Comments on Errors of DFT Spectrum

2014 ◽  
Vol 568-570 ◽  
pp. 189-192
Author(s):  
Feng Han ◽  
Yao Lin Liu ◽  
Zhen Liu ◽  
Hai Dong Zeng
Keyword(s):  
Fourier Transform ◽  
Error Analysis ◽  
Energy Loss ◽  
Discrete Fourier Transform ◽  
Spectrum Analysis ◽  
Analytical Error ◽  
Error Estimators ◽  
Asynchronous Sampling ◽  
Parseval’S Theorem ◽  
Further Development

Discrete Fourier transform (DFT/FFT) spectrums contain a variety of inherent errors in asynchronous sampling. Spectrum analysis with the accuracy above 10-3 are generally challenging issues. This work divides the DFT procedure into four signal transforms and exams six spectrum errors originated from these distortions. Besides the review of traditional errors, a so-called energy loss-gain (ELG) error is briefly introduced, which is proved to be a considerable error on the basis of Parseval's theorem. With the help of full error analysis mentioned here and the further development of analytical error estimators, it is expectable to obtain a DFT spectrum with a specified accuracy.

The Role of the Discrete Fourier Transform in the Contribution to Gear Transmission Error Spectra from Tooth-Spacing Errors

1987 ◽  
Vol 201 (3) ◽  
pp. 227-229 ◽  
Author(s):  
W D Mark
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Error Analysis ◽  
Discrete Fourier Transform ◽  
Transmission Error ◽  
Spur Gear ◽  
Contact Ratio ◽  
Elastic Deformations ◽  
Exact Agreement

An expression is derived for the Fourier series spectrum of the transmission error arising from tooth-spacing errors on a single spur gear meshing with a perfect involute mating gear for the case of a contact ratio of unity and no elastic deformations present. This expression is found to be in exact agreement with previously derived results. The expression illustrates the role of the discrete Fourier transform in transmission error analysis and interpretation.

Analysis of Energy Loss-Gain Error in Discrete Fourier Transform

2014 ◽  
Vol 568-570 ◽  
pp. 172-175 ◽  
Author(s):  
Yao Lin Liu ◽  
Feng Han ◽  
Zhen Liu ◽  
Min Chen Zhai
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Discrete Fourier Transform ◽  
Energy Difference ◽  
Half Cycle ◽  
Cycle Number ◽  
Conservation Of Energy ◽  
Rectangular Window ◽  
Continuous Signal ◽  
Gain Error

In asynchronous sampling, discrete Fourier transform (DFT) spectrum involves errors. Scholars have done great investigations on the correction techniques of DFT spectrum, but the errors have not been completely eliminated all along. In this paper, spectrums were examined from the principle of conservation of energy. It is unnoticed before that the energy of the digital signal, which is the analysis object of DFT, isn't equal to that of the finite continuous signal truncated by rectangular window. Thus the energy of their spectrums are different according to the Parseval's theorem. The Energy Loss-Gain (ELG) error was introduced to express the energy difference between these two spectrums. The ELG error is zero if the observed continuous signal is truncated in integral multiple of half cycle and it is related to the cycle number and sampling number in one cycle. Analysis show that the ELG error decreases with the increment of these two parameters, which are helpful to the engineering.

The Research and Improvement of APFFT Algorithm

2012 ◽  
Vol 532-533 ◽  
pp. 1841-1845
Author(s):  
Jie Tao Diao ◽  
Jing Guo ◽  
Nan Li ◽  
Wei Yi ◽  
Sen Liu ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Discrete Fourier Transform ◽  
Spectrum Analysis ◽  
Spectral Leakage ◽  
Overlapping Data

Discrete Fourier Transform (DFT) is widely used in spectrum analysis. All Phase Fast Fourier Transform (APFFT) is proposed to improve FFT. It has merits such as phase invariance and small spectral leakage. But APFFT is a concrete algorithm. There are no parameters to adjust the result. The paper researches APFFT by changing the length of the overlapping data segments of APFFT. This change gives APFFT some new characteristic and some meaningful results are got.

scholarly journals Non-Distorted Optimization Spectrum Analysis

Energies ◽  
2018 ◽  
Vol 11 (7) ◽  
pp. 1841
Author(s):  
Rong-Ching Wu ◽  
Li-Ju Huang
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Spectrum Analysis ◽  
Time Frequency ◽  
Frequency Scale ◽  
Signal Parameters ◽  
Signal Characteristics ◽  
Sampled Signal ◽  
Three Stages ◽  
The Ideal

The discrete Fourier transform is extensively applied in spectrum analysis. However, the sampled signal is random, and the discrete Fourier transform has its own specific limitations. Thus, errors will inevitably occur in time–frequency transformation work. The most common are the leakage effects of the spectrum that are caused from the scale of the spectrum not being able to match the characteristics of the signal. The optimal spectrum is proposed to overcome this defect by adjusting the frequency scale to fit signal characteristics. This includes three stages whereby frequency scale can match signal characteristics. Firstly, the signal parameters must be found. Secondly, the frequency scale can be determined from these signal parameters. Finally, the optimal spectrum can be realized using the adjustable spectrum with the new frequency scale. After processing the optimal spectrum, the leakage effects of the signal will be decreased to a minimum. This method preserves signal characteristics in the optimization process, which reaches the ideal of non-distortion.

Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

1990 ◽  
Vol 48 (2) ◽  
pp. 64-65
Author(s):  
L. Reimer ◽  
R. Oelgeklaus
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Electron Energy ◽  
Rotational Symmetry ◽  
Mean Free Path ◽  
Single Scattering ◽  
Specimen Thickness ◽  
Two Dimensional ◽  
Image Position ◽  
Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

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