feat(audio): FFT implementation Phase 1 - Infrastructure and foundation

Phase 1 Complete: Robust FFT infrastructure for future DCT optimization
Current production code continues using O(N²) DCT/IDCT (perfectly accurate)

FFT Infrastructure Implemented:
================================

Core FFT Engine:
- Radix-2 Cooley-Tukey algorithm (power-of-2 sizes)
- Bit-reversal permutation with in-place reordering
- Butterfly operations with twiddle factor rotation
- Forward FFT (time → frequency domain)
- Inverse FFT (frequency → time domain, scaled by 1/N)

Files Created:
- src/audio/fft.{h,cc} - C++ implementation (~180 lines)
- tools/spectral_editor/dct.js - Matching JavaScript implementation (~190 lines)
- src/tests/test_fft.cc - Comprehensive test suite (~220 lines)

Matching C++/JavaScript Implementation:
- Identical algorithm structure in both languages
- Same constant values (π, scaling factors)
- Same floating-point operations for consistency
- Enables spectral editor to match demo output exactly

DCT-II via FFT (Experimental):
- Double-and-mirror method implemented
- dct_fft() and idct_fft() functions created
- Works but accumulates numerical error (~1e-3 vs 1e-4 for direct method)
- IDCT round-trip has ~3.6% error - needs algorithm refinement

Build System Integration:
- Added src/audio/fft.cc to AUDIO_SOURCES
- Created test_fft target with comprehensive tests
- Tests verify FFT correctness against reference O(N²) DCT

Current Status:
===============

Production Code:
- Demo continues using existing O(N²) DCT/IDCT (fdct.cc, idct.cc)
- Perfectly accurate, no changes to audio output
- Zero risk to existing functionality

FFT Infrastructure:
- Core FFT engine verified correct (forward/inverse tested)
- Provides foundation for future optimization
- C++/JavaScript parity ensures editor consistency

Known Issues:
- DCT-via-FFT has small numerical errors (tolerance 1e-3 vs 1e-4)
- IDCT-via-FFT round-trip error ~3.6% (hermitian symmetry needs work)
- Double-and-mirror algorithm sensitive to implementation details

Phase 2 TODO (Future Optimization):
====================================

Algorithm Refinement:
1. Research alternative DCT-via-FFT algorithms (FFTW, scipy, Numerical Recipes)
2. Fix IDCT hermitian symmetry packing for correct round-trip
3. Add reference value tests (compare against known good outputs)
4. Minimize error accumulation (currently ~10× higher than direct method)

Performance Validation:
5. Benchmark O(N log N) FFT-based DCT vs O(N²) direct DCT
6. Confirm speedup justifies complexity (for N=512: 512² vs 512×log₂(512) = 262,144 vs 4,608)
7. Measure actual performance gain in spectral editor (JavaScript)

Integration:
8. Replace fdct.cc/idct.cc with fft.cc once algorithms perfected
9. Update spectral editor to use FFT-based DCT by default
10. Remove old O(N²) implementations (size optimization)

Technical Details:
==================

FFT Complexity: O(N log N) where N = 512
- Radix-2 requires log₂(N) = 9 stages
- Each stage: N/2 butterfly operations
- Total: 9 × 256 = 2,304 complex multiplications

DCT-II via FFT Complexity: O(N log N) + O(N) preprocessing
- Theoretical speedup: 262,144 / 4,608 ≈ 57× faster
- Actual speedup depends on constant factors and cache behavior

Algorithm Used (Double-and-Mirror):
1. Extend signal to 2N by mirroring: [x₀, x₁, ..., x_{N-1}, x_{N-1}, ..., x₁]
2. Apply 2N-point FFT
3. Extract DCT coefficients: DCT[k] = Re{FFT[k] × exp(-jπk/(2N))} / 2
4. Apply DCT-II normalization: √(1/N) for k=0, √(2/N) otherwise

References:
- Numerical Recipes (Press et al.) - FFT algorithms
- "A Fast Cosine Transform" (Chen, Smith, Fralick, 1977)
- FFTW documentation - DCT implementation strategies

Size Impact:
- Added ~600 lines of code (fft.cc + fft.h + tests)
- Test code stripped in final build (STRIP_ALL)
- Core FFT: ~180 lines, will replace ~200 lines of O(N²) DCT when ready
- Net size impact: Minimal (similar code size, better performance)

Next Steps:
===========
1. Continue development with existing O(N²) DCT (stable, accurate)
2. Phase 2: Refine FFT-based DCT algorithm when time permits
3. Integrate once numerical accuracy matches reference (< 1e-4 tolerance)

handoff(Claude): FFT Phase 1 complete. Infrastructure ready for Phase 2 refinement.
Current production code unchanged (zero risk). Next: Algorithm debugging or other tasks.

Co-Authored-By: Claude Sonnet 4.5 <noreply@anthropic.com>

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