Relevant bibliographies by topics / Parseval's Theorem

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'Parseval's Theorem'

Author: Grafiati
Published: 4 June 2021
Last updated: 7 September 2023

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Journal articles on the topic "Parseval's Theorem"

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FINE, JONATHAN. "A NOTE ON BRAIDS AND PARSEVAL'S THEOREM." Journal of Knot Theory and Its Ramifications 21, no. 05 (2012): 1250024. http://dx.doi.org/10.1142/s0218216511009546.

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In 1986 Falk and Randell, based on Arnold's 1969 paper on braids, proved that the pure braid groups are residually nilpotent. They also proved that the quotients in the lower central series are free abelian groups. This brief note uses an example to provide evidence for a much stronger conjectural statement: That each braid b can be written as an infinite sum [Formula: see text], where each bi is a linear function of the ith Vassiliev–Kontsevich Zi(b) invariant of b. The example is pure braids on two strands. This leads to solving eτ = q for τ a Laurent series in q. We set [Formula: see text] and use Fourier series and Parseval's theorem to prove eτ = q. For more than two strands the stronger statement seems to rely on an as yet unstated Plancherel theorem for braid groups, which is likely both to be deep and to have deep consequences.
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Melin, Jan O. "Interpreting ISAR Images by Means of Parseval's Theorem." IEEE Transactions on Antennas and Propagation 55, no. 2 (2007): 498–500. http://dx.doi.org/10.1109/tap.2006.889993.

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Bonatti, Ivanil S., Pedro L. D. Peres, and Amauri Lopes. "Velocity of Propagation in Transmission Lines." International Journal of Electrical Engineering & Education 35, no. 1 (1998): 79–86. http://dx.doi.org/10.1177/002072099803500107.

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This paper discusses the skin effect on lossy transmission lines in the context of undergraduate electrical engineering courses. A new definition for propagation time derived from Parseval's theorem is proposed. In lossless transmission lines the proposed definition produces the conventional results and for lossy lines it matches quite exactly with the time simulation results, as shown by an illustrative example.
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Han, Feng, Yao Lin Liu, Zhen Liu, and Hai Dong Zeng. "Comments on Errors of DFT Spectrum." Applied Mechanics and Materials 568-570 (June 2014): 189–92. http://dx.doi.org/10.4028/www.scientific.net/amm.568-570.189.

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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.
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Urbancic, T. I., C. I. Trifu, R. A. Mercer, A. J. Feustel, and J. A. G. Alexander. "Automatic time-domain calculation of source parameters for the analysis of induced seismicity." Bulletin of the Seismological Society of America 86, no. 5 (1996): 1627–33. http://dx.doi.org/10.1785/bssa0860051627.

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Abstract A time-domain methodology for the automatic (on-line) calculation of source parameters is introduced. The method is based on applying Parseval's theorem to the spectral equations proposed by Andrews (1986). The appropriateness of the technique for the analysis of induced seismicity was tested by comparing on-line values of seismic moment, seismic energy, and static stress drop with those obtained in the frequency domain (off-line) for an aftershock sequence of events recorded underground at Creighton mine, Ontario. Our analysis shows that the calculated on-line and off-line source parameters are in good agreement, with an average slope offset of 4% from the 1:1 line, suggesting that the proposed method is reliable. Additionally, by applying empirical on-line attenuation corrections (coda Q), similar results are obtained to those calculated using off-line corrections (spectral decay), improving the average slope offset to 1%.
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Liu, Yao Lin, Feng Han, Zhen Liu, and Min Chen Zhai. "Analysis of Energy Loss-Gain Error in Discrete Fourier Transform." Applied Mechanics and Materials 568-570 (June 2014): 172–75. http://dx.doi.org/10.4028/www.scientific.net/amm.568-570.172.

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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.
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Karpov, Eduard G., Larry A. Danso, and John T. Klein. "Anomalous strain energy transformation pathways in mechanical metamaterials." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2226 (2019): 20190041. http://dx.doi.org/10.1098/rspa.2019.0041.

