Please check this video presentation for a simple overview of Laplace Transform [1]. Basically Fourier Transform (FT) is a special case (subset) of Laplace Transform (LT) where the signal waveform revolves around a unit circle (real power of exponents) . Similarly, LT is basically a generalization (superset) of FT that the signal waveforms revolve off the unit circle (complex power of exponents).

The discrete FT or DFT, however, as the name clearly implied, is the discrete version of FT and similarly discrete Laplace Transform (DLT) is the discrete version of LT. The main difference is that DFT covers finite sum but DLT covers infinite sum.

The faster version of DFT (without compromising the resolution accuracy) is called FFT and it is probably the most useful and important algorithm in the 21st century! The inverse FFT is called IFFT and it was discovered around the same time of FFT. The faster version of DLT is interestingly called Chirp-Z Transform (CZT) and somehow its inverse (ICZT) discovery is at a much later date as has been reported recently [2] and also featured in HN [3]. This much later date of discovery is mainly due to the complexity of complex power exponents (pardon the pun but cannot resist).

Fun fact, CT was discovered by Lawrence Rebinar who was working at AT&T's speech processing lab (SPL) [4]. The lab is so well funded that Kernighan and Ritchie who were belong to the other lab has to scrap by the older computer of the SDL (the infamous PDP-7) where Unix was originally developed on when Multics project got canceled.

[1]https://youtu.be/n2y7n6jw5d0

[2]https://www.electronicsweekly.com/news/research-news/dsp-inv...

[3]https://news.ycombinator.com/item?id=21230757

[4]https://ieeexplore.ieee.org/document/1276120