On: “Computing the Fourier transform in geophysics with the transform decomposition DFT,” by Michael J. Reed, Hung V. Nguyen, and Ronald E. Chambers (November 1993 GEOPHYSICS, 58, p. 1701–1709).

Geophysics ◽  
1994 ◽  
Vol 59 (8) ◽  
pp. 1308-1309
Author(s):  
Eugene Lichman
Keyword(s):  
Fourier Transform ◽  
Frequency Spectrum ◽  
Time Domain ◽  
Computation Technique ◽  
Domain Sequence ◽  
Zero Values ◽  
Input Time ◽  
The Fourier Transform

The authors recently presented the illustration of the frequency spectrum computation technique which allows a reduction of the overall amount of computations (compared to a conventional FFT method) when the input time‐domain sequence contains significant number of zero values.

A Deep Neural Network Model for Learning Runtime Frequency Response Function Using Sensor Measurements

2021 ◽  
Author(s):  
Yongzhi Qu ◽  
Gregory W. Vogl ◽  
Zechao Wang
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Fourier Transform ◽  
Time Domain ◽  
Frequency Response Function ◽  
Frequency Component ◽  
Operating Conditions ◽  
Input Output ◽  
The Fourier Transform ◽  
The Time Domain

Abstract The frequency response function (FRF), defined as the ratio between the Fourier transform of the time-domain output and the Fourier transform of the time-domain input, is a common tool to analyze the relationships between inputs and outputs of a mechanical system. Learning the FRF for mechanical systems can facilitate system identification, condition-based health monitoring, and improve performance metrics, by providing an input-output model that describes the system dynamics. Existing FRF identification assumes there is a one-to-one mapping between each input frequency component and output frequency component. However, during dynamic operations, the FRF can present complex dependencies with frequency cross-correlations due to modulation effects, nonlinearities, and mechanical noise. Furthermore, existing FRFs assume linearity between input-output spectrums with varying mechanical loads, while in practice FRFs can depend on the operating conditions and show high nonlinearities. Outputs of existing neural networks are typically low-dimensional labels rather than real-time high-dimensional measurements. This paper proposes a vector regression method based on deep neural networks for the learning of runtime FRFs from measurement data under different operating conditions. More specifically, a neural network based on an encoder-decoder with a symmetric compression structure is proposed. The deep encoder-decoder network features simultaneous learning of the regression relationship between input and output embeddings, as well as a discriminative model for output spectrum classification under different operating conditions. The learning model is validated using experimental data from a high-pressure hydraulic test rig. The results show that the proposed model can learn the FRF between sensor measurements under different operating conditions with high accuracy and denoising capability. The learned FRF model provides an estimation for sensor measurements when a physical sensor is not feasible and can be used for operating condition recognition.

scholarly journals Extracting Features of Transient Electric Fields With Fourier And Wavelet Transform–A Case Study Of Lightning Positive Return Stroke

2021 ◽  
Vol 14 (14) ◽  
pp. 44-50
Author(s):  
Shriram Sharma
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Spectral Density ◽  
Frequency Spectrum ◽  
Electric Fields ◽  
Frequency Range ◽  
Return Stroke ◽  
The Fourier Transform ◽  
Positive Return ◽  
Domain Information

Frequency domain information were extracted from the time domain electric fields pertinent to the lightning positive return strokes applying Fourier transform and Wavelet transform. The electric field radiated by positive ground flashes striking the sea were recorded at 10 ns resolution at a coastal station to minimize the propagation effects. The frequency spectrum of the electric field of positive return strokes were computed applying the Fourier transform technique in the range of 10 kHz to 20 MHz owing to the fact that this range of frequency is of very much interest to the researchers and design engineers. The amplitude of the energy spectral density decreases nearly as ƒ-1 from 10 kHz to about 0.1 MHz and drops nearly as ƒ-2 up to 8 MHz.  Applying the wavelet transform technique, the same positive return strokes are found to radiate in the frequency range of 5.5 to 81 kHz with the average spread distribution of 13.6 kHz to about 30 kHz. From frequency spectrum obtained from the Fourier transform it is difficult to identify as which phase of the return stroke radiates in the higher frequency range and that in the lower frequency range, whereas, one can easily identify from the frequency spectrum obtained with the wavelet transform that ramp portion of the positive return stroke radiates in the larger spectral range as compared to that of initial peak of the return stroke.  Also, from the spectral density map obtained from wavelet transform one can easily observe the contribution of each phase in a range of frequency, which is not possible from the Fourier transform technique. Clearly, the wavelet transform is much more powerful tool to extract the frequency domain information of a non-stationary signal as compared to that of Fourier transform.

Discrete Time, Discrete Frequency

2021 ◽  
pp. 106-155
Author(s):  
Victor Lazzarini
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Discrete Time ◽  
Time Domain ◽  
Continuous Time ◽  
Time Varying ◽  
Discrete Frequency ◽  
The Fourier Transform ◽  
Definition Of ◽  
The Time Domain

This chapter is dedicated to exploring a form of the Fourier transform that can be applied to digital waveforms, the discrete Fourier transform (DFT). The theory is introduced and discussed as a modification to the continuous-time transform, alongside the concept of windowing in the time domain. The fast Fourier transform is explored as an efficient algorithm for the computation of the DFT. The operation of discrete-time convolution is presented as a straight application of the DFT in musical signal processing. The chapter closes with a detailed look at time-varying convolution, which extends the principles developed earlier. The conclusion expands the definition of spectrum once more.

