DOI: 10.14489/vkit.2014.12.pp.003-009

Самойленко М. В.
ТОМОГРАФИЧЕСКИЙ МЕТОД ДИСКРЕТНОГО ПРЕОБРАЗОВАНИЯ ФУРЬЕ
(с. 3-9)

Аннотация. Предложен новый метод, позволяющий осуществлять преобразование Фурье по «разреженной» дискретной выборке временнóй функции и имеющий преимущества по сравнению с классическими методами при анализе сигналов с ограниченным линейчатым спектром. По результатам компьютерных экспериментов проведен сравнительный анализ томографического метода, классического дискретного и быстрого преобразований Фурье.

Ключевые слова: преобразование Фурье; спектр; томографический подход; восстановление.

Samoilenko M. V.
TOMOGRAPHY METHOD OF DISCRETE FOURIER TRANSFORM
(pp. 3-9)

Abstract. The article presents a new method of spectrum analysis, which restore the spectrum by the temporal signal sampling. The well known method of Discrete Fourier Transform (DFT) and its fast implementation called Fast Fourier Transform (FFT) also solve this problem, but the proposed tomography method is principally different and has certain advantages. Tomography method of DFT is based on the spec-trum restoration as on the restoration of the integrand in the inverse Fourier transform by the set of values of its integral. The temporal signal sampling is just the set of values of the integral, which are using to restore the spectrum. The proposed method has significant advantages over the DFT and FFT analysis in the analysis of signals with the limited spectrum. First of all, tomography method is free from such phenomenon as the power leakage to the neighborhood field of frequencies due to a finite interval of temporal sampling, which is inevitably inherent to DFT when analyzing of signals with the limited spectrum. Secondly, in the case of line limited spectrum tomography method is theoretically possible to restore the exact spectrum. The condition of the exact restoration is the superposition of the frequencies of the signal with the discretization frequencies and the absence of noise. Thirdly, the method makes it possible to receive high accuracy of the spectrum restoration by the rarefied temporal sampling. This gives an opportunity to analyze high frequency signals with no transfer their spectrum at low frequencies, that is impossible to do with known methods due to the limited perfor-mance of the analog-to-digital converter. With regard to the implementation, tomography DFT method consists in the multiplication of a vector composed of the temporal sampling of the signal with the matrix recovery. This matrix can be calculated a priori that potentially guarantees a high performance of the method. By changing the matrix recovery, you can realize spectral analysis with different precision by one and the same temporal sampling. Moreover, when analyzing the signals with the limited line spectrum, you can “select” such matrix of the recovery, which in the ab-sence of noise will provide practically exact restoration of the spectrum. Correctness of the method and its advantages are shown in computer experiments.

Keywords: Fourier transform; Spectrum; Tomography approach; Permission element.