# Difference between Laplace Transform and Fourier Transform

Signals and SystemsElectronics & ElectricalDigital Electronics

## Laplace Transform

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

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 −

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

## Comparison between Laplace Transform and Fourier Transform

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

Laplace TransformFourier 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.