What does the Laplace Transform tell us? A visual explanation [video]
CONTROL THEORY NERD ALERT! This is hands-down the best video on control theory I've ever seen, clearly depicting the relationship of the Laplace Transform to the Fourier Transform.The Fourier Transform decomposes a signal into its sinusoidal frequency components. It is used in a bunch of everyday appliances like your guitar tuner, JPEG images (wavelet compression) and speech recognition on your smartphone or smart speaker.The Laplace Transform, on the other hand, decomposes a signal into both its exponential factors (decaying or rising) AND its sinusoidal components. So the FT is just one slice of the Laplace Transform where the input signal has no exponential rise or decay.In the electrical and mechanical domains, spring mass damper systems are super common, even your car's suspension! And to analyse and control them, engineers apply the Laplace Transform.