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zugtMR9&hBdc{nlmdnum%7OoAqOZk&T&f2EK{6NavoMV9IQCk5!>)~?yGyhj3x|<{~ zdBaiP&h~SNy4vBqR2j z=0h0&`tD>8)E_RgKs2^&e`FV#X2tWsh`-0b3RQ9h9Zr@gh>unvO24PL>{|&#jZs-azyQ0!4TryA18%tNgRGJl!Y* z6|SJ`E|PaB!yPCG#T7+IC&kI>{gl&qC(IwezD+Pu{{j#1 B_i_LL literal 0 HcmV?d00001 diff --git a/chapter_02.tex b/chapter_02.tex index 09f421d..5cb1502 100644 --- a/chapter_02.tex +++ b/chapter_02.tex @@ -49,12 +49,32 @@ During the description of transfer functions, the term ''filter'' was used but n \label{fig:fig_lowpass} \end{figure} Examples for an adaptive filter is the Least-Mean-Square-Algorithm used for adaptive noise reduction, which will be introduced in the following chapters. -\subsection{Finite Impulse Response- and Infinite Impulse Response-filters} -Before we continue with the introduction to the actual topic of this thesis, ANR, two very essential filter designs need further explanation - the Finite Impulse Response- (FIR) and Infinite Impulse Response-filters (IIR). +\subsection{Filter designs} +Before we continue with the introduction to the actual topic of this thesis, ANR, two very essential filter designs need further explanation - the Finite Impulse Response- (FIR) and Infinite Impulse Response-filters (IIR). \subsubsection{Finite Impulse Response-filters} -A Finite Impulse Respone Filter, commonly referred to as a ''Feedforward Filter'' is defined through the property, that it uses only present and past input values and not feedback from output samples - therefore the response of a FIR-filter reaches zero after a finite number of samples. Due to the fact, that there is no feedback, a FIR-filter offers unconditional stability, meaning that the filter response can´t diverge in any case. A disatavante to the FIR-filter desgin is the relatively slow frequency reaction compared to its IIR counterpart. +A Finite Impulse Respone filter, commonly referred to as a ''Feedforward Filter'' is defined through the property, that it uses only present and past input values and not feedback from output samples - therefore the response of a FIR-filter reaches zero after a finite number of samples. Due to the fact, that there is no feedback, a FIR-filter offers unconditional stability, meaning that the filter response converges, no matter how the coefficients are set. A disadvantage to the FIR-desgin is the relatively slow frequency response compared to its IIR counterpart. +\begin{equation} +\label{equation_fir} + y[n] = \sum_{k=1}^{M} b_kx[n-k] +\end{equation} +\begin{figure}[H] + \centering + \includegraphics[width=0.8\linewidth]{Bilder/fig_fir.jpg} + \caption{FIR-filter} + \label{fig:fig_fir} +\end{figure} \subsubsection{Infinite Impulse Response-filters} -A Ininite Impulse Respone Filter, commonly referred to as a ''Feedback Filter'' does, in contrary to its FIR-counterpart, use past output samples in addition to current and past input samples - hterefore the response of a IIR-filter theoretically continues indefinitely. +A Ininite Impulse Respone filter, commonly referred to as a ''Feedback Filter'' does, in contrary to its FIR-counterpart, use past output samples in addition to current and past input samples - therefore the response of an IIR-filter theoretically continues indefinitely. This recursive nature allows IIR filter to achieve a sharp frequency response with significantly fewer coefficients than an equivalent FIR filter but it also opens up the possibility, that the filter-response diverges, depending on the set coefficients. +\begin{equation} +\label{equation_iir} + y[n] = \sum_{k=1}^{M} b_kx[n-k] - \sum_{k=1}^{N} a_ky[n-k] +\end{equation} +\begin{figure}[H] + \centering + \includegraphics[width=0.8\linewidth]{Bilder/fig_iir.jpg} + \caption{IIR-filter} + \label{fig:fig_iir} +\end{figure} \subsection{Introduction to Adaptive Nose Reduction} diff --git a/drawio/FIR.drawio b/drawio/FIR.drawio index ba2fc24..1ee190a 100644 --- a/drawio/FIR.drawio +++ b/drawio/FIR.drawio @@ -1,14 +1,14 @@ - + - + - + - + @@ -25,9 +25,9 @@ - + - + @@ -36,7 +36,7 @@ - + @@ -45,7 +45,7 @@ - + @@ -54,9 +54,9 @@ - + - + @@ -71,12 +71,24 @@ - - + + + + + + + + + + + + + + diff --git a/drawio/IIR.drawio b/drawio/IIR.drawio index 39843ac..bef0c41 100644 --- a/drawio/IIR.drawio +++ b/drawio/IIR.drawio @@ -1,17 +1,17 @@ - + - + - + - - + + - + @@ -22,17 +22,6 @@ - - - - - - - - - - - @@ -40,24 +29,24 @@ - + - + - - + + - + - - + + - + - + @@ -66,13 +55,32 @@ - - - + + + + + + + + + + + + + + + + - - + + + + + + + +