3.1,3.2,3.3

chapter_02.tex

@@ -266,17 +266,16 @@ The LMS algorithm therefore updates the filter coefficients $w[n]$ after every s
     \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 target signal $d[n]$ and the reference noise signal $x[n]$ are formed. Figure \ref{fig:fig_anr_implant} now shows a more complete and 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 receives the target- and noise signal over their respective transfer functions and outputs the corrupted target signal $d[n]$, which consists out of the recorded target signal $s[n]$ and the recorded corruption noise signal $n[n]$, whereas the noise signal sensor aims to receive (ideally) only the noise signal $v[n]$ over its transfer function and outputs the reference noise signal $x[n]$, which then feeds the adaptive filter.\\ \\
-Additionally, 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 $C_n$ with $A_n$ describe the path from the signal sources to their respective sensors inside the cochlear implant system. As the sources and the relative location of the user to the sources can vary, these transfer functions are time-variant and unknown. In the case of the noise signal, we establish the possibility, that the noise signal reaches the target signal sensor not only through the air, represented through the transfer function $C_n + A_n$, but also through mechanical vibrations, represented through the transfer function $C_n + B$. This circumstance, together with the fact, that the mechanical properties of CI system are fixed and therefore can be seen as time-invariant and known, allows us to apply an hybrid static/adaptive filter design for the ANR implementation, as described in chapter 2.5.2.\\ \\ 
-The corrupted target signal $d[n]$ can therefore be mathematically described as: 
+Additionally, now the relevant transfer functions of the overall system are illustrated in Figure \ref{fig:fig_anr_implant}. The transfer functions $C_n$, $D_n$, and $E_n$ describe the path from the signal sources to the cochlear implant system. As the sources, the relative location of the user to the sources and the medium bewteen them can vary, these transfer functions are time-variant and unknown. After the signals reached the implant systems, we establish the possibility, that the remaining path of the signals to their respective sensors inside the CI system is fixed and therefore can be seen as time-invariant and known. This knowen transfer functions, which are titled $A$ and $B$, allow us to apply an hybrid static/adaptive filter design for the ANR implementation, as described in chapter 2.5.2.\\ \\ 
 \begin{equation}
 \label{equation_dn}
- d[n] = s[n] + n[n] = t[n] * (D_n) + v[n] * (F_n)
+ d[n] = s[n] + n[n] = t[n] * (C_nA) + v[n] * (D_nA)
 \end{equation}
 where  $t[n]$ and $v[n]$ are the target- and noise signals at their respective source, $s[n]$ is the recorded target signal and $v[n]$ is the recorded corruption noise 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_n * (A_n + B))
+x[n] = v[n] * (E_nB)
 \end{equation}
 where $v[n]$ is the noise signal at its source and $x[n]$ is the recorded reference 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 the partial removal of the target signal from the output signal. This case is not illustrated in Figure \ref{fig:fig_anr_implant} as it won't be further evaluated in this thesis, but shall be mentioned for the sake of completeness.\\ \\