Kalid from BetterExplained here, my quick intuition:
The Fourier Transform breaks a signal into its "cycle recipe" (what circular paths are present?). The LaPlace Transform breaks a signal into its "spiral recipe".
Circles are made from a type of exponential (given by e^ix), and spirals are the more general version, where the radius changes (if s=a+bi, then e^is = e^a * e^bi, aka a circular path where the radius changes exponentially).
The LaPlace transform actually deals with decaying spirals (negative s) -- why is this useful?
Well, perfect circles that never decay (the Fourier Transform) are nice for analyzing audio samples, as in music. (Repeated drumbeat throughout the song.)
Decaying spirals model things in the real world, where friction, etc. dampen the signal over time. The Laplace transform can cleanly represent this scenario, whereas you need an infinite number of cancelling terms in the Fourier Transform to represent the "decays over time" setup.
Engineering applications prefer Laplace, compression mechanisms may prefer Fourier. (Separately, Laplace/Fourier make differential equations easier to solve by writing functions in terms of exponentials, which are easy to derive/integrate. The Laplace transform is more general and powerful in this regard, since it can handle any rate of decay, including 0. The Fourier Transform is embedded within the Laplace.)
Just some quick thoughts from an amateur on this :).
The Fourier Transform breaks a signal into its "cycle recipe" (what circular paths are present?). The LaPlace Transform breaks a signal into its "spiral recipe".
Circles are made from a type of exponential (given by e^ix), and spirals are the more general version, where the radius changes (if s=a+bi, then e^is = e^a * e^bi, aka a circular path where the radius changes exponentially).
The LaPlace transform actually deals with decaying spirals (negative s) -- why is this useful?
Well, perfect circles that never decay (the Fourier Transform) are nice for analyzing audio samples, as in music. (Repeated drumbeat throughout the song.)
Decaying spirals model things in the real world, where friction, etc. dampen the signal over time. The Laplace transform can cleanly represent this scenario, whereas you need an infinite number of cancelling terms in the Fourier Transform to represent the "decays over time" setup.
Engineering applications prefer Laplace, compression mechanisms may prefer Fourier. (Separately, Laplace/Fourier make differential equations easier to solve by writing functions in terms of exponentials, which are easy to derive/integrate. The Laplace transform is more general and powerful in this regard, since it can handle any rate of decay, including 0. The Fourier Transform is embedded within the Laplace.)
Just some quick thoughts from an amateur on this :).