Fast Fourier Transform

TL;DR

• Efficient algorithm to compute the discrete Fourier transform (DFT) of a sequence much faster than the direct method.
• Converts time-domain data into frequency-domain components for analysis or manipulation (e.g., filtering, compression).
• Uses a divide-and-conquer approach to reduce complexity from O(n^2) to O(n log n), enabling practical real-time processing of large datasets.

Definition

Fast Fourier Transform (FFT) is an efficient algorithm for calculating the discrete Fourier transform (DFT) of a sequence of data points. The FFT exploits symmetries in the Fourier coefficients to reduce the computational complexity of calculating the DFT from O(n^2) to O(n log n), where n is the number of data points.

Explanation

• The discrete Fourier transform (DFT) converts a time-domain signal into its frequency-domain representation, revealing the contributions of different frequencies to the signal.
• The DFT computes coefficients (Fourier coefficients) that represent a periodic signal as a sum of sinusoids of different frequencies, amplitudes, and phases.
• The direct (naive) calculation of the DFT computes each Fourier coefficient by summing over all input samples, which yields O(n^2) computational complexity. The DFT for frequency index k is given by:

$$F[k] = \sum_{n=0}^{n-1} x[n] \cdot e^{-i \, 2 \pi k n / n}$$

where x[n] is the time-domain signal and the exponential is a complex sinusoid at frequency k.

• The FFT uses a divide-and-conquer strategy: it splits the time-domain signal into two shorter signals (each containing half the data points), computes the DFT of each shorter signal, and then combines those results. This recursive decomposition continues until the subproblems are small enough to compute directly, yielding an overall complexity of O(n log n).
• This reduction in computational cost makes FFT valuable in applications requiring real-time or large-scale frequency-domain processing.

Examples

Audio signal processing

In audio processing, the time-domain signal represents the amplitude of a sound wave over time; the frequency-domain representation shows the contributions of different frequencies to the overall sound. Using FFT, specific frequency components can be isolated and manipulated, for example to remove unwanted noise or emphasize certain frequencies to improve perceived sound quality.

Image processing

For images, the time-domain signal corresponds to pixel intensities and the frequency-domain representation shows spatial frequency content. By manipulating frequency components via FFT, operations such as edge detection, sharpening, and blurring can be performed on an image.

Use cases

• Signal filtering
• Compression
• Spectral analysis
• Image processing (including edge detection, sharpening, blurring)
• Real-time processing of large amounts of data

Related terms

• Discrete Fourier Transform (DFT)
• Fourier coefficients
• Divide-and-conquer