# FFT Module Performance Analysis

## Initial Benchmark Results

### 1D FFT Operations (Time in microseconds)

| Size  | FFT    | RFFT   | FrFT    | Ratio (FFT/RFFT) |
|-------|--------|--------|---------|------------------|
| 64    | 66.2   | 2.0    | 64.8    | 33.1×            |
| 256   | 34.4   | 4.1    | 113.9   | 8.4×             |
| 1024  | 107.3  | 11.4   | 408.5   | 9.4×             |
| 4096  | 376.9  | 49.8   | 1336.2  | 7.6×             |

### Memory-Efficient Comparison

For a signal of size 4096:
- Regular FFT: 89.7µs
- In-place FFT: 181.6µs
- **Regular FFT is 2.02× faster**

## Key Findings

1. **Real FFT Optimization**: RFFT consistently outperforms regular FFT by 7-33×, which is expected since it exploits the symmetry properties of real-valued signals.

2. **Fractional Fourier Transform**: FrFT is significantly slower than regular FFT:
   - 1-4× slower for small sizes
   - 3-4× slower for larger sizes
   - This is expected due to the additional chirp multiplications and complex calculations

3. **Memory-Efficient Operations**: Surprisingly, the in-place FFT is slower than the regular FFT. This could be due to:
   - Additional overhead in the implementation
   - Suboptimal memory access patterns
   - Extra copying between input and output buffers

## Performance Characteristics

### Algorithmic Complexity
- FFT: O(n log n)
- RFFT: O(n log n) with constant factor improvement
- FrFT: O(n log n) with larger constant factor due to chirp multiplications

### Memory Usage
- FFT: Requires O(n) additional memory for output
- RFFT: Requires O(n/2 + 1) additional memory
- In-place FFT: Uses O(1) additional memory but has performance overhead

## Recommendations

1. **Use RFFT for Real Data**: Always use RFFT when working with real-valued signals for significant performance gains.

2. **Optimize FrFT**: The current FrFT implementation has both performance and numerical stability issues. Consider:
   - Implementing the Ozaktas-Kutay algorithm
   - Using SIMD optimizations for chirp calculations
   - Pre-computing chirp values when possible

3. **Review Memory-Efficient Implementation**: The current in-place FFT is slower than expected. Consider:
   - Profiling to identify bottlenecks
   - Optimizing memory access patterns
   - Reducing unnecessary copying

4. **Benchmark Against SciPy**: Need to run comparative benchmarks against SciPy's FFT implementation to understand relative performance.

## Next Steps

1. Profile the in-place FFT to understand performance bottlenecks
2. Implement optimized FrFT algorithms
3. Add SIMD optimizations where applicable
4. Create comprehensive benchmarks against SciPy
5. Add GPU acceleration for large transforms