Difference Between Fourier Transform vs Laplace Transform

The uncertainty principle describes how functions that are localised in the time domain have Fourier transformations that are spread out over the frequency domain and vice versa. The Gaussian function, which is important in probability theory and statistics as well as the study of physical events with normal distributions, is a fundamental instance for this idea (e.g., diffusion). 

A Gaussian function’s Fourier transform is another Gaussian function. In his study of heat transmission, Joseph Fourier introduced the transform, which shows Gaussian functions as solutions to the heat equation.

Although this definition is useful for many applications needing a more advanced integration theory, the Fourier transform can be formally described as an improper Riemann integral, making it an integral transform. Laplace also realised that Joseph Fourier’s Fourier series approach for solving the diffusion problem could only be used in a small area of space since the solutions were periodic. Laplace used his transform to identify infinitely distributed solutions in space in 1809.

Fourier Transform vs Laplace Transform

The Fourier transform is only specified for functions that are defined for all real numbers, but the Laplace transform does not require that the function be defined for a set of negative real numbers. 

A specific case of the Laplace transform is the Fourier transform. Both coincide for non-negative real numbers, as can be seen. (i.e., in the Laplace equation, s = I +, where and are real numbers, and e = 1/(2)). Every function with a Fourier transform also has a Laplace transform, but not the other way around.

Unstable systems can be studied using the Laplace transform. In order to analyse unstable systems, the Fourier transform cannot be utilised. Because the Laplace transform exists even for signals for which the Fourier transform does not exist, it is commonly utilised to solve differential equations. Due to the fact that the Fourier transform does not exist for many signals, it is rarely employed to solve differential equations. 

What is a Laplace Transform?

The Laplace transform was named after Pierre-Simon Laplace, a mathematician and astronomer who employed a similar transform in his work on probability theory. The integral form of the Laplace transform arose organically as a result of Laplace’s extensive usage of generating functions in Essai philosophique sur les probabilités (1814). 

Laplace’s usage of generating functions was akin to what is now known as the z-transform, and he paid little heed to Niels Henrik Abel’s discussion of the continuous variable situation.Mathias Lerch, Oliver Heaviside, and Thomas Bromwich advanced the theory in the 19th and early 20th centuries. The transform’s current widespread application (mostly in engineering) began during and shortly after World War II, when it replaced the previous Heaviside operational calculus.

When someone mentions “the Laplace transform” without qualifier, they usually mean the unilateral or one-sided transform. By extending the bounds of integration to the entire real axis, the Laplace transform can be characterised as the bilateral Laplace transform, or two-sided Laplace transform. 

Define the Fourier analysis

Fourier analysis is a broad topic that covers a wide range of mathematics. Fourier analysis is the technique of dissecting a function into oscillatory components, and Fourier synthesis is the process of reconstructing the function from these parts in science and engineering. Computing the Fourier transform of a sampled musical note, for example, would be used to determine what component frequencies are present in a musical note.  

Fourier analysis is a term used in mathematics to describe the study of both operations. A Fourier transformation is the name for the decomposition process. The Fourier transform, which is its output, is given a more precise name depending on the context.

Data must be evenly spaced to use Fourier analysis. For analysing unequally spaced data, various methodologies have been developed, including least-squares spectral analysis (LSSA) methods, which apply a least squares fit of sinusoids to data samples, comparable to Fourier analysis. Long-periodic noise in long gapped records is often boosted by Fourier analysis.

Conclusion

The Fourier transform is only specified for functions that are defined for all real numbers, but the Laplace transform does not require that the function be defined for a set of negative real numbers. A specific case of the Laplace transform is the Fourier transform. Because the Laplace transform exists even for signals for which the Fourier transform does not exist, it is commonly utilised to solve differential equations. By extending the bounds of integration to the entire real axis, the Laplace transform can be characterised as the bilateral Laplace transform, or two-sided Laplace transform. Fourier analysis is a term used in mathematics to describe the study of both operations. The Fourier transform, which is its output, is given a more precise name depending on the context.