1260 lines
57 KiB
TeX
1260 lines
57 KiB
TeX
\chapter{Proximal Decoding}%
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\label{chapter:proximal_decoding}
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In this chapter, the proximal decoding algorithm is examined.
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First, the algorithm itself is described.
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Then, some interesting ideas concerning the implementation are presented.
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Simulation results are shown, on the basis of which the behaviour of the
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algorithm is investigated for different codes and parameters.
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Finally, an improvement on proximal decoding is proposed.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Decoding Algorithm}%
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\label{sec:prox:Decoding Algorithm}
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Proximal decoding was proposed by Wadayama et. al as a novel formulation of
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optimization-based decoding \cite{proximal_paper}.
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With this algorithm, minimization is performed using the proximal gradient
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method.
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In contrast to \ac{LP} decoding, the objective function is based on a
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non-convex optimization formulation of the \ac{MAP} decoding problem.
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In order to derive the objective function, the authors begin with the
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\ac{MAP} decoding rule, expressed as a continuous maximization problem%
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\footnote{The expansion of the domain to be continuous doesn't constitute a
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material difference in the meaning of the rule.
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The only change is that what previously were \acp{PMF} now have to be expressed
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in terms of \acp{PDF}.}
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over $\boldsymbol{x}$:%
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%
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\begin{align}
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\hat{\boldsymbol{x}} = \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}}
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f_{\tilde{\boldsymbol{X}} \mid \boldsymbol{Y}}
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\left( \tilde{\boldsymbol{x}} \mid \boldsymbol{y} \right)
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= \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}} f_{\boldsymbol{Y}
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\mid \tilde{\boldsymbol{X}}}
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\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
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f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)%
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\label{eq:prox:vanilla_MAP}
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.\end{align}%
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%
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The likelihood $f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
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\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) $ is a known function
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determined by the channel model.
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The prior \ac{PDF} $f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)$ is also
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known, as the equal probability assumption is made on
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$\mathcal{C}$.
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However, since the considered domain is continuous,
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the prior \ac{PDF} cannot be ignored as a constant during the minimization
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as is often done, and has a rather unwieldy representation:%
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%
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\begin{align}
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f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right) =
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\frac{1}{\left| \mathcal{C} \right| }
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\sum_{\boldsymbol{c} \in \mathcal{C} }
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\delta\big( \tilde{\boldsymbol{x}} - \left( -1 \right) ^{\boldsymbol{c}}\big)
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\label{eq:prox:prior_pdf}
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.\end{align}%
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%
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In order to rewrite the prior \ac{PDF}
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$f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)$,
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the so-called \textit{code-constraint polynomial} is introduced as:%
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%
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\begin{align*}
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h\left( \tilde{\boldsymbol{x}} \right) =
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\underbrace{\sum_{i=1}^{n} \left( \tilde{x_i}^2-1 \right) ^2}_{\text{Bipolar constraint}}
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+ \underbrace{\sum_{j=1}^{m} \left[
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\left( \prod_{i\in N_c \left( j \right) } \tilde{x_i} \right)
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-1 \right] ^2}_{\text{Parity constraint}}%
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.\end{align*}%
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%
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The intention of this function is to provide a way to penalize vectors far
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from a codeword and favor those close to one.
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In order to achieve this, the polynomial is composed of two parts: one term
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representing the bipolar constraint, providing for a discrete solution of the
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continuous optimization problem, and one term representing the parity
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constraints, accommodating the role of the parity-check matrix $\boldsymbol{H}$.
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The prior \ac{PDF} is then approximated using the code-constraint polynomial as:%
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%
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\begin{align}
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f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)
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\approx \frac{1}{Z}\mathrm{e}^{-\gamma h\left( \tilde{\boldsymbol{x}} \right) }%
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\label{eq:prox:prior_pdf_approx}
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.\end{align}%
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%
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The authors justify this approximation by arguing, that for
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$\gamma \rightarrow \infty$, the approximation in equation
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(\ref{eq:prox:prior_pdf_approx}) approaches the original function in equation
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(\ref{eq:prox:prior_pdf}).
