cleanfm

Implementation of the Dirichlet Kernel Deconvolution algorithm (parallel and single-threaded) for precise FFT parameter estimation and spectral leakage elimination. Repository for code contained in Matthew McQuistion's Applied Mathematics Honors Thesis at Brown University

CleanFM: A Windowless Approach to Spectral Leakage and the Frequency Modulation Transform

CleanFM is a novel approach to spectral analysis that directly addresses the fundamental limitations of traditional windowing techniques in digital signal processing. The primary contributions of this project are:

1. Dirichlet Kernel Deconvolution (DKD): A method for extracting precise frequency components from discrete spectra by modeling and removing the intrinsic spectral leakage pattern, preserving both resolution and information content.

2. Parallel DKD Implementation: An efficient parallel version of DKD that maintains accuracy while significantly reducing computational requirements.

3. Applications of DKD: Demonstration of DKD's utility in tasks such as idealized signal resampling and creating a "cleaned" Short-Time Fourier Transform (CSTFT) with superior resolution.

4. Frequency Modulation Transform (FMT): A novel mathematical operator that reveals harmonic relationships within complex signals by analyzing the "frequencies of frequencies."

Dependencies

This project requires the FFTW3 library for Fast Fourier Transform computations.

Installing FFTW3

• On macOS (using Homebrew): brew install fftw3
• On Linux (using apt): sudo apt-get install libfftw3-dev
• On Windows: Download the precompiled FFTW3 libraries from http://www.fftw.org/install/windows.html
  • Extract the downloaded file and add the lib directory to your system's PATH environment variable

Repository Structure

• CMakeLists.txt: CMake configuration file for building the project.
• constants.h: Header file defining various inline constants used throughout the project.
• main.cpp: Main driver program for the CleanDFT (DKD) spectral analysis tool.
• deconvolve/cleandft.cpp, deconvolve/cleandft.h: Implementation and declaration of the DKD algorithm.
• deconvolve/dirichletkernel.cpp, deconvolve/dirichletkernel.h: Implementation and declaration of the Dirichlet kernel calculations.
• deconvolve/threadpool.h: Defines helper structures and a basic ThreadPool class for parallel processing.
• fft/waveprocessor.cpp, fft/waveprocessor.h: Implements functions for reading, processing (FFT/IFFT), and writing WAV audio files.
• cstft/cstft.sh: Bash script to run the Cleaned STFT (DKD + STFT) algorithm.
• fmt/fmt.cpp: FMT code that reads in 5-th order modulation index approximation from fmt/fmt_lookup_table.cpp

Building and Running

1. Ensure you have CMake, a C++20 compatible compiler, and FFTW3 installed.

2. If the FFTW3 library is not in a standard system path, you might need to update the include path in CMakeLists.txt and the source files:

• In CMakeLists.txt, update the INTERFACE_INCLUDE_DIRECTORIES property for the FFTW3::FFTW3 target.
• In waveprocessor.h, update the #include directive for the FFTW3 header.

3. Create a build directory: mkdir build && cd build

4. Configure the project: cmake ..

5. Build the project: make

6. Run the compiled executable: ./cleanfm <options>

Use ./cleanfm --help to see available command line options.

Applications and Future Work

In addition to the core DKD and FMT algorithms, this project explores several promising applications:

• Idealized frequency domain resampling that preserves spectral characteristics
• A "Cleaned STFT" with dramatically improved frequency resolution
• Preliminary work on source separation using the FMT representation

Future work will focus on completing the C++ implementation of FMTSS, further optimizing computational efficiency, and exploring applications of these techniques to areas such as audio analysis, feature extraction, sound transformation, and unsupervised learning.

License

This project is released under the Apache 2.0 License.