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/***********************************************/
/**
* @file digitalFilterIntegral.h
*
* @brief Numerical integration using polynomial approximation.
*
* @author Andreas Kvas
* @date 2016-06-21
*
*/
/***********************************************/
#ifndef __GROOPS_DIGITALFILTERINTEGRAL__
#define __GROOPS_DIGITALFILTERINTEGRAL__
// Latex documentation
#ifdef DOCSTRING_DigitalFilter
static const char *docstringDigitalFilterIntegral = R"(
\subsection{Integral}
Numerical integration using polynomial approximation.
The input time series is approximated by a moving polynomial of degree \config{polynomialDegree}
by solving
\begin{equation}
\begin{bmatrix} x(t_k+\tau_0) \\ \vdots \\ x(t_k+\tau_M) \end{bmatrix}
=
\begin{bmatrix}
1 & \tau_0 & \tau_0^2 & \cdots & \tau_0^M \\
\vdots & \vdots & \vdots & & \vdots \\
1 & \tau_M & \tau_M^2 & \cdots & \tau_M^M \\
\end{bmatrix}%^{-1}
\begin{bmatrix}
a_0 \\ \vdots \\ a_M
\end{bmatrix}
\qquad\text{with}\quad
\tau_j = (j-M/2)\cdot \Delta t,
\end{equation}
for each time step $t_k$ ($\Delta t$ is the \config{sampling} of the time series).
The numerical integral for each time step $t_k$ is approximated by the center interval of the estimated polynomial.
\fig{!hb}{0.7}{DigitalFilter_integral}{fig:DigitalFilterIntegral}{Numerical integration by polynomial approximation.}
\config{polynomialDegree} should be even to avoid a phase shift.
)";
#endif
/***********************************************/
#include "classes/digitalFilter/digitalFilter.h"
/***** CLASS ***********************************/
/** @brief Numerical integration using polynomial approximation.
* @ingroup digitalFilterGroup
* @see DigitalFilter */
class DigitalFilterIntegral : public DigitalFilterARMA
{
public:
DigitalFilterIntegral(Config &config);
};
/***********************************************/
/***** Inlines *********************************/
/***********************************************/
inline DigitalFilterIntegral::DigitalFilterIntegral(Config &config)
{
try
{
UInt degree;
Double dt;
readConfig(config, "polynomialDegree", degree, Config::MUSTSET, "7", "degree of approximation polynomial");
readConfig(config, "sampling", dt, Config::DEFAULT, "1.0", "assumed time step between points");
readConfig(config, "padType", padType, Config::MUSTSET, "", "");
if(isCreateSchema(config)) return;
Matrix F(degree+1, degree+1);
Matrix D(degree+1, degree+1);
Vector v(degree+1);
for(UInt j=0; j<F.rows(); j++)
{
D(j, j) = 1.0/std::pow(dt, j);
v(j) = dt/(j+1.);
F(j,0) = 1.;
for(UInt n=1; n<F.columns(); n++)
F(j,n) = std::pow(0.5*degree-j, n);
}
inverse(F);
bnStartIndex = bn.rows()/2;
bn = Vector(degree+1);
matMult(1.0, v.trans(), D*F, bn.trans());
an = Vector(2);
an(0) = 1.0; an(1) = -1.0;
}
catch(std::exception &e)
{
GROOPS_RETHROW(e)
}
}
/***********************************************/
#endif
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