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Difference between Laplace Transform and Fourier Transform
In engineering analysis, a complex mathematically modelled physical system is converted into a simpler, solvable model by employing an integral transform. Once the model is solved, the inverse integral transform is used to provide the solution in the original form.
There are two most commonly used integral transforms namely – Laplace Transform and Fourier Transform. In both these transforms, a physical system represented in differential equations is converted into algebraic equations or in easily solvable differential equations of lower degree. Thus, Laplace Transform and Fourier Transform make the problem easier to solve.
In this article, we will learn the important differences between Laplace Transform and Fourier Transform. Let's start with some basics so that it becomes easier to understand the differences between them.
What is 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 $\mathrm{x\:(\:t\:)}$ is defined as −
$$\mathrm{L\:[x\:(t)]\:=\:X\:(s)\:=\:\int_{-\:\infty}^{\infty}\:x\:(t)\:e^{-st}\:dt}$$
Where, s is a complex variable and it is given by,
$$\mathrm{s\:=\:\sigma\:+\:j\:\omega}$$
The operator L is called the Laplace transform operator which transforms the time domain function $\mathrm{x\:(\:t\:)}$ into the frequency domain function $\mathrm{X\:(\:s\:)}$.
What is Fourier Transform?
Fourier Transform is a transformation technique which transforms signals from continuoustime domain to the corresponding frequency domain and vice-versa. Mathematically, the Fourier transform of a continuous-time signal $\mathrm{x\:(\:t\:)}$ is defined as −
$$\mathrm{F\:[x\:(t)]\:=\:X\:(\omega)\:=\:\int_{-\:\infty}^{\infty}\:x\:(t)\:e^{-j\:\omega\:t}\:dt}$$
Hence, the Fourier Transform is used to analyze a function in frequency domain. However, the Fourier Transform is only defined for functions that are defined for all real numbers. Also, it cannot be used to analyze unstable systems.
Difference between Laplace Transform and Fourier Transform
The following table highlights the major differences between Laplace Transform and Fourier Transform −
| Laplace Transform | Fourier Transform |
|---|---|
| The Laplace transform of a function x(t) can be represented as a continuous sum of complex exponential damped waves of the form est. | The Fourier transform of a function x(t) can be represented by a continuous sum of exponential functions of the form of ejωt. |
| 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 analyze unstable systems. | Fourier transform cannot be used to analyze unstable systems. |
| Laplace Transform does not require that the function is defined for a set of negative real numbers. | Fourier Transform is only defined for functions that are defined for all real numbers. |
| Laplace transform exists for every function with a Fourier Transform. | On the other hand, it is not always true that every function with a Laplace Transform has a Fourier Transformer. |
| 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 |x(t)| 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 x(t) is equivalent to the Fourier transform of the signal x(t)e-σt. | The Fourier transform is equivalent to the Laplace transform evaluated along the imaginary axis of the s-plane. |
Conclusion
The most significant difference between Laplace Transform and Fourier Transform is that the Laplace Transform converts a time-domain function into an s-domain function, while the Fourier Transform converts a time-domain function into a frequency-domain function. Also, the Fourier Transform is only defined for functions that are defined for all real numbers, but the Laplace Transform does not require that the function is defined for a set of negative real numbers.
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