Discrete smoothing splines and digital filtration. Theory and applications

Jiří Hřebíček; František Šik; Vítězslav Veselý

Aplikace matematiky (1990)

  • Volume: 35, Issue: 1, page 28-50
  • ISSN: 0862-7940

Abstract

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Two universally applicable smoothing operations adjustable to meet the specific properties of the given smoothing problem are widely used: 1. Smoothing splines and 2. Smoothing digital convolution filters. The first operation is related to the data vector r = ( r 0 , . . . , r n - 1 ) T with respect to the operations 𝒜 , and to the smoothing parameter α . The resulting function is denoted by σ α ( t ) . The measured sample r is defined on an equally spaced mesh Δ = { t i = i h } i = 0 n - 1 T = n h . The smoothed data vector y is then y = { σ α ( t i ) } i = 0 n - 1 . The other operation gives y E n computed by 𝐲 = 𝐡 * 𝐫 , where * stands for the discrete convolution, the running weighted mean by h . The main aims of the present contribution: to prove the existence of close interconnection between the two smoothing approaches (Cor. 2.6 and [11]), to develop the transfer function, which characterizes the smoothing spline as a filter in terms of α and λ i k (the eigenvalues of the discrete analogue of C a l L ) (Th. 2.5), to develop the reduction ratio between the original and the smoothed data in the same terms (Th. 3.1).

How to cite

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Hřebíček, Jiří, Šik, František, and Veselý, Vítězslav. "Discrete smoothing splines and digital filtration. Theory and applications." Aplikace matematiky 35.1 (1990): 28-50. <http://eudml.org/doc/15608>.

@article{Hřebíček1990,
abstract = {Two universally applicable smoothing operations adjustable to meet the specific properties of the given smoothing problem are widely used: 1. Smoothing splines and 2. Smoothing digital convolution filters. The first operation is related to the data vector $r=\{(r_0,..., r_\{n-1\})\}^T$ with respect to the operations $\mathcal \{A\}$, $\mathcal \{L\}$ and to the smoothing parameter $\alpha $. The resulting function is denoted by $\sigma _\alpha (t)$. The measured sample $r$ is defined on an equally spaced mesh $\Delta =\lbrace t_i=ih\rbrace ^\{n-1\}_\{i=0\}$$T=nh$. The smoothed data vector $y$ is then $y=\lbrace \sigma _\alpha (t_i)\rbrace ^\{n-1\}_\{i=0\}$. The other operation gives $y\in E^n$ computed by $\mathbf \{y=h*r\}$, where $\mathbf \{*\}$ stands for the discrete convolution, the running weighted mean by $h$. The main aims of the present contribution: to prove the existence of close interconnection between the two smoothing approaches (Cor. 2.6 and [11]), to develop the transfer function, which characterizes the smoothing spline as a filter in terms of $\alpha $ and $\lambda _\{ik\}$ (the eigenvalues of the discrete analogue of $Cal \{L\}$) (Th. 2.5), to develop the reduction ratio between the original and the smoothed data in the same terms (Th. 3.1).},
author = {Hřebíček, Jiří, Šik, František, Veselý, Vítězslav},
journal = {Aplikace matematiky},
keywords = {discrete smoothing spline CDS-spline; smoothing parameter; digital convolution filter; transfer function; sinusoidal wave; saw-like waves; rectangular pulse train; smoothing parameter; digital convolution filter; discrete smoothing splines; digital filtration; transfer function; sinusoidal wave; saw-like waves; rectangular pulse train},
language = {eng},
number = {1},
pages = {28-50},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Discrete smoothing splines and digital filtration. Theory and applications},
url = {http://eudml.org/doc/15608},
volume = {35},
year = {1990},
}

TY - JOUR
AU - Hřebíček, Jiří
AU - Šik, František
AU - Veselý, Vítězslav
TI - Discrete smoothing splines and digital filtration. Theory and applications
JO - Aplikace matematiky
PY - 1990
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 35
IS - 1
SP - 28
EP - 50
AB - Two universally applicable smoothing operations adjustable to meet the specific properties of the given smoothing problem are widely used: 1. Smoothing splines and 2. Smoothing digital convolution filters. The first operation is related to the data vector $r={(r_0,..., r_{n-1})}^T$ with respect to the operations $\mathcal {A}$, $\mathcal {L}$ and to the smoothing parameter $\alpha $. The resulting function is denoted by $\sigma _\alpha (t)$. The measured sample $r$ is defined on an equally spaced mesh $\Delta =\lbrace t_i=ih\rbrace ^{n-1}_{i=0}$$T=nh$. The smoothed data vector $y$ is then $y=\lbrace \sigma _\alpha (t_i)\rbrace ^{n-1}_{i=0}$. The other operation gives $y\in E^n$ computed by $\mathbf {y=h*r}$, where $\mathbf {*}$ stands for the discrete convolution, the running weighted mean by $h$. The main aims of the present contribution: to prove the existence of close interconnection between the two smoothing approaches (Cor. 2.6 and [11]), to develop the transfer function, which characterizes the smoothing spline as a filter in terms of $\alpha $ and $\lambda _{ik}$ (the eigenvalues of the discrete analogue of $Cal {L}$) (Th. 2.5), to develop the reduction ratio between the original and the smoothed data in the same terms (Th. 3.1).
LA - eng
KW - discrete smoothing spline CDS-spline; smoothing parameter; digital convolution filter; transfer function; sinusoidal wave; saw-like waves; rectangular pulse train; smoothing parameter; digital convolution filter; discrete smoothing splines; digital filtration; transfer function; sinusoidal wave; saw-like waves; rectangular pulse train
UR - http://eudml.org/doc/15608
ER -

References

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