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{{Short description|Multidimensional fast Fourier transform 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
'''Vector-radix FFT algorithm''' can reduce the number of complex multiplications significantly, compared to row-vector algorithm. For example, for a <math>N^M</math> element matrix (M dimensions, and size N on each dimension), the number of complex multiples of vector-radix FFT algorithm for radix-2 is <math>\frac{2^M -1}{2^M} N^M \log_2 N</math>, meanwhile, for row-column algorithm, it is <math>\frac{M N^M} 2 \log_2 N</math>. And generally, even larger savings in multiplies are obtained when this algorithm is operated on larger radices and on higher dimensional arrays.<ref name=Harris77/><br />▼
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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,<ref name="Buijs74">{{cite journal|last1=Buijs|first1=H.|last2=Pomerleau|first2=A.|last3=Fournier|first3=M.|last4=Tam|first4=W.|title=Implementation of a fast Fourier transform (FFT) for image processing applications|journal=IEEE Transactions on Acoustics, Speech, and Signal Processing|date=Dec 1974|volume=22|issue=6|pages=420–424|doi=10.1109/TASSP.1974.1162620}}</ref> and high speed FFT processor designing.<ref name="Badar15">{{cite book|last1=Badar|first1=S.|last2=Dandekar|first2=D.|title=2015 International Conference on Industrial Instrumentation and Control (ICIC) |chapter=High speed FFT processor design using radix −<sup>4</sup> pipelined architecture |pages=1050–1055|doi=10.1109/IIC.2015.7150901|year=2015|isbn=978-1-4799-7165-7|s2cid=11093545 }}</ref>
== 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
A decimation-in-time ('''DIT''') algorithm means the decomposition is based on time ___domain <math>x</math>, see more in [[Cooley–Tukey FFT algorithm]].
We suppose the 2-D
:<math>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}, </math>
where <math>k_1 = 0,\dots,N_1-1</math>, and <math>k_2 = 0,\dots,N_2-1</math>, and <math>x[n_1, n_2]</math> is
For simplicity, let us assume that <math>N_1=N_2=N</math>, and the radix-<math>(r\times r)</math>
Using the change of variables:
* <math>n_i=rp_i+q_i</math>, where <math>p_i=0,\ldots,(N/r)-1; q_i = 0,\ldots,r-1;</math>
* <math>k_i=u_i+v_i N/r</math>, where <math>u_i=0,\ldots,(N/r)-1; v_i = 0,\ldots,r-1;</math>
where <math>i = 1</math> or <math>2</math>, then the two dimensional DFT can be written as:<ref name="Chan92">{{cite journal|last1=Chan|first1=S. C.|last2=Ho|first2=K. L.|title=Split vector-radix fast Fourier transform|journal=IEEE Transactions on Signal Processing|volume=40|issue=8|pages=2029–2039|doi=10.1109/78.150004|
:<math> 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,
[[File:
The equation above defines the basic structure of the 2-D DIT radix-<math>(r\times r)</math> "butterfly". (See 1-D "butterfly" in [[Cooley–Tukey FFT algorithm]])
When <math>r=2</math>, the equation can be broken into four summations
:<math> 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}</math> for <math>0\leq k_1, k_2 < \frac{N}{2}</math>,
where <math>S_{ij}(k_1,k_2)=\sum_{n_1=0}^{N/2-1} \sum_{n_2=0}^{N/2-1} x[2 n_1 + i, 2 n_2 + j] \cdot W_{N/2}^{n_1 k_1} W_{N/2}^{n_2 k_2}
The <math>S_{ij}</math> can be viewed as the <math>N/2</math>-dimensional DFT, each over a subset of the original sample:
* <math>S_{00}</math> is the DFT over those samples of <math>x</math> for which both <math>n_1</math> and <math>n_2</math> are even;
* <math>S_{01}</math> is the DFT over the samples for which <math>n_1</math> is even and <math>n_2</math> is odd;
* <math>S_{10}</math> is the DFT over the samples for which <math>n_1</math> is odd and <math>n_2</math> is even;
* <math>S_{11}</math> is the DFT over the samples for which both <math>n_1</math> and <math>n_2</math> are odd.
Thanks to the [[List of trigonometric identities#Shifts and periodicity|periodicity of the complex exponential]], we can obtain the following additional identities, valid for <math>0\leq k_1, k_2 < \frac{N}{2}</math>:
* <math>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}</math>;
* <math>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}</math>;
* <math>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}</math>.
== 2-D DIF case ==
Similarly, a decimation-in-frequency ('''DIF''', also called the
Using the change of variables:
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* <math>k_i=r u_i+v_i</math>, where <math>u_i=0,\ldots,(N/r)-1; v_i = 0,\ldots,r-1;</math>
where <math>i = 1</math> or <math>2</math>, and the DFT equation can be written as:<ref name=Chan92/>
:<math> X(r u_1+v_1,r u_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,
== Other approaches ==
The [[
In conventional 2-D vector-radix algorithm, we decompose the indices <math>k_1,k_2</math> into 4 groups:
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</math>
That means the fourth term in 2-D DIT radix-<math>(2\times 2)</math> equation, <math>S_{11}(k_1,k_2) W_{N}^{k_1+k_2}</math> becomes:<ref name="Wu89">{{cite journal|last1=Wu|first1=H.|last2=Paoloni|first2=F.|title=On the two-dimensional vector split-radix FFT algorithm|journal=IEEE Transactions on Acoustics, Speech, and Signal Processing|date=Aug 1989|volume=37|issue=8|pages=1302–1304|doi=10.1109/29.31283
: <math> A_{11}(k_1,k_2) W_N^{k_1+k_2} + A_{13}(k_1,k_2) W_N^{k_1+3 k_2} +A_{31}(k_1,k_2) W_N^{3 k_1+k_2} + A_{33}(k_1,k_2) W_N^{3(k_1+k_2)},</math>
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where <math>A_{ij}(k_1,k_2)=\sum_{n_1=0}^{N/4-1} \sum_{n_2=0}^{N/4-1} x[4 n_1 + i, 4 n_2 + j] \cdot W_{N/4}^{n_1 k_1} W_{N/4}^{n_2 k_2}</math>
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 <math>1024\times 1024</math> array, compared with the vector-radix algorithm.<ref name=Pei87/>
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