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This approximation can then be plugged into equation (\ref{eq:prox:vanilla_MAP})
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and the likelihood can be rewritten using the negative log-likelihood
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$L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) = -\ln\left(
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f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}\left(
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\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) \right) $:%
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%
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\begin{align*}
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\hat{\boldsymbol{x}} &= \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}}
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\mathrm{e}^{- L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) }
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\mathrm{e}^{-\gamma h\left( \tilde{\boldsymbol{x}} \right) } \\
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&= \argmin_{\tilde{\boldsymbol{x}} \in \mathbb{R}^n} \big(
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L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
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+ \gamma h\left( \tilde{\boldsymbol{x}} \right)
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\big)%
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.\end{align*}%
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%
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Thus, with proximal decoding, the objective function
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$g\left( \tilde{\boldsymbol{x}} \right)$ considered is%
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%
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\begin{align}
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g\left( \tilde{\boldsymbol{x}} \right) = L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}}
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\right)
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+ \gamma h\left( \tilde{\boldsymbol{x}} \right)%
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\label{eq:prox:objective_function}
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\end{align}%
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%
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and the decoding problem is reformulated to%
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%
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\begin{align*}
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\text{minimize}\hspace{2mm} &L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
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+ \gamma h\left( \tilde{\boldsymbol{x}} \right)\\
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\text{subject to}\hspace{2mm} &\tilde{\boldsymbol{x}} \in \mathbb{R}^n
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.\end{align*}
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%
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For the solution of the approximate \ac{MAP} decoding problem, the two parts
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of equation (\ref{eq:prox:objective_function}) are considered separately:
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the minimization of the objective function occurs in an alternating
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fashion, switching between the negative log-likelihood
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$L\left( \boldsymbol{y} \mid \boldsymbol{x} \right) $ and the scaled
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code-constraint polynomial $\gamma h\left( \boldsymbol{x} \right) $.
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Two helper variables, $\boldsymbol{r}$ and $\boldsymbol{s}$, are introduced,
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describing the result of each of the two steps.
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The first step, minimizing the log-likelihood, is performed using gradient
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descent:%
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%
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\begin{align}
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\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \nabla
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L\left( \boldsymbol{y} \mid \boldsymbol{s} \right),
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\hspace{5mm}\omega > 0
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\label{eq:prox:step_log_likelihood}
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.\end{align}%
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%
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For the second step, minimizing the scaled code-constraint polynomial, the
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proximal gradient method is used and the \textit{proximal operator} of
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$\gamma h\left( \tilde{\boldsymbol{x}} \right) $ has to be computed.
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It is then immediately approximated with gradient-descent:%
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%
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\begin{align*}
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\textbf{prox}_{\gamma h} \left( \tilde{\boldsymbol{x}} \right) &\equiv
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\argmin_{\boldsymbol{t} \in \mathbb{R}^n}
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\left( \gamma h\left( \boldsymbol{t} \right) +
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\frac{1}{2} \lVert \boldsymbol{t} - \tilde{\boldsymbol{x}} \rVert \right)\\
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&\approx \tilde{\boldsymbol{x}} - \gamma \nabla h \left( \tilde{\boldsymbol{x}} \right),
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\hspace{5mm} \gamma > 0, \text{ small}
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.\end{align*}%
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%
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The second step thus becomes%
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%
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\begin{align*}
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\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right),
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\hspace{5mm}\gamma > 0,\text{ small}
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.\end{align*}
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%
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While the approximation of the prior \ac{PDF} made in equation (\ref{eq:prox:prior_pdf_approx})
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theoretically becomes better
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with larger $\gamma$, the constraint that $\gamma$ be small is important,
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as it keeps the effect of $h\left( \tilde{\boldsymbol{x}} \right) $ on the landscape
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of the objective function small.
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Otherwise, unwanted stationary points, including local minima, are introduced.
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The authors say that ``in practice, the value of $\gamma$ should be adjusted
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according to the decoding performance.'' \cite[Sec. 3.1]{proximal_paper}.
