Radix-4 FFT algorithms with ordered input and output data

In this work we derive two families of radix-4 factorizations for the FFT (fast Fourier transform) that have the property that both inputs and outputs are addressed in natural order. These factorizations are obtained from another two families of radix-2 algorithms that have the same property. The radix-4 algorithms obtained have the same mathematical complexity (number of multiplications and additions) that Cooley-Tukey radix-4 algorithms but avoid de bit-reversal ordering applied to the input or at the output.