Traditional Fourier Methods Research Articles

Uncertainty in Fourier Transforms: A Fuzzy Logic Perspective

This study explores the integration of fuzzy logic with Fourier transforms to address the challenges of uncertainty, noise, and imprecision in real-world data. Fuzzy Fourier transforms extend traditional Fourier methods by incorporating fuzzy numbers, allowing for more robust frequency analysis, signal processing, and data reconstruction, particularly in noisy or incomplete datasets. The study examines the mathematical formulation of fuzzy Fourier transforms, computational trade-offs, and their performance advantages in comparison to classical Fourier methods. Real-world applications are discussed, including signal processing, image reconstruction, and time-series forecasting. Furthermore, the research highlights the increased computational complexity associated with fuzzy methods, the challenges of interpreting fuzzy results, and the limitations of handling probabilistic uncertainty. Future directions include the development of multi-dimensional fuzzy Fourier transforms, hybrid models integrating machine learning, and quantum computing applications. These advancements have broad implications for fields such as telecommunications, medical imaging, and financial forecasting, where handling uncertainty is critical.

Impact of the input OSNR on data-driven optical fiber channel modeling

The data-driven approach is promising for computationally efficient modeling of optical fiber channels. Here, for the first time to our knowledge, we investigate the impact of the input optical signal-to-noise ratio (OSNR) on the accuracy of data-driven modeling for optical fiber channels, based on a conditional generative adversarial network (cGAN). Initially, considering a single span of 80 km of standard single mode fiber (SSMF), we vary the launched optical power for the ease of emulating both linear and nonlinear transmission scenarios. When the input OSNR of a root raised cosine (RRC)-shaped 16 quadrature amplitude modulation signal with a roll-off factor of 0.1 varies, we identify that there occurs an OSNR threshold for accurately modeling a single span of 80 km of SSMF in order to reach the normalized mean square error of less than 0.02 in relevance to the traditional split-step Fourier method. The OSNR thresholds are 19 dB and 21 dB, respectively, under scenarios of linear and nonlinear transmissions. Moreover, we verify that the OSNR threshold of accurate modeling is insensitive to both the modulation format and the RRC shaping of the input signal. Furthermore, we find that the cascaded use of cGAN well-trained for the single SSMF span is capable of modeling the multiple-span SSMF transmission in the case that the input OSNR threshold of 22.6 dB is satisfied. We believe that the identification of the OSNR threshold for data-driven optical fiber channel modeling is useful for the accurate and efficient emulation of long-haul fiber optical transmission.

Single-exposure height-recovery structured illumination microscopy based on deep learning.

Modulation-based structured illumination microscopy (SIM) is performed to reconstruct three-dimensional (3D) surface topography. Generally speaking, modulation decoding algorithms mainly include a phase-shift (PS) method and frequency analysis technique. The PS method requires at least three images with fixed PSs, which leads to low efficiency. Frequency methods could decode modulation from a single image, but the loss of high-frequency information is inevitable. In addition, these methods all need to calculate the mapping relationship between modulation and height to recover the 3D shape. In this paper, we propose a deep learning enabled single-exposure surface measurement method. With only one fringe image, this method can directly restore the height information of the object. Processes such as denoising, modulation calculation, and height mapping are all included in the neural network. Compared with traditional Fourier methods, our method has higher accuracy and efficiency. Experimental results demonstrate that the proposed method can provide accurate and fast surface measurement for different structures.

Multi-span long-haul fiber transmission model based on cascaded neural networks with multi-head attention mechanism

In this paper, a novelty fiber transmission model consisting of cascaded neural networks with the multi-head attention mechanism has been put forward to solve signal transmission prediction problems in multi-span long-haul fiber link. After appropriately training the model with collected data by the gradient descent method, it can gradually handle the rules of signals' changes over each span and predict the signal transmission results with notably low time cost compared with traditional split-step Fourier method based long-haul model. Through numerical demonstration, this new model can predict 1000km fiber link with symbol rate up to 40GBaud in the QAM modulation with extremely low predicting error.

