Wrote conclusion

latex/thesis/chapters/comparison.tex

@@ -270,6 +270,11 @@ to the constraints never being quite satisfied.
 With \ac{LP} decoding using \ac{ADMM},
 the constraints are fulfilled for each parity check individualy after each
 iteration of the decoding process.
+It should be noted that while in this thesis proximal decoding was
+examined with respect to its performance in \ac{AWGN} channels, in
+\cite{proximal_paper} it is presented as a method applicable to non-trivial
+channel models such as \ac{LDPC}-coded massive \ac{MIMO} channels, perhaps
+broadening its usefulness beyond what is shown here.
 
 The timing requirements of the decoding algorithms are visualized in figure
 \ref{fig:comp:time}.

latex/thesis/chapters/conclusion.tex

@@ -1,8 +1,55 @@
 \chapter{Conclusion}%
 \label{chapter:conclusion}
 
-\begin{itemize}
+In the context of this thesis, two decoding algorithms were considered:
-    \item Summary of results
+proximal decoding and \ac{LP} decoding using \ac{ADMM}.
-    \item Future work
+The two algorithms were first analyzed individually, before comparing them
-\end{itemize}
+based on simulation results as well as their theoretical structure.
+
+For proximal decoding, the effect of each parameter on the behavior of the
+decoder was examined, leading to an approach to choosing the value of each
+of the parameters.
+The convergence properties of the algorithm were investigated in the context
+of the relatively high decoding failure rate, to derive an approach to correct
+possible wrong componets of the estimate.
+Based on this approach, an improvement over proximal decoding was suggested,
+leading to a decoding gain of up to $\SI{1}{dB}$, depending on the code and
+the parameters considered.
+
+For \ac{LP} decoding using \ac{ADMM}, the circumstances brought about via the
+relaxation while formulating the \ac{LP} decoding problem were first explored.
+The decomposable nature arising from the relocation of the constraints into
+the objective function itself was recognized as the major driver in enabling
+the efficent implementation of the decoding algorithm.
+Based on simulation results, general guidelines for choosing each parameter
+were again derived.
+The decoding performance, in form of the \ac{FER}, of the algorithm was
+analyzed, observing that \ac{LP} decoding using \ac{ADMM} nearly reaches that
+of \ac{BP}, staying within approximately $\SI{0.5}{dB}$ depending on the code
+in question.
+
+Finally, strong parallells were discovered with regard to the theoretical
+structure of the two algorithms, both in the constitution of their respective
+objective functions as in the iterative approaches used to minimize them.
+One difference noted was the approximate nature of the minimization in the
+case of proximal decoding, leading to the constraints never being truly
+satisfied.
+In conjunction with the alternating minimization with respect to the same
+variable leading to oscillatory behavior, this was identified as the
+root cause of its comparatively worse decoding performance.
+Furthermore, both algorithms were expressed as message passing algorithms,
+justifying their similar computational performance.
+
+While the modified proximal decoding algorithm presented in section
+\ref{sec:prox:Improved Implementation} shows some promising results, further
+investigation is required to determine how different choices of parameters
+affect the decoding performance.
+Additionally, a more mathematically rigorous foundation for determining the
+potentially wrong components of the estimate is desirable.
+Another area benefiting from future work is the expantion of the \ac{ADMM}
+based \ac{LP} decoder into a decoder approximating \ac{ML} performance,
+using \textit{adaptive \ac{LP} decoding}.
+With this method, the successive addition of redundant parity checks is used
+to mitigate the decoder becoming stuck in erroneous solutions introduced due
+the relaxation of the constraints of the \ac{LP} decoding problem \cite{alp}.
 

latex/thesis/chapters/discussion.tex

@@ -33,13 +33,6 @@ examined with respect to its performance in \ac{AWGN} channels, in
 channel models such as \ac{LDPC}-coded massive \ac{MIMO} channels, perhaps
 broadening its usefulness beyond what is shown here.
 
-While the modified proximal decoding algorithm presented in section
-\ref{sec:prox:Improved Implementation} shows some promising results, further
-investigation is required to determine how different choices of parameters
-affect the decoding performance.
-Additionally, a more mathematically rigorous foundation for determining the
-potentially wrong components of the estimate is desirable.
-
 Another interesting approach might be the combination of proximal and \ac{LP}
 decoding.
 Performing an initial number of iterations using proximal decoding to obtain

latex/thesis/chapters/lp_dec_using_admm.tex

@@ -1370,11 +1370,6 @@ of one another.
     \label{fig:admm:results}
 \end{figure}%
 %
-
-\footnotetext{; $K=200, \mu = 3.3, \rho=1.9,
-        \epsilon_{\text{pri}} = 10^{-5}, \epsilon_{\text{dual}} = 10^{-5}$
-}%
-%
 In figure \ref{fig:admm:ber_fer}, the \ac{BER} and \ac{FER} for \ac{LP} decoding
 using\ac{ADMM} and \ac{BP} are shown for a (3, 6) regular \ac{LDPC} code with 
 $n=204$.

latex/thesis/chapters/proximal_decoding.tex

@@ -1195,17 +1195,10 @@ $\SI{2.80}{GHz}$ and utilizing all cores.
         \end{axis}
     \end{tikzpicture}
 
-    \caption{Timing requirements of the proximal decoding imlementation%
+    \caption{Timing requirements of the proximal decoding imlementation}
-        \protect\footnotemark{}}
     \label{fig:prox:time_comp}
 \end{figure}%
 %
-\footnotetext{The datapoints depicted were calculated by evaluating the
-    metadata of \ac{FER} simulation results from the following codes:
-    BCH (31, 11); BCH (31, 26); \cite[\text{96.3.965; 204.33.484; 204.55.187;
-    408.33.844; PEGReg252x504}]{mackay_enc}
-}%
-%
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1499,16 +1492,10 @@ theoretical considerations.
     \end{tikzpicture}
 
     \caption{Comparison of the timing requirements of the implementations of proximal
-        decoding and the improved algorithm\protect\footnotemark{}}
+        decoding and the improved algorithm}
     \label{fig:prox:time_complexity_comp}
 \end{figure}%
 %
-\footnotetext{The datapoints depicted were calculated by evaluating the
-    metadata of \ac{FER} simulation results from the following codes:
-    BCH (31, 11); BCH (31, 26); \cite[\text{96.3.965; 204.33.484; 204.55.187;
-    408.33.844; PEGReg252x504}]{mackay_enc}
-}%
-%
 
 In conclusion, the decoding performance of proximal decoding can be improved
 by appending an ML-in-the-List step when the algorithm does not produce a

latex/thesis/thesis.tex

@@ -218,7 +218,7 @@
     \include{chapters/proximal_decoding}
     \include{chapters/lp_dec_using_admm}
     \include{chapters/comparison}
-    \include{chapters/discussion}
+%    \include{chapters/discussion}
     \include{chapters/conclusion}
     \include{chapters/appendix}