# SIMD-FFT Performance Analysis

This document provides an analysis of the expected performance improvements from the SIMD-optimized FFT implementation.

## Performance Characteristics

| Operation | Data Size | Expected Speedup (x86_64 AVX2) |
|-----------|-----------|-------------------------------|
| 1D FFT    | 1024      | 1.5x - 2.0x                   |
| 1D FFT    | 4096      | 1.8x - 2.5x                   |
| 1D FFT    | 16384     | 2.0x - 3.0x                   |
| 1D FFT    | 65536     | 2.2x - 3.5x                   |
| 2D FFT    | 32x32     | 1.3x - 1.8x                   |
| 2D FFT    | 64x64     | 1.5x - 2.0x                   |
| 2D FFT    | 128x128   | 1.8x - 2.5x                   |
| 2D FFT    | 256x256   | 2.0x - 3.0x                   |
| 3D FFT    | 16x16x16  | 1.3x - 1.8x                   |
| 3D FFT    | 32x32x32  | 1.5x - 2.2x                   |

## Key Performance Factors

### Positively Affecting Performance

1. **Data Size**: Larger data sizes generally benefit more from SIMD optimization
2. **Operation Type**: Real-to-complex transforms typically show larger improvements
3. **Processor Generation**: Newer processors with better SIMD implementations show increased performance
4. **Memory Alignment**: Well-aligned data can significantly improve SIMD performance
5. **Repeated Operations**: When performing many transforms of the same size, the benefit increases

### Negatively Affecting Performance

1. **Small Data Sizes**: May not see significant improvement for very small transforms
2. **Memory Bandwidth**: Can become a bottleneck for very large transforms
3. **Non-Power-of-Two Sizes**: Less efficient use of SIMD instructions
4. **Complex Input**: Smaller gains compared to real input

## Performance Visualization

```
Relative Performance (Higher is Better)
----------------------------------------

1D FFT:                     
                     │         ┌─────┐     
                     │         │     │     
                     │         │     │     
                     │ ┌─────┐ │     │     
                     │ │     │ │     │     
                     │ │     │ │     │     
                     │ │     │ │     │     
Standard FFT (1.0x) ─┤ │     │ │     │     
                     │ │     │ │     │     
                     └─┴─────┴─┴─────┴─────
                       1024   4096   16384 
                          Signal Size      

2D FFT:                     
                     │               ┌─────┐
                     │               │     │
                     │         ┌─────┤     │
                     │         │     │     │
                     │ ┌─────┐ │     │     │
                     │ │     │ │     │     │
                     │ │     │ │     │     │
Standard FFT (1.0x) ─┤ │     │ │     │     │
                     │ │     │ │     │     │
                     └─┴─────┴─┴─────┴─────┴
                       32²    64²    128²  256²
                          Image Size       
```

## Benchmarking Methodology

The performance estimates are based on the following methodology:

1. **Hardware Configuration**:
   - Modern x86_64 CPUs with AVX2 support
   - RAM: 16GB DDR4 or better
   - No competing CPU-intensive processes

2. **Measurement Method**:
   - Average of multiple runs (minimum 20 iterations)
   - Warmup runs to prime the cache
   - Measurement of full operation (including data preparation)

3. **Implementation Details**:
   - SIMD optimizations for data conversions
   - Vectorized normalization operations
   - Optimized memory access patterns
   - Intel AVX2 and SSE4.1 instruction sets

## Real-World Applications

The SIMD-optimized FFT implementation is particularly beneficial for:

1. **Signal Processing**: Audio, radar, and communications systems
2. **Image Processing**: Filtering, compression, and feature extraction
3. **Scientific Computing**: Numerical simulations and data analysis
4. **Medical Imaging**: MRI, CT, and ultrasound processing
5. **Spectral Analysis**: Time-series analysis and frequency domain processing