Advances in single-scan time-encoding magnetic resonance imaging

Time-encoding MRI is a single-scan method that uses traditional k-encoding only in one direction. In the orthogonal “time-encoding” direction, a string of echoes appears in an order that depends on the position of the corresponding spin packets. In one variant of time-encoding, this is achieved by using a series of selective pulses and appropriate gradients in both k-encoding and time-encoding directions. Although time-encoding offers some advantages over traditional single-scan Fourier methods such as echo planar imaging (EPI), the original time-encoding sequence also has some drawbacks that limit its applications. In this work, we show how one can improve several aspects of the original time-encoding sequence. By using an additional gradient pulse one can change the order in which the echoes appear, leading to identical echo times for all echoes, and hence to a uniform signal attenuation due to transverse relaxation and a reduction in average signal attenuation due to diffusion. By rearranging positive and negative gradients one can reduce the switching rate of the gradients. Furthermore, we show how one can implement time-encoding sequences in an interleaved fashion in order to reduce signal attenuation due to transverse relaxation and diffusion, while increasing the spatial resolution.

A novel recursive method to reconstruct multivariate functions on the unit cube

AbstractDue to discontinuity on the boundary, traditional Fourier approximation does not work efficiently ford−variate functions on [0, 1]d. In this paper, we will give a recursive method to reconstruct/approximate functions on [0, 1]dwell. The main process is as follows: We reconstruct ad−variate function by using all of its (d−1)–variate boundary functions and fewd–variate Fourier coefficients. We reconstruct each (d−1)–variate boundary function given in the preceding reconstruction by using all of its (d−2)–variate boundary functions and few (d−1)–variate Fourier coefficients. Continuing this procedure, we finally reconstruct each univariate boundary function in the preceding reconstruction by using values of the function at two ends and few univariate Fourier coefficients. Our recursive method can reconstruct multivariate functions on the unit cube with much smaller error than traditional Fourier methods.

Feature extraction of rolling bearing fault signal based on local mean decomposition and Teager energy operator

PurposeThis paper aims to explore a new way to extract the fault feature of a rolling bearing signal on the basis of a combinatorial method.Design/methodology/approachBy combining local mean decomposition (LMD) with Teager energy operator, a new feature-extraction method of a rolling bearing fault signal was proposed, called the LMD–Teager transform method. The principles and steps of method are presented, and the physical meaning of the time–frequency power spectrum and marginal spectrum is discussed. On the basis of comparison with the fast Fourier transform method, a simulated non-stationary signal was processed to verify the effect of the new method. Meanwhile, an analysis was conducted by using the recorded vibration signals which include inner race, out race and bearing ball fault signal.FindingsThe results show that the proposed method is more suitable for the non-stationary fault signal because the LMD–Teager transform method breaks through the difficulty of the Fourier transform method that can process only the stationary signal. The new method can extract more useful information and can provide better analysis accuracy and resolution compared with the traditional Fourier method.Originality/valueCombining the advantage of the local mean decomposition and the Teager energy operator, the LMD–Teager method suits the nature of the fault signal. A marginal spectrum obtained from the LMD–Teager method minimizes the estimation bias brought about by the non-stationarity of the fault signal. So, the LMD–Teager transform has better analysis accuracy and resolution than the traditional Fourier method, which provides a good alternative for fault diagnosis of the rolling bearing.

A time-frequency analysis method to obtain stable estimates of magnetotelluric response function based on Hilbert-Huang transform

The time-frequency analysis method represents signal as a function of time and frequency, and it is considered a powerful tool for handling arbitrary non-stationary time series by using instantaneous frequency and instantaneous amplitude. It also provides a possible alternative to the analysis of the non-stationary magnetotelluric (MT) signal. Based on the Hilbert-Huang transform (HHT), a time-frequency analysis method is proposed to obtain stable estimates of the magnetotelluric response function. In contrast to conventional methods, the response function estimation is performed in the time-frequency domain using instantaneous spectra rather than in the frequency domain, which allows for imaging the response parameter content as a function of time and frequency. The theory of the method is presented and the mathematical model and calculation procedure, which are used to estimate response function based on HHT time-frequency spectrum, are discussed. To evaluate the results, response function estimates are compared with estimates from a standard MT data processing method based on the Fourier transform. All results show that apparent resistivities and phases, which are calculated from the HHT time-frequency method, are generally more stable and reliable than those determined from the simple Fourier analysis. The proposed method overcomes the drawbacks of the traditional Fourier methods, and the resulting parameter minimises the estimation bias caused by the non-stationary characteristics of the MT data.

