FFT Study Repository

A comprehensive collection of Fast Fourier Transform (FFT) algorithms and implementations in C, designed for educational purposes and performance comparison.

📚 Table of Contents

• Overview
• Features
• Repository Structure
• Algorithms Implemented
• Getting Started
• Building and Running
• Code Quality
• Documentation
• Contributing
• License

Overview

This repository serves as a comprehensive study resource for understanding FFT algorithms, their implementations, optimizations, and applications. It includes various FFT algorithms from basic to advanced, real-world applications, and performance optimizations.

Key Features

• Educational Focus: Each implementation includes detailed documentation explaining the algorithm, its complexity, and use cases
• Production Quality: Professional coding standards with proper error handling, memory management, and testing
• Performance Analysis: Comprehensive benchmarking suite for comparing different implementations
• Real Applications: Practical examples showing how FFT is used in audio processing, filtering, and more
• Multiple Optimizations: SIMD, parallel, and fixed-point implementations for different platforms

Repository Structure

FFT-implementation-in-C/
├── include/               # Header files
│   ├── fft_common.h      # Common utilities and definitions
│   └── fft_algorithms.h  # FFT function declarations
├── algorithms/           # Core algorithm implementations
│   ├── core/            # FFT algorithms
│   └── dft/             # DFT implementations
├── applications/         # Real-world applications
├── optimizations/        # Performance optimizations
├── benchmarks/          # Performance testing
├── tests/               # Test suites
├── examples/            # Example programs
├── utils/               # Utility functions
└── lib/                 # Compiled library files

Algorithms Implemented

✅ Core FFT Algorithms

Algorithm	Description	Complexity	Status
Radix-2 DIT	Decimation-in-Time FFT	O(n log n)	✅
Radix-2 DIF	Decimation-in-Frequency FFT	O(n log n)	✅
Radix-4	Base-4 FFT algorithm	O(n log n)	✅
Split-Radix	Optimal operation count	O(n log n)	✅
Bluestein	Arbitrary size FFT	O(n log n)	✅
Mixed-Radix	Composite size FFT	O(n log n)	✅
Recursive	Educational implementation	O(n log n)	✅
Iterative	Cache-efficient	O(n log n)	✅

🔧 Optimizations

Optimization	Target Platform	Speedup	Status
SIMD (SSE/AVX)	x86/x64	2-4x	✅
OpenMP Parallel	Multi-core	~Nx cores	✅
Fixed-Point	Embedded/DSP	Platform-specific	✅

📊 Applications

• Audio Spectrum Analyzer: Real-time frequency analysis
• Digital Filtering: Low-pass, high-pass, band-pass filters
• Convolution: Fast convolution using FFT
• Power Spectrum: Signal power analysis
• 2D Image FFT: Image processing applications
• Pitch Detection: Musical note detection

Getting Started

Prerequisites

• GCC compiler (or compatible C compiler)
• Make build system
• POSIX-compliant system (Linux, macOS, WSL)
• Optional: OpenMP support for parallel implementations

Quick Start

1. Clone the repository:

   git clone https://github.com/muditbhargava66/FFT-implementation-in-C.git
   cd FFT-implementation-in-C

2. Run the quick start script:

   chmod +x quickstart.sh
   ./quickstart.sh

3. Or build manually:

   make all

Building and Running

Build Commands

# Build everything
make all

# Build specific components
make algorithms      # Core FFT algorithms
make applications   # Application examples
make optimizations  # Optimized versions
make benchmarks     # Benchmark suite
make tests          # Test suite

# Build with debug symbols
make debug

# Build with profiling
make profile

# Clean build artifacts
make clean

Running Examples

# Run a specific algorithm
./bin/radix2_dit

# Run benchmarks
./bin/benchmark_all

# Run tests
./bin/test_all

# Run demonstrations
make demo

Library Usage

The project builds a static library libfft.a that can be linked with your programs:

#include "fft_common.h"
#include "fft_algorithms.h"

int main() {
    int n = 1024;
    complex_t* signal = allocate_complex_array(n);
    
    // Generate signal...
    
    // Apply FFT
    radix2_dit_fft(signal, n, FFT_FORWARD);
    
    // Process spectrum...
    
    free_complex_array(signal);
    return 0;
}

Compile with:

gcc -I/path/to/include myprogram.c -L/path/to/lib -lfft -lm

Code Quality

Coding Standards

• C99 Standard: Modern C features for better code clarity
• Consistent Style: 4-space indentation, clear naming conventions
• Comprehensive Documentation: Doxygen-style comments for all functions
• Error Handling: Proper validation and error reporting
• Memory Safety: Careful allocation/deallocation, no memory leaks

Algorithm Implementation

Each algorithm implementation includes:

1. Detailed Header Documentation: Mathematical background, complexity analysis, references
2. Step-by-Step Comments: Clear explanation of each algorithmic step
3. Input Validation: Checking for valid sizes, null pointers, etc.
4. Test Cases: Built-in testing in main() function
5. Performance Measurement: Timing and complexity verification

Example structure:

/**
 * @brief Algorithm name and brief description
 * 
 * @details
 * Detailed explanation of the algorithm, including:
 * - Mathematical foundation
 * - Step-by-step process
 * - Data structures used
 * 
 * @param x Input/output array
 * @param n Array length
 * @param dir Transform direction
 * 
 * Time Complexity: O(n log n)
 * Space Complexity: O(1)
 */
void algorithm_fft(complex_t* x, int n, fft_direction dir) {
    /* Input validation */
    if (!x || n <= 0) {
        fprintf(stderr, "Error: Invalid input\n");
        return;
    }
    
    /* Algorithm implementation with clear comments */
    // Step 1: ...
    // Step 2: ...
}

Documentation

Algorithm Documentation

Each algorithm includes:

• Mathematical derivation
• Complexity analysis
• Implementation notes
• Usage examples
• Performance characteristics

API Documentation

See include/fft_common.h and include/fft_algorithms.h for the complete API reference.

Key functions:

• radix2_dit_fft() - Standard FFT for power-of-2 sizes
• bluestein_fft() - FFT for arbitrary sizes
• allocate_complex_array() - Memory allocation helper
• compute_magnitude() - Convert to magnitude spectrum

Performance

Benchmark results on typical hardware (Intel i7, 3.6GHz):

Algorithm	N=1024	N=4096	N=16384
Radix-2 DIT	0.08ms	0.4ms	1.8ms
Split-Radix	0.06ms	0.3ms	1.4ms
SIMD Optimized	0.03ms	0.15ms	0.7ms
Parallel (4 cores)	0.02ms	0.1ms	0.5ms

Contributing

We welcome contributions! Please see CONTRIBUTING.md for guidelines.

Areas for contribution:

• Additional algorithms (Prime Factor, Winograd)
• GPU implementations (CUDA/OpenCL)
• More applications
• Performance optimizations
• Documentation improvements

References

1. Cooley, J. W., & Tukey, J. W. (1965). "An algorithm for the machine calculation of complex Fourier series"
2. Duhamel, P., & Vetterli, M. (1990). "Fast Fourier transforms: a tutorial review"
3. Frigo, M., & Johnson, S. G. (2005). "The design and implementation of FFTW3"
4. Van Loan, C. (1992). "Computational frameworks for the fast Fourier transform"

License

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

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Note: This is an educational repository. For production use, consider established libraries like FFTW, Intel MKL, or cuFFT.