The Winograd Fourier transform algorithm (WFTA) uses a very powerful property of the Type-1 index map and the DFT to give a further reduction of the number of multiplications in the PFA. Using an operator notation where $$F_1$$ represents taking row DFT's and $$F_2$$ represents column DFT's, the two-factor PFA of the equation is represented by
$X=F_2F_1x$
It has been shown that if each operator represents identical operations on each row or column, they commute. Since $$F_1$$ and $$F_2$$ represent length $$N_1$$ and $$N_2$$ DFT's, they commute and the equation can also be written