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This discussion starts with a mechanics version of Parseval's energy theorem applicable to any discrete lattice material with periodic internal structure: a microtruss, grid, frame, origami or tessellation. It provides a simple relationship between the strain energy volumetric/usual and spectral distributions in the reciprocal space. The spectral energy distribution leads directly to a spectral entropy of lattice deformation (Shannon's type), whose variance with a material coordinate represents the decrease of information about surface loads in the material interior. Spectral entropy is also a basic measure of complexity of mechanical responses of metamaterials to surface and body loads. Considering transformation of the energy volumetric and spectral distributions with a material coordinate pointed away from a surface load, several interesting anomalies are seen even for simple lattice materials, when compared to continuum materials. These anomalies include selective filtering of surface Raleigh waves (sinusoidal pressure patterns), Saint–Venant effect inversion illustrated by energy spectral distribution contours, occurrence of ‘hiding pockets’ of low deformation, and redirection of strain energy maximum away from axis of a concentrated surface load. The latter phenomenon can be significant for impact protection applications of mechanical metamaterials.
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PUTHANKATTIL, SUBHA D., and PAUL K. JOSEPH. "CLASSIFICATION OF EEG SIGNALS IN NORMAL AND DEPRESSION CONDITIONS BY ANN USING RWE AND SIGNAL ENTROPY." Journal of Mechanics in Medicine and Biology 12, no. 04 (2012): 1240019. http://dx.doi.org/10.1142/s0219519412400192.

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EEG is useful for the analysis of the functional activity of the brain and a detailed assessment of this non-stationary waveform can provide crucial parameters indicative of the mental state of patients. The complex nature of EEG signals calls for automated analysis using various signal processing methods. This paper attempts to classify the EEG signals of normal and depression patients using well-established signal processing techniques involving relative wavelet energy (RWE) and artificial feedForward neural network. High frequency noise present in the recorded signal is removed using total variation filtering (TVF). Classification of the frequency bands of EEG signals into appropriate detail levels and approximation level is carried out using an eight-level multiresolution decomposition method of discrete wavelet transform (DWT). Parseval's theorem is used for calculating the energy at different resolution levels. RWE analysis gives information about the signal energy distribution at different decomposition levels. Both RWE and feedforward Network are used to classify the signals from normal controls and depression patients. The performance of the artificial neural network was evaluated using the classification accuracy and its value of 98.11% indicates a great potential for classifying normal and depression signals.
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Bhatnagar, R. M. "Noise reduction in linear variable differential transformer data of recoil motion measurement by numerical methods." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 220, no. 2 (2006): 159–66. http://dx.doi.org/10.1243/09544062jmes140.

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The measurement of recoil distance versus time by various methods such as the recoil potentiometer, photo electric transducer, slide wire, accelerometer, revolving drum system, and linear variable differential transformer (LVDT) has been used for gross muzzle brake efficiency measurements and recoil system performance evaluation by the calculation of recoil velocities. For a long recoil-length gun system, a combination of recoil potentiometer and LVDT is used extensively. In order to dispense with the use of recoil potentiometer in the above combination, the article proposes the use of the least-square-fit-based Richardson's extrapolation method and mean square velocity calculation for the accurate determination of free recoil velocity. The mean square velocity calculation is based on Parseval's theorem. The proposed method is based on the comparative evaluation of second- and third-order finite difference method, Richardson's fourth-order method, and the least-square-fit-based Richardson's extrapolation. The least-square-fit-based Richardson's extrapolation gives the lowest value of residual entropy. This is because the maximum likelihood estimators for Gauss probability distribution function and least-square estimators for the coefficients of polynomial representing recoil velocity time curve are coincident. The results of each of the four methods combined with the mean square velocity method were compared, and the least-square-fit-based Richardson's extrapolation was found to be accurate and consistent. The method can be used even when low pass filter is included in the LVDT circuit for stand-alone use.
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Hassanzadeh, Mohammad, and Behnam Shahrrava. "Linear Version of Parseval’s Theorem." IEEE Access 10 (2022): 27230–41. http://dx.doi.org/10.1109/access.2022.3157736.

