Control Systems

Basically you got some system. And when you "bang" it, or tap it or whatever, it will respond in a certain way. Like a bell will ring at some frequency and decay after some time.

If it is linear time invariant, it will respond the same way regardless of when you bang it.

Because linearaity, if you bang it twice as hard it will ring twice as loud and for twice longer.

and also because linearity, if you bang it a bunch of times, you can assume the response will be the same as if you just added up the effects.

whats really nice about the frequnecy and laplace domain is that those transforms can turn convolutions into multiplication, and multiplying things is easy. Turns really hard calculus into algebra.

the bell is the system in this case and the displacement of it would be the thing that changes with respect to time after being impacted, or affected by the impulse response.

for power supplies, it is common to measure the step response, which is quite similar. Basically just step the current draw from something low like 10mA to something like 1A as instantaneously as possible and see how the voltage responds. It might dip initially then the controller responds and it might overshoot as it attempts to recover or whatever
if it is underdamped it might oscillate a bit before settling down, if it is overdamped it may take some time to slowely recover. Critically damped would be ideal because it recovers the fastest and without overshoot or ugly ringing

the laplace domain is just a different way of looking at the same things. Fourier is a bit easier to understand. If you have a time domain signal, you basically just multiply that time domain signal from the begining to end of time, to a sine wave and find the average area under the curve of that signal.

do this for every possible frequency and then plot those averages.

that is the fourier transform in a nutshell, and the cool thing about it is that it tells you how much of each frequency component exists in your signal.

the cochlea in your ear does this! it is filled with various lengths of hair that respond only to specific frequencies. the hairs connect to nerves which respond to the displacement and vibration of those hairs.
when they do cochleo implants, they have a set of electrodes that they spiral down into it. And the implant will record the sound of the enviroment, and compute an FFT essentially.

once you have a model of how all your systems are in the laplace domain, when you string them together in series they just multiply!

so like if you have this system A that then controls system B which finally controls system C, in the laplace domain the transfer function for the whole contrived thing is A(s)B(s)C(s). But in the time domain you would have to do A(t)⁕B(t)⁕C(t) and the ⁕ operand is shorthand for convolution intergral.

if you want a simple multiply or amplify some input terracore signal, just convolve that signal with an impulse multiplied to the scaling factor you want!
if you want to take your signal and move it forward or backward in time, simply move your impulse right or left in the time domain! keep in mind the flipping part of the convolution.