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chapter_02.tex
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@@ -5,7 +5,7 @@ Filters are used in various functional designs, therefore a short explanation in
At this point an introduction into adaptive noise reduction follows, including a short overview of the most important steps in history, the general concept of \ac{ANR}, its design possibilities and its optimization possibilities in regard of error calculation.\\
With this knowledge covered, a realistic signal flow diagram of an implanted \ac{CI} system with corresponding transfer functions is designed, essential to implement \ac{ANR} on a low-power digital signal processor.\\
At the end of chapter two, high-level Python simulations shall function as a practical demonstration of the recently presented theoretical background.\\ \\
Throughout this thesis, sampled signals are denoted in lowercase with square brackets (e.g. {x[n]}) to distinguish them from time-continuous signals (e.g. {x(t)}). Vectors are notaded in lowercase bold font, whereas matrix are notaded in uppercase bold font. Scalars are notated in normal lowercase font.\\
Throughout this thesis, sampled signals are denoted in lowercase with square brackets (e.g. {x[n]}) to distinguish them from time-continuous signals (e.g. {x(t)}). Vectors are notated in lowercase bold font, whereas matrix are notated in uppercase bold font. Scalars are notated in normal lowercase font.\\
\subsection{Signals and signal interference}
A signal is a physical parameter (e.g. pressure, voltage) changing its value over time. Whereas in nature, a signal is always analog, meaning continuous in both time and amplitude, a digital signal is represented in a discrete form, being sampled at specific time intervals and quantized to finite amplitude levels.\\ \\
The term "signal interference" describes the overlapping of unwanted signals or noise with the desired signal, degrading the overall quality and intelligibility of the processed information. A simple example of signal interference is shown in Figure \ref{fig:fig_interference} - the noisy signal (top) consists out of several signals of different frequencies, representing both the desired signal and unwanted noise. The cleaned signal (bottom) shows the desired signal after unwanted frequencies has been cut off by a filter.\\ \\
@@ -81,7 +81,7 @@ As there are three operators present in the filter, three samples are needed bef
\label{fig:fig_fir}
\end{figure}
\subsubsection{Infinite Impulse Response filters}
An \ac{IIR} filter, commonly referred to as a ``Feedback Filter'' can be seen as an extension of the \ac{FIR} filter. In contrary to its counterpart, it also uses past output samples in addition to current input samples to adapt its filtering behavior - therefore the response of an \ac{IIR} filter theoretically continues indefinitely, even if the input signal is reduced to zero.\\ \\
A \ac{IIR} filter, commonly referred to as a ``Feedback Filter'' can be seen as an extension of the \ac{FIR} filter. In contrary to its counterpart, it also uses past output samples in addition to current input samples to adapt its filtering behavior - therefore the response of a \ac{IIR} filter theoretically continues indefinitely, even if the input signal is reduced to zero.\\ \\
Equation \ref{equation_iir} specifies the input-output relationship of a \ac{IIR} filter. In addition to Equation \ref{equation_fir} there is now a second term included, where $a_0$ to $a_N$ are the feedback coefficients with their own filter length N.
\begin{equation}
\label{equation_iir}
@@ -96,7 +96,7 @@ Figure \ref{fig:fig_iir} visualizes a simple \ac{IIR} filter with two feedforwar
\end{figure}
\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.\\ \\
The recursive nature of a \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.
\subsection{Introduction to Adaptive Noise Reduction}
@@ -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 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.
\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 function, 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.
@@ -277,7 +277,7 @@ The following definitions of the involved signals shall help to better understan
\item Filter output / Cleaned signal: The output signal of the \ac{ANR} algorithm, representing the desired signal after noise reduction. This signal also equals the error signal of the adaptive filter.
\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.\\ \\
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 between 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 depended 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 a 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)