Equivalent relationship and unified indexing of FFT algorithm

The twiddle factor matrix of the discrete Fourier transform can be recursively factorized into the cascading of the basic butterfly stage matrices. It is shown that the matrix can be further partitioned into three matrices practically specifying the input data, twiddle factor, and output data sequences of the fast Fourier transform (FFT). The equivalent relationship of these matrices is introduced. Thus, the equivalent relationship for a variety of the FFT algorithms can be obtained by equivalent matrix transformation. It is shown that the multidimensional (M-D) FFT can be represented by the same vector-matrix form as the 1-D FFT. The addressing sequences of the M-D FFT is the subset of the 1-D FFT. Therefore, the signal flow graph of the 1-D FFT can be used to describe that of the M-D FFT and the 1-D FFT addressing sequences can be employed to implement the M-D FFT. The simulation results of the proposed FFT approach are given