2.2

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}