2.6
This commit is contained in:
Patrick Hangl
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@@ -122,7 +122,7 @@ Although active noise cancellation and adaptive noise reduction share obvious si
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\caption{The basic idea of an adaptive filter design for noise reduction.}
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\label{fig:fig_anr}
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\end{figure}
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\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 recieve the 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 recieve (ideally) only the noise signal $x[n]$, which then feeds the adaptive filter. 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 will be illustrated in a more realistic signal flow diagram of an implanted CI system.
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\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 recieve 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 recieve (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 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 will be illustrated in a more realistic signal flow diagram of an implanted CI system.
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\subsubsection{Fully adaptive vs. hybrid filter design}
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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 sample.\\ \\
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To reduce the required computing power, a hybrid static/adaptive filter design can be taken into consideration instead. In this approach, the inital 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.
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@@ -141,13 +141,13 @@ Adaptive filters rely on an error metric to self-reliantely evaluate their perfo
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\label{equation_error}
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e[n] = d[n] - y[n] = š[n]
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\end{equation}
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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 target signal $d[n]$ and the output signal of the filter $y[n]$.
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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 $y[n]$.
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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:
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\begin{itemize}
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\item The error signal $e[n]$ is not a perfect representation of the clean speech signal $s[n]$ present in the 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.
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\item Altough, the corruption-noise signal $n[n]$ and the 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.
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\item The clean speech signal $s[n]$ is not directly available, as it is contaminated with the corruption-noise signal and 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 speech signal and is used as an indirect measure of the filter's performance, guiding the adaptation process by its own stepwise minimization.
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\item The noise signal $x[n]$ fed into the adaptive filter could also contaminated with parts of the target signal. If this circumstance occurs, it can lead to undesired effects like signal distortion if not handled properly.
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\item The error signal $e[n]$ is not a perfect representation of the recorded target signal $s[n]$ present in the corrputed 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.
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\item Altough, 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.
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\item The recorded target signal $s[n]$ is not directly available, as it is only available contaminated with the corruption noise signal $n[n]$ in the form of $d[n]$ and 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 speech signal and is used as an indirect measure of the filter's performance, guiding the adaptation process by its own stepwise minimization.
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\item The reference noise signal $x[n]$ fed into the adaptive filter could also contaminated with parts of the target signal. If this circumstance occurs, it can lead to undesired effects if not handled properly.
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\end{itemize}
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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 it´s noise-component.\\
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The minimization of the error signal $e[n]$ can by achieved by applying different error metrics used to evaluate the performance of an adaptive filter, including:
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@@ -172,7 +172,7 @@ Before the Least Mean Squares algorithm can be explained in detail, the Wiener f
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\label{equation_wien}
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y[n] = w_0x[n] + w_1x[n-1] + ... + w_Mx[n-M] = \sum_{k=0}^{M} w_kx[n-k]
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\end{equation}
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The Wiener filter aims to adjust it´s coefficients to generate a filter output, which resembles the corruption noise signal $n[n]$ contained in the target signal $d[n]$ as close as possible. After the filter output is substracted from the target signal, we recvieve the error signal $e[n]$, which represents the cleaned signal $š[n]$ after the noise component has been removed. For better unsderstanding, 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.
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The Wiener filter aims to adjust it´s 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 substracted from the corrupted target signal, we recvieve 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.
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\begin{equation}
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\label{equation_wien_error}
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e[n] = d[n] - y[n] = d[n] - wx[n]
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@@ -185,8 +185,8 @@ If we square the error signal and calculate the expected value, we receive the M
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The terms contained in Equation \ref{equation_j} can be further be defined as:
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\begin{itemize}
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\item $\sigma^2$ = $E(d^2[n])$: The expected value of the squared corrupted target signal - a constant term independent of the filter coefficients $w$.
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\item P = $E(d[n]x[n])$: The cross-correlation between the corrupted target signal and the noise reference signal - a measure of how similar these two signals are.
