\section{Theoretical Background}
The following subchapters shall equip the reader with the theoretical foundations of digital signal processing to better understand the following implementation of ANR on a low-power signal processor.\\ \\
We will beginn with the fundamentals of digital signal processing in general, covering the transfer-funtions and filters.\\
To fully understand ANR, a short deep-dive into the concepts of Finite Impulse Respone (FIR) and Infinite Impulse Respone (IIR) filters is indispensable.\\
From this point we will continue into the history and the mathematical concepts of ANR, its realtime feedback possibilities and its use of the Least Mean Square (LMS) Algorithm.\\
With this knowledge covered, we will design a realstic signal flow diagram and  the corresponding transfer functions, of an implanted CI system essential to implement a functioning ANR on a low-power DSP.\\
At the end of chapter two, high-level Python simualtions shall function as a practical demonstrations of the recently presented thereotical background.\\ \\
Chapter 2 is relying on the textbook ''Digital Signal Processing Fundamentals and Applications 2nd Ed'' by Tan and Jiang \cite{source_dsp1}.
\subsection{Fundamentals of digital signal processing, transfer functions and filters}
Digital Signal Processing (DSP) describes the manipulation of an analog signal trough mathematical approaches after it has been recorded and converted into a digital form. Nearly every part of the modern daily live, be it communication via cellphones, X-Ray imaging or picture editing, is affected by DSP.\\ \\
\begin{figure}[H]
    \centering
    \includegraphics[width=0.8\linewidth]{Bilder/fig_dsp.jpg}
    \caption{Block diagram of processing an analog input signal to an analog output signal with digital signal processing in between  \cite{source_fig_dsp}}
    \label{fig:fig_dsp}
\end{figure}
Before digital signal processing can be applied to an analog signal like voice, several steps are required beforehand. An analog signal, continous in both time and amplitude, is passed through a initial filter, which limits the freqency bandwith. An analog-digital converter then samples and quantizies the signal into a digital form, now discrete in time and amplitude. This digital signal can now be processed, before (possibly) being converted to a analog signal again. (refer to figure \ref{fig:fig_dsp}).\\ \\
A signal (either analog or digital) can be displayed and analyzed in two ways: the time spectrum and the freqency spectrum. The time spectrum shows the amplitude of the signal over time - like the sine waves from figure XXX. If a fast fourier transformation (FFT) is applied to the signal in the time spectrum, we recieve the same signal in the frequency spectrum, now showing the frequencies present in the signal (refer to figure \ref{fig:fig_fft}).\\ \\
\begin{figure}[H]
    \centering
    \includegraphics[width=0.8\linewidth]{Bilder/fig_fft.jpg}
    \caption{Sampled digital signal in the time spectrum and in the frequency spectrum  \cite{source_fig_fft}}
    \label{fig:fig_fft}
\end{figure}

\subsection{Explanation of Finite Impulse Response- and Infinite Impulse Response-filters}
\subsection{Introduction to Adaptive Nose Reduction}

\subsection{Introduction to the Least Mean Square algorithm for adaptive filtering}
\subsection{Signal flow diagram showing the origin of the useful signal,
noise signal, and their coupling}
\subsection{Derivation of the system’s transfer function based on the problem setup}
\subsection{Example applications and high-level simulations using Python}