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Dissertations / Theses on the topic "Parseval's Theorem"

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Grunden, Beverly K. "On the Characteristics of a Data-driven Multi-scale Frame Convergence Algorithm." Wright State University / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=wright1622208959661057.

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Hedayati, Mohammad Hassan. "Integrated CM Filter for Single-Phase and Three-Phase PWM Rectifiers." Thesis, 2015. http://etd.iisc.ac.in/handle/2005/3947.

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The use of insulated-gate bipolar transistor (IGBT)-based power converters is increasing exponentially. This is due to high performance of these devices in terms of efficiency and switching speed. However, due to the switching action, high frequency electromagnetic interference (EMI) noises are generated. Design of a power converter with reduced EMI noise level is one of the primary objectives of this research. The first part of the work focuses on designing common-mode (CM) filters, which can be integrated with differential-mode (DM) filters for three-phase pulse-width modulation (PWM) rectifier-based motor drives. This work explores the filter design based on the CM equivalent circuit of the drive system. Guidelines are provided for selection of the filter components. Different variants of the filter topology are evaluated to establish the effectiveness of the proposed topology. Analytical results based on Bode plot of the transfer functions are presented, which suggest effective EMI reduction. Experimental results based on EMI measurement on the grid side and CM current measurement on the motor side are presented. These results validate the effectiveness of the filter. In the second part of the work, it is shown that inclusion of CM filters into DM filters results in resonance oscillations in the CM circuit. An active damping strategy is proposed to damp the oscillations in both line-to-line and line-to-ground ac voltages and currents. An approach based on pole placement by state feedback is used to actively damp both the DM and CM filter oscillations. Analytical expressions for state-feedback controller gains are derived for both continuous-and discrete-time models of the filter. Trade-off in selection of the active damping gain on the lower-order grid current harmonics is analysed using a weighted admittance function method. In the third part of the work, single-phase grid-connected power converters are considered. An integrated CM filter with DM LCL filter is proposed. The work explores the suitability of PWM methods for single-phase and parallel single-phase grid-connected power converters. It is found that bipolar PWM and unipolar PWM with 180◦interleaving angle are suitable for single-phase and parallel single-phase power converters, respectively. The proposed configuration along with the PWM methods reduces the CM voltage, CM current, and EMI noise level effectively. It is also shown that the suggested circuit is insensitive to nonidealities of the power converter such as dead-time mismatch, mismatch in converter-side inductors, unequal turn on and turn off of the switches, and propagation delays. In the fourth part of the work, the inter-phase inductor in parallel interleaved power converters is integrated with LCL filter boost inductor. Different variant designs are presented and compared with the proposed structure. It is shown that the proposed structure makes use of standard core geometries and consumes lesser core material as well as copper wire. Hence, it reduces the overall size and cost of the power converter. In the present work, a 10kVA three-phase back-to-back connected with input LCL filter and output dv/dt filter, a 5kVA single-phase grid-connected power converter with LCL filter, and a 7.5kVA parallel single-phase grid-connected power converter with LCL filter are fabricated in the laboratory to evaluate and validate the proposed methods. The experimental results validate the proposed methods that result in significant EMI performance improvement of grid-connected power converters.
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Hedayati, Mohammad Hassan. "Integrated CM Filter for Single-Phase and Three-Phase PWM Rectifiers." Thesis, 2015. http://etd.iisc.ernet.in/2005/3947.

