Vector-radix FFT algorithm

The vector-radix FFT algorithm, is a multidimensional fast Fourier transform (FFT) algorithm, which is a generalization of the ordinary Cooley–Tukey FFT algorithm that divides the transform dimensions by arbitrary radices. It breaks a multidimensional (MD) discrete Fourier transform (DFT) down into successively smaller MD DFTs until, ultimately, only trivial MD DFTs need to be evaluated.^[1]

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,^[2] 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,^[3] 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 ${\displaystyle N^{M}}$ element matrix (M dimensions, and size N on each dimension), the number of complex multiples of vector-radix FFT algorithm for radix-2 is ${\displaystyle {\frac {2^{M}-1}{2^{M}}}N^{M}\log _{2}N}$, meanwhile, for row-column algorithm, it is ${\displaystyle {\frac {MN^{M}}{2}}\log _{2}N}$. And generally, even larger savings in multiplies are obtained when this algorithm is operated on larger radices and on higher dimensional arrays.^[3]

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,^[4] and high speed FFT processor designing.^[5]

2-D DIT case

As with the Cooley–Tukey FFT algorithm, the two dimensional vector-radix FFT is derived by decomposing the regular 2-D DFT into sums of smaller DFT's multiplied by "twiddle" factors.

A decimation-in-time (DIT) algorithm means the decomposition is based on time domain ${\displaystyle x}$, see more in Cooley–Tukey FFT algorithm.

We suppose the 2-D DFT is defined

${\displaystyle X(k_{1},k_{2})=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}x[n_{1},n_{2}]\cdot W_{N_{1}}^{k_{1}n_{1}}W_{N_{2}}^{k_{2}n_{2}},}$

where ${\displaystyle k_{1}=0,\dots ,N_{1}-1}$,and ${\displaystyle k_{2}=0,\dots ,N_{2}-1}$, and ${\displaystyle x[n_{1},n_{2}]}$ is an ${\displaystyle N_{1}\times N_{2}}$ matrix, and ${\displaystyle W_{N}=\exp(-j2\pi /N)}$.

For simplicity, let us assume that ${\displaystyle N_{1}=N_{2}=N}$, and the radix-${\displaystyle (r\times r)}$ is such that ${\displaystyle N/r}$ is an integer.

Using the change of variables:

• ${\displaystyle n_{i}=rp_{i}+q_{i}}$, where ${\displaystyle p_{i}=0,\ldots ,(N/r)-1;q_{i}=0,\ldots ,r-1;}$
• ${\displaystyle k_{i}=u_{i}+v_{i}N/r}$, where ${\displaystyle u_{i}=0,\ldots ,(N/r)-1;v_{i}=0,\ldots ,r-1;}$

where ${\displaystyle i=1}$ or ${\displaystyle 2}$, then the two dimensional DFT can be written as:^[6]

${\displaystyle X(u_{1}+v_{1}N/r,u_{2}+v_{2}N/r)=\sum _{q_{1}=0}^{r-1}\sum _{q_{2}=0}^{r-1}\left[\sum _{p_{1}=0}^{N/r-1}\sum _{p_{2}=0}^{N/r-1}x[rp_{1}+q_{1},rp_{2}+q_{2}]W_{N/r}^{p_{1}u_{1}}W_{N/r}^{p_{2}u_{2}}\right]\cdot W_{N}^{q_{1}u_{1}+q_{2}u_{2}}W_{r}^{q_{1}v_{1}}W_{r}^{q_{2}v_{2}},}$

One stage "butterfly" for DIT vector-radix 2x2 FFT

The equation above defines the basic structure of the 2-D DIT radix-${\displaystyle (r\times r)}$ "butterfly". (See 1-D "butterfly" in Cooley–Tukey FFT algorithm)

When ${\displaystyle r=2}$, the equation can be broken into four summations, and this leads to:^[1]

${\displaystyle X(k_{1},k_{2})=S_{00}(k_{1},k_{2})+S_{01}(k_{1},k_{2})W_{N}^{k_{2}}+S_{10}(k_{1},k_{2})W_{N}^{k_{1}}+S_{11}(k_{1},k_{2})W_{N}^{k_{1}+k_{2}}}$ for ${\displaystyle 0\leq k_{1},k_{2}<{\frac {N}{2}}}$,

where ${\displaystyle S_{ij}(k_{1},k_{2})=\sum _{n_{1}=0}^{N/2-1}\sum _{n_{2}=0}^{N/2-1}x[2n_{1}+i,2n_{2}+j]\cdot W_{N/2}^{n_{1}k_{1}}W_{N/2}^{n_{2}k_{2}}}$.