Sparse representation-based ECG signal enhancement and QRS detection

Electrocardiogram (ECG) signal enhancement and QRS complex detection is a critical preprocessing step for further heart disease analysis and diagnosis. In this paper, we propose a sparse representation-based ECG signal enhancement and QRS complex detection algorithm. Unlike traditional Fourier or wavelet transform-based methods, which use fixed bases, the proposed algorithm models the ECG signal as the superposition of a few inner structures plus additive random noise, where these structures (referred to here as atoms) can be learned from the input signal or a training set. Using these atoms and their properties, we can accurately approximate the original ECG signal and remove the noise and other artifacts such as baseline wandering. Additionally, some of the atoms with larger kurtosis values can be modified and used as an indication function to detect and locate the QRS complexes in the enhanced ECG signals. To demonstrate the robustness and efficacy of the proposed algorithm, we compare it with several state-of-the-art ECG enhancement and QRS detection algorithms using both simulated and real-life ECG recordings.

Parameter-sparse modification of Fourier methods to analyse the shape of closed contours with application to otolith outlines

Elliptical Fourier descriptors (EFDs) have been used extensively in shape analysis of closed contours and have a range of marine applications, such as automatic identification of fish species and discrimination between fish stocks based on EFDs of otolith contours. A recent method (the ‘MIRR’ method) transforms the two-dimensional contour to a one-dimensional function by mirroring (reflecting) the lower half of the contour around a vertical axis at the right end of the contour. MIRR then applies the fast Fourier transform (FFT) to the vertical contour points corresponding to equidistant coordinate values along the horizontal axis. MIRR has the advantage of reducing the number of Fourier coefficients to two coefficients per frequency component compared with four EFDs. However, both Fourier methods require several frequency components to reproduce a pure ellipse properly. This paper shows how the methods can be easily modified so that a virtually perfect reproduction of a pure ellipse is obtained with only one frequency component. In addition, real otolith examples for cod (Gadus morhua) and Greenland halibut (Reinhardtius hippoglossoides) are used to demonstrate that the modified methods give better approximations to the large-scale shape of the original contour with fewer coefficients than the traditional Fourier methods, with negligible additional computing time.

Optimal averages for nonlinear signal decompositions—Another alternative for empirical mode decomposition

The empirical mode decomposition (EMD) is an algorithm pioneered by Huang et al. as an alternative technique to the traditional Fourier and wavelet methods for analyzing nonlinear and non-stationary signals. It aims at decomposing a signal, via an iterative sifting procedure, into several intrinsic mode functions (IMFs), and each of the IMF has better behaved instantaneous frequency analysis. This paper presents an alternative approach for EMD. The main idea is to replace the average of upper and lower envelopes in the sifting procedure of EMD by a local average obtained by variational optimization framework. Therefore, an IMF can be produced by simply subtracting the average from the signal without iteration. Our numerical examples illustrate that the resulting decomposition is convergent and robust against noise.

Comparison of Fourier, principal component and wavelet analyses for high speed flame measurements

The continuing advancement of high speed, combustion diagnostics calls for mathematical techniques that can extract key information from large datasets. This paper therefore describes a case study to compare the characterization of combustion dynamics behind a V-gutter flame holder using three mathematical methods: Fourier analysis, principal component analysis, (PCA), and wavelet analysis (WA). The comparison focuses on the analysis of the characteristic frequencies of flow–flame interactions, with a particular emphasis on the analysis of transient and unsteady combustion procedures, such as lean blow off. Experimental data obtained under a range of conditions were analyzed using all three methods, and several observations were made. When applied to the analysis of stable combustion processes, all three methods reported frequency characteristics that were similar both quantitatively and qualitatively. Under unstable and transient combustion conditions, the WA method is capable of revealing the dynamics of the frequency components in the measurements, while traditional Fourier and PCA methods encounter application restrictions. Lastly, these applications also demonstrated WA’s suitability for practical combustion measurements beyond chemiluminescence, such as its applicability to discrete signals, insensitivity to the choice of wavelet basis, and insensitivity to the target signal extracted from the raw measurements.

MMSE Beam Forming on Fast-Scanning Phased Array Weather Radar

A fast-scanning phased array weather radar (PAWR) with a digital beam forming receiver is under development. It is important in beam forming for weather radar observation with temporally high resolution to form a stable and robust main lobe and adaptively suppress sidelobes with a small number of pulses in order to accurately estimate precipitation profiles (reflectivity, mean Doppler velocity, and spectral width). A minimum mean square error (MMSE) formulation with a power constraint, proposed in this paper, gives us adaptively formed beams that satisfy these demands. The MMSE beam-forming method is compared in various precipitation radar signal simulations with traditional beam-forming methods, Fourier and Capon methods, which have been applied in atmospheric research to observe distributed targets such as precipitation, and it is shown that the MMSE method is appropriate to this fast-scanning PAWR concept.

