From d4dc490e494c34462cfcc24bd63795a0d378c94f Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Wed, 8 Mar 2023 13:55:20 +0100 Subject: [PATCH] Fixed typos; Added citation --- latex/thesis/chapters/decoding_techniques.tex | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/latex/thesis/chapters/decoding_techniques.tex b/latex/thesis/chapters/decoding_techniques.tex index b15dc54..2ee9a2c 100644 --- a/latex/thesis/chapters/decoding_techniques.tex +++ b/latex/thesis/chapters/decoding_techniques.tex @@ -189,7 +189,7 @@ making the \ac{ML} and \ac{MAP} decoding problems equivalent.}% .\end{align}% % Assuming a memoryless channel, equation (\ref{eq:lp:ml}) can be rewritten in terms -of the \acp{LLR} $\gamma_i$ \cite[Sec 2.5]{feldman_thesis}:% +of the \acp{LLR} $\gamma_i$ \cite[Sec. 2.5]{feldman_thesis}:% % \begin{align*} \hat{\boldsymbol{c}} = \argmin_{\boldsymbol{c}\in\mathcal{C}} @@ -236,6 +236,7 @@ decoding, redefining the constraints in terms of the \text{codeword polytope} % which represents the \textit{convex hull} of all possible codewords, i.e., the convex set of linear combinations of all codewords. +This corresponds to simply lifting the integer requirement. However, since the number of constraints needed to characterize the codeword polytope is exponential in the code length, this formulation is relaxed further. By observing that each check node defines its own local single parity-check @@ -249,8 +250,7 @@ This consideration leads to constraints, that can be described as follows \boldsymbol{T}_j \tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j} \hspace{5mm}\forall j\in \mathcal{J} ,\end{align*}% -\todo{Explicitly state that the first relaxation is essentially just lifing the integer -requirement}% +% where $\mathcal{P}_{d_j}$ is the \textit{check polytope}, the convex hull of all binary vectors of length $d_j$ with even parity% \footnote{Essentially $\mathcal{P}_{d_j}$ is the set of vectors that satisfy @@ -827,7 +827,7 @@ It is then immediately approximated with gradient-descent:% \begin{align*} \text{prox}_{\gamma h} \left( \boldsymbol{x} \right) &\equiv \argmin_{\boldsymbol{t} \in \mathbb{R}^n} - \left( \gamma h\left( \boldsymbol{x} \right) + + \left( \gamma h\left( \boldsymbol{t} \right) + \frac{1}{2} \lVert \boldsymbol{t} - \boldsymbol{x} \rVert \right)\\ &\approx \boldsymbol{r} - \gamma \nabla h \left( \boldsymbol{r} \right), \hspace{5mm} \gamma > 0, \text{ small} @@ -847,7 +847,7 @@ as it keeps the effect of $h\left( \boldsymbol{x} \right) $ on the landscape of the objective function small. Otherwise, unwanted stationary points, including local minima, are introduced. The authors say that in practice, the value of $\gamma$ should be adjusted -according to the decoding performance. +according to the decoding performance \cite[Sec. 3.1]{proximal_paper}. %The components of the gradient of the code-constraint polynomial can be computed as follows:% %%