Rewrote introduction and conclusion

latex/thesis/chapters/conclusion.tex

@@ -1,4 +1,4 @@
-\chapter{Conclusion}%
+\chapter{Conclusion and Outlook}%
 \label{chapter:conclusion}
 
 In the context of this thesis, two decoding algorithms were considered:
@@ -11,16 +11,16 @@ 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.
+possible wrong components 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.
+For \ac{LP} decoding using \ac{ADMM}, the circumstances brought about by the
+\ac{LP} relaxation 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.
+an efficient 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
@@ -28,15 +28,15 @@ 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
+Finally, strong parallels 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.
+variable, leading to oscillatory behavior, this was identified as
+a possible cause of its comparatively worse decoding performance.
 Furthermore, both algorithms were expressed as message passing algorithms,
 justifying their similar computational performance.
 
@@ -46,7 +46,7 @@ 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}
+Another area benefiting from future work is the expansion 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

latex/thesis/chapters/introduction.tex

@@ -19,7 +19,7 @@ linear codes was conducted in Feldman's 2003 Ph.D. thesis and subsequent paper,
 establishing the field of \ac{LP} decoding \cite{feldman_thesis}, \cite{feldman_paper}.
 There, the \ac{ML} decoding problem is approximated by a \textit{linear program},
 a linear, convex optimization problem, which can subsequently be solved using
-a number of different algorithms \cite{alp}, \cite{interior_point},
+several different algorithms \cite{alp}, \cite{interior_point},
 \cite{original_admm}, \cite{pdd}.
 More recently, novel approaches such as \textit{proximal decoding} have been
 introduced. Proximal decoding is based on a non-convex optimization formulation
@@ -40,4 +40,6 @@ the existing literature.
 Specifically, the proximal decoding algorithm and \ac{LP} decoding using
 the \ac{ADMM} \cite{original_admm} are explored within the context of
 \ac{BPSK} modulated \ac{AWGN} channels.
+Implementations of both decoding methods are produced, and based on simulation
+results from those implementations the algorithms are examined and compared.