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%The components of the gradient of the code-constraint polynomial can be computed as follows:%
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%%
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%\begin{align*}
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% \frac{\partial}{\partial x_k} h\left( \boldsymbol{x} \right) =
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% 4\left( x_k^2 - 1 \right) x_k + \frac{2}{x_k}
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% \sum_{i\in \mathcal{B}\left( k \right) } \left(
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% \left( \prod_{j\in\mathcal{A}\left( i \right)} x_j\right)^2
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% - \prod_{j\in\mathcal{A}\left( i \right) }x_j \right)
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%.\end{align*}%
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%\todo{Only multiplication?}%
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%\todo{$x_k$: $k$ or some other indexing variable?}%
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%%
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In the case of \ac{AWGN}, the likelihood
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$f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
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\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)$
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is%
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%
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\begin{align*}
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f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
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\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
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= \frac{1}{\sqrt{2\pi\sigma^2}}\mathrm{e}^{
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-\frac{\lVert \boldsymbol{y}-\tilde{\boldsymbol{x}}
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\rVert^2 }
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{2\sigma^2}}
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.\end{align*}
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%
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Thus, the gradient of the negative log-likelihood becomes%
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\footnote{For the minimization, constants can be disregarded. For this reason,
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it suffices to consider only proportionality instead of equality.}%
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%
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\begin{align*}
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\nabla L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
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&\propto -\nabla \lVert \boldsymbol{y} - \tilde{\boldsymbol{x}} \rVert^2\\
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&\propto \tilde{\boldsymbol{x}} - \boldsymbol{y}
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,\end{align*}%
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%
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allowing equation (\ref{eq:prox:step_log_likelihood}) to be rewritten as%
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%
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\begin{align*}
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\boldsymbol{r} \leftarrow \boldsymbol{s}
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- \omega \left( \boldsymbol{s} - \boldsymbol{y} \right)
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.\end{align*}
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%
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One thing to consider during the actual decoding process, is that the gradient
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of the code-constraint polynomial can take on extremely large values.
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To avoid numerical instability, an additional step is added, where all
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components of the current estimate are clipped to $\left[-\eta, \eta \right]$,
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where $\eta$ is a positive constant slightly larger than one:%
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%
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\begin{align*}
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\boldsymbol{s} \leftarrow \Pi_{\eta} \left( \boldsymbol{r}
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- \gamma \nabla h\left( \boldsymbol{r} \right) \right)
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,\end{align*}
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%
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$\Pi_{\eta}\left( \cdot \right) $ expressing the projection onto
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$\left[ -\eta, \eta \right]^n$.
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The iterative decoding process resulting from these considerations is shown in
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figure \ref{fig:prox:alg}.
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\begin{figure}[H]
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\centering
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\begin{genericAlgorithm}[caption={}, label={}]
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$\boldsymbol{s} \leftarrow \boldsymbol{0}$
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for $K$ iterations do
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$\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \left( \boldsymbol{s} - \boldsymbol{y} \right) $
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$\boldsymbol{s} \leftarrow \Pi_\eta \left(\boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) \right)$
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$\boldsymbol{\hat{x}} \leftarrow \text{sign}\left( \boldsymbol{s} \right) $
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if $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ do
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return $\boldsymbol{\hat{c}}$
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end if
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end for
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return $\boldsymbol{\hat{c}}$
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\end{genericAlgorithm}
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\caption{Proximal decoding algorithm for an \ac{AWGN} channel}
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\label{fig:prox:alg}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Implementation Details}%
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\label{sec:prox:Implementation Details}
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The algorithm was first implemented in Python because of the fast development
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process and straightforward debugging ability.
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It was subsequently reimplemented in C++ using the Eigen%
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\footnote{\url{https://eigen.tuxfamily.org}}
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linear algebra library to achieve higher performance.
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The focus has been set on a fast implementation, sometimes at the expense of
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memory usage.
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The evaluation of the simulation results has been wholly realized in Python.
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The gradient of the code-constraint polynomial \cite[Sec. 2.3]{proximal_paper}
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is given by%
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%
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\begin{align*}
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\nabla h\left( \tilde{\boldsymbol{x}} \right) &= \begin{bmatrix}
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\frac{\partial}{\partial \tilde{x}_1}h\left( \tilde{\boldsymbol{x}} \right) &
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\ldots &
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\frac{\partial}{\partial \tilde{x}_n}h\left( \tilde{\boldsymbol{x}} \right) &
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\end{bmatrix}^\text{T}, \\[1em]
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\frac{\partial}{\partial \tilde{x}_k}h\left( \tilde{\boldsymbol{x}} \right)
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&= 4\left( \tilde{x}_k^2 - 1 \right) \tilde{x}_k
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+ \frac{2}{\tilde{x}_k} \sum_{j\in N_v\left( k \right) }\left(
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\left( \prod_{i \in N_c\left( j \right)} \tilde{x}_i \right)^2
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- \prod_{i\in N_c\left( j \right) } \tilde{x}_i \right)
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.\end{align*}
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%
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Since the products
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$\prod_{i\in N_c\left( j \right) } \tilde{x}_i,\hspace{2mm}j\in \mathcal{J}$
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are the same for all components $\tilde{x}_k$ of $\tilde{\boldsymbol{x}}$, they can be
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precomputed.
