Vector-radix FFT algorithm: Difference between revisions

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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 [[DFT]]sDFTs until, ultimately, only trivial MD [[DFT]]sDFTs need to be evaluated.<ref name="Dudgeon83">{{cite book|last1=Dudgeon|first1=Dan|last2=Russell|first2=Mersereau|title=Multidimensional Digital Signal Processing|date=September 1983|publisher=Prentice Hall|isbn=0136049591|pages=76}}</ref><br />

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,<ref name="Rivard77">{{cite journal|last1=Rivard|first1=G.|title=Direct fast Fourier transform of bivariate functions|journal=IEEE Transactions on Acoustics, Speech, and Signal Processing|volume=25|pages=250–252|doi=10.1109/TASSP.1977.1162951|url=http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1162951&isnumber=26125}}</ref> 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,<ref name="Harris77">{{cite journal|last1=Harris|first1=D.|last2=McClellan|first2=J.|last3=Chan|first3=D.|last4=Schuessler|first4=H.|title=Vector radix fast Fourier transform|journal=IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP '77|volume=2|pages=548–551|doi=10.1109/ICASSP.1977.1170349|url=http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1170349&isnumber=26347}}</ref> which is the general Vector-Radix algorithm. <br />

'''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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== 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.<br />

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 [[DFT]]

:<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 a <math>N_1 \times N_2</math> matrix, and <math>W_N^{k n} = \exp(-j 2\pi /N)</math>.