Mathematical definition of Laplace transform

The mathematical definitions of Laplace transform and inverse transform are as follows (in the case of mathematical integration), where (1) is Laplace transform, which converts time domain signals into frequency domain signals. Equation (2) is an inverse Laplace transform, which converts a frequency domain signal into a time domain signal.

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Related instructions in matlab

Matlab provides related function instructions Laplace and ilaplace for Laplace transform and inverse transform. The specific calling syntax and functions are as follows:

Fs = Laplace (ft, t, s)% function: Find Laplace transform fs of "time domain" function ft;

Ft=ilaplace(Fs, s, t)% function: Find Laplace transform ft of "frequency domain" function Fs.

The input ft and Fs in the instruction are time domain function with t as independent variable and frequency domain function with complex frequency s as independent variable, respectively. The following figure gives the help information of Laplace function that comes with matlab.

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Laplace instruction is a simple application, and the function exp(-a*t)*sin(b*t) is Laplace transform. The results are as follows:

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Influence of parameter a in time domain function u(t-a) on laplace transform. The specific impact is shown in the following figure. It is not difficult to see from the results that when the positive and negative parameters of parameter A are uncertain, the conversion fails, and when the positive and negative parameters of parameter A are redefined, the conversion succeeds.

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Influence of parameter b in time domain function δ(t-b) on Laplace transform. It can be seen from the results that Laplace transform gives two kinds of results without limiting the positive or negative of parameter b, and gives the correct result when the parameter is limited to a positive number. Specific code as shown in the figure below:

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Influence of parameter n in time domain function t n on laplace transform. Similarly, it can be seen from the results that different definitions of parameter n will lead to different results, so we must pay attention to the definition of parameters in the function when using Laplace transform. It can be seen from the results that when n is defined, the transformation is successful.