Fast Fourier Transfor (FFT) - dewabuanam/algostable GitHub Wiki

Overview

FFT (Fast Fourier Transform) is an efficient algorithm used to compute the Discrete Fourier Transform (DFT) and its inverse. The DFT is a mathematical operation that converts a signal from the time domain to the frequency domain. This transformation is widely used in signal processing, image processing, audio analysis, and various other applications.

Equation of the DFT

The Discrete Fourier Transform of a sequence of N complex numbers {x[0], x[1], ..., x[N-1]} is given by the following equation:

$$X[k] = \sum_{n=0}^{N-1} x[n] \cdot e^{-j(2\pi/N) \cdot kn}$$

Where:

  • X[k] represents the k-th frequency component (bin) of the signal.
  • x[n] is the n-th element of the input sequence.
  • j is the imaginary unit (√(-1)).
  • N is the total number of samples in the sequence.

Usage of FFT

The FFT algorithm significantly reduces the computational complexity of the DFT from O(N^2) to O(N log N), making it much faster for large datasets (where N is a power of 2). It finds applications in various fields, including:

  • Signal processing: Filtering, convolution, and spectral analysis.
  • Image processing: Image enhancement, compression, and pattern recognition.
  • Audio analysis: Speech recognition, audio compression, and music analysis.
  • Communications: Modulation/demodulation, channel equalization, and echo cancellation.
  • Numerical analysis: Solving partial differential equations and other mathematical problems.

How the FFT Algorithm Works

The Cooley-Tukey algorithm is the most common and efficient FFT algorithm. It follows a divide-and-conquer approach and can be summarized in three main steps:

  1. Data Reordering: Rearrange the input sequence by separating the even-indexed samples from the odd-indexed samples. This is often done using a technique called "bit-reversal permutation."

  2. Recursive Computations: Recursively compute the DFT of the even-indexed and odd-indexed subsequences.

  3. Combine Results: Combine the results of the sub-problems to get the final DFT of the original sequence.

The algorithm continues to divide the problem into smaller sub-problems until it reaches the base case of a sequence with just one element. At this point, the DFT of a single element is trivially computed. Then, the results are combined back up the recursion until the full DFT is obtained.

Conclusion

FFT is a fundamental algorithm with wide-ranging applications in modern computing. By efficiently computing the Discrete Fourier Transform, it enables us to analyze and manipulate signals and data in the frequency domain, providing valuable insights and solutions in numerous fields.