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A Fast Fourier Transform Compiler

A Fast Fourier Transform Compiler. Silvio D Carnevali. Contents. FFTW and genfft : an introduction genfft: How it works 1.) DAG Creation 2.) Simplifier 3.) Scheduler 4.) Unparsing Conclusion: similar applications. genfft. special purpose compiler objective Camelot

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A Fast Fourier Transform Compiler

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  1. A Fast Fourier Transform Compiler Silvio D Carnevali

  2. Contents • FFTW and genfft: an introduction • genfft: How it works 1.) DAG Creation 2.) Simplifier 3.) Scheduler 4.) Unparsing • Conclusion:similarapplications

  3. genfft • special purpose compiler • objective Camelot • produces DFT subroutines • Outputs C code • parameterized according to: - Input length - Data type

  4. FFTW • Collection of “Codelets” • Codelets: fragments of C code • Generated by genfft • plan: optimal composition of codelets  depends on input size and HW  automatically selected by FFTW (FJ98)

  5. Performance of FFTW Powers of 2 Any powers of 2, 3, 5, 7

  6. genfft: creation of the codelet’s DAG • Nodes: data types  Encode arithmetic expressions  Use real numbers for C compatibility • Generic node = operator • Children = operands • DAG Algorithm depends on input size

  7. DAG creation Algorithms

  8. FT Equation • X = input vector • Y = FT of X • wn = nth root of unity

  9. genfft: DAG Simplifier • Bottom-up traversal of DAG • local improvements:  Algebraic transformations (constant folding, +/* simplification)  CSE: eliminate existing + create new ones  DFT-specific improvements

  10. Algebraic transformations • Simplifies multiplication by 1, 0 or -1 • Simplifies addition by 0 • Distribution: kx + ky = k(x + y)

  11. DFT-Specific improvements • Numeric constants made positive (Local)  Constants: generally k and -k  Reduces number of loads • DAG transposition (for Linear Function)  Simplifies DAG, transpose + simplify, transpose + simplify  Reduces number of multiplications only

  12. 5 X A 2 3 Y B 4 5 X A 2 3 Y B 4 5 X A 2 3 Y B 4 DFT-Specific improvements Simplify DAG E DAG D Transpose Simplify DAG FT DAG ET Transpose Simplify DAG F DAG E

  13. genfft: DAG Scheduler • Goal: minimize use of regs • No instruction scheduling • Partitions DAG in 2 recursively  register mapping  Optimal for n = 2k  Partitioning heuristics • Optimality? Not for n != 2k

  14. genfft: Unparsing • Schedule unparsed to C • Pipeline usage managed by C compiler • genfft + C compiler: performance problems  egcs “optimizer”

  15. Conclusion & future work • FFTW: The best of the best of the best… • Over 100 downloads every week! • genfft: specialized for linear functions  Crystallographic FT  FIR & IIR filters  Image processing (JPEG discrete cosine transform)

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