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chapter_02.tex
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@@ -94,7 +94,7 @@ Figure \ref{fig:fig_iir} visualizes a simple \ac{IIR} filter with two feedforwar
\caption{\ac{IIR} filter example with two feedforward operators and two feedback operators.}
\label{fig:fig_iir}
\end{figure}
\subsubsection{\ac{FIR}- vs. \ac{IIR}-filters}
\subsubsection{FIR- vs. IIR-filters}
Due to the fact, that there is no feedback, a \ac{FIR} filter offers unconditional stability, meaning that the filter response always converges, no matter how the coefficients are set. The disadvantages of the \ac{FIR} design is the relatively flat frequency response and the higher number of needed coefficients needed to achieve a sharp frequency response compared to its Infinite Impulse Response counterpart.\\ \\
The recursive nature of an \ac{IIR} filter, in contrary, allows achieving a sharp frequency response with significantly fewer coefficients than an equivalent \ac{FIR} filter, but it also opens up the possibility, that the filter response diverges, depending on the set coefficients.\\ \\
A higher number of needed coefficients implies, that the filter itself needs more time to complete its signal response, as the group delay is increased.
@@ -122,7 +122,7 @@ Although active noise cancellation and adaptive noise reduction share obvious si
\caption{The basic idea of an adaptive filter design for noise reduction.}
\label{fig:fig_anr}
\end{figure}
\noindent Figure \ref{fig:fig_anr} shows the basic concept of an adaptive filter design, represented through a feedback filter application. The primary sensor (top) aims to receive the desired signal and outputs the corrupted signal $d[n]$, which consists out of the recorded desired signal $s[n]$ and the recorded corruption noise signal $n[n]$, whereas the secondary signal sensor aims to receive (ideally) only the noise signal and outputs the recorded reference noise signal $x[n]$, which then feeds the adaptive filter. We assume at this point, that the corruption noise signal is uncorrelated to the desired signal, and therefore separable from it. In addition, we assume, that the corruption noise signal is correlated to the reference noise signal, as it originates 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 approximated desired signal $š[n]$. In reality, a signal contamination of the two sensors has to be expected, which will be illustrated in a more realistic signal flow diagram of an implanted \ac{CI} system in chapter 2.6.
\noindent Figure \ref{fig:fig_anr} shows the basic concept of an adaptive filter design, represented through a feedback filter application. The primary sensor (top) aims to receive the desired signal and outputs the corrupted signal $d[n]$, which consists out of the desired signal $s[n]$ and the corruption noise signal $n[n]$ after passing the transfer function of the sensor, whereas the secondary signal sensor aims to receive (ideally) only the noise signal and outputs the reference noise signal $x[n]$ after also passing the respective transfer funtion, which then feeds the adaptive filter. We assume at this point, that the corruption noise signal is uncorrelated to the desired signal, and therefore separable from it. In addition, we assume, that the corruption noise signal is correlated to the reference noise signal, as it originates 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 approximated desired signal $š[n]$. In reality, a signal contamination of the two sensors has to be expected, which will be illustrated in a more realistic signal flow diagram of an implanted \ac{CI} system in chapter 2.6.
\subsubsection{Fully adaptive vs. hybrid filter design}
The basic \ac{ANR} concept illustrated in Figure \ref{fig:fig_anr} can be understood as a fully adaptive variant. A fully adaptive filter design works with a fixed number of coefficients of which everyone is updated after every sample processing. Even if this approach features the best performance in noise reduction, it also requires a relatively high amount of computing power, as every coefficient has to be re-calculated after every evaluation step.\\ \\
To reduce the required computing power, a hybrid static/adaptive filter design can be taken into consideration instead (refer to Figure \ref{fig:fig_anr_hybrid}). In this approach, the initial fully adaptive filter is split into a fixed and an adaptive part - the static filter removes a certain, known, or estimated, frequency portion of the noise signal, whereas the adaptive part only has to adapt to the remaining, unforecastable, noise parts. This approach reduces the number of coefficients required to be adapted, therefore lowering the required computing power.
@@ -265,21 +265,32 @@ The \ac{LMS} algorithm therefore updates the filter coefficients $w[n]$ after ev
\caption{Realistic implant design.}
\label{fig:fig_anr_implant}
\end{figure}
\noindent Figure \ref{fig:fig_anr_hybrid} showed us the basic concept of an \ac{ANR} implementation, without a detailed description how the corrupted 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 primary sensor receives the desired- and noise signal over their respective transfer functions and outputs the corrupted signal $d[n]$, which consists out of the recorded desired 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.\\ \\
\noindent Figure \ref{fig:fig_anr_hybrid} showed us the basic concept of an \ac{ANR} implementation, without a detailed description how the corrupted 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 following definitions of the involved signals shall help to better understand the involved signals and their interactions:
\begin{itemize}
\item Desired signal: The wanted signal, like human voice, which shall be preserved.
\item Noise signal: The unwanted signal, like background noise, which shall be reduced.
\item Recorded desired signal: The desired signal after passing the transfer function to the primary sensor.
\item Corruption noise signal: The noise signal after passing the transfer function to the primary sensor.
\item Reference noise signal: The noise signal after passing the transfer function to the secondary sensor.
\item Corrupted signal: The combination of the recorded desired signal and the corruption noise signal
\item Filter output / Cleaned signal: The output signal of the \ac{ANR} system, representing the desired signal after noise reduction.
\end{itemize}
The primary sensor receives the desired- and noise signal over their respective transfer functions and outputs the corrupted signal $d[n]$, which consists out of the recorded desired signal $s[n]$ and the 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 $C_n$, $D_n$, 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 is mainly depented on the sensitivity curve of the respective sensors and therefore can be seen as time-invariant and known. This known transfer functions, which are titled $A$ and $B$, allow us to apply an hybrid static/adaptive filter design for the \ac{ANR} implementation, as described in chapter 2.5.2.\\ \\
\begin{equation}
\label{equation_dn}
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 desired signal and $n[n]$ is the recorded corruption noise after passing the transfer functions.\\ \\
The recorded noise reference signal $x[n]$ can be mathematically described as:
where $t[n]$ and $v[n]$ are the desired- and noise signals at their respective source, $s[n]$ is the recorded desired signal and $n[n]$ is the 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] * (E_nB)
\end{equation}
where $v[n]$ is the noise signal at its source.\\ \\
Another possible signal interaction could be the leakage of the desired signal into the secondary sensor, leading to the partial removal of the desired 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.\\ \\
At this point, the theoretical background and the fundamentals of adaptive noise reduction have been adequately introduced and explained as necessary for the understanding of the following chapters of this thesis. The next chapter will now focus on practical high level simulations of different filter concepts and \ac{LMS} algorithm variations to evaluate their performance in regard of noise reduction quality before the actual implementation on a low-power digital signal processor is conducted.
At this point, the theoretical background and the fundamentals of adaptive noise reduction have been adequately introduced and explained as necessary for the understanding of the following chapters of this thesis. The next chapter will now focus on practical high level simulations of the \ac{ANR} algorithm under different circumstances to evaluate their performance in regard of noise reduction quality before the actual implementation on a low-power digital signal processor is conducted.