scholarly journals Testing Stability of Digital Filters Using Optimization Methods with Phase Analysis

Energies ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1488
Author(s):  
Damian Trofimowicz ◽  
Tomasz P. Stefański
Keyword(s):  
Unit Circle ◽  
Phase Analysis ◽  
Characteristic Equation ◽  
Digital Filter ◽  
Digital Filters ◽  
Computing Time ◽  
Complex Function ◽  
Stochastic Methods ◽  
Final Population ◽  
Candidate Regions

In this paper, novel methods for the evaluation of digital-filter stability are investigated. The methods are based on phase analysis of a complex function in the characteristic equation of a digital filter. It allows for evaluating stability when a characteristic equation is not based on a polynomial. The operation of these methods relies on sampling the unit circle on the complex plane and extracting the phase quadrant of a function value for each sample. By calculating function-phase quadrants, regions in the immediate vicinity of unstable roots (i.e., zeros), called candidate regions, are determined. In these regions, both real and imaginary parts of complex-function values change signs. Then, the candidate regions are explored. When the sizes of the candidate regions are reduced below an assumed accuracy, then filter instability is verified with the use of discrete Cauchy’s argument principle. Three different algorithms of the unit-circle sampling are benchmarked, i.e., global complex roots and poles finding (GRPF) algorithm, multimodal genetic algorithm with phase analysis (MGA-WPA), and multimodal particle swarm optimization with phase analysis (MPSO-WPA). The algorithms are compared in four benchmarks for integer- and fractional-order digital filters and systems. Each algorithm demonstrates slightly different properties. GRPF is very fast and efficient; however, it requires an initial number of nodes large enough to detect all the roots. MPSO-WPA prevents missing roots due to the usage of stochastic space exploration by subsequent swarms. MGA-WPA converges very effectively by generating a small number of individuals and by limiting the final population size. The conducted research leads to the conclusion that stochastic methods such as MGA-WPA and MPSO-WPA are more likely to detect system instability, especially when they are run multiple times. If the computing time is not vitally important for a user, MPSO-WPA is the right choice, because it significantly prevents missing roots.

scholarly journals Reducing the Impact of the Frequency Change on the Formation of Orthogonal Components of the Relay Protection Input Signals

2020 ◽  
Vol 63 (1) ◽  
pp. 42-54 ◽  
Author(s):  
E. A. Romaniuk ◽  
V. Yu. Rumiantsev ◽  
Yu. V. Rumiantsev ◽  
A. A. Dziaruhina
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Digital Filter ◽  
Input Signal ◽  
Digital Filters ◽  
Signal Frequency ◽  
Filter Model ◽  
Input Signal Frequency ◽  
Orthogonal Components ◽  
The Impact

Digital filters made with the use of discrete Fourier Transform are applied in most microprocessor protections produced both in the home country and abroad. When the input signal frequency deviates from the value to which these filters are configured, a signal is generated at their output with oscillation amplitude that is proportional to the deviation of the signal frequency from the specified one. The article proposes an algorithm for compensating the oscillations of orthogonal components of the output signals of digital filters implemented on the basis of a discrete Fourier transform, when the input signal frequency deviates from the nominal one. A mathematical model of the proposed digital filter with an algorithm for compensating the oscillations of its orthogonal components, as well as a signal model for reproducing input effects, is implemented in the MatLab-Simulink dynamic modeling environment. The digital filter model is provided with two channels, viz. a current channel and a voltage channel, which makes it possible to simulate their operation in relation to protections that use one or two input values, for example, for current and remote protection. Verification of the functioning of the digital filter model with compensation for fluctuations in its output signal was carried out with the use of two types of test effects, viz. a sinusoidal signal with a frequency of 48–51 Hz (idealized effect), and the effects that are close to the real secondary signals of measuring current transformers and voltage transformers in case of short circuits accompanied by a decrease in frequency. The conducted computational experiments with deviation of frequency from the nominal one, revealed the presence of undamped oscillations at the output of standard digital Fourier filters and their almost complete absence in the proposed digital filters. This makes us possible to recommend digital filters based on a discrete Fourier transform supplemented by an algorithm for compensation of fluctuations in the amplitudes of the output signals for the use in microprocessor protection.

