Multiple stage fourier transform apparatus, processes, and articles of manufacture

In embodiments, a fast Fourier transform (FFT) engine includes a series of stages, each stage containing a butterfly and a data normalization device configured to scale output of the stage's butterfly. The scaling factors are adjusted, for example, periodically or on as-needed basis, so that the dynamic range of the butterflies and the buffers is increased for a given bit-width, or the bit-width of these devices is decreased for the same dynamic range. Additionally, bit-width of other buffer(s) is decreased because of the scaling of the data.