Nessun titolo diapositiva

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A Fast Fourier Transform Compiler
Silvio D Carnevali
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Contents

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

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genfft

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

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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)

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Performance of FFTW
Powers of 2
Any powers of 2, 3, 5, 7
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genfft creation of the codelets DAG

• Nodes data types
• ? Encode arithmetic expressions
• ? Use real numbers for C compatibility
• Generic node operator
• Children operands
• DAG Algorithm depends on input size

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DAG creation Algorithms
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FT Equation

• X input vector
• Y FT of X
• wn nth root of unity

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genfft DAG Simplifier

• Bottom-up traversal of DAG
• local improvements
• ? Algebraic transformations (constant folding,
  / simplification)
• ? CSE eliminate existing create new ones
• ? DFT-specific improvements

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Algebraic transformations

• Simplifies multiplication by 1, 0 or -1
• Simplifies addition by 0
• Distribution kx ky k(x y)

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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

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DFT-Specific improvements
Simplify
DAG E
DAG D
Transpose
Simplify
DAG FT
DAG ET
Transpose
Simplify
DAG F
DAG E
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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

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genfft Unparsing

• Schedule unparsed to C
• Pipeline usage managed by C compiler
• genfft C compiler performance problems
• ? egcs optimizer

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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)