Unified commutation-pruning technique for efficient computation of composite DFTs

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 DFT COMM 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 DFT COMM 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 DFT COMM 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 DFT COMM technique manifests robustness against such model uncertainties in the sense of insensitivity for sparsity/non-sparsity restrictions and the variability of the operating parameters.

Optimising Linear Key Recovery Attacks with Affine Walsh Transform Pruning

“…The overhead computations were reduced by Singh and Srinivasan [29]. Pruning has been recently generalised to mixed-radix and composite length DFTs in works such as [33,14].…”

The JM-Filter to Detect Specific Frequency in Monitored Signal

“…Goertzel and JM-Filter are dedicated to compute an arbitrary specific frequency (a single frequency) and not a subset of consecutive K frequencies as usually performed by input/output pruning FFT methods [22][23] [24]. According to [22] and [23] the only frequency that could be detected or monitored is the first one X (0) (k=0) and that is why input/output pruning FFTs will be excluded from our performance comparison study due to the increasing complexity associated with the computation of the desired k th frequency that is obtained by computing the first k outputs.…”

“…The Table I summarize the complexity operations for the proposed JM-Filter radices-2, 4, 8 and r, Goertzel algorithm and the previously published RDFT algorithms [23]- [27].…”

“…Input/output pruning FFTs are efficient fast Fourier transform (FFT), where the efficiency can be increased by removing operations on consecutive input values which are zero [22] [23]. Furthermore, output pruning FFT is a method used to compute a discrete Fourier transform (DFT) where only a subset of the consecutive outputs is needed.…”