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Signals and Systems β Properties of Laplace Transform
Laplace Transform
The Laplace transform is a mathematical tool which is used to convert the differential equations in time domain into the algebraic equations in the frequency domain or s-domain.
Mathematically, the Laplace transform of a time-domain function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is defined as −
$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{\mathrm{0} }^{\mathrm{\infty} }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{st}}\:\mathit{dt}}}$$
Where, s is a complex variable and it is given by,
$$\mathrm{\mathit{s}\:\mathrm{=}\:\sigma \:\mathrm{+}\:\mathit{j\omega}}$$
And the operator L is called the Laplace transform operator which transforms the domain function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ into the frequency domain function X(s).
Properties of Laplace Transform
The following table highlights some of the important properties of Laplace transform −
| Property | Function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ | Laplace Transform $\mathit{X}\mathrm{\left(\mathit{s}\right)}$ |
|---|---|---|
| Notation | $\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{t}\right)}}$ | $\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{s}\right)}}$ |
| $\mathrm{\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{t}\right)}}$ | $\mathrm{\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{s}\right)}}$ | |
| Scalar Multiplication | $\mathrm{\mathit{k}\mathit{x}\mathrm{\left(\mathit{t}\right)}}$ | $\mathrm{\mathit{k}\mathit{X}\mathrm{\left(\mathit{s}\right)}}$ |
| Linearity | $\mathrm{\mathit{a}\mathit{x}_{\mathrm{1}}\mathrm{\left( \mathit{t}\right)}\:\mathrm{+}\:\mathit{b}\mathit{x}_{\mathrm{2}}\mathrm{\left( \mathit{t}\right)}}$ | $\mathrm{\mathit{a}\mathit{X}_{\mathrm{1}}\mathrm{\left( \mathit{s }\right)}\:\mathrm{+}\:\mathit{b}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{s}\right)}}$ |
| Time Shifting | $\mathrm{\mathit{x}\mathrm{\left(\mathit{t-t_{\mathrm{0}}}\right)}}$ | $\mathrm{\mathit{e}^{- \mathit{t}_{\mathrm{0}}\mathit{s}}\mathit{X}\mathrm{\left(\mathit{s}\right)}}$ |
| Frequency Shifting | $\mathrm{\mathit{e}^{\mathit{-at}}\mathit{x}\mathrm{\left(\mathit{t}\right)}}$ | $\mathrm{\mathit{X}\mathrm{\left(\mathit{s\mathrm{+}}\mathit{a} \right)}}$ |
| Time Scaling | $\mathrm{\mathit{x}\mathrm{\left(\mathit{at}\right)}}$ | $\mathrm{\frac{1}{\left| \mathit{a}\right|}\mathit{X}\mathrm{\left ( \frac{\mathit{s}}{\mathit{a}}\right)}}$ |
| Differentiation in Time Domain | $\mathrm{\frac{\mathit{d}}{\mathit{dt}}\:\mathit{x}\mathrm{\left(\mathit{t}\right)}}$ | $\mathrm{\mathit{s}\mathit{X}\mathrm{\left(\mathit{s}\right)}-\mathit{x}\mathrm{\left(\mathrm{0}^{-}\right)}}$ |
| $\mathrm{\frac{\mathit{d}^{\mathrm{2}}}{\mathit{dt}^{\mathrm{2}}}\:\mathit{x}\mathrm{\left(\mathit{t}\right)}}$ | $\mathrm{\mathit{s}^{\mathrm{2}}\mathit{X}\mathrm{\left(\mathit{s}\right)}-\mathit{s}\mathit{x}\mathrm{\left(\mathrm{0}^{-}\right)}-\frac{\mathit{d}}{\mathit{dt}}\:\mathit{x}\mathrm{\left(\mathrm{0}^{-}\right)}}$ | |
