Numerical error analysis of the ICZT algorithm for chirp contours on the unit circle

Sukhoy, Vladimir; Stoytchev, Alexander

doi:10.1038/s41598-020-60878-7

Cited by 8 publications

(5 citation statements)

References 29 publications

(45 reference statements)

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“…The best approximations for irrational numbers can be found using the Stern-Brocot tree [7]. Recent research has connected the components of a Farey sequence to the singularities of the Inverse Chirp Z-Transform [14], which is a generalization of the Inverse Fast Fourier Transform [15].The Farey sequence can be shown using Ford circles. The Stern-Brocot tree, which was created by removing unnecessary branches, has a subtree defined by the Farey sequence 𝐹𝐹 � [7].…”

Section: Resultsmentioning

confidence: 99%

Some Characteristics of the Farey Sequences with Ford Circles

et al. 2023

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Farey sequence is a pattern of rational numbers that approximates irrational numbers. In this paper, we use the Farey sequence to describe the Ford Circles. Its results and applications are equally fascinating as its pattern. Hurwitz Theorem is the main outcome of the approximation of irrational by rational numbers. Also, we examine the relationship between the Ford circle and the Farey sequence for rational values between 0 and 1.

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Section: Resultsmentioning

confidence: 99%

Some Characteristics of the Farey Sequences with Ford Circles

et al. 2023

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“…Traditional numerical analysis methods examine numeric errors and algorithm stability problems [172]. Many researchers reviewed various numeric analysis studies and how algorithms produce varying results due to numeric errors [172][173][174][175]. The major challenge lies in fundamental theories for capturing random numerical errors with unknown factors underlying hardware and software designs [176][177][178].…”

Section: Computational Reliabilitymentioning

confidence: 99%

A Critical Review for Trustworthy and Explainable Structural Health Monitoring and Risk Prognosis of Bridges with Human-In-The-Loop

et al. 2023

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Trustworthy and explainable structural health monitoring (SHM) of bridges is crucial for ensuring the safe maintenance and operation of deficient structures. Unfortunately, existing SHM methods pose various challenges that interweave cognitive, technical, and decision-making processes. Recent development of emerging sensing devices and technologies enables intelligent acquisition and processing of massive spatiotemporal data. However, such processes always involve human-in-the-loop (HITL), which introduces redundancies and errors that lead to unreliable SHM and service safety diagnosis of bridges. Comprehending human-cyber (HC) reliability issues during SHM processes is necessary for ensuring the reliable SHM of bridges. This study aims at synthesizing studies related to HC reliability for supporting the trustworthy and explainable SHM of bridges. The authors use a bridge inspection case to lead a synthesis of studies that examined techniques relevant to the identified HC reliability issues. This synthesis revealed challenges that impede the industry from monitoring, predicting, and controlling HC reliability in bridges. In conclusion, a research road map was provided for addressing the identified challenges.

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“…For the special case when the magnitudes of both A and W are equal to 1, the contour is restricted to lie on the unit circle in the complex plane. In this case, the ICZT has a singularity 5 if and only if the polar angle of W can be expressed as where is an element of . Consequently, the numerical error profile for the ICZT of size n is determined 5 by the elements of .…”

Section: Related Workmentioning

confidence: 99%

“…In this case, the ICZT has a singularity 5 if and only if the polar angle of W can be expressed as where is an element of . Consequently, the numerical error profile for the ICZT of size n is determined 5 by the elements of . Therefore, the number of possible values of the parameter W for which the transform is singular is equal to the length of .…”

Section: Related Workmentioning

confidence: 99%

“…These sequences are also tied to the Stern–Brocot tree, which could be used to find the best rational approximations for irrational numbers 4 . Recently, the elements of a Farey sequence have been linked to the singularities 5 of the Inverse Chirp Z-Transform (ICZT), which is a generalization 6 of the Inverse Fast Fourier Transform (IFFT).…”

Section: Introductionmentioning

confidence: 99%

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Formulas and algorithms for the length of a Farey sequence

Sukhoy

Stoytchev

2021

Sci Rep

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This paper proves several novel formulas for the length of a Farey sequence of order n. The formulas use different trade-offs between iteration and recurrence and they range from simple to more complex. The paper also describes several iterative algorithms for computing the length of a Farey sequence based on these formulas. The algorithms are presented from the slowest to the fastest in order to explain the improvements in computational techniques from one version to another. The last algorithm in this progression runs in $$O(n^{2/3})$$ O ( n 2 / 3 ) time and uses only $$O(\sqrt{n})$$ O ( n ) memory, which makes it the most efficient algorithm for computing $$|F_n|$$ | F n | described to date. With this algorithm we were able to compute the length of the Farey sequence of order $$10^{18}$$ 10 18 .

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Numerical error analysis of the ICZT algorithm for chirp contours on the unit circle

Cited by 8 publications

References 29 publications

Some Characteristics of the Farey Sequences with Ford Circles

Some Characteristics of the Farey Sequences with Ford Circles

A Critical Review for Trustworthy and Explainable Structural Health Monitoring and Risk Prognosis of Bridges with Human-In-The-Loop

Formulas and algorithms for the length of a Farey sequence

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