Explain in simple terms how Fast Fourier Transform works
The fast Fourier transform (FFT) is a mathematical algorithm that allows us to analyze the frequency content of data. It takes as input an array of samples, typically taken from sound or image data, and outputs the corresponding frequencies in another array. The FFT can be used for many different types of analysis, including signal processing, time-series analysis, and computer graphics.

To perform an FFT, we first split our input data into overlapping windows of length N, where N is the number of samples. We then calculate the discrete Fourier transform (DFT) of each window using a fast algorithm such as the Householder algorithm. We then merge the DFTs from all the windows together to form the FFT output.

The FFT has several advantages over other algorithms for analyzingfrequency content:

It is efficient : The FFT requires only O(N log N) computational complexity, which is much less than the O(N^2) required by some other frequency analysis methods. This makes it suitable for processing large datasets.

It's stable : The FFT is robust against noise and errors in the input data, because it operates on signals whose Fourier components are evenly distributed across the frequency spectrum.

It's flexible : The FFT can be adapted to handle various types of data, including audio files, images, and time series. It can also be combined with other algorithms to form more complex analysis techniques.

Overall, the FFT is a powerful tool for exploring the frequency content of data and solving many problems in science, engineering, and technology.