0. Rader’s FFT algorithm acceleration using Maxeler
Author: Tadej Matek

1. Fourier Transform
Fourier transform decomposes a signal into its frequency components
Used in telecommunications, data compression, digital signal processing, fast multiplication of polynomials ...
Tadej Matek
Source:
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2. Fourier Transform and computers
Transformation: Discrete Fourier Transform
Time: O(n2)
Algorithm(s): Fast Fourier Transform (FFT)
(Cooley-Tukey, Bruun’s FFT, Rader’s FFT, Bluestein’s FFT …)
Time: O(nlogn)
Tadej Matek
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3. Why is FFT faster than DFT
Divide & conquer + properties of primitive roots
Primitive root of unity:
Conquer step (butterfly):
Source:
Tadej Matek
3/17

4. Rader’s FFT algorithm overview
Primitive root defined as:
Bit reversal revk(i):
rev4(3): 3(10) = 0011(2) → 1100(2) = 12(10)
Tadej Matek
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5. Example of calculation
n = k = log(n) = z = p = 13
8, 2, 2, 4
i = 0
s = revk(i) = 2
s = revk(i) = 0
i = 1
10, 6
6, 11
8+z0*2 % 13 = 10
8+z2*2 % 13 = 6
2+z0*4 % 13 = 6
2+z2*4 % 13 = 11
i = 0
i = 1
i = 2
i = 3
s = 0
s = 3
s = 2
s = 1 3 4 9 3
Tadej Matek
5/17

6. Example: fast multiplication
How to multiply two large polynomials?
Basic approach: multiply each component of 1st with each component of 2nd -> O(n2)
Using FFT: compute DFT transform of both polynomials, multiply in O(n) time and do inverse FFT -> O(nlogn)
Tadej Matek
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7. Dataflow implementation (1)
8, 2, 2, 4
Data
dependency!
10, 6
6, 11
Kernel needs updated data for each level!
Solution: LMem
7/17
Tadej Matek

8. Dataflow implementation (2)
Input sequence
Call kernel k times
CPU
(1)
(3)
(2)
...
Output sequence
Kernel
Manager
Manager streams data in and out of Kernel
LMem
Tadej Matek
8/17

9. Dataflow implementation (3)
LMem works in bursts (example: 384 B, but depends on DFE)
Good for consecutive calculations
zs are calculated on CPU and written to LMem
Tadej Matek
9/17

10. Performance & results (1)
CPU used for testing: Intel Core2 Quad Processor Q GHz
Maxeler card of type MAX2336B was used for DFE testing
Tadej Matek
10/17

11. Performance & results (2)
Conditions: BIG data, 95% run time in loops
Type of experiments: consecutive calculations starting from 10K and up to 10M
Consecutive calculations for input sequences of length 32, 64, 128 and 256
Tadej Matek
11/17

12. Performance & results (3)
Execution time, N = 32, for CPU and DFE
Tadej Matek
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13. Performance & results (4)
Speedup according to the number of consecutive calculations for N = 32
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Tadej Matek

14. Performance & results (5)
Speedup according to the number of consecutive calculations for N = 64
Tadej Matek
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15. Performance & results (6)
Speedup according to the number of consecutive calculations for N = 256
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Tadej Matek

16. Performance & results (7)
Speedup according to the size of input sequence (for 100K calculations)
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Tadej Matek

17. Conclusion FFTs are one of the most used algorithms today
There can be massive speedup but the requirement are consecutive calculations
Power usage: reduced due to lower frequency (200Mhz vs 2.86GHz)
Tadej Matek
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