Is F Better than D

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Is F Better than D

• David Hansen and James Michelussi

2
Introduction

• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• FFT Algorithm Applying the Mathematics
• Implementations of DFT and FFT
• Hardware Benchmarks
• Conclusion

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DFT

• In 1807 introduced by Jean Baptiste Joseph
  Fourier.
• allows a sampled or discrete signal that is
  periodic to be transformed from the time domain
  to the frequency domain
• Correlation between the time domain signal and N
  cosine and N sine waves

X(k) DFT Frequency Signal N Number of Sample
Points X(n) Time Domain Signal WN Twiddle
Factor
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DFT (Walking Speed)

• Why is this important? Where is this used?
• allows machines to calculate the frequency domain
• allows for the convolution of signals by just
  multiplying them together
• Used in digital spectral analysis for speech,
  imaging and pattern recognition as well as signal
  manipulation using filters
• But the DFT requires N2 multiplications!

5
FFT (Jet Speed)

• J. W. Cooley and J. W. Tukey are given credit for
  bringing the FFT to the world in the 1960s
• Simply an algorithm for more efficiently
  calculating the DFT
• Takes advantage of symmetry and periodicity in
  the twiddle factors as well as uses a divide and
  conquer method
• Symmetry WNr N/2 -WNr
• Periodicity WNrN WNr
• Requires only (N/2)log2(N) multiplications !
• Faster computation times
• More precise results due to less round-off error

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FFT Algorithm

• Several different types of FFT Algorithms
  (Radix-2, Radix-4, DIT DIF)
• Focus on Radix-2 using Decimation in Time (DIT)
  method
• Breaks down the DFT calculation into a number of
  2-point DFTs
• Each 2-point DFT uses an operation called the
  Butterfly
• These groups are then re-combined with another
  group of two and so on for log2(N) stages
• Using the DIT method the input time domain points
  must be reordered using bit reversal

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Butterfly Operation
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Bit Reversal
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8-Point Radix-2 FFT Example
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8-Point Radix-2 FFT Example
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Implementations of DFT and FFT

• David Hansen

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DFT Implementation
for (r0 rltsamples/2 r) float re 0.0f,
im 0.0f float part (float)r -2.0f PI /
(float)samples for (k0 kltsamples
k) float theta part (float)k re
data_ink cos(theta) im data_ink
sin(theta)

• Nested For Loop, (N/2)N Iterations O(N2)
• 63027.41 Cycles / Sample (123 cycles per inner
  loop iteration)
• Obvious Inefficiencies, cos and sin math.h
  functions
• Efficient assembly coding could reduce the inner
  loop to 3 cycles per iteration (1,536 cycles /
  sample)

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C FFT Implementation
void fft_float (unsigned NumSamples, float
RealIn, float ImagIn, float RealOut,
float ImagOut ) for ( i0 i lt NumSamples
i ) // Iterate over the samples and
perform the bit-reversal j ReverseBits
( i, NumBits ) BlockEnd 1 //
Following loop iterates Log2(NumSamples) for
( BlockSize 2 BlockSize lt NumSamples
BlockSize ltlt 1 ) // Perform Angle
Calculations (Using math.h sin/cos) //
Following 2 loops iterate over NumSamples/2
for ( i0 i lt NumSamples i BlockSize )
for ( ji, n0 n lt BlockEnd
j, n ) // Perform
butterfly calculations
BlockEnd BlockSize
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C FFT Implementation

• Bit-Reverse For Loop N iterations
• Nested For Loops
• First Outer Loop Log2(N) iterations
• Made use of sin/cos math.h functions
• Second Outer Loop N / BlockSize iterations
• Inner Loop BlockSize/2 iterations
• O(N Log2(N) N/BlockSize BlockSize/2)
• O(NNLog2(N))
• 193.84 Cycles / Sample

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Assembly FFT Implementation

• Bit-Reverse Address Generation
• Hide Bit-Reverse operation inside first and
  second FFT Stages
• Sin and Cos values stored in a Look-Up-Table
• 256 Kbyte LUT added to Data1
• Needed to grow Data1 Memory Space using LDF file
• Interleaved Real and Imaginary Arrays
• Quad Reads Loads 2 Complex Points per Cycle
• Supports the Real FFT for input signals with no
  Imaginary component
• 40 Algorithm-based Savings

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Assembly FFT Implementation

• Special Butterfly Instruction
• Can perform addition/subtraction in parallel in
  one compute block
• Speeds up the inner-most loop
• VLIW and SIMD Operations
• Performs simultaneous operations in both compute
  blocks
• Loop unrolling and instruction scheduling keeps
  the entire processor busy with instructions.
• 11.35 Cycles per Sample

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Assembly FFT Implementation
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DC FFT Test

• FFT Source Array

• FFT Output Magnitude

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Audio FFT Test

• FFT Source Array

• FFT Output Magnitude

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1024 Point DFT / FFT Comparison
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1024 Point Radix-2 FFT Hardware Comparison
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Conclusion

• The FFT algorithm is very useful when computing
  the frequency domain on a DSP.
• FFT is much faster than a regular DFT algorithm
• FFT is more precise by having less errors
  created due to round off.
• The timed coding examples further support this
  claim and demonstrate how to code the algorithm.
  
• The Radix-2 FFT isnt the fastest but it uses a
  less complex addressing and twiddle factor
  routine
• In this case (unlike in school) F is better then
  D.