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Fast Fourier Transform (FFT) Algorithms Project

Overview

This project focuses on the implementation, comparison, and analysis of two distinct Fast Fourier Transform (FFT) algorithms with varying complexities. Each algorithm is chosen to demonstrate different computational trade-offs and use-cases in FFT processing. Our goal is to provide insights into the applicability and performance of these algorithms in diverse scenarios.

Algorithms

1. Cooley-Tukey FFT Algorithm (Radix-2)

Complexity:

  • Sequential: O(N log N)
  • Parallel: OpenMP can enhance parallel complexity to approximately O(log² N) using multi-core processors, MPI has the potential to reduce complexity to near O(log N) based on inter-node communication efficiency, while CUDA can significantly lower complexity, potentially close to O(log N), thanks to extensive GPU parallelism.

Description:

The Cooley-Tukey algorithm, specifically its Radix-2 variant, is a widely-used FFT algorithm. It's a divide-and-conquer approach that efficiently handles sequences of lengths that are powers of 2. This algorithm offers a good balance between ease of implementation and computational efficiency.

Use-case:

Ideal for general-purpose applications and educational purposes, particularly when dealing with data sizes that are powers of two.

2. Bluestein's FFT Algorithm (Chirp Z-Transform)

Complexity:

  • Sequential: O(N log N) but with a higher constant factor than Cooley-Tukey
  • Parallel: In Bluestein's FFT algorithm, OpenMP can achieve parallel complexity around O(log² N); MPI could lower it to near O(log N), and CUDA might significantly reduce it to close to O(log N), utilizing the power of GPU parallelism.

Description:

This algorithm is tailored for computing DFTs of arbitrary lengths, not just powers of two. It cleverly transforms a DFT into a convolution problem, which is then solved using an FFT of a suitable size. This approach is more intricate than the basic Cooley-Tukey algorithm but provides flexibility in handling diverse data sizes.

Use-case:

Applicable in situations where the data length is not a power of two, offering versatility in data processing.

Project Structure

  • src/: Source code for each FFT algorithm implementation.
  • docs/: Documentation and theoretical background of each FFT algorithm.
  • tests/: Unit tests to ensure the reliability and accuracy of implementations.

Notes

  • OpenMP/MPI: More suitable for systems with multiple CPUs or CPU clusters, where inter-process or inter-thread communication is key.
  • CUDA: Highly efficient for algorithms that can be parallelized at a fine-grained level, like FFT, especially on large data sets.

License

This project is licensed under the MIT License - see the LICENSE file for details.