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The use of insulated-gate bipolar transistor (IGBT)-based power converters is increasing exponentially. This is due to high performance of these devices in terms of efficiency and switching speed. However, due to the switching action, high frequency electromagnetic interference (EMI) noises are generated. Design of a power converter with reduced EMI noise level is one of the primary objectives of this research. The first part of the work focuses on designing common-mode (CM) filters, which can be integrated with differential-mode (DM) filters for three-phase pulse-width modulation (PWM) rectifier-based motor drives. This work explores the filter design based on the CM equivalent circuit of the drive system. Guidelines are provided for selection of the filter components. Different variants of the filter topology are evaluated to establish the effectiveness of the proposed topology. Analytical results based on Bode plot of the transfer functions are presented, which suggest effective EMI reduction. Experimental results based on EMI measurement on the grid side and CM current measurement on the motor side are presented. These results validate the effectiveness of the filter. In the second part of the work, it is shown that inclusion of CM filters into DM filters results in resonance oscillations in the CM circuit. An active damping strategy is proposed to damp the oscillations in both line-to-line and line-to-ground ac voltages and currents. An approach based on pole placement by state feedback is used to actively damp both the DM and CM filter oscillations. Analytical expressions for state-feedback controller gains are derived for both continuous-and discrete-time models of the filter. Trade-off in selection of the active damping gain on the lower-order grid current harmonics is analysed using a weighted admittance function method. In the third part of the work, single-phase grid-connected power converters are considered. An integrated CM filter with DM LCL filter is proposed. The work explores the suitability of PWM methods for single-phase and parallel single-phase grid-connected power converters. It is found that bipolar PWM and unipolar PWM with 180◦interleaving angle are suitable for single-phase and parallel single-phase power converters, respectively. The proposed configuration along with the PWM methods reduces the CM voltage, CM current, and EMI noise level effectively. It is also shown that the suggested circuit is insensitive to nonidealities of the power converter such as dead-time mismatch, mismatch in converter-side inductors, unequal turn on and turn off of the switches, and propagation delays. In the fourth part of the work, the inter-phase inductor in parallel interleaved power converters is integrated with LCL filter boost inductor. Different variant designs are presented and compared with the proposed structure. It is shown that the proposed structure makes use of standard core geometries and consumes lesser core material as well as copper wire. Hence, it reduces the overall size and cost of the power converter. In the present work, a 10kVA three-phase back-to-back connected with input LCL filter and output dv/dt filter, a 5kVA single-phase grid-connected power converter with LCL filter, and a 7.5kVA parallel single-phase grid-connected power converter with LCL filter are fabricated in the laboratory to evaluate and validate the proposed methods. The experimental results validate the proposed methods that result in significant EMI performance improvement of grid-connected power converters.
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Guerra, Rita Catarina Correia. "Generalizations of the Fourier transform and their applications." Doctoral thesis, 2019. http://hdl.handle.net/10773/29813.

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In this thesis, we consider a new generalization of the Fourier transform, depending on four complex parameters and all the powers of the Fourier transform. This new transform is studied in some Lebesgue spaces. In fact, taking into account the values of the parameters of the operator, we can have very different kernels and so, the corresponding operator is studied in different Lebesgue spaces, accordingly with its kernel. We begin with the characterization of each operator by its characteristic polynomial. This characterization serves as a basis for the study of the forthcoming properties. Following this, we present, for each case, the spectrum of the corresponding operator, necessary and sufficient conditions for which the operator is invertible, Parseval-type identities and conditions for which the operator is unitary and an involution of order n. After this, we contruct new convolutions associated with those operators and obtain the corresponding factorization identities and some norm inequalities. By using these new operators and convolutions, we construct new integral equations and study their solvability. In this sense, we have equations generated by the studied operators and also a class of equations of convolution-type depending on multi-dimensional Hermite functions. Furthermore, we study the solvability of classical integral equations, using the new operators and convolutions, namely a class of Wiener-Hopf plus Hankel equations, whose solution is written in terms of a Fourier-type series. For one case of this generalization of the Fourier transform, that only depends on the cosine and sine Fourier transforms, we obtain PaleyWiener and Wiener’s Tauberian results, using the associated convolution and a new translation induced by that convolution. Heisenberg uncertainty principles for the one-dimensional case and for the multi-dimensional case are obtained for a particular case of the introduced operator. At the end, as an application outside of mathematics, we obtain a new result in signal processing, more properly, in a filtering processing, by applying one of our new convolutions.<br>Nesta tese, consideramos uma nova generalização da transformação de Fourier, dependente de quatro parâmetros complexos e de todas as potências da transformação de Fourier. Esta nova transformação é estudada em alguns espaços de Lebesgue. De facto, tendo em conta os valores dos parâmetros, podemos ter núcleos muito diferentes e assim, o correspondente operador é estudado em diferentes espaços de Lebesgue, de acordo com o seu núcleo. Começamos com a caracterização de cada operador pelo seu polinómio característico. Esta caracterização serve de base para o estudo das propriedades seguintes. Seguindo isto, apresentamos, para cada caso, o espetro do correspondente operador, condições necessárias e suficientes para as quais o operador é invertível, identidades do tipo de Parseval e condições para as quais o operador é unitário e uma involução de ordem n. Depois disto, construímos novas convoluções associadas àqueles operadores e obtemos as correspondentes identidades de factorização e algumas desigualdades da norma. Usando estes novos operadores e convoluções, construímos novas equações integrais e estudamos a sua solvabilidade. Neste sentido, temos equações geradas pelos operadores estudados e também uma classe de equações do tipo de convolução dependendo de funções de Hermite multidimensionais. Além disso, estudamos a solvabilidade de equações integrais clássicas, usando os novos operadores e convoluções, nomeadamente uma classe de equações de Wiener-Hopf mais Hankel, cuja solução é escrita em termos de uma série do tipo de Fourier. Para um caso desta generalização da transformação de Fourier, que depende apenas das transformações de Fourier do cosseno e do seno, obtemos resultados de Paley-Wiener e resultados Tauberianos de Wiener, usando a convolução associada e uma nova translação induzida por essa convolução. Princípios de incerteza de Heisenberg para os casos unidimensional e multidimensional são obtidos para um caso particular do operador introduzido. No final, como uma aplicação fora da matemática, obtemos um novo resultado em processamento de sinal, mais propriamente, num processo de filtragem, por aplicação de uma das nossas novas convoluções.<br>Programa Doutoral em Matemática Aplicada
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Book chapters on the topic "Parseval's Theorem"

