ramp Function

The ramp function is defined as,

$$\mathrm{\mathit{x\left ( t \right )}\mathrm{=}\mathit{t\, u\left ( t \right )}}$$

Thus, from the definition of Laplace transform, we have,

$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{=}L\left [ t\, u\left ( t \right ) \right ]\mathrm{=}\int_{\mathrm{0} }^{\infty }t\, u\left ( t \right )e^{-st}\; dt}}$$

$$\mathrm{\Rightarrow \mathit{L\left [ t\, u\left ( t \right ) \right ]\mathrm{=}\int_{\mathrm{0} }^{\infty }t\, e^{-st}\; dt}}$$

$$\mathrm{\Rightarrow \mathit{L\left [ t\, u\left ( t \right ) \right ]\mathrm{=}\left [ \frac{t\, e^{-st}}{-s} \right ]_{\mathrm{0}}^{\infty }-\int_{\mathrm{0}}^{\infty }\left ( \mathrm{1} \right )\frac{e^{-st}}{-s}dt }}$$

$$\mathrm{\Rightarrow \mathit{L\left [ t\, u\left ( t \right ) \right ]\mathrm{=}\mathrm{0}-\left [ \frac{e^{-st}}{s^{\mathrm{2}}} \right ]_{\mathrm{0}}^{\infty }\mathrm{=}\left ( \mathrm{0} -\frac{\mathrm{1}}{s^{\mathrm{2}}}\right )\mathrm{=}\frac{\mathrm{1}}{s^{\mathrm{2}}}}}$$

The region of convergence (ROC) of the Laplace transform of the ramp function $\mathrm{\mathit{\left [ tu\left ( t \right ) \right ]}}$ is 𝑅𝑒(𝑠) > 0 as shown in Figure-1. Hence, the Laplace transform of the ramp function along with its ROC is,

$$\mathrm{\mathit{t\, u\left ( t \right )\overset{LT}{\leftrightarrow}\frac{\mathrm{1}}{s^{\mathrm{2}}} }\;\;\;and\;\;\;ROC\rightarrow Re\left ( \mathit{s} \right )>\mathrm{0}}$$

Laplace Transform of Parabolic Function

The parabolic function is defined as,

$$\mathrm{\mathit{x\left ( t \right )\mathrm{=}t^{\mathrm{2}}u\left ( t \right )}}$$

Now, from the definition of the Laplace transform, we have,

$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{=}L\left [ t^{\mathrm{2}}u\left ( t \right ) \right ]\mathrm{=}\int_{\mathrm{0}}^{\infty }t^{\mathrm{2}}u\left ( t \right )e^{-st}\: dt}}$$

$$\mathrm{\Rightarrow \mathit{L\left [ t^{\mathrm{2}}u\left ( t \right ) \right ]\mathrm{=}\int_{\mathrm{0}}^{\infty }t^{\mathrm{2}}\, e^{-st}\: dt\mathrm{=}\left [ \frac{t^{\mathrm{2}}e^{-st}}{-s} \right ]_{\mathrm{0}}^{\infty }-\int_{\mathrm{0}}^{\infty }\left ( \mathrm{2}t \right )\frac{e^{-st}}{-s}dt }}$$

$$\mathrm{\Rightarrow \mathit{L\left [ t^{\mathrm{2}}u\left ( t \right ) \right ]\mathrm{=}\mathrm{0}\mathrm{\mathrm{+}}\frac{\mathrm{2}}{s}\int_{\mathrm{0}}^{\infty }t\, e^{-st}\: dt}}$$

$$\mathrm{\Rightarrow \mathit{L\left [ t^{\mathrm{2}}u\left ( t \right ) \right ]\mathrm{=}\frac{\mathrm{2}}{s}\left\{\left [ \frac{te^{-st}}{-s} \right ]_{\mathrm{0}}^{\infty }-\int_{\mathrm{0}}^{\infty }\left ( \mathrm{1} \right )\frac{e^{-st}}{-s} \: dt\right\}}}$$

$$\mathrm{\Rightarrow \mathit{L\left [ t^{\mathrm{2}}u\left ( t \right ) \right ]\mathrm{=}\frac{\mathrm{2}}{s}\left\{\mathrm{0}-\left [ \frac{e^{-st}}{s^{\mathrm{2}}} \right ]_{\mathrm{0}}^{\infty } \right\}\mathrm{=}\frac{\mathrm{2}}{s^{\mathrm{3}}}\left [ e^{-st} \right ]_{\mathrm{0}}^{\infty }}}$$

$$\mathrm{\therefore \mathit{L\left [ t^{\mathrm{2}}u\left ( t \right ) \right ]\mathrm{=}\frac{\mathrm{2}}{s^{\mathrm{3}}}}}$$

The ROC of Laplace transform of the parabolic function $\mathrm{\mathit{\left [ t^{\mathrm{2}}u\left ( t \right ) \right ]}}$ is also 𝑅𝑒(𝑠) > 0, which is shown in Figure-1. Therefore, the Laplace transform of the parabolic function along with its ROC is,

$$\mathrm{ \mathit{t^{\mathrm{2}}u\left ( t \right )\overset{LT}{\leftrightarrow}\frac{\mathrm{2}}{s^{\mathrm{3}}}\; \; \; \mathrm{and\; \; \; ROC\to Re\left ( \mathit{s} \right )>0}}}$$