Abstract
An efficient computation of a composite length discrete Fourier transform (DFT), as well as a fast Fourier transform (FFT) of both time and space data sequences in uncertain (non-sparse or sparse) computational scenarios, requires specific processing algorithms. Traditional algorithms typically employ some pruning methods without any commutations, which prevents them from attaining the potential computational efficiency. In this paper, we propose an alternative unified approach with automatic commutations between three computational modalities aimed at efficient computations of the pruned DFTs adapted for variable composite lengths of the non-sparse input-output data. The first modality is an implementation of the direct computation of a composite length DFT, the second one employs the second-order recursive filtering method, and the third one performs the new pruned decomposed transform. The pruned decomposed transform algorithm performs the decimation in time or space (DIT) data acquisition domain and, then, decimation in frequency (DIF). The unified combination of these three algorithms is addressed as the DFTCOMM technique. Based on the treatment of the combinational-type hypotheses testing optimization problem of preferable allocations between all feasible commuting-pruning modalities, we have found the global optimal solution to the pruning problem that always requires a fewer or, at most, the same number of arithmetic operations than other feasible modalities. The DFTCOMM method outperforms the existing competing pruning techniques in the sense of attainable savings in the number of required arithmetic operations. It requires fewer or at most the same number of arithmetic operations for its execution than any other of the competing pruning methods reported in the literature. Finally, we provide the comparison of the DFTCOMM with the recently developed sparse fast Fourier transform (SFFT) algorithmic family. We feature that, in the sensing scenarios with sparse/non-sparse data Fourier spectrum, the DFTCOMM technique manifests robustness against such model uncertainties in the sense of insensitivity for sparsity/non-sparsity restrictions and the variability of the operating parameters.
Highlights
1.1 Motivation Many signal processing applications require computation of the so-called pruned discrete Fourier transform (DFT), i.e., an efficient alternative to compute the required DFT when the input sequence and/or the required output sequences are smaller than the length of the full DFT; in the literature those are referred to as pruned fast Fourier transforms (FFTs) or pruned DFTs [1]
DFT means that all the output components are to be computed, and all the input elements are used to compute the transform); in the literature those are referred to as pruned fast Fourier transforms (FFTs) or pruned DFTs [1]
It is considered that the DFTs of length P required by the intermediate stage of the pruned decomposed transform have been implemented by applying the split-radix FFT, e.g., [26]
Summary
1.1 Motivation Many signal processing applications require computation of the so-called pruned discrete Fourier transform (DFT), i.e., an efficient alternative to compute the required DFT when the input sequence and/or the required output sequences are smaller than the length of the full DFT We demonstrate that both decomposed transforms (DFTDIF−DIT−Pr and DFTDIT−DIF−Pr) can be obtained from a general decomposition methodology It manifests the robustness in sparse and non-sparse sensing scenarios (i.e., operability for an arbitrary number of consecutive input elements (Li), the number of consecutive outputs that should be computed (Lo), and the length of the full transform (N)) in contrast to the recently developed most prominent SFFT family-related methods [20, 21] operable in sparse scenarios only. The output stage of both pruned decomposed transform modalities can be computed by the direct addition of complex multiplications or a kind of recursive algorithm as those proposed in [23] (referred to as the 2BF filtering method), which reduces the number of required multiplications by about half. The decomposed transform algorithm always selects a pair (Dip, Dop) for which the largest DFTs could be successfully pruned, or equivalently, a pair (Dip, Dop) for which the intermediate stage results in the smallest dimension DFTs
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