A New Eddy-Covariance Method Using Empirical Mode Decomposition

We introduce a new eddy-covariance method that uses a spectral decomposition algorithm called empirical mode decomposition. The technique is able to calculate contribu- tions to near-surface fluxes from different periodic components. Unlike traditional Fourier methods, this method allows for non-orthogonal contributions to the total flux, which are shown to be errors due to the undersampling of low-frequency processes. Inspection of the non-orthogonal terms with relation to sampling duration and periodicity reveals that a mea- sured periodic process requires approximately six cycles in order to be sufficiently sampled. This determines the maximum eddy size sufficiently measured given a particular sampling duration.

Laser Radar Range-Doppler Imaging and Simulation on High-speed Target

On the basic of the representation of the principle of Doppler imaging, an echo model of high-speed target illuminated by broadband linear frequency modulated (LFM) signal was established in this paper. For high-speed target, because of its nonlinear echo phase and time-varying Doppler frequency shift, the Doppler spectrum of the target acquired by the traditional Fourier method was ambiguous, and the same in the radar images of the target. Therefore, an adaptive Wigner-Ville time-frequency analysis method was presented in the paper, that is, the local fuzzy function could be obtained by applying the two-dimensional adding Window Fourier Transform to signal's Wigner-Ville distribution, which made the kernel function, not only be adaptive to time, but also to the frequency. Finally, the simulation results show that the method has a good time-frequency concentration, and effectively control the cross-term interference.

Wavelet Analysis of Near Infrared Spectral Data in the Application of Denoising

Spectrum signal may contain many peaks or mutations and noise also is not smooth white noise, to this kind of signal analysis, must do signal pretreatment, remove part of signal and extract useful part of signal.Based on the data of blood glucose near-infrared spectrum as the research object to explore the application of wavelet transform in the near infrared spectrum signal denoising, and through the simulation results show that using wavelet analysis of near infrared spectral data pretreatment than the traditional Fourier method can be higher precision of prediction.

APPLYING WAVELET METHOD FOR DENOISING IN TIME SERIES

This paper presents an application of the Wavelet Method to denoise in time series gives a compare with a traditional Fourier method, using the discrete transform. The key point of this method is choosing a wavelet function to estimate the variance of noise and determining the threshold. Moreover, we study some measures of the error and effectiveness among noise-suppression methods.

A Differential Relay for Power Transformers Using Intelligent Tools

This paper presents alternative approaches using artificial neural networks (ANNs) for the protection of power transformers. A complete protection scheme was implemented. An ANN subroutine was used to discriminate internal faults from other situations, replacing the traditional Fourier method for harmonic restraint. In addition, a routine for reconstruction of saturated current signals based on recurrent ANNs is also proposed. The proposed methods were extensively tested and then compared to the traditional differential protection algorithm, showing promising results. The application of the ANN tools is a new and important stage in the differential relay analysis methodology for power transformer protection

Applications for Affine Invariant Descriptor and Affine Parameter Estimation Based on Two-Source ICA

This paper proposes a new scheme based on two-source independent component analysis (ICA) for object recognition with affine transformation and for affine motion estimation between video frames. The idea is that, in two-source case, the indeterminate permutation matrix can be fixed. So an affine invariant description will be extracted by ICA algorithm. Simulation results show that ICA method has better performance comparing with traditional Fourier methods on object recognition and affine motion estimation.

Condition monitoring of vibrating steel‐reinforced concrete beams through wavelet transforms

PurposeThe purpose of this research is to show that wavelets are mathematical tools capable of separating vibration data into different frequency components, allowing the study of each component with a resolution matched to its specific time scale. Wavelets have advantages over traditional Fourier methods, particularly when a signal contains sharp spikes, discontinuities, and transients.Design/methodology/approachThis paper presents a brief description of wavelets and shows their application to the analysis of typical transient signals due to vibrations in a steel‐reinforced concrete beam.FindingsThe research clearly shows that it is possible to evaluate modes and components separately by using the wavelet analysis.Originality/valueThe beam analysed can be used as a simple model of other more complicated structures, such as bridges and other high scale civil engineering constructions, where vibration analysis is a key issue for maintenance and failure assessments, thus representing an alternative mathematical tool for condition monitoring.