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The Laplace transform is a mathematical tool which is used to convert the differential equations representing a linear time invariant system in time domain into algebraic equations in the frequency domain.

Mathematically, the Laplace transform of a time domain function 𝑥(𝑡) is defined as −

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

Where, 𝑠 is a complex variable and it is given by,

$$\mathrm{\mathit{s\mathrm{=}\sigma \mathrm{+}j\omega }}$$

The operator L is called the Laplace transform operator which transforms the time domain function 𝑥(𝑡) into the frequency domain function 𝑋(𝑠).

Fourier transform is a transformation technique which transforms signals from continuous-time domain to the corresponding frequency domain and viceversa. Mathematically, the Fourier transform of a continuous-time signal 𝑥(𝑡) is defined as −

$$\mathrm{\mathit{F\left [ x\left ( t \right ) \right ]= X\left ( \omega \right )=\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}dt }}$$

The following table highlights the major differences between Laplace Transform and Fourier Transform −

Laplace Transform | Fourier Transform |
---|---|

The Laplace transform of a function $\mathrm{\mathit{x\left ( t \right )}}$ can be represented as a continuous sum of complex exponential damped waves of the form $\mathrm{\mathit{e^{st}}}$. | The Fourier transform of a function $\mathrm{\mathit{x\left ( t \right )}}$ can be represented by a continuous sum of exponential functions of the form of $\mathrm{\mathit{e^{j\, \omega t}}}$. |

The Laplace transform is applied for solving the differential equations that relate the input and output of a system. | The Fourier transform is also applied for solving the differential equations that relate the input and output of a system. |

The Laplace transform can be used to analyse unstable systems. | Fourier transform cannot be used to analyse unstable systems. |

The Laplace transform is widely used for solving differential equations since the Laplace transform exists even for the signals for which the Fourier transform does not exist. | The Fourier transform is rarely used for solving the differential equations since the Fourier transform does not exists for many signals. For example |$\mathrm{\mathit{x\left ( t \right )}}$| as it is not absolutely integrable. |

The Laplace transform has a convergence factor and hence it is more general. | The Fourier transform does not have any convergence factor. |

The Laplace transform of a signal $\mathrm{\mathit{x\left ( t \right )}}$ is equivalent to the Fourier transform of the signal $\mathrm{\mathit{x\left ( t \right )e^{-\sigma t}}}$. | The Fourier transform is equivalent to the Laplace transform evaluated along the imaginary axis of the s-plane. |

- Related Questions & Answers
- Relation between Laplace Transform and Fourier Transform
- Difference between Z-Transform and Laplace Transform
- Difference between Fourier Series and Fourier Transform
- Signals and Systems – Relation between Laplace Transform and Z-Transform
- Signals and Systems – Relation between Discrete-Time Fourier Transform and Z-Transform
- Laplace Transform of Periodic Functions (Time Periodicity Property of Laplace Transform)
- Common Laplace Transform Pairs
- Discrete-Time Fourier Transform
- Circuit Analysis with Laplace Transform
- Laplace Transform of Sine and Cosine Functions
- Derivation of Fourier Transform from Fourier Series
- Modulation Property of Fourier Transform
- Fourier Transform of Rectangular Function
- Fourier Transform of Signum Function
- Inverse Discrete-Time Fourier Transform

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