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]

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

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 ${\displaystyle 0\leq k_{1},k_{2}<{\frac {N}{2}}}$ :

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

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]

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

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]