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Weik, Martin H. "Parseval's theorem." In Computer Science and Communications Dictionary. Springer US, 2000. http://dx.doi.org/10.1007/1-4020-0613-6_13642.

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Weinmann, Alexander. "Optimierung eines linearen Reglers mittels des Parseval-Theorems." In Regelungen Analyse und technischer Entwurf. Springer Vienna, 1986. http://dx.doi.org/10.1007/978-3-7091-6994-0_13.

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Gutiérrez, A., E. Marcault, C. Alonso, J. P. Laur, and D. Trémouilles. "Parseval’s Theorem Used for the Inductor Analysis in High-Frequency Boost Converters." In Lecture Notes in Electrical Engineering. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-56970-9_26.

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Dwivedi, Ankita, S. K. Singh, and R. K. Srivastava. "Analysis of Permanent Magnet Brushless AC Motor Using Two Dimensional Fourier Transform-Parseval’s Theorem." In Theory and Applications of Applied Electromagnetics. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-17269-9_20.

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Al-Sa’di, Sa’ud, and Eric S. Weber. "On Parseval Frames of Kernel Functions in de Branges Spaces of Entire Vector Valued Functions." In New Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-76473-9_1.

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"Parseval’s Theorem." In Optimal Reference Shaping for Dynamical Systems. CRC Press, 2009. http://dx.doi.org/10.1201/9781439805633.ax3.

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Berber, Stevan. "Transforms of Deterministic Continuous-Time Signals." In Discrete Communication Systems. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198860792.003.0012.

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Chapter 12 presents a detailed analysis of continuous-time signals and systems in the frequency domain, including the theory of Fourier series and Fourier transforms, and key examples relevant for the analysis and synthesis of signals processed in the digital transceiver blocks of a communication system. The amplitude, magnitude, phase, and power spectra are defined and calculated for typical signals. In particular, the Fourier transform of periodic signals is presented, due to its importance in communication systems theory and practice. Using a unique notation that distinguishes energy and power signals, the correlation, power, and energy spectral density functions are inter-related by proving the Wiener–Khintchine theorem. A comprehensive analysis of a linear-time-invariant system, using the concepts of impulse response, system correlation function, and power spectral density, both for power signals and energy signals, is presented. In addition, Parseval’s theorem and the Rayleigh theorem are proven.
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Allahverdiev, Bilender P., and Hüseyin Tuna. "The Parseval Equality and Expansion Formula for Singular Hahn-Dirac System." In Emerging Applications of Differential Equations and Game Theory. IGI Global, 2020. http://dx.doi.org/10.4018/978-1-7998-0134-4.ch010.

