small lit fix
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@@ -5,7 +5,7 @@
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% Repetitive Box filter to approx Gauss
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% Simple multipass, n/m approach, extended box filter
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Digital filters are implemented by convolving the input signal with a filter kernel, i.e. the digital filter's impulse response.
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Consequently, the filter kernel of a Gaussian filter is a Gaussian with finite support \cite[120]{dspGuide1997}.
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Consequently, the filter kernel of a Gaussian filter is a Gaussian with finite support \cite{dspGuide1997}.
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Assuming a finite-support Gaussian filter kernel of size $M$ and a input signal $x$, discrete convolution produces the smoothed output signal
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\begin{equation}
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\label{eq:gausFilt}
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@@ -42,7 +42,7 @@ In general, it is desirable to omit a grid, as the data points do not necessary
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However, reducing the sample size by distributing the data on a equidistant grid can significantly reduce the computation time, if an approximative KDE is acceptable.
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Silverman \cite{silverman1982algorithm} originally suggested to combine adjacent data points into data bins, which results in a discrete convolution structure of the KDE.
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Allowing to efficiently compute the estimate using a FFT algorithm.
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This approximation scheme was later called binned KDE (BKDE) and was extensively studied \cite{fan1994fast} \cite{wand1994fast} \cite{hall1996accuracy} \cite{holmstrom2000accuracy}.
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This approximation scheme was later called binned KDE (BKDE) and was extensively studied \cite{fan1994fast} \cite{wand1994fast} \cite{hall1996accuracy}.
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While the FFT algorithm poses an efficient algorithm for large sample sets, it adds an noticeable overhead for smaller ones.
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The idea to approximate a Gaussian filter using several box filters was first formulated by Wells \cite{wells1986efficient}.
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