Determining the Envelope of Real Finite Discrete Signal via Parametric Discrete Fourier Transform

the purpose of the work is to develop a novel method for determining the envelope of a real finite discrete vibro-acoustic signal (RFDVS), given on a finite interval. The novel method must be free from the disadvantages of methods based on discrete Fourier transform and discrete-time Fourier transform. A brief review of the advantages of vibro-acoustic functional diagnostics over other diagnostic methods of the technical condition of machines is carried out. The role of envelopes of vibro-acoustic signal in the functional diagnosis of objects is considered. The article investigates the methods for determining the envelope (instantaneous amplitude) of RFDVS, both in the time and in the frequency domain. Analysis of the advantages and disadvantages of the Hilbert envelopes, determined respectively on the basis of discrete Fourier transform (DFT) and discrete-time Fourier transform (DTFT) is given. The main reason for the large relative error in measuring the Hilbert envelope by DFT method in comparison with the measurement of Hilbert envelopes by DTFT method is revealed. A generalization of DFT in the form of parametric discrete Fourier transform (DFT-P) is given, and the analytical properties of DFT and DFT-P are compared. Fast algorithms for computing DFT-P are proposed parametric fast Fourier transform (FFT-P) algorithms. Based on DFT-P, implemented by the FFT-P, a novel method for determining the envelope of a compact discrete vibro-acoustic signal has been developed. The effectiveness and efficiency of the developed method of extracting the envelope is proved. Application of the proposed method allows one to reduce the relative error of envelope extracting by an order of magnitude, and reduce the computational cost and the required memory of the processor facility instrumentation by an order of magnitude. The theoretical results obtained in the work are confirmed by the results of numerical simulation of the measurement of the envelope of a real finite discrete signal by the known and proposed method, as well as by experiment on real vibro-acoustic signal.