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This work studies the singular Hahn-Dirac system given by Here 𝜇 is a complex spectral parameter, p(.) and r(.) are real-valued continuous functions at 𝜔0, defined on [𝜔0,∞) and q∈(0,1), , 𝜔&gt;0, x∈[𝜔0,∞). The existence of a spectral function for this system is proved. Further, a Parseval equality and an expansion formula in eigenfunctions are proved in terms of the spectral function.
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Conference papers on the topic "Parseval's Theorem"

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Frieden, B. Roy. "Probability-law estimation by a principle of minimum physical information." In OSA Annual Meeting. Optica Publishing Group, 1992. http://dx.doi.org/10.1364/oam.1992.tudd5.

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A principle I–J = min. of minimum Fisher information I = ∫dxp'2(x)/p(x) is often used to estimate an unknown probability density function p(x) in the presence of insufficient information J. The latter is conventionally a Lagrange constraint term. The resulting estimate p(x) may or may not be correct (where there is a "correct" answer), although it is smooth. But suppose the correct answer is required and the user has the resources to choose the particular information J to input. What form J should be chosen, and what form should I – J ≡K take, where K is the "physical" information? The choice of J (and K) follows from three reasonable axioms: (i) K is linear in I, i.e., K measures disorder, (ii) K is a universal value, irrespective of physical phenomenon, i.e., all phenomena p(x) are equivalent in their information content; (iii) there is no preferred space, direct or Fourier, for representing K. The solution is that K = 0 or, numerically, J = 1. However, functionally J depends upon p(x) differently than does I; this dependence, which often arises out of Parseval's theorem, is the required input of information. We show that many of the known laws p(x) of physics may be "estimated" in this way.
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BOZOVIC, DUBRAVKA, and NENAD POPOVICH. "Submarine Optimal Depth Control applying Parseval s Theorem." In Fourth International Conference on Advances in Mechanical and Automation Engineering - MAE 2016. Institute of Research Engineers and Doctors, 2016. http://dx.doi.org/10.15224/978-1-63248-102-3-41.

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Malhotra, Hari Krishan, and Lalit Kumar Vashisht. "Construction of Non-Uniform Parseval Wavelet Frames for L2 (R) via UEP." In 2019 13th International conference on Sampling Theory and Applications (SampTA). IEEE, 2019. http://dx.doi.org/10.1109/sampta45681.2019.9030867.

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Clark, J. P., and E. A. Grover. "Assessing Convergence in Predictions of Periodic-Unsteady Flowfields." In ASME Turbo Expo 2006: Power for Land, Sea, and Air. ASMEDC, 2006. http://dx.doi.org/10.1115/gt2006-90735.

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Predictions of time-resolved flowfields are now commonplace within the gas-turbine industry, and the results of such simulations are often used to make design decisions during the development of new products. Hence it is necessary for design engineers to have a robust method to determine the level of convergence in design predictions. Here we report on a method developed to determine the level of convergence in a predicted flowfield that is characterized by periodic-unsteadiness. The method relies on fundamental concepts from digital signal processing including the discrete Fourier transform, cross-correlation, and Parseval’s theorem. Often in predictions of vane-blade interaction in turbomachines, the period of the unsteady fluctuations is expected. In this method, the development of time-mean quantities. Fourier components (both magnitude and phase), cross-correlations, and integrated signal power are tracked at locations of interest from one period to the next as the solution progresses. Each of these separate quantities yields some relative measure of convergence that is subsequently processed to form a fuzzy set. Thus the overall level of convergence in the solution is given by the intersection of these sets. Examples of the application of this technique to several predictions of unsteady flows from two separate solvers are given. These include a prediction of hot-streak migration as well as more typical cases. It is shown that the method yields a robust determination of convergence. Also, the results of the technique can guide further analysis and/or post-processing of the flowfield. Finally, the method is useful for the detection of inherent unsteadiness in the flowfield, and as such it can be used to prevent design escapes.
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