Research on Digital Filters for Si Wafer Surface Profile Measurement - Design of Filters by Total Variation

2012 ◽  
Vol 565 ◽  
pp. 656-661
Author(s):  
Hirotaka Ojima ◽  
Kazutaka Nonomura ◽  
Li Bo Zhou ◽  
Jun Shimizu ◽  
Teppei Onuki
Keyword(s):  
Total Variation ◽  
Digital Filter ◽  
Digital Filters ◽  
Large Population ◽  
Surface Profile ◽  
Wafer Surface ◽  
Profile Measurement ◽  
Actual Measurement ◽  
True Value ◽  
New Tv

The underlying data form of a wafer is a matrix of length (or height) measurements. In the presence of noise, evaluation parameters are normally biased. The expectation value such as peak-to-valley and GBIR (global backside ideal range) is systematically larger than the “true” value. Correction and compensation need a large population of measurements to analytically estimate both bias and the uncertainty. In this study, approach to obtain the true value is to extract a “true” profile by filtering noise from the measured data. In previous paper, the digital filter with wavelet transformation (WT) is proposed and efficiency to remove the noise, however, the method is introduced the pseudo-Gibbs effect. Then, we propose the digital filter with new algorithm of total variation (TV). In this paper, the new algorithm of TV is proposed and the digital filter by new TV indicate that data is filtered without the pseudo-Gibbs effect. The digital filters by WT and new TV are applied on the sample data of actual measurement system to investigate their performance of noise reduction.

Stability-Guaranteed Two-Phase Design of Odd-Order Variable-Magnitude Digital Filters

2016 ◽  
Vol 26 (02) ◽  
pp. 1750033
Author(s):  
Tian-Bo Deng
Keyword(s):  
Transfer Function ◽  
Digital Filter ◽  
Digital Filters ◽  
Tuning Parameter ◽  
Second Phase ◽  
Variable Model ◽  
Odd Order ◽  
Order Variable ◽  
The Stability ◽  
Fifth Order

Guaranteeing the stability is one of the most critical issues in designing a variable recursive digital filter. In this paper, we first present an odd-order recursive variable model (transfer function) that is used for designing an odd-order variable-magnitude (VM) digital filter, and then we replace the original coefficients of the denominator of the odd-order transfer function with a set of new parameters. These new parameters can ensure that they can take arbitrary values without incurring instability of the designed odd-order VM filter. To make the VM filter coefficients variable, we find all the VM filter coefficients as polynomial functions of the tuning parameter, which includes two phases. The first phase designs a set of recursive digital filters with fixed coefficients (constant filters), and the second phase utilizes a curve-fitting scheme to represent each coefficient as a polynomial function. As a result, the VM filter coefficients become variable, and the proposed parameter-substitution-based denominator coefficients ensure the filter stability. This is the most important contribution of the parameter-substitution-based design scheme. This paper uses the fifth-order demonstrative example to verify the stability guarantee as well as the design accuracy of the obtained the fifth-order VM filter.

scholarly journals A Method of Increasing Digital Filter Performance Based on Truncated Multiply-Accumulate Units

2020 ◽  
Vol 10 (24) ◽  
pp. 9052
Author(s):  
Pavel Lyakhov ◽  
Maria Valueva ◽  
Georgii Valuev ◽  
Nikolai Nagornov
Keyword(s):  
Signal Processing ◽  
Theoretical Analysis ◽  
Digital Signal Processing ◽  
Digital Filter ◽  
Digital Filters ◽  
Digital Signal ◽  
Digital Filtering ◽  
Filter Performance ◽  
Filter Architecture ◽  
Hardware Costs