Efficient 3D frequency response modeling with spectral accuracy by the rapid expansion method

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. T117-T123 ◽  
Author(s):  
Chunlei Chu ◽  
Paul L. Stoffa
Keyword(s):  
Fourier Transform ◽  
Frequency Response ◽  
Time Domain ◽  
Wave Equations ◽  
Time Integration ◽  
Expansion Method ◽  
Rapid Expansion ◽  
The Fourier Transform ◽  
The Time Domain ◽  
Response Modeling

Frequency responses of seismic wave propagation can be obtained either by directly solving the frequency domain wave equations or by transforming the time domain wavefields using the Fourier transform. The former approach requires solving systems of linear equations, which becomes progressively difficult to tackle for larger scale models and for higher frequency components. On the contrary, the latter approach can be efficiently implemented using explicit time integration methods in conjunction with running summations as the computation progresses. Commonly used explicit time integration methods correspond to the truncated Taylor series approximations that can cause significant errors for large time steps. The rapid expansion method (REM) uses the Chebyshev expansion and offers an optimal solution to the second-order-in-time wave equations. When applying the Fourier transform to the time domain wavefield solution computed by the REM, we can derive a frequency response modeling formula that has the same form as the original time domain REM equation but with different summation coefficients. In particular, the summation coefficients for the frequency response modeling formula corresponds to the Fourier transform of those for the time domain modeling equation. As a result, we can directly compute frequency responses from the Chebyshev expansion polynomials rather than the time domain wavefield snapshots as do other time domain frequency response modeling methods. When combined with the pseudospectral method in space, this new frequency response modeling method can produce spectrally accurate results with high efficiency.

scholarly journals Wave splitting for first-order systems of equations

2002 ◽  
Vol 32 (6) ◽  
pp. 371-381 ◽  
Author(s):  
G. Caviglia ◽  
A. Morro
Keyword(s):  
Fourier Transform ◽  
Partial Differential Equations ◽  
Time Domain ◽  
Space Variable ◽  
Transform Domain ◽  
Systems Of Equations ◽  
First Order ◽  
Wave Splitting ◽  
The Fourier Transform ◽  
Fourier Transform Domain

Systems of first-order partial differential equations are considered and the possible decomposition of the solutions in forward and backward propagating is investigated. After a review of a customary procedure in the space-time domain (wave splitting), attention is addressed to systems in the Fourier-transform domain, thus considering frequency-dependent functions of the space variable. The characterization is given for the direction of propagation and applications are developed to some cases of physical interest.

Simultaneous Determination of the Carrier Concentration, Mobility and Thickness of SiC Homo-Epilayers Using Terahertz Reflectance Spectroscopy

2007 ◽  
Vol 556-557 ◽  
pp. 423-426 ◽  
Author(s):  
Shingo Oishi ◽  
Yasuto Hijikata ◽  
Hiroyuki Yaguchi ◽  
Sadafumi Yoshida
Keyword(s):  
Fourier Transform ◽  
Electrical Properties ◽  
Carrier Concentration ◽  
Curve Fitting ◽  
Time Domain ◽  
Reflectance Spectroscopy ◽  
Free Carrier ◽  
Ftir Spectrometer ◽  
The Fourier Transform

We have simultaneously determined the carrier concentration, mobility, and thickness of 4H-SiC homo-epilayers with carrier concentration of 1016–1018 cm-3 from reflectance spectroscopy in the wavenumber range of 20–2000 cm-1. The spectra at 20–100 cm-1 and at 80–2000 cm-1 were measured by using the terahertz time domain spectrometer (THz-TDS) and the Fourier-transform infrared (FTIR) spectrometer, respectively. A modified classical dielectric function (MDF) model was employed for the curve fitting. We have compared the values of free carrier concentrations estimated from the reflectance spectroscopy with the net doping concentrations obtained from C–V measurements, and have discussed the validity of the electrical properties estimated from the reflectance spectroscopy.

scholarly journals Frequency Domain System Identification With Instrumental Variable Based Subspace Algorithm

1997 ◽  
Author(s):  
Tomas McKelvey
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Time Domain ◽  
Instrumental Variable ◽  
Theoretical Discussion ◽  
Subspace Algorithm ◽  
Input And Output ◽  
Rank Constraint ◽  
The Fourier Transform ◽  
The Time Domain

Abstract In this paper we discuss how the time domain subspace based identification algorithms can be modified in order to be applicable when the primary measurements are given as samples of the Fourier transform of the input and output signals or alternatively samples of the transfer function. An instrumental variable (IV) based subspace algorithm is presented. We show that this method is consistent if a certain rank constraint is satisfied and the frequency domain noise is zero mean with bounded covariances. An example is presented which illuminates the theoretical discussion.

A NEURAL NETWORK FOR POSITION INVARIANT PATTERN RECOGNITION COMBINING SPIKING NEURONS WITH THE FOURIER-TRANSFORM

1996 ◽  
Vol 07 (06) ◽  
pp. 727-733 ◽  
Author(s):  
MICHAEL STOECKER ◽  
HERBERT J. REITBOECK
Keyword(s):  
Neural Network ◽  
Fourier Transform ◽  
Time Domain ◽  
Amplitude Spectrum ◽  
Dynamic Network ◽  
Spiking Neurons ◽  
Invariant Representation ◽  
Activity Distribution ◽  
The Fourier Transform ◽  
The Time Domain

We present an approach for position invariant recognition of individual objects in composite scenes, combining neural networks and algorithmic methods. A dynamic network of spiking neurons is used to generate object definition and figure/ground separation via temporal signal correlations. A shift invariant representation of the network spike activity distribution is subsequently realized via the amplitude spectrum of the Fourier-transform. Objects and their transformed representations are therefore linked in the time domain. The model segregates scenes and classifies individual patterns independent of their position in the input scene.

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