In this paper, a new radix-2 2-D decimation-in-time discrete Fourier transform algorithm is presented. This algorithm uses an approach employing vectorized data to reduce the structural overhead of the algorithm compared with the Cooley-Tukey radix-2 2-D algorithm. Computational complexity and run-time comparison of the algorithm are also provided.

Since the invention of fast algorithms for the computation of the discrete Fourier transform (DFT) by Cooley and Tukey (1965), the DFT has been widely used for the frequency-domain analysis and design of signals and systems in communications, digital signal processing, and in many other areas of science and engineering. While the Cooley-Tukey algorithms are simple, regular, and efficient, they have the drawback of requiring a significant amount of overhead operations such as bit-reversal, data-swapping, etc. In this paper, we develop a generalized version of a new family of DFT algorithms by decomposing a form of the DFT relation in which the input data and transform quantities are represented as vectors. These algorithms have the features that eliminate or reduce the drawbacks of the Cooley-Tukey algorithms while improving the simplicity and regularity of their implementations. The generalized version makes it easier to deduce a large family of algorithms with different features. The relative merits of the various algorithms with different vector lengths are discussed and the optimum vector length for DFT computation is pointed out.

Fast computation of the two-dimensional discrete Fourier transform

Vector computation of the discrete Fourier transform

http://ieeexplore.ieee.org/iel3/4494/12742/00588018.pdf

The problem of comparing different algorithms for the execution of the fast Fourier transform (FFT) is considered. Instead of counting the required arithmetic operations, the necessary number of instruction cycles for an FFT implementation on different digital signal processors (DSPs) is used as a measure. It turns out that this more practical figure of merit yields a rather different valuation of the algorithms. Furthermore, a method to halve the table size for the radix-2 twiddle factors is described. Some new FFT programs for execution on DSPs are compared with programs provided by the manufacturers. >

FFT implementation on DSP-chips-theory and practice

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. >

Fast computation of the two-dimensional discrete Fourier transform

Citations

Vector computation of the discrete Fourier transform

References

FFT implementation on DSP-chips-theory and practice

Equivalent relationship and unified indexing of FFT algorithm

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