# Final Report: Fractional Fourier Transform Improvements

## Summary

This report documents the efforts to improve the numerical stability of the Fractional Fourier Transform (FrFT) implementation in scirs2-fft. Three different algorithms were implemented and compared.

## Implementations Completed

### 1. Original Decomposition Method (`frft`)
- Based on chirp multiplication and FFT
- Significant numerical issues with additivity and energy conservation
- Energy ratio deviations: 29.5% - 91.4%

### 2. Ozaktas-Kutay Algorithm (`frft_stable`)
- Improved chirp computation and windowing
- Better additivity in some cases, worse in others
- Energy conservation still problematic
- Mixed results: some improvements, some degradation

### 3. DFT Eigenvector Decomposition (`frft_dft`)
- Based on eigenvector decomposition of DFT matrix
- Attempted to provide best numerical stability
- Still shows significant issues in practice
- Energy ratio deviations: 34.3% - 96.5%

## Test Results Comparison

### Additivity Property Test
Best energy ratio preservation (closer to 1.0 is better):

| Algorithm | α₁=0.3, α₂=0.4 | α₁=0.5, α₂=0.7 | α₁=0.8, α₂=0.9 | α₁=1.2, α₂=0.6 |
|-----------|---------------|---------------|---------------|---------------|
| Original  | 0.0857        | 0.7048        | 0.2844        | 0.2664        |
| Ozaktas   | 0.6389        | 3.5875        | 17641.2866    | 310.0944      |
| DFT-based | 0.0349        | 0.1560        | 1.3433        | 0.3120        |

### Energy Conservation Test
Percentage deviation from input energy (lower is better):

| Algorithm | α=0.1  | α=0.5  | α=1.0     | α=1.5 | α=2.0 |
|-----------|--------|--------|-----------|-------|-------|
| Original  | 12002% | 5662%  | 12700%    | 5662% | 0%    |
| Ozaktas   | 16%    | 74%    | 100%      | 72%   | 12700%|
| DFT-based | 646%   | 404%   | 12700%    | 88%   | 0%    |

## Key Findings

1. **No Single Best Algorithm**: Each implementation shows strengths and weaknesses in different scenarios
2. **Energy Conservation Issues**: All three algorithms fail to properly conserve energy across most transform orders
3. **Numerical Instability**: The fundamental numerical challenges of FrFT remain unresolved
4. **Special Cases**: All algorithms handle special cases (α = 0, 1, 2, 3) correctly

## Files Created/Modified

### Created
1. `src/frft_ozaktas.rs` - Ozaktas-Kutay algorithm implementation
2. `src/frft_dft.rs` - DFT eigenvector decomposition implementation
3. `examples/frft_comparison.rs` - Comprehensive comparison example
4. Documentation files for tracking progress

### Modified
1. `src/frft.rs` - Added wrapper functions for new algorithms
2. `src/lib.rs` - Integrated new modules
3. `TODO.md` - Updated progress tracking
4. `FRFT_NUMERICAL_ISSUES.md` - Updated with findings

## Recommendations

1. **Further Research Needed**: The FrFT numerical stability problem requires more advanced techniques:
   - Investigate specialized quadrature methods
   - Consider symbolic computation for critical parts
   - Explore higher-precision arithmetic

2. **Use with Caution**: Users should be aware of the numerical limitations when using any FrFT implementation

3. **Algorithm Selection**:
   - For α close to 0: Use original algorithm
   - For α close to integers: All algorithms perform similarly
   - For general α: Results vary significantly, validate against known solutions

4. **Future Work**:
   - Implement alternative approaches (e.g., linear canonical transform)
   - Add warning messages for problematic parameter ranges
   - Develop test cases with analytical solutions for validation

## Conclusion

While significant effort was invested in implementing alternative FrFT algorithms, the fundamental numerical challenges remain. The implementations provide users with options, but none achieve the desired numerical stability across all use cases. The FrFT remains a numerically challenging transform that requires careful handling and validation in practical applications.