1, integer domain and transformation kernel
Fourier transform and Laplace transform belong to integral transform and are two common mathematical transformation methods. The so-called integral transformation is to transform one function into another through integral operation, and its function is to turn complex function operation into simple function operation. When different integration domains and transformation kernels are selected, different names of integral transformations are obtained. Fourier transform and Laplace transform are obtained by taking different integration domains and transform kernels.
2. Frequency domain and complex frequency domain
Fourier transform is a special case of Laplace transform. Laplace transform transforms the time domain signal into "complex frequency domain", which is different from the transformed "frequency domain". ?
Application:
1, Laplace transform is mainly used in circuit analysis as a powerful tool for solving differential equations (calculus operation is converted into multiplication and division operation).
2. Fourier transform is widely used in physics, electronics, number theory, combinatorial mathematics, signal processing, probability theory, statistics, cryptography, acoustics, optics, oceanography, structural dynamics and other fields (for example, in signal processing, the typical use of Fourier transform is to decompose the signal into amplitude spectrum-display the amplitude corresponding to the frequency). With the development of FFT, it has become the most important mathematical tool in the field of digital signal processing.
Fourier transform means that functions that meet certain conditions can be expressed as trigonometric functions (sine and/or cosine functions) or linear combinations of their integrals. In different research fields, Fourier transform has many different variants, such as continuous Fourier transform and discrete Fourier transform. Firstly, Fourier analysis is proposed as a tool for thermal process analysis.
Laplace transform is suitable for continuous time function x(t) with t & gt=0 function value not zero.
(where -st is the exponent of natural logarithm base E) The function X(s) converted into complex variable S is also the "complex frequency domain" representation of time function x(t).
References:
Baidu Encyclopedia Fourier Transform-Laplace Transform