The ${\displaystyle S_{ij}}$ can be viewed as the ${\displaystyle N/2}$-dimensional DFT, each over a subset of the original sample:

• ${\displaystyle S_{00}}$ is the DFT over those samples of ${\displaystyle x}$ for which both ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ are even;
• ${\displaystyle S_{01}}$ is the DFT over the samples for which ${\displaystyle n_{1}}$ is even and ${\displaystyle n_{2}}$ is odd;
• ${\displaystyle S_{10}}$ is the DFT over the samples for which ${\displaystyle n_{1}}$ is odd and ${\displaystyle n_{2}}$ is even;
• ${\displaystyle S_{11}}$ is the DFT over the samples for which both ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ are odd.

Thanks to the periodicity of the complex exponential, we can obtain the following additional identities, valid for ${\displaystyle 0\leq k_{1},k_{2}<{\frac {N}{2}}}$:

• ${\displaystyle X{\biggl (}k_{1}+{\frac {N}{2}},k_{2}{\biggr )}=S_{00}(k_{1},k_{2})+S_{01}(k_{1},k_{2})W_{N}^{k_{2}}-S_{10}(k_{1},k_{2})W_{N}^{k_{1}}-S_{11}(k_{1},k_{2})W_{N}^{k_{1}+k_{2}}}$;
• ${\displaystyle X{\biggl (}k_{1},k_{2}+{\frac {N}{2}}{\biggr )}=S_{00}(k_{1},k_{2})-S_{01}(k_{1},k_{2})W_{N}^{k_{2}}+S_{10}(k_{1},k_{2})W_{N}^{k_{1}}-S_{11}(k_{1},k_{2})W_{N}^{k_{1}+k_{2}}}$;
• ${\displaystyle X{\biggl (}k_{1}+{\frac {N}{2}},k_{2}+{\frac {N}{2}}{\biggr )}=S_{00}(k_{1},k_{2})-S_{01}(k_{1},k_{2})W_{N}^{k_{2}}-S_{10}(k_{1},k_{2})W_{N}^{k_{1}}+S_{11}(k_{1},k_{2})W_{N}^{k_{1}+k_{2}}}$.

2-D DIF case

Similarly, a decimation-in-frequency (DIF, also called the Sande–Tukey algorithm) algorithm means the decomposition is based on frequency domain ${\displaystyle X}$, see more in Cooley–Tukey FFT algorithm.

Using the change of variables:

• ${\displaystyle n_{i}=p_{i}+q_{i}N/r}$, where ${\displaystyle p_{i}=0,\ldots ,(N/r)-1;q_{i}=0,\ldots ,r-1;}$
• ${\displaystyle k_{i}=ru_{i}+v_{i}}$, where ${\displaystyle u_{i}=0,\ldots ,(N/r)-1;v_{i}=0,\ldots ,r-1;}$

where ${\displaystyle i=1}$ or ${\displaystyle 2}$, and the DFT equation can be written as:^[6]

${\displaystyle X(ru_{1}+v_{1},ru_{2}+v_{2})=\sum _{p_{1}=0}^{N/r-1}\sum _{p_{2}=0}^{N/r-1}\left[\sum _{q_{1}=0}^{r-1}\sum _{q_{2}=0}^{r-1}x[p_{1}+q_{1}N/r,p_{2}+q_{2}N/r]W_{r}^{q_{1}v_{1}}W_{r}^{q_{2}v_{2}}\right]\cdot W_{N}^{p_{1}v_{1}+p_{2}v_{2}}W_{N/r}^{p_{1}u_{1}}W_{N/r}^{p_{2}u_{2}},}$

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.^[6][7]

In conventional 2-D vector-radix algorithm, we decompose the indices ${\displaystyle k_{1},k_{2}}$ into 4 groups:

${\displaystyle {\begin{array}{lcl}X(2k_{1},2k_{2})&:&{\text{even-even}}\\X(2k_{1},2k_{2}+1)&:&{\text{even-odd}}\\X(2k_{1}+1,2k_{2})&:&{\text{odd-even}}\\X(2k_{1}+1,2k_{2}+1)&:&{\text{odd-odd}}\end{array}}}$

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:

${\displaystyle {\begin{array}{lcl}X(2k_{1},2k_{2})&:&{\text{even-even}}\\X(2k_{1},2k_{2}+1)&:&{\text{even-odd}}\\X(2k_{1}+1,2k_{2})&:&{\text{odd-even}}\\X(4k_{1}+1,4k_{2}+1)&:&{\text{odd-odd}}\\X(4k_{1}+1,4k_{2}+3)&:&{\text{odd-odd}}\\X(4k_{1}+3,4k_{2}+1)&:&{\text{odd-odd}}\\X(4k_{1}+3,4k_{2}+3)&:&{\text{odd-odd}}\end{array}}}$

That means the fourth term in 2-D DIT radix-${\displaystyle (2\times 2)}$ equation, ${\displaystyle S_{11}(k_{1},k_{2})W_{N}^{k_{1}+k_{2}}}$ becomes:^[8]

${\displaystyle A_{11}(k_{1},k_{2})W_{N}^{k_{1}+k_{2}}+A_{13}(k_{1},k_{2})W_{N}^{k_{1}+3k_{2}}+A_{31}(k_{1},k_{2})W_{N}^{3k_{1}+k_{2}}+A_{33}(k_{1},k_{2})W_{N}^{3(k_{1}+k_{2})},}$

where ${\displaystyle A_{ij}(k_{1},k_{2})=\sum _{n_{1}=0}^{N/4-1}\sum _{n_{2}=0}^{N/4-1}x[4n_{1}+i,4n_{2}+j]\cdot W_{N/4}^{n_{1}k_{1}}W_{N/4}^{n_{2}k_{2}}}$

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 ${\displaystyle 1024\times 1024}$ array, compared with the vector-radix algorithm.^[7]

References

1. Dudgeon, Dan; Russell, Mersereau (September 1983). Multidimensional Digital Signal Processing. Prentice Hall. p. 76. ISBN 0136049591.
2. Rivard, G. (1977). "Direct fast Fourier transform of bivariate functions". IEEE Transactions on Acoustics, Speech, and Signal Processing. 25 (3): 250–252. doi:10.1109/TASSP.1977.1162951.
3. Harris, D.; McClellan, J.; Chan, D.; Schuessler, H. (1977). "Vector radix fast Fourier transform". ICASSP '77. IEEE International Conference on Acoustics, Speech, and Signal Processing. Vol. 2. pp. 548–551. doi:10.1109/ICASSP.1977.1170349.
4. Buijs, H.; Pomerleau, A.; Fournier, M.; Tam, W. (Dec 1974). "Implementation of a fast Fourier transform (FFT) for image processing applications". IEEE Transactions on Acoustics, Speech, and Signal Processing. 22 (6): 420–424. doi:10.1109/TASSP.1974.1162620.
5. Badar, S.; Dandekar, D. (2015). "High speed FFT processor design using radix −⁴ pipelined architecture". 2015 International Conference on Industrial Instrumentation and Control (ICIC). pp. 1050–1055. doi:10.1109/IIC.2015.7150901. ISBN 978-1-4799-7165-7. S2CID 11093545.
6. Chan, S. C.; Ho, K. L. (1992). "Split vector-radix fast Fourier transform". IEEE Transactions on Signal Processing. 40 (8): 2029–2039. Bibcode:1992ITSP...40.2029C. doi:10.1109/78.150004.
7. Pei, Soo-Chang; Wu, Ja-Lin (April 1987). "Split vector radix 2D fast Fourier transform". ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing. Vol. 12. pp. 1987–1990. doi:10.1109/ICASSP.1987.1169345. S2CID 118173900.
8. Wu, H.; Paoloni, F. (Aug 1989). "On the two-dimensional vector split-radix FFT algorithm". IEEE Transactions on Acoustics, Speech, and Signal Processing. 37 (8): 1302–1304. doi:10.1109/29.31283.