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Defining%
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%
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\begin{align*}
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\boldsymbol{p} := \begin{bmatrix}
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\prod_{i\in N_c\left( 1 \right) } \tilde{x}_i \\
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\vdots \\
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\prod_{i\in N_c\left( m \right) } \tilde{x}_i \\
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\end{bmatrix}
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\hspace{5mm}
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\text{and}
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\hspace{5mm}
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\boldsymbol{v} := \boldsymbol{p}^{\circ 2} - \boldsymbol{p}
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,\end{align*}
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%
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the gradient can be written as%
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%
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\begin{align*}
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\nabla h\left( \tilde{\boldsymbol{x}} \right) =
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4\left( \tilde{\boldsymbol{x}}^{\circ 3} - \tilde{\boldsymbol{x}} \right)
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+ 2\tilde{\boldsymbol{x}}^{\circ -1} \circ \boldsymbol{H}^\text{T}
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\boldsymbol{v}
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,\end{align*}
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%
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enabling the computation of the gradient primarily with element-wise
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operations and matrix-vector multiplication.
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This is beneficial, as the libraries used for the implementation are
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heavily optimized for such calculations (e.g., through vectorization of the
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operations).
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\todo{Note about how the equation with which the gradient is calculated is
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itself similar to a message-passing rule}
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The projection $\prod_{\eta}\left( . \right)$ also proves straightforward to
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compute, as it amounts to simply clipping each component of the vector onto
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$[-\eta, \eta]$ individually.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Analysis and Simulation Results}%
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\label{sec:prox:Analysis and Simulation Results}
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In this section, the general behaviour of the proximal decoding algorithm is
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analyzed.
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The impact of the parameters $\gamma$, as well as $\omega$, $K$ and $\eta$ is
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examined.
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The decoding performance is assessed on the basis of the \ac{BER} and the
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\ac{FER} as well as the \textit{decoding failure rate} - the rate at which
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the algorithm produces invalid results.
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The convergence properties are reviewed and related to the decoding
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performance.
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Finally, the computational performance is examined on a theoretical basis
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as well as on the basis of the implementation completed in the context of this
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work.
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All simulation results presented hereafter are based on Monte Carlo
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simulations.
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The \ac{BER} and \ac{FER} curves in particular have been generated by
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producing at least 100 frame-errors for each data point, unless otherwise
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stated.
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\todo{Mention number of datapoints from which each graph was created for
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non ber and fer curves}
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\subsection{Choice of Parameters}
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First, the effect of the parameter $\gamma$ is investigated.
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Figure \ref{fig:prox:results} shows a comparison of the decoding performance
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of the proximal decoding algorithm as presented by Wadayama et al. in
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\cite{proximal_paper} and the implementation realized for this work.
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\noindent The \ac{BER} curves for three different choices of the parameter
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$\gamma$ are shown, as well as the \ac{BER} curve resulting from decoding
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using \ac{BP}, as a reference.
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The results from Wadayama et al. are shown with solid lines,
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while the newly generated ones are shown with dashed lines.
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\begin{axis}[grid=both, grid style={line width=.1pt},
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xlabel={$E_b / N_0$ (dB)}, ylabel={BER},
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ymode=log,
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legend style={at={(0.5,-0.7)},anchor=south},
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width=0.6\textwidth,
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height=0.45\textwidth,
