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@@ -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 target signal sensor (top) aims to receive the target signal and outputs the corrupted target signal $d[n]$, which consists out of the recorded target signal $s[n]$ and the corruption noise signal $n[n]$, whereas the noise 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 recorded target 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 target 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 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 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 CI system in chapter 2.6.
\subsubsection{Fully adaptive vs. hybrid filter design}
The basic 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.
@@ -141,13 +141,13 @@ Adaptive filters rely on an error metric to self-reliantely evaluate their perfo
\label{equation_error}
e[n] = d[n] - y[n] = š[n]
\end{equation}
The error signal $e[n]$, already illustrated in Figure \ref{fig:fig_anr} and \ref{fig:fig_anr_hybrid}, is calculated as the difference between the corrupted target signal $d[n]$ and the output signal of the filter $y[n]$.
The error signal $e[n]$, already illustrated in Figure \ref{fig:fig_anr} and \ref{fig:fig_anr_hybrid}, is calculated as the difference between the corrupted signal $d[n]$ and the output signal of the filter $y[n]$.
As we will see in the following chapters, a real world application of an adaptive filter system poses several challenges, which have to be taken into consideration when designing the filter. These challenges include:
\begin{itemize}
\item The error signal $e[n]$ is not a perfect representation of the recorded target signal $s[n]$ present in the corrupted target signal $d[n]$, as the adaptive filter can only approximate the noise signal based on its current coefficients, which in general, do not represent the optimal solution at that given time.
\item The error signal $e[n]$ is not a perfect representation of the desired signal $s[n]$ present in the corrupted signal $d[n]$, as the adaptive filter can only approximate the noise signal based on its current coefficients, which in general, do not represent the optimal solution at that given time.
\item Although, the corruption noise signal $n[n]$ and the reference noise signal $x[n]$ are correlated, they are not identical, as they take different signal paths from the noise source to their respective sensors. This discrepancy can lead to imperfect noise reduction, as the adaptive filter has to estimate the relationship between these two signals.
\item The recorded target signal $s[n]$ is not directly available, as it is only available combined with the corruption noise signal $n[n]$ in the form of $d[n]$ while there is no reference available. Therefore, the error signal $e[n]$, respectively $š[n]$, of the adaptive filter serves as an approximation of the clean target signal and is used as an indirect measure of the filter's performance, guiding the adaptation process by its own stepwise minimization.
\item The reference noise signal $x[n]$ fed into the adaptive filter could also be contaminated with parts of the target signal. If this circumstance occurs is not handled properly, it could lead to the undesired removal of parts of the target signal from the output signal $š[n]$.
\item The desired signal $s[n]$ is not directly available, as it is only available combined with the corruption noise signal $n[n]$ in the form of $d[n]$ while there is no reference available. Therefore, the error signal $e[n]$, respectively $š[n]$, of the adaptive filter serves as an approximation of the desired signal and is used as an indirect measure of the filter's performance, guiding the adaptation process by its own stepwise minimization.
\item The reference noise signal $x[n]$ fed into the adaptive filter could also be contaminated with parts of the desired signal. If this circumstance occurs is not handled properly, it could lead to the undesired removal of parts of the desired signal from the output signal $š[n]$.
\end{itemize}
The goal of the adaptive filter is therefore to minimize this error signal over time, thereby improving the quality of the output signal by reducing it by its noise-component.\\
The minimization of the error signal $e[n]$ can be achieved by applying different error metrics and algorithms used to evaluate the performance of an adaptive filter, including:
@@ -167,12 +167,12 @@ Before the Least Mean Squares algorithm can be explained in detail, the Wiener f
\caption{Simple implementation of a Wiener filter.}
\label{fig:fig_wien}
\end{figure}
\noindent The Wiener filter, the base of many adaptive filter designs, is a statistical filter used to minimize the Mean Squares Error between a target signal and the output of a linear filter. The output $y[n]$ of the Wiener filter is the sum of the weighted input samples, where the weights are represented by the filter coefficients.
\noindent The Wiener filter, the base of many adaptive filter designs, is a statistical filter used to minimize the Mean Squares Error between a desired signal and the output of a linear filter. The output $y[n]$ of the Wiener filter is the sum of the weighted input samples, where the weights are represented by the filter coefficients.
\begin{equation}
\label{equation_wien}
y[n] = w_0x[n] + w_1x[n-1] + ... + w_Mx[n-M] = \sum_{k=0}^{M} w_kx[n-k]
\end{equation}
The Wiener filter aims to adjust its coefficients to generate a filter output, which resembles the corruption noise signal $n[n]$ contained in the corrupted target signal $d[n]$ as close as possible. After the filter output is subtracted from the corrupted target signal, we receive the error signal $e[n]$, which represents the cleaned signal $š[n]$ after the corruption noise component has been removed. For better understanding, a simple Wiener filter with one coefficient shall be illustrated in the following mathematical approach, before the generalization to an n-dimensional filter is made.
The Wiener filter aims to adjust its coefficients to generate a filter output, which resembles the corruption noise signal $n[n]$ contained in the corrupted signal $d[n]$ as close as possible. After the filter output is subtracted from the corrupted signal, we receive the error signal $e[n]$, which represents the cleaned signal $š[n]$ after the corruption noise component has been removed. For better understanding, a simple Wiener filter with one coefficient shall be illustrated in the following mathematical approach, before the generalization to an n-dimensional filter is made.
\begin{equation}
\label{equation_wien_error}
e[n] = d[n] - y[n] = d[n] - wx[n]
@@ -184,8 +184,8 @@ If we square the error signal and calculate the expected value, we receive the M
\end{equation}
The terms contained in Equation \ref{equation_j} can be further be defined as:
\begin{itemize}
\item $\sigma^2$ = $E(d^2[n])$: The expected value of the squared corrupted target signal - a constant term independent of the filter coefficients $w$.
\item \textbf{P} = $E(d[n]x[n])$: The cross-correlation between the corrupted target signal and the reference noise signal - a measure of how similar these two signals are.
\item $\sigma^2$ = $E(d^2[n])$: The expected value of the squared corrupted signal - a constant term independent of the filter coefficients $w$.
\item \textbf{P} = $E(d[n]x[n])$: The cross-correlation between the corrupted signal and the reference noise signal - a measure of how similar these two signals are.
\item \textbf{R} = $E(x^2[n])$: The auto-correlation (or serial-correlation) of the reference noise signal - a measure of the similarity of a signal with it's delayed copy and therefore of the signal's spectral power.
\end{itemize}
Equation {\ref{equation_j}} can therefore be further simplified and written as:
@@ -256,7 +256,7 @@ The result of Equation \ref{equation_j_lms_final} can now be inserted into Equat
\label{equation_lms}
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 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.\\ \\
The LMS algorithm therefore updates the filter coefficients $w[n]$ after every sample by adding a correction term, which 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 desired 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 with 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 digital signal processor.
\begin{figure}[H]
@@ -265,20 +265,20 @@ The LMS algorithm therefore updates the filter coefficients $w[n]$ after every s
\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 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 $C_n$, $D_n$, 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.\\ \\
\noindent Figure \ref{fig:fig_anr_hybrid} showed us the basic concept of an 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.\\ \\
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 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 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:
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:
\begin{equation}
\label{equation_xn}
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.\\ \\
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 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.