| $\mathrm{\frac{\mathit{d}^{\mathit{n}}}{\mathit{dt}^{\mathit{n}}}\:\mathit{x}\mathrm{\left(\mathit{t}\right)}}$ | $\mathrm{\mathit{s}^{\mathit{n}}\mathit{X}\mathrm{\left(\mathit{s}\right)}-\mathit{s}^\mathrm{\left(\mathit{n-\mathrm{1}}\right)}\mathit{x}\mathrm{\left(\mathrm{0}^{-}\right)}-...-\frac{\mathit{d^{\mathrm{\left ( \mathit{n-\mathit{1}} \right )}}}}{\mathit{dt}^{\mathrm{\left (\mathit{n-\mathit{1}} \right)}} } \mathit{x}\mathrm{\left(\mathrm{0}^{-}\right)}}$ | |
| Integration in Time Domain | $\mathrm{\int_{\mathrm{0}^{-}}^{\mathit{t}}\mathit{x}\mathrm{\left(\mathit{\tau}\right)}\:\mathit{d\tau}}$ | $\mathrm{\frac{\mathit{X}\mathrm{\left(\mathit{s}\right)}}{\mathit{s}}}$ |
| $\mathrm{\int_{\mathrm{-\infty }}^{\mathit{t}}\mathit{x}\mathrm{\left(\mathit{\tau}\right)}\:\mathit{d\tau}}$ | $\mathrm{\frac{\mathit{X}\mathrm{\left(\mathit{s}\right)}}{\mathit{s}}\:\mathrm{+}\:\frac{1}{\mathit{s}}\int_{-\infty }^{\mathrm{0}^{-}}\mathit{x}\mathrm{\left(\mathit{\tau}\right)}\:\mathit{d\tau}}$ | |
| Differentiation in Frequency Domain | $\mathrm{\mathit{t}\mathit{x}\mathrm{\left(\mathit{t}\right)}}$ | $\mathrm{-\frac{\mathit{d}}{\mathit{ds}}\mathit{X}\mathrm{\left(\mathit{s}\right)}}$ |
| $\mathrm{\mathit{t}^{\mathit{n}}\mathit{x}\mathrm{\left(\mathit{t}\right)}}$ | $\mathrm{\mathrm{\left ( -\mathrm{1} \right )}^{\mathit{n}}\frac{\mathit{d}^{\mathit{n}}}{\mathit{ds}^{\mathit{n}}}\mathit{X}\mathrm{\left(\mathit{s}\right)}}$ | |
| Integration in Frequency Domain | $\mathrm{\frac{\mathit{x}\mathrm{\left(\mathit{t}\right)}}{\mathit{t}}}$ | $\mathrm{\int_{\mathit{s}}^{\infty}\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathit{ds}}$ |
| Time Convolution | $\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{t}\right)}*\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{t}\right)}}$ | $\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{s}\right)}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{s}\right)}}$ |
| Convolution in Frequency Domain | $\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{t}\right)}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{t}\right)}}$ | $\mathrm{\frac{1}{2\pi \mathit{j}}\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{s}\right)}*\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\frac{1}{2\pi \mathit{j}}\int_{\mathrm{\left ( \mathit{c-\mathit{j\infty}}\right)}}^{\mathrm{\left ( \mathit{c\mathrm{+}\mathit{j\infty}}\right)}}\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{p}\right)}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{s-p}\right)}\:\mathit{dp}}$ |
| Time Periodicity | $\mathrm{\mathit{x}\mathrm{\left(\mathit{t}\right)}\:\mathrm{=}\:\mathit{x}\mathrm{\left( \mathit{t}\mathrm{+}\mathit{nT}\right)}\:\mathrm{where},\mathit{n}\:\mathrm{=}\mathrm{1,2,3,}...}$ | $\mathrm{\frac{1}{\mathrm{\left( \mathrm{1}-\mathit{e^{-st}}\right)}}\mathit{X}^{'}\mathrm{\left(\mathit{s}\right)};\\mathrm{where},\mathit{X}^{'}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{\mathrm{0}}^{\mathit{T}}\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-st}}\:\mathit{dt}}$ |
| Initial Value Theorem | $\mathrm{\mathit{x}\mathrm{\left(\mathrm{0}^{-}\right)}}$ | $\mathrm{\displaystyle \lim_{\mathit{s} \to \infty }\mathit{s}\mathit{X}\mathrm{\left(\mathit{s}\right)}}$ |
| Final Value Theorem | $\mathrm{\mathit{x}\mathrm{\left(\mathit{\infty }\right)}}$ | $\mathrm{\displaystyle \lim_{\mathit{s} \to 0}\mathit{s}\mathit{X}\mathrm{\left(\mathit{s}\right)}}$ |
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