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\item R = $E(x^2[n])$: The auto-correlation (or serial-correlation) of the noise reference signal - a measure of the similarity of a signal with it´s delayed copy and therefore of the signal's spectral power.
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\item 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.
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\item 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.
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\end{itemize}
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Equation {\ref{equation_j}} can therefore be further simplified and written as:
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\begin{equation}
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@@ -265,21 +265,21 @@ The LMS algorithm therefore updates the filter coefficients $w[n]$ after every s
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\caption{Realstic implant design.}
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\label{fig:fig_anr_implant}
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\end{figure}
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\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.\\ \\
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\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 $d[n]$ and the reference noise signal $x[n]$ is 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 recieves the target signal and the 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.\\ \\
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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.\\ \\
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The corrupted target signal $d[n]$ can thereforebe mathematically described as:
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\begin{equation}
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\label{equation_dn}
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d[n] = s[n] + n[n] = t[n] * (D_nG) + v[n] * ((F_nG) + (C_nA))
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\end{equation}
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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.\\ \\
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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.\\ \\
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The noise reference signal $x[n]$ can be mathematically described as:
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\begin{equation}
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\label{equation_xn}
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x[n] = v[n] * (C_nB)
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\end{equation}
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where $v[n]$ is the noise signal at its source and $x[n]$ is the uncorrupted noise signal after passing the transfer functions.\\ \\
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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.\\ \\
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where $v[n]$ is the noise signal at its source and $x[n]$ is the recorded reference noise signal after passing the transfer functions.\\ \\
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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 at this point.\\ \\
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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.
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\subsection{Derivation of the system’s transfer function based on the problem setup}
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</mxCell>
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<mxCell id="uwalZ4GZDBHuo1GNIbxM-2" value="<div><span style="font-size: 20px;">Target</span></div><span style="font-size: 20px;">signal <br>&nbsp; &nbsp; +<br>Noise</span><div><span style="font-size: 20px;">signal</span></div>" style="text;whiteSpace=wrap;html=1;align=center;" parent="1" vertex="1">
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<mxGeometry x="170" y="203.75" width="80" height="132.5" as="geometry" />
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<mxCell id="uwalZ4GZDBHuo1GNIbxM-2" value="<div><span style="font-size: 20px;">Recorded target</span></div><span style="font-size: 20px;">signal</span><div><span style="font-size: 20px;">+<br>Recorded noise</span><div><span style="font-size: 20px;">signal</span></div></div>" style="text;whiteSpace=wrap;html=1;align=center;" parent="1" vertex="1">
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<mxGeometry x="80" y="203.75" width="170" height="132.5" as="geometry" />
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</mxCell>
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<mxCell id="ngw2JN3qWJ9t0b-mEEB_-1" value="<span style="font-size: 20px;">d[n] = s[n]+n[n]</span>" style="text;whiteSpace=wrap;html=1;" parent="1" vertex="1">
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<mxGeometry x="270" y="227.5" width="180" height="50" as="geometry" />
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</mxCell>
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<mxCell id="ngw2JN3qWJ9t0b-mEEB_-3" value="<span style="font-size: 20px;">Noise<br>signal</span>" style="text;whiteSpace=wrap;html=1;align=center;" parent="1" vertex="1">
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<mxGeometry x="170" y="430" width="80" height="50" as="geometry" />
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<mxCell id="ngw2JN3qWJ9t0b-mEEB_-3" value="<span style="font-size: 20px;">Recorded noise<br>signal</span>" style="text;whiteSpace=wrap;html=1;align=center;" parent="1" vertex="1">
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<mxGeometry x="85" y="430" width="160" height="50" as="geometry" />
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</mxCell>
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<mxCell id="ngw2JN3qWJ9t0b-mEEB_-4" value="<span style="font-size: 20px;">x[n]</span>" style="text;html=1;align=center;verticalAlign=middle;resizable=0;points=[];autosize=1;strokeColor=none;fillColor=none;" parent="1" vertex="1">
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<mxGeometry x="300" y="420" width="60" height="40" as="geometry" />
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|
||||
Reference in New Issue
Block a user