How exactly does a Transform “gets” the frequency components of a system or signal?


*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.

In: 0

It looks like you’re missing that by replacing s with jw you’re effectively doing a Fourier transform. And the whole point of them is frequency analysis.

There are many books written about Fourier transforms, but the relevant point is that the integral of e^ja e^-jb is zero unless a=b. So if you consider your input as the sum of waves, you can use the Fourier transform to pull it apart.

If you think that is amazing, wait till you learn about wavelets.

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.