Add Fast Fourier Transform (FFT) in R

[nachiket2005]

Overview

The fft_recursive function decomposes a DFT of size N into smaller DFTs of even and odd indexed elements, recursively combining their results.
If the input vector’s length is not a power of two, it is zero-padded to the next power of two for optimal performance.

This implementation serves as a clear, educational example of FFT internals while maintaining compatibility with complex input data.

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Features

• Fully recursive Cooley–Tukey FFT implementation
• Accepts both numeric and complex input vectors
• Automatically zero-pads input to the next power of two
• Returns complex output representing frequency-domain components
• Compact and easy to understand — ideal for teaching and research use
• Includes example usage block for quick verification

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Complexity

• Time Complexity: O(N log N)
• Space Complexity: O(N) (due to recursive calls and intermediate storage)

[Copilot]

Pull Request Overview

Adds a recursive Cooley–Tukey FFT implementation in R with zero-padding to next power-of-two and accompanying documentation.

• Introduces fft_recursive with power-of-two padding and complex output
• Adds documentation and usage examples
• Includes an interactive example block in the R script

Reviewed Changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 8 comments.

File	Description
mathematics/fast_fourier_transform.r	Adds the recursive FFT implementation, zero-padding helper, and an interactive example block
documentation/fast_fourier_transform.md	Documents the FFT function, usage, and complexity notes

[siriak]

Please remove this file, we don't add documentation separately from the source code