Fast Fourier Transform:
Transform (DFT) of a sequence, which is a widely used tool in signal processing and various other applications The Fast Fourier Transform (FFT) is an algorithm for efficiently computing the Discrete Fourier. The FFT was developed to overcome the computational complexity of directly calculating the DFT using the standard formula.
The key idea behind the FFT is to exploit the symmetry properties of the complex exponentials in the DFT computation to reduce the number of arithmetic operations required. The standard DFT formula involves\(N^{2}\) complex multiplications N(Nāˆ’1) complex additions for a sequence of length N. In contrast, the Cooley-Tukey algorithm, one of the most famous FFT algorithms, significantly reduces the number of operations to O (N log N ) , (ā€Oā€ is used to describe the upper bound or worst-case performance of an algorithm in terms of its input size.), making it much more efficient for large values of N.
There are several FFT algorithms, and they differ in their approaches to decomposing the DFT computation into smaller, more manageable subproblems.