This paper proposes new digital filter architecture based on a modified multiply-accumulate (MAC) unit architecture called truncated MAC (TMAC), with the aim of increasing the performance of digital filtering. This paper provides a theoretical analysis of the proposed TMAC units and their hardware simulation. Theoretical analysis demonstrated that replacing conventional MAC units with modified TMAC units, as the basis for the implementation of digital filters, can theoretically reduce the filtering time by 29.86%. Hardware simulation showed that TMAC units increased the performance of digital filters by up to 10.89% compared to digital filters using conventional MAC units, but were associated with increased hardware costs. The results of this research can be used in the theory of digital signal processing to solve practical problems such as noise reduction, amplification and suppression of the frequency spectrum, interpolation, decimation, equalization and many others.

scholarly journals DETECTION OF CHAOS IN SOME LOCAL REGIONS OF PHASE PORTRAITS USING SHANNON ENTROPIES

2004 ◽  
Vol 14 (04) ◽  
pp. 1493-1499 ◽  
Author(s):  
BINGO WING-KUEN LING ◽  
CHARLOTTE YUK-FAN HO ◽  
PETER KWONG-SHUN TAM
Keyword(s):  
Unit Circle ◽  
Digital Filters ◽  
Second Order ◽  
The State ◽  
Phase Portraits ◽  
Type Ii ◽  
State Variables ◽  
Type Iii ◽  
Special Values ◽  
Symbolic Sequences

This letter demonstrates the use of Shannon entropies to detect chaos exhibited in some local regions on the phase portraits. When both eigenvalues of the second-order digital filters with two's complement arithmetic are outside the unit circle, the Shannon entropies of the state variables are independent of the initial conditions and the filter parameters, except for some special values of the filter parameters. At these special values, the Shannon entropies of the state variables are relatively small. The state trajectories corresponding to these filter parameters either exhibit random-like chaotic behaviors in some local regions or converge to some fixed points on the phase portraits. Hence, by measuring the Shannon entropies of the state variables, these special state trajectory patterns can be detected. For completeness, we extend the investigation to the case when the eigenvalues of the second-order digital filters with two's complement arithmetic are complex and are inside or on the unit circle. It is found that the Shannon entropies of the symbolic sequences for the type II trajectories may be higher than that for the type III trajectories, even though the symbolic sequences of the type II trajectories are periodic and have limit cycle behaviors, while that of the type III trajectories are aperiodic and have chaotic behaviors.

The Application of Digital Filters to the Analysis of Ge and Si(Li) Detector X-Ray Spectra

1979 ◽  
Vol 23 ◽  
pp. 125-131
Author(s):  
L. A. Rayburn
Keyword(s):  
Mathematical Model ◽  
Least Squares ◽  
Digital Filter ◽  
Digital Filters ◽  
Least Squares Method ◽  
Linear Least Squares ◽  
Raw Data ◽  
X Ray ◽  
Peak Structure

AbstractOne of the uncertain aspects in the analysis of x-ray spectra is the determination of the proper background to subtract from the raw data. In those cases where the background is a smoothly varying funct ion of the x-ray energy, the application of a digital filter to the raw data will effectively remove the background leaving only the filtered peak information. These filtered peaks can then be fit by using a non-linear least squares method in conjunction with a suitably chosen mathematical model of the peak structure.

Composition Operators

1968 ◽  
Vol 20 ◽  
pp. 442-449 ◽  
Author(s):  
Eric A. Nordgren
Keyword(s):  
Holomorphic Function ◽  
Unit Circle ◽  
Composition Operators ◽  
Complex Function ◽  
Inner Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Inner Functions ◽  
Almost Everywhere ◽  
Image Position

The object of this note is to report on some of the properties of a class of operators induced by inner functions. If m is normalized Lebesgue measure on the unit circle X in the complex plane and Cϕ is an inner function (a complex function on X of unit modulus almost everywhere whose Poisson integral is a non-constant holomorphic function in the open unit disk), then an operator Cϕ on L2(m) is defined by

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