\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 topics like signals, transfer-functions 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 real-time feedback possibilities and its use of the Least Mean Square (LMS) Algorithm.\\
With this knowledge covered, we will design a realistic 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 simulations shall function as a practical demonstration of the recently presented theoretical 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}
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.
\subsubsection{Signals}
\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_dsp_ch1}}
    \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, continuous in both time and amplitude, is passed through an initial filter, which limits the frequency bandwidth. An analog-digital converter then samples and quantities the signal into a digital form, now discrete in time and amplitude. This digital signal can now be processed, before (possibly) being converted to an analog signal again. (refer to Figure \ref{fig:fig_dsp}). The sampling rate definies, in how many samples per secotnd are taken from the analog signal - a higher sample rate delivers a more accurate digital representation of the signal but also uses more ressources. According to the Nyquist–Shannon sampling theorem, the sample rate must be at least twice the highest frequency component present in the signal to avoid distorions of the signal.\\ \\
Throughout this thesis, sampled signals are denoted in lowercase with square brackets (e.g. {x[n]}) to distinguish them from continuous-time signals 
(e.g. {x(t)}).\\ 
The discrete digital signal can be viewed as a sequence of finite samples with  it´s amplitude being a discrete value, like a 16- or 32-bit integer. A signal vector of the length N, containing N samples, is therefore notated as
\begin{equation}
\label{equation1}
 x[n] = [x[n-N+1],x[n-N+2],...,x[n-1],x[n]]
\end{equation}
where x[n] is the current sample and x[n-1] is the preceding sample. 
\subsubsection{Time domain vs. frequency domain}
A signal (either analog or digital) can be displayed and analyzed in two ways: the time spectrum and the frequency spectrum. The time spectrum shows the amplitude of the signal over time - like the sine waves from Figure \ref{fig:fig_interference}. If a fast Fourier transformation (FFT) is applied to the signal in the time spectrum, we receive 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_dsp_ch1}}
    \label{fig:fig_fft}
\end{figure}
\subsubsection{Transfer Functions and filters}
When we discuss signals in a mathematical way, we need to explain the term ''transfer function''. A transfer function is a mathematical representation of an abstract system that describes how an input signal is transformed into an output signal. This could mean a simple amplification or a phase shift applied to an input signal.
\begin{figure}[H]
    \centering
    \includegraphics[width=0.8\linewidth]{Bilder/fig_transfer.jpg}
    \caption{Simple representation of a transfer function taking a noisy input signal and delivering a clean output signal  \cite{source_dsp_ch1}}
    \label{fig:fig_transfer}
\end{figure}
In digital signal processing, especially in the design of a noise reduction algorithm, transfer functions are essential for modeling and analyzing filters, amplifiers, and the pathway of the signal itself. By understanding a system’s transfer function, one can predict how sound signals are altered and therefore how filter parameters can be adapted to deliver the desired output signal.\\ \\
During the description of transfer functions, the term ''filter'' was used but not yet defined. A filter can be understood as a component in signal processing, designed to modify or extract specific parts of a signal by selectively allowing certain frequency ranges to pass while attenuating others. Filters can be static, meaning they always extract the same portion of a signal, or adaptive, meaning they change their filtering-behavior over time according to their environment. Examples for static filter include low-pass-, high-pass-, band-pass- and band-stop filters, each tailored to isolate or remove particular frequency content (refer to Figure \ref{fig:fig_lowpass}).
\begin{figure}[H]
    \centering
    \includegraphics[width=0.8\linewidth]{Bilder/fig_lowpass.jpg}
    \caption{Behavior of a low-pass-filter.\cite{source_dsp_ch2}}
    \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).
\subsubsection{Finite Impulse Response-filters}
\subsubsection{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}
