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

[Copilot]

The path includes an extra 'R/' segment. Update to mathematics/fast_fourier_transform.r to match the actual file location.

Suggested change

	This file documents the recursive Cooley-Tukey FFT implementation added to `R/mathematics/fast_fourier_transform.r`.
	This file documents the recursive Cooley-Tukey FFT implementation added to `mathematics/fast_fourier_transform.r`.

[Copilot]

The source path should not include 'R/'. Change to: Rscript -e "source('mathematics/fast_fourier_transform.r'); print(fft_recursive(c(0,1,2,3)))".

Suggested change

	Rscript -e "source('R/mathematics/fast_fourier_transform.r'); print(fft_recursive(c(0,1,2,3)))"
	Rscript -e "source('mathematics/fast_fourier_transform.r'); print(fft_recursive(c(0,1,2,3)))"

[Copilot]

The time complexity is not dominated by padding; the runtime is O(M log M) where M is the next power of two (padding itself is O(M−N)). Suggest: 'Time complexity: O(M log M), where M is the next power of two at least as large as the input length.'

Suggested change

	Time complexity: O(n log n) for inputs with length a power of two; otherwise dominated by padding to next power of two.
	Time complexity: O(M log M), where M is the next power of two at least as large as the input length.

[Copilot]

This comment does not match the condition below (which only runs in interactive sessions). Update to 'Example usage in an interactive R session'.

Suggested change

	# Example usage when run directly with Rscript
	# Example usage in an interactive R session

[Copilot]

[nitpick] Avoid relying on a non-standard environment variable and simplify the guard; use only interactive(): if (interactive()) {.

Suggested change

	# Example usage when run directly with Rscript
	if (identical(Sys.getenv("R_SCRIPT_NAME"), "") && interactive()) {
	# Example usage when run directly in an interactive R session
	if (interactive()) {

[Copilot]

[nitpick] The docs state the function accepts numeric or complex input, but as.complex() will coerce other types silently. Add validation, e.g., before conversion: if (!is.numeric(x) && !is.complex(x)) stop('fft_recursive expects a numeric or complex vector').

Suggested change

	fft_recursive <- function(x) {
	fft_recursive <- function(x) {
	# Validate input type
	if (!is.numeric(x) && !is.complex(x)) stop('fft_recursive expects a numeric or complex vector')

[Copilot]

Please add tests that validate correctness against base::fft for multiple cases (power-of-two length, non-power-of-two input with zero-padding, and complex-valued inputs). This repository requires adding a test or example demonstrating correctness.

[Copilot]

The path should match the file location. Update to: # When running via Rscript, users can call: Rscript -e "source('mathematics/fast_fourier_transform.r'); print(fft_recursive(c(0,1,2,3)))".

Suggested change

	# When running via Rscript, users can call: Rscript -e "source('R/mathematics/fast_fourier_transform.r'); print(fft_recursive(c(0,1,2,3)))"
	# When running via Rscript, users can call: Rscript -e "source('mathematics/fast_fourier_transform.r'); print(fft_recursive(c(0,1,2,3)))"