In signals & systems, control theory, we use laplace for LTI etc..What is the difference instead of Fourier transform?
Laplace transform is normally used for system Analysis,where as Fourier transform is used for Signal Analysis.
IN case of laplace transform you need to have real function where as in case of fourier transform you can have complex as well as real function moreover in case of fourier transform you can control the limit of integration
IN case of laplace transform you need to have real function where as in case of fourier transform you can have complex as well as real function moreover in case of fourier transform you can control the limit of integration
Fourier transform could not manage polynomial expressions and and unstable systems.
Dear Regan D,
If we put the exp(-sigma t) to multiply the Fourier transform, then it becomes the Laplace transform. We can easily see that what is written in the Fourier transform as exp(-jwt), now becomes exp(-st) where s = sigma + jw. The sigma is the real or called as the exponential part, where the jw is the imaginary or called sinusoidal part. The Fourier transform does not really care on the changing magnitudes of a signal, whereas the Laplace transform 'care' both the changing magnitudes (exponential) and the oscillation (sinusoidal) parts. We can say that Fourier transform is a subset of Laplace transform.
The Laplace transform is essentially helpful for solving differential equations, since most of any differential equation's solution will contain exponential and sinusoidal parts. The solution can be more easily express and understand in the s domain.
Please have a look the the following video.
https://www.youtube.com/watch?v=ZGPtPkTft8g
I hope this would help.
Laplace transform transforms a signal to a complex plane s. Fourier transform transforms the same signal into the jw plane and is a special case of Laplace transform where the real part is 0. In Laplace domain, s=r+jw where r is the real part and the imaginary part depicts the oscillatory component.
Dear D. Regan sir,
1. Fourier transform is the special case of laplace transform which is evaluated keeping the real part zero.
2. Fourier transform is generally used for analysis in frequency domain whereas laplace transform is generally used for analysis in s-domain(it's not frequency domain).
3. Fourier transform helps us to study anything in the frequency domain whereas laplace transform is usually done for complex analysis (when anything is not easier to analyse in time domain, we convert it into s domain and then take the inverse laplace transform to complete the analysis).
PS: I have answered this question with respect to electronics engineering domain.
Dear D. Regan ,
welcome!
The La Place transform is a tool to solve the linear differential equation in an easy and elegant way since it transforms a linear differential equation into algebraic equation.
In order to solve for the response of a system in a time domain upon an application of time domain stimuli, one must solve the linear system of the differential equations mathematically modelling the system together with the initial and boundary conditions. Such solution will be normally tedious.
But using the La Place transform to transform the equations from time domain into the S-domain, which is called the complex frequency domain, one can convert the time domain differential equations into S-domain algebraic equations.
Naturally simultaneously solving algebraic equations is much easier than solving simultaneous differential equations. This is the power of this mathematical transformation.
As for the Fourier transform, it is a tool for transforming time domain functions into frequency domain function. Physically any time domain function can be thought of a summation of complex exponential signals. And this what the Fourier transform yield. So, it is an analysis tool of time domain signals.
The Fourier transform is a subset of the La Place transform as the colleagues above pointed.
Best wishes
Laplace transform is simply a generalization of Fourier Transform. If you notice computing Laplace transform is simpler that Fourier, however, the inverse Laplace transform is much harder to compute compared to Fourier Inverse transform because the integrals are over complex limits.
I highly recommend the videos of Zach Star such as this one: https://www.youtube.com/watch?v=n2y7n6jw5d0
Laplace Transform is in the Complex domain (s=R*jw) while Fourier transform is in jw plane. in simple words, Fourier transform is the special case of Laplace transform.
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