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chapter_02.tex
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@@ -254,9 +254,33 @@ The given approach of the steepest decent algorithm in the subchapter above stil
The result of Equation \ref{equation_j_lms_final} can now be inserted into Equation \ref{equation_gradient} to receive the LMS update rule for the filter coefficients:
\begin{equation}
\label{equation_lms}
w[n+1] = w[n] + 2\mu e[n]x[n]
w[n+1] = w[n] - 2\mu e[n]x[n]
\end{equation}
The LMS algorithm therefore updates the filter coefficients $w[n]$ after every sample by adding a correction term, which is is calculated by the error signal $e[n]$ and the reference noise signal $x[n]$, scaled by the constant step size $\mu$. By iteratively applying the LMS algorithm, the filter coefficients converge towards the optimal values that minimize the mean squared error between the target signal and the filter output. When a predefined acceptable error level is reached, the adaptation process can be stopped to save computing power.\\ \\
\subsection{Signal flow diagram of an implanted cochlear implant system}
Now equipped with the necessary theoretical background about signal processing, adaptive noise reduction and the LMS algorithm, a realistic signal flow diagram wwith the relevant transfer functions of an implanted cochlear implant system can be designed, which will serve as the basis for the implementation of ANR on a low-power digitial signal processor.
\begin{figure}[H]
\centering
\includegraphics[width=1.1\linewidth]{Bilder/fig_anr_implant.jpg}
\caption{Realstic implant design.}
\label{fig:fig_anr_implant}
\end{figure}
\noindent Figure \ref{fig:fig_anr_hybrid} showed us the basic concept of an ANR implementation, without a detailed description how the corrupted targed signal and the reference noise signal is formed. Figure \ref{fig:fig_anr_implant} now shows a more realistic signal flow diagram of an implanted cochlear implant system, with two signal sensors and an adaptive noise reduction circuit afterwards. The target signal sensor recieves the corrupted target signal $d[n]$, which consists out of the speech signal $s[n]$ and the corruption-noise signal $n[n]$, whereas the noise signal sensor aims to receive (ideally) only the noise signal $x[n]$, which then feeds the adaptive filter.\\ \\
AAdittionaly, now the relevant transfer functions of the overall system are illustrated in Figure \ref{fig:fig_anr_implant}. The transfer functions $D_n$, $F_n$, and $C_n$ describe the path from the signal sources to the chasis of the cochlear implant, where the sensors are located. As the sources and the relative location of the user to the sources can vary, these transfer functions are time-variant and unknown. From the chasis, there are two options for continuing the signal path - either directly to the microphone membranes of the respective sensors, represented through the transfer function $G$, or through mechanical vibrations of the implant´s chasis, represented through the transfer functions $A$ and $B$. As the mechanical properties of the implanted cochlear systems are fixed, these transfer functions do not change over time, so they can be seen as time-invariant and known.\\ \\
The corrupted target signal $d[n]$ can thereforebe mathematically described as:
\begin{equation}
\label{equation_dn}
d[n] = s[n] + n[n] = t[n] * (D_nG) + v[n] * ((F_nG) + (C_nA))
\end{equation}
where $t[n]$ and $v[n]$ are the target- and noise signals at their respective source and $s[n]$ and $v[n]$ are the uncorrupted target- and noise signals after passing the transfer functions.\\ \\
The noise reference signal $x[n]$ can be mathematically described as:
\begin{equation}
\label{equation_xn}
x[n] = v[n] * (C_nB)
\end{equation}
where $v[n]$ is the noise signal at its source and $x[n]$ is the uncorrupted noise signal after passing the transfer functions.\\ \\
Another possible signal interaction could be the leakage of the target signal into the noise signal sensor, leading to undesired effects. This case is not illustrated in Figure \ref{fig:fig_anr_implant} as it wont be further evaluated in this thesis, but shall be mentioned for the sake of completeness.\\ \\
We assume at this point, that the corruption-noise signal is uncorellated to the speech signal, and therefore seperable from it. In addition, we asume, that the corruption-noise signal is correlated to the noise signal, as it originitaes from the same source, but takes a different signal path. \\ \\ The adaptive filter removes a certain, noise-related, frequency part of the input signal and re-evaluates the output through its feedback design. The filter parameters are then adjusted and applied to the next sample to minimize the observed error $e[n]$, which also represents the aproximated speech signal $š[n]$. In reality, a signal contamination of the two sensors has to be expected, which is represented through the transfer functions $H_{sd}$ and $H_{sx}$ in Figure \ref{fig:fig_anr_implant}. These transfer functions describe how much of the speech signal leaks into the target signal sensor and into the noise signal sensor respectively. This contamination can lead to undesired effects like signal distortion if not handled properly. Therefore, these transfer functions have to be taken into consideration when deriving the overall system´s transfer function.
\subsection{Derivation of the system’s transfer function based on the problem setup}
\subsection{Example applications and high-level simulations using Python}

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chapter_03.tex
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\section{Hardware and low-level simulation of different algorithm approaches}
\subsection{Hardware description}
\subsection{System setup}
\subsection{Low-level simulations of different algorithm approaches}
\section{High level simulations}

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1
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\include{chapter_01}
\include{chapter_02}
\include{chapter_03}
\include{chapter_04}
\printbibliography
\listoffigures