*Note: this is definitely NOT a 5 years old’s question, but given my current understanding of signal analysis and control systems, I need a 5 years old’s explanation. Also, bear with me as english is not my first language, so I might use some terms that are not that common or used at all in the math vocabulary of the english speaking world.*
Let’s take Laplace’s Transform, for example, where we have the term `exp(-st)` multiplying our original function. If we take `s` to be `jw`, then by Euler’s formula `exp(-st)` can be written as `cos(wt) + jsin(wt)`, so clearly `s` can be viewed as some kind of frequency (a complex frequency). But we are talking about the exponential term.
Now let’s take a simple RC low-pass filter. Its _transfer function_ can be written as `1 / (1 + RCs)`. Again, we can have `s` be `jw` and, given some values for our resistor and capacitor, we can plot the absolute value of our H(jw) and it will be the system’s response for that specific frequency `w`. Why?
What really puzzles me is how integrating the product of a signal with some decaying exponential term gives us back some frequency informations about our system.
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I have been directed by /u/yargleisheretobargle to an amazing video by 3Blue1Brown (that I somehow missed) titled “But what is the Fourier Transform? A visual introduction.”, where he visually explains what the Fourier Transform is, how it works and how it is used as a tool for diverse purposes. I highly recommend that anybody interest in these subjects have a look at that video.
Given that the Fourier Transform is basically a special case of the Laplace Transform and both exist in the frequency domain, the explanation given in the video answered my question.
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