Principles of Fourier Analysis

Howell, Kenneth B.

doi:10.1201/9781420036909

Cited by 57 publications

(35 citation statements)

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“…For the case of Condition (i) in Lemma 3, the boundedness of the first-order moment guarantees that is also absolutely integrable, and therefore, both Fourier transforms are uniformly continuous functions and the proof is complete. Although Condition (ii) is weaker in the sense that is not continuous and has a singularity at , it still implies that is finite, which is again a sufficient condition [32]. …”

Section: Appendix B Proof Of Lemmamentioning

confidence: 99%

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On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

Amini

Thévenaz

Ward

et al. 2013

IEEE Trans. Inform. Theory

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Abstract-Bayesian estimation problems involving Gaussian distributions often result in linear estimation techniques. Nevertheless, there are no general statements as to whether the linearity of the Bayesian estimator is restricted to the Gaussian case. The two common strategies for non-Gaussian models are either finding the best linear estimator or numerically evaluating the Bayesian estimator by Monte Carlo methods. In this paper, we focus on Bayesian interpolation of non-Gaussian first-order autoregressive (AR) processes where the driving innovation can admit any symmetric infinitely divisible distribution characterized by the Lévy-Khintchine representation theorem. We redefine the Bayesian estimation problem in the Fourier domain with the help of characteristic forms. By providing analytic expressions, we show that the optimal interpolator is linear for all symmetric -stable distributions. The Bayesian interpolator can be expressed in a convolutive form where the kernel is described in terms of exponential splines. We also show that the limiting case of Lévy-type AR(1) processes, the system of which has a pole at the origin, always corresponds to a linear Bayesian interpolator made of a piecewise linear spline, irrespective of the innovation distribution. Finally, we show the two mentioned cases to be the only ones within the family for which the Bayesian interpolator is linear.

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Section: Appendix B Proof Of Lemmamentioning

confidence: 99%

“…If is such that either i) the white innovation has a finite first-order moment , or ii) and the pdf of is continuous, with being bounded for some , then we have that (32) where stands for the characteristic function of the random variable (Fourier transform of its pdf) and denotes its derivative in the Fourier domain.…”

Section: Lemma 2: Letmentioning

confidence: 99%

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

Amini

Thévenaz

Ward

et al. 2013

IEEE Trans. Inform. Theory

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“…Under large-signal excitation with one or more signals at integers of a fundamental frequency lω 0 , i.e. there is no intermodulation, the nonlinear two-port's currents and voltages can be formulated in terms of complex Fourier-series (Howell, 2001) assuming their periodic form and neglecting sub-harmonics…”

Section: Introductionmentioning

confidence: 99%

Behavioral modeling of nonlinear transfer systems with load-dependent <i>X</i>-parameters

2017

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Abstract. In this contribution, different approaches based on the X-parameters TM to model the behavior of mismatched nonlinear transfer systems are examined. The X-parameters based on the PHD 1 -principle introduced by Verspecht and Root (2006) as an extension of the well-known S-parameters describe nonlinear microwave 2-port-networks under large signal conditions. Using load-pull measurement techniques they can be used for arbitrary load situations. Beside this load-pull approach, in the work of , it is stated that it is sufficient to use one optimized X-parameter set for each value of the load reflection coefficient without introducing a large error. In another contribution of Cai and Yu (2015), this approach is extended to cover the whole smith chart with one optimized X-parameter set instead. In this work, these different approaches are compared and brought into question.

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“…In fact, meaningful Fourier analysis is relies heavily on the nature of the data to be analyzed. Howell [9] deals comprehensively with the mathematics of Fourier analysis. Most of these theories are derived from the assumptions of continuity, smoothness and periodicity or stationarity.…”

Section: Introductionmentioning

confidence: 99%

The combined use of order tracking techniques for enhanced Fourier analysis of order components

Wang

Heyns

2011

Mechanical Systems and Signal Processing

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Order tracking is one of the most important vibration analysis techniques for diagnosing faults in rotating machinery. It can be performed in many different ways, each of these with distinct advantages and disadvantages. However, in the end the analyst will often use Fourier analysis to transform the data from a time series to frequency or order spectra. It is therefore surprising that the study of the Fourier analysis of order-tracked systems seems to have been largely ignored in the literature. This paper considers the frequently used Vold-Kalman filter-based order tracking and computed order tracking techniques. The main pros and cons of each technique for Fourier analysis are discussed and the sequential use of Vold-Kalman filtering and computed order tracking is proposed as a novel idea to enhance the results of Fourier analysis for determining the order components. The advantages of the combined use of these order tracking techniques are demonstrated numerically on an SDOF rotor simulation model. Finally, the approach is also demonstrated on experimental data from a real rotating machine.

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Principles of Fourier Analysis

Cited by 57 publications

References 0 publications

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

Behavioral modeling of nonlinear transfer systems with load-dependent <i>X</i>-parameters

The combined use of order tracking techniques for enhanced Fourier analysis of order components

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Principles of Fourier Analysis

Cited by 57 publications

References 0 publications

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

Behavioral modeling of nonlinear transfer systems with load-dependent &lt;i&gt;X&lt;/i&gt;-parameters

The combined use of order tracking techniques for enhanced Fourier analysis of order components

Contact Info

Product

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Behavioral modeling of nonlinear transfer systems with load-dependent <i>X</i>-parameters