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Mathematical Preliminaries
Published in Basuraj Bhowmik, Budhaditya Hazra, Vikram Pakrashi, Real-Time Structural Health Monitoring of Vibrating Systems, 2022
Basuraj Bhowmik, Budhaditya Hazra, Vikram Pakrashi
Parseval’s theorem in the context of Fourier transform is an important result that loosely states that the average power of the signal x(t) is equal to the sum of the power associated with individual frequency components. Mathematically, this statement translates to [5]: 1T∫0TX(t)2dt|Cn2|
Time and Frequency Representation of Continuous Time Signals
Published in Afshin Samani, An Introduction to Signal Processing for Non-Engineers, 2019
This intuition is correct. Parseval’s theorem mathematically shows that the calculated signal energy in the time domain is directly related to its energy in the frequency domain. This is quite interesting because the energy of a signal in a certain frequency band may reflect a specific phenomenon. For example, in the context of and electroencephalogram (EEG) representing the electrical activity of the brain, alpha waves have a frequency band of 8 to 12.99 Hz, and they are dominant in the wakeful state, but in a coma state, they are diffused (Rana, Ghouse, and Govindarajan, 2017). Thus, the average signal energy in a unit of time is called the signal power.
Frequency Domain Analysis
Published in Anastasia Veloni, Nikolaos I. Miridakis, Erysso Boukouvala, Digital and Statistical Signal Processing, 2018
Anastasia Veloni, Nikolaos I. Miridakis, Erysso Boukouvala
Parseval’s theorem refers to the conservation of energy during the transition from the time domain to the frequency domain. The quantity |X(ejω)|2 is called energy-density spectrum of the discrete signal, x[n], and describes the distribution of the energy of the discrete signal, x[n], over a frequency range.
Energy based denoising convolutional neural network for image enhancement
Published in The Imaging Science Journal, 2023
V. Karthikeyan, E. Raja, D. Pradeep
Energy conservation is at the core of energy analysis. Calculate the detailed component (DC) and approximate component (AC) coefficients to determine the energy percentage. Energy conservation in the wavelet domain is reflected by Parseval’s theorem. The squared aggregate of the spectral components of every frequency band in the discrete wavelet transform equals the fundamental consumptive energy. The input image I is fed into DWT. Computing the detailed and approximate parameters, which offer the fundamental framework for energy analysis based on intelligent band selection (l), allows us to determine the band energy. Using a finite impulse response filter, the selected band was separated into numerous subbands of equal length. To acquire precise energy dissemination, DWT was utilized to analyze the experimental hazy images and calculate the proportion of energy in every frequency band [48]. Foremost, this work utilized DB1 wavelets to determine the energy of the detailed as well as analysis components. After appealing DWT 10 times (with K set to 1), the subband energy percent of the selected band can be computed. Doing the energy normalization of the band produces the percentage calculation formula that is defined as follows: Here represents the sub-band energy % of the particular band, and K = 1.
Evaluation of the stress gradient of the superficial layer in ferromagnetic components based on sub-band energy of magnetic Barkhausen noise
Published in Nondestructive Testing and Evaluation, 2022
Jingyu Di, Cunfu He, Yung-Chun Lee, Xiucheng Liu
where x(t) and X(f) are, respectively, the time domain form and frequency domain form of the magnetic Barkhausen noise signal. Parseval’s theorem indicates that the total energy of signal in time domain is equal to the total energy of signal in frequency domain, that is, the total energy of the signal after the Fourier transform remains unchanged, in accordance with the law of conservation of energy.
Analysis of the consistency of the Sperling index for rail vehicles based on different algorithms
Published in Vehicle System Dynamics, 2021
Chenxin Deng, Jinsong Zhou, David Thompson, Dao Gong, Wenjing Sun, Yu Sun
In this paper, the Sperling index calculated using different algorithms is shown to give inconsistent results in dynamic simulations and performance measurements of rail vehicles. The main conclusions are as follows: Calculating the Sperling index in the frequency domain, different algorithms can be summarised into a unified formula. If the parameter m, representing the power of the weighted acceleration, is greater than 2, the calculation result of the Sperling index decreases as the sample time length in the time domain increases or the sample interval in the frequency domain decreases. Only if the parameter m is equal to 2, that is, the algorithm is based on r.m.s. values, is the Sperling index result stable with varying sample length.When comparing results between those obtained from the formulae based on the second and third powers of acceleration, the Sperling index result based on the third power is around 15-20% smaller than that based on the second power when a 20 s sample time is used.With the stable frequency-domain algorithm based on the r.m.s. values, a time-domain algorithm is derived based on Parseval’s theorem. Its time-frequency consistency is also verified with the time-domain data from dynamic simulations of a rail vehicle. However, the time-frequency consistency of the algorithm with m equal to 3 cannot be proved and the results from time-domain and frequency-domain analysis are found to differ by around 3%. Therefore, it is recommended that the algorithm based on the r.m.s. in the time domain should be selected for rail vehicle measurements.With the stable and time-frequency consistent algorithm based on the r.m.s., an accurate evaluation based on the Sperling index can be made in dynamic simulations and performance measurements of rail vehicles. The research in this paper can guide the calculation of ride comfort index and improve the dynamic performance evaluation system of rail vehicles.