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Split-radix FFT Algorithm

Module by: Douglas L. Jones. E-mail the author

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The split-radix algorithm has the lowest M+A M A operations of any known power-of-two algorithm. Most FFT experts believe it to be optimum in this respect.

Figure 1
Motivation for S-R algorithm
radix-2radix-4
(a) (b)
radix-2 (image1.bmp)radix-4 (image2.bmp)
Figure 2: Alternatively, note that these two butterflies are equivalent
(a) (b)
Figure 2(a) (image3.bmp)Figure 2(b) (image4.bmp)

Split-radix algorith muses part radix-2 decomposition and part radix-4 decomposition.

DIT Split-radix derivation

Xk= n =0N21x2ne(i2π×(2n)kN)+ n =0N21x4n+1e(i2π(4n+1)kN)+ n =0N21x4n+3e(i2π(4n+3)kN)= DFT N 2 x2n+ W N k DFTx4n+1+ W N 3 k DFTx4n+3 X k n N 2 1 0 x 2 n 2 2 n k N n N 2 1 0 x 4 n 1 2 4 n 1 k N n N 2 1 0 x 4 n 3 2 4 n 3 k N DFT N 2 x 2 n W N k DFT x 4 n 1 W N 3 k DFT x 4 n 3
(1)
Figure 3
Butterfly
Butterfly (image5.bmp)
Figure 4: Transform has L-shaped butterflies
Figure 4 (image6.bmp)
Table 1: Operation Counts
Complex M/As Real M/As (4/2) Real M/As (3/3)
ON3log 2 N O N 3 2 N 43Nlog 2 N389N+6+291M 4 3 N 2 N 38 9 N 6 2 9 1 M Nlog 2 N3N+4 N 2 N 3 N 4
ONlog 2 N O N 2 N 83Nlog 2 N169N+2+291M 8 3 N 2 N 16 9 N 2 2 9 1 M 3Nlog 2 N3N+4 3 N 2 N 3 N 4

Comments

  • Split-radix algorithm has an irregular structure. Successful progams have been written (Sorensen) for uni-processor machines, but its not good for multi processors.
  • D. Braun's algorithm has split-radix counts N2 N 2 , has a regular structure, and is thus appealing for multiprocessors or special-purpose hardware. (Contact me if you ever need the references)

References

  1. H.V. Sorensen, M.T. Heideman, and C.S. Burrus. (1986). On computing the split-radix FFT. ASSP, 34(1), 152-156.
  2. P. Duhamel and H. Hollman. (1984, Jan 5). Split-radix FFT algorithms. Electronics Letters, 20, 14-16.

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