Why the Fourier transform is so important compared to other?

[ramdas]

I am engineering student and studying signal processing. The term Fourier transform comes in the discussion several times. There are many transforms like Laplace transform,Z transform,Wavelet transform.But as per my view ,Fourier transform is mostly used compared to others in general.

My question is why Fourier Transform is so important compared to others?I request you all while answering please add any link or video or figure for reference to understand the concept easily.Thank you in advance

 

[meBigGuy]

I'll assume you have or will read the wikipedia article.

Mathematically the Fourier transform is just another transform to convert between numerical domains. It happens to have a useful physical significance since it converts between time and frequency domain, which is extremely useful.

But I can't say it is any more important that the others. For example digital signal processing relies on the Z transform. . No one says "take a z transform of the signal", they just convert the analog signal to digital (with an ADC, for example) and then they can run an FFT if they want to determine the spectrum.

Again, all transforms are important. But the Fourier transform happens to be a useful (well compartmentalized) function that can do something important that is easily understood in a physical system.

 

[Hechima]

One way of thinking about this is that the Fourier Transform is a certain special case of the Laplace Transform, namely by a few assumptions and the substitution s=jω. Time-frequency is something that can be measured, and we have a certain intuition about it because of our senses- we can hear tones. Whereas the kernals of the Z transform or the Laplace transform are on a domain that is more abstract, and their use is more generalized.

It seems like there is a certain trade off between keeping things general and having them be intuitive. For example, an extremely general 'integral' transform would be something like: Find the coefficients a_n such that

f(t) = Σa_ntⁿ

Which is almost just a Taylor series. Mathematically simple, but what kind of physical intuition do we get from a_n? Would someone build a machine that takes the time series and tells us the coefficients, like we can with a spectrum analyzer? Probably not. With wavelet transforms, it is general to the degree that you even need to specify what wavelet shape to use before you can begin.

My point is that the importance is mostly accessibility.