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Fast Transforms

  • Leonid Yaroslavsky
Chapter

Abstract

Fast Fourier Transforms (FFT) are algorithms for fast computation of the DFT. The principle of the FFT can easily be understood if one compares 1-D and 2-D DFT. Let { a n (1) } and { a k,l (2) } be 1-D and 2-D arrays with the same number N = N 1 N 2 of samples: n = 0,1 N 1 N 2 − 1; k = 0,1 N 1 − 1; l = 0,1 N 2 − 1. Direct computing of the 1-D DFT of array { a n (1) }
$$ {\alpha _r} = \frac{1}{{\sqrt N }}\sum\limits_{n = 0}^{N - 1} {a_n^{\left( 1 \right)}\exp \left( {i2\pi \frac{{nr}}{N}} \right)} $$
(6.1.1)
requires N 2 = N 1 2 N 2 2 operations with complex numbers while computing the 2-D DFT of array { a k,l (2) }
$$ {\alpha _{r,s}} = \frac{1}{{\sqrt N }}\sum\limits_{k = 0}^{{N_1} - 1} {\sum\limits_{l = 0}^{{N_2} - 1} {a_{k,l}^{\left( 2 \right)}\exp \left[ {i2\pi \left( {\frac{{kr}}{{{N_1}}} + \frac{{ls}}{{{N_2}}}} \right)} \right]} } $$
(6.1.2)
requires only N 1 2 N 2 + N 1 N 2 2 = N 1 N 2 ( N 1 + N 2 ) operations as the 2-D DFT is separable into two 1-D DFTs:
$$ {\alpha _{r,s}} = \frac{1}{{\sqrt N }}\sum\limits_{k = 0}^{{N_1} - 1} {\exp \left[ {i2\pi \frac{{kr}}{{{N_1}}}} \right]} \sum\limits_{l = 0}^{{N_2} - 1} {{a_n}\exp \left[ {i2\pi \frac{{ls}}{{{N_2}}}} \right]} $$
(6.1.3)
Therefore one can accelerate computing the DFT by representing it in a separable multi-dimensional form. One can do it if the size of the array is a composite number. Let, as in the above example, N = N 1 N 2 . Represent indices k and r of signal and its transform samples as two-dimensional ones:
$$ k = {k_2}{N_1} + {k_1};{k_1} = 0,1,...,{N_1} - 1;{k_2} = 0,1,...,{N_2} - 1; $$
$$ r = {r_1}{N_2} + {r_2};{r_1} = 0,1,...,{N_1} - 1;{r_2} = 0,1,...,{N_2} - 1; $$
(6.1.4)
Then the 2-D DFT in Eq. 6.1.2 can be split into two successive 1-D DFTs:
$$ \begin{gathered} {\alpha _{{r_1},{r_2}}} = \frac{1}{{\sqrt {{N_1}{N_2}} }}\sum\limits_{{k_2} = 0}^{{N_2} - 1} {\sum\limits_{{k_1} = 0}^{{N_1} - 1} {{a_{{k_1},{k_2}}}\exp \left[ {i2\pi \frac{{\left( {{k_2}{N_1} + {k_1}} \right)\left( {{r_1}{N_2} + {r_2}} \right)}}{{{N_2}{N_1}}}} \right]} } = \hfill \\ \frac{1}{{\sqrt N }}\sum\limits_{{k_1} = 0}^{{N_1} - 1} {\exp \left( {i2\pi \frac{{{k_1}{r_1}}}{{{N_1}}}} \right)\left[ {\exp \left( {i2\pi \frac{{{k_1}{r_2}}}{N}} \right)\sum\limits_{{k_2} = 0}^{{N_2} - 1} {{a_{{k_1},{k_2}}}\exp \left( {i2\pi \frac{{{k_2}{r_2}}}{{{N_2}}}} \right)} } \right] = } \hfill \\ DF{T_{{N_1}}}\left\{ {\exp \left( {i2\pi \frac{{{k_1}{r_2}}}{N}} \right) \cdot DF{T_{{N_2}}}\left\{ {{a_{{k_1},{k_2}}}} \right\}} \right\}. \hfill \\ \end{gathered} $$
(6.1.5)
This computation scheme is illustrated in flow diagram of Fig. 6-1.

Keywords

Fast Fourier Transform Binary Code Quantization Level Gray Code Hadamard Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Leonid Yaroslavsky
    • 1
  1. 1.Tel Aviv UniversityIsrael

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