The vector-radix FFT algorithm, is a multidimensional fast Fourier transform algorithm, which is a generalization of the ordinary Cooley–Tukey FFT algorithm that divides the transform dimensions by arbitrary radices. It breaks a multidimensional discrete Fourier transform down into successively smaller MD DFTs until, ultimately, only trivial MD DFTs need to be evaluated. The most common multidimensional FFT algorithm is the row-column algorithm, which means transforming the array first in one index and then in the other, see more in FFT. Then a radix-2 direct 2-D FFT has been developed, and it can eliminate 25% of the multiplies as compared to the conventional row-column approach. And this algorithm has been extended to rectangular arrays and arbitrary radices, which is the general vector-radix algorithm. Vector-radix FFT algorithm can reduce the number of complex multiplications significantly, compared to row-vector algorithm. For example, for a element matrix, the number of complex multiples of vector-radix FFT algorithm for radix-2 is, meanwhile, for row-column algorithm, it is. And generally, even larger savings in multiplies are obtained when this algorithm is operated on larger radices and on higher dimensional arrays. Overall, the vector-radix algorithm significantly reduces the structural complexity of the traditional DFT having a better indexing scheme, at the expense of a slight increase in arithmetic operations. So this algorithm is widely used for many applications in engineering, science, and mathematics, for example, implementations in image processing, and high speed FFT processor designing.
2-D DIT case
As with Cooley–Tukey FFT algorithm, two dimensional vector-radix FFT is derived by decomposing the regular 2-D DFT into sums of smaller DFT's multiplied by "twiddle" factor. A decimation-in-time algorithm means the decomposition is based on time domain, see more in Cooley–Tukey FFT algorithm. We suppose the 2-D DFT where,and, and is a matrix, and. For simplicity, let us assume that, and radix-. Using the change of variables:
, where
, where
where or, then the two dimensional DFT can be written as: The equation above defines the basic structure of the 2-D DIT radix- "butterfly". When, the equation can be broken into four summations: one over those samples of x for which both and are even, one for which is even and is odd, one of which is odd and is even, and one for which both and are odd, and this leads to: where
2-D DIF case
Similarly, a decimation-in-frequency algorithm means the decomposition is based on frequency domain, see more in Cooley–Tukey FFT algorithm. Using the change of variables:
, where
, where
where or, and the DFT equation can be written as:
Other approaches
The split-radix FFT algorithm has been proved to be a useful method for 1-D DFT. And this method has been applied to the vector-radix FFT to obtain a split vector-radix FFT. In conventional 2-D vector-radix algorithm, we decompose the indices into 4 groups: By the split vector-radix algorithm, the first three groups remain unchanged, the fourth odd-odd group is further decomposed into another four sub-groups, and seven groups in total: That means the fourth term in 2-D DIT radix- equation, becomes: where The 2-D N by N DFT is then obtained by successive use of the above decomposition, up to the last stage. It has been shown that the split vector radix algorithm has saved about 30% of the complex multiplications and about the same number of the complex additions for typical array, compared with the vector-radix algorithm.
