# 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.

## References

1. 1.
L. Yaroslaysky, M. Eden, Fundamentals of Digital Optics, Birkhauser, Boston, 1996Google Scholar
2. 2.
I. J. Good, J. Roy, Statist. Soc. Ser. B 20, 362, 1958Google Scholar
3. 3.
A. M. Trakhtman, V. A. Trakhtman, Fundamentals of the Theory of Discrete Signals Defined of Finite Intervals, Soy. Radio, Moscow, 1975, (In Russian)Google Scholar
4. 4.
N. Ahmed, K. R. Rao, Orthogonal Transforms for Digital Signal Processing, Springer Verlag, 1975Google Scholar
5. 5.
L. Yaroslaysky, Digital Picture Processing, An Introduction, Springer Verlag, Heidelberg, 1985
6. 6.
L. Yaroslayskii, N. Merzlyakov, Methods of Digital Holography, Plenum Press, N.Y., 1980Google Scholar
7. 7.
L. Yaroslaysky, A. Happonen, Y.Katiyi, Signal Discrete Signal Sinc–interpolation in DCT Domain: Fast Algorithms, in: Proceedings of International TISCP workshop on Spectral Methods and Multirate Signal Processing, SMMSP’02, Toulouse — France, 07.09.2002–08.09.2002, Ed. T Saramaki, K. Egiazarian J. Astola, p. 179–185, TICSP Series #17, ISBN 952–15–08881–7, ISSN 1456–2774, TTKK Monistamo 2002Google Scholar
8. 8.
Y. Katiyi, L. Yaroslaysky, V/HS Structure for Transforms and their Fast Algorithms, 3 Int. Symposium, Image and Signal Processing and Analysis, Sept. 18–20, 2003, Rome, ItalyGoogle Scholar
9. 9.
Y. Katiyi, L. Yaroslaysky, Regular Matrix Methods for Synthesis of Fast Transforms: General Pruned and Integer-to-Integer Transforms, in Proc. Of International Workshop on Spectral Methods and Multirate Signal Processing, ed. By T. Saramäki, K. Egiazarian J. Astola, SMMSP’2001, Pula, Croatia, June 16–18, 2001Google Scholar