The Fourier Transform is a mathematical operation that transforms a function of time (or space) into a function of frequency. It decomposes a complex signal into its constituent sinusoidal components, each with a specific frequency, amplitude, and phase. This is particularly useful in many fields, such as signal processing, physics, and engineering, because it allows for analysing the frequency characteristics of signals. The Fourier Transform provides a bridge between the time and frequency domains, enabling the analysis and manipulation of signals in more intuitive and computationally efficient ways. The result of applying a Fourier Transform is often represented as a spectrum, showing how much of each frequency is present in the original signal.

$$\boxed{{\displaystyle \boxed{{\displaystyle \hat{f}(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-i2\pi \xi x}\mathrm{d}x,\mathrm{\forall }\xi \in \mathbb{R}.}}}}$$

Inverse Fourier Transform:
$$\boxed{{\displaystyle \boxed{{\displaystyle f(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\xi )e^{i2\pi \xi x}\mathrm{d}\xi ,\mathrm{\forall }x\in \mathbb{R}.}}}}$$

The equation allows us to listen to mp3s today. Digital Music Couldn’t Exist Without the Fourier Transform: http://bit.ly/22kbNfi

These women fell in love with an #AI-voiced #chatbot. Then it died | @restofworld@restof.social

An AI-voiced virtual lover called users every morning. Its shutdown left them heartbroken.

"Chinese #AI voice startup #Timedomain's "Him" created virtual companions that called users and left them messages.

Some #users were so moved by the messages, written by humans but voiced by AI, that they considered "#Him" to be their romantic partner.

Timedomain shut down "Him", leaving these users distraught."

#China #ArtificialIntelligence #romance

https://restofworld.org/2023/boyfriend-chatbot-ai-voiced-shutdown/