118 research outputs found
The fractional orthogonal derivative
This paper builds on the notion of the so-called orthogonal derivative, where
an n-th order derivative is approximated by an integral involving an orthogonal
polynomial of degree n. This notion was reviewed in great detail in a paper in
J. Approx. Theory (2012) by the author and Koornwinder. Here an approximation
of the Weyl or Riemann-Liouville fractional derivative is considered by
replacing the n-th derivative by its approximation in the formula for the
fractional derivative. In the case of, for instance, Jacobi polynomials an
explicit formula for the kernel of this approximate fractional derivative can
be given. Next we consider the fractional derivative as a filter and compute
the transfer function in the continuous case for the Jacobi polynomials and in
the discrete case for the Hahn polynomials. The transfer function in the Jacobi
case is a confluent hypergeometric function. A different approach is discussed
which starts with this explicit transfer function and then obtains the
approximate fractional derivative by taking the inverse Fourier transform. The
theory is finally illustrated with an application of a fractional
differentiating filter. In particular, graphs are presented of the absolute
value of the modulus of the transfer function. These make clear that for a good
insight in the behavior of a fractional differentiating filter one has to look
for the modulus of its transfer function in a log-log plot, rather than for
plots in the time domain.Comment: 32 pages, 7 figures. The section between formula (4.15) and (4.20) is
correcte
Improved IIR Low-Pass Smoothers and Differentiators with Tunable Delay
Regression analysis using orthogonal polynomials in the time domain is used
to derive closed-form expressions for causal and non-causal filters with an
infinite impulse response (IIR) and a maximally-flat magnitude and delay
response. The phase response of the resulting low-order smoothers and
differentiators, with low-pass characteristics, may be tuned to yield the
desired delay in the pass band or for zero gain at the Nyquist frequency. The
filter response is improved when the shape of the exponential weighting
function is modified and discrete associated Laguerre polynomials are used in
the analysis. As an illustrative example, the derivative filters are used to
generate an optical-flow field and to detect moving ground targets, in real
video data collected from an airborne platform with an electro-optic sensor.Comment: To appear in Proc. International Conference on Digital Image
Computing: Techniques and Applications (DICTA), Adelaide, 23rd-25th Nov. 201
Recent history of fractional calculus
This survey intends to report some of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date
A signal processing view of differintegration
Proceedings of the European Control Conference, ECC’01, Porto, Portugal, September 2001The fractional differintegration problem is treated from the Signal Processing point of view. A brief review of the Laplace transform approach to differintegration is done. The continuoustime/ discrete-time system conversion is discussed and presented a Grünwald-Letnikov integration
Design of digital differentiators
A digital differentiator simply involves the derivation of an input signal. This work includes the presentation of first-degree and second-degree differentiators, which are designed as both infinite-impulse-response (IIR) filters and finite-impulse-response (FIR) filters. The proposed differentiators have low-pass magnitude response characteristics, thereby rejecting noise frequencies higher than the cut-off frequency. Both steady-state frequency-domain characteristics and Time-domain analyses are given for the proposed differentiators. It is shown that the proposed differentiators perform well when compared to previously proposed filters. When considering the time-domain characteristics of the differentiators, the processing of quantized signals proved especially enlightening, in terms of the filtering effects of the proposed differentiators. The coefficients of the proposed differentiators are obtained using an optimization algorithm, while the optimization objectives include magnitude and phase response. The low-pass characteristic of the proposed differentiators is achieved by minimizing the filter variance. The low-pass differentiators designed show the steep roll-off, as well as having highly accurate magnitude response in the pass-band. While having a history of over three hundred years, the design of fractional differentiator has become a ‘hot topic’ in recent decades. One challenging problem in this area is that there are many different definitions to describe the fractional model, such as the Riemann-Liouville and Caputo definitions. Through use of a feedback structure, based on the Riemann-Liouville definition. It is shown that the performance of the fractional differentiator can be improved in both the frequency-domain and time-domain. Two applications based on the proposed differentiators are described in the thesis. Specifically, the first of these involves the application of second degree differentiators in the estimation of the frequency components of a power system. The second example concerns for an image processing, edge detection application
Fractional - order system modeling and its applications
In order to control or operate any system in a closed-loop, it is important to know its behavior in the form of
mathematical models. In the last two decades, a fractional-order model has received more attention in system identification instead of classical integer-order model transfer function. Literature shows recently that some techniques on fractional calculus and fractional-order models have been presenting valuable contributions to real-world processes and achieved better results. Such new developments have impelled research into extensions of the classical identification techniques to advanced fields of science and engineering. This article surveys the recent methods in the field and other related challenges to implement the fractional-order derivatives and miss-matching with conventional science. The comprehensive discussion on available literature would help the readers to grasp the concept of fractional-order modeling and can facilitate future investigations. One can anticipate manifesting recent advances in fractional-order modeling in this paper and unlocking more opportunities for research
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