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ymax=1.2, ymin=0.8e-4,
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xtick={1, 2, ..., 5},
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xmin=0.9, xmax=5.6,
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legend columns=2,]
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\addplot [ForestGreen, mark=*, line width=1pt]
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table [x=SNR, y=gamma_0_15, col sep=comma] {res/proximal/ber_paper.csv};
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\addlegendentry{$\gamma = 0.15$ (Wadayama et al.)}
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\addplot [ForestGreen, mark=triangle, dashed, line width=1pt]
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table [x=SNR, y=BER, col sep=comma,
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discard if not={gamma}{0.15},
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discard if gt={SNR}{5.5},]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.15$ (Own results)}
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\addplot [NavyBlue, mark=*, line width=1pt]
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table [x=SNR, y=gamma_0_01, col sep=comma] {res/proximal/ber_paper.csv};
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\addlegendentry{$\gamma = 0.01$ (Wadayama et al.)}
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\addplot [NavyBlue, mark=triangle, dashed, line width=1pt]
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table [x=SNR, y=BER, col sep=comma,
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discard if not={gamma}{0.01},
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discard if gt={SNR}{5.5},]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.01$ (Own results)}
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|
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\addplot [RedOrange, mark=*, line width=1pt]
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table [x=SNR, y=gamma_0_05, col sep=comma] {res/proximal/ber_paper.csv};
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\addlegendentry{$\gamma = 0.05$ (Wadayama et al.)}
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\addplot [RedOrange, mark=triangle, dashed, line width=1pt]
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table [x=SNR, y=BER, col sep=comma,
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discard if not={gamma}{0.05},
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discard if gt={SNR}{5.5},]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.05$ (Own results)}
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\addplot [RoyalPurple, mark=*, line width=1pt]
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table [x=SNR, y=BP, col sep=comma] {res/proximal/ber_paper.csv};
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\addlegendentry{BP (Wadayama et al.)}
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\end{axis}
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|
\end{tikzpicture}
|
|
|
|
\caption{Comparison of datapoints from Wadayama et al. with own simulation results%
|
|
\protect\footnotemark{}}
|
|
\label{fig:prox:results}
|
|
\end{figure}
|
|
%
|
|
\footnotetext{(3,6) regular LDPC code with $n = 204$, $k = 102$
|
|
\cite[\text{204.33.484}]{mackay_enc}; $\omega = 0.05, K=200, \eta=1.5$
|
|
}%
|
|
%
|
|
\noindent It is noticeable that for a moderately chosen value of $\gamma$
|
|
($\gamma = 0.05$) the decoding performance is better than for low
|
|
($\gamma = 0.01$) or high ($\gamma = 0.15$) values.
|
|
The question arises if there is some optimal value maximazing the decoding
|
|
performance, especially since the decoding performance seems to dramatically
|
|
depend on $\gamma$.
|
|
To better understand how $\gamma$ and the decoding performance are
|
|
related, figure \ref{fig:prox:results} was recreated, but with a considerably
|
|
larger selection of values for $\gamma$.
|
|
In this new graph, shown in figure \ref{fig:prox:results_3d}, instead of
|
|
stacking the \ac{BER} curves on top of one another in the same plot, the
|
|
visualization is extended to three dimensions.
|
|
The previously shown results are highlighted.
|
|
|
|
Evidently, while the decoding performance does depend on the value of
|
|
$\gamma$, there is no single optimal value offering optimal performance, but
|
|
rather a certain interval in which it stays largely unchanged.
|
|
When examining a number of different codes (figure
|
|
\ref{fig:prox:results_3d_multiple}), it is apparent that while the exact
|
|
landscape of the graph depends on the code, the general behaviour is the same
|
|
in each case.
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
|
|
\begin{axis}[view={75}{30},
|
|
zmode=log,
|
|
xlabel={$E_b / N_0$ (dB)},
|
|
ylabel={$\gamma$},
|
|
zlabel={BER},
|
|
%legend pos=outer north east,
|
|
legend style={at={(0.5,-0.7)},anchor=south},
|
|
ytick={0, 0.05, 0.1, 0.15},
|
|
width=0.6\textwidth,
|
|
height=0.45\textwidth,]
|
|
|
|
\addplot3[surf,
|
|
mesh/rows=17, mesh/cols=14,
|
|
colormap/viridis] table [col sep=comma,
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = \left[ 0\text{:}0.01\text{:}0.16 \right] $}
|
|
\addplot3[NavyBlue, line width=1.5] table [col sep=comma,
|
|
discard if not={gamma}{0.01},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.01$}
|
|
\addplot3[RedOrange, line width=1.5] table [col sep=comma,
|
|
discard if not={gamma}{0.05},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.05$}
|
|
\addplot3[ForestGreen, line width=1.5] table [col sep=comma,
|
|
discard if not={gamma}{0.15},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.15$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Visualization of relationship between the decoding performance\protect\footnotemark{}
|
|
and the parameter $\gamma$}
|
|
\label{fig:prox:results_3d}
|
|
\end{figure}%
|
|
%
|
|
\footnotetext{(3,6) regular LDPC code with n = 204, k = 102 \cite[\text{204.33.484}]{mackay_enc};
|
|
$\omega = 0.05, K=200, \eta=1.5$
|
|
}%
|
|
%
|
|
\noindent This indicates \todo{This is a result fit for the conclusion}
|
|
that while the choice of the parameter $\gamma$ significantly
|
|
affects the decoding performance, there is not much benefit attainable in
|
|
undertaking an extensive search for an exact optimum.
|
|
Rather, a preliminary examination providing a rough window for $\gamma$ may
|
|
be sufficient.
|
|
|
|
TODO: $\omega, K$
|
|
|
|
Changing the parameter $\eta$ does not appear to have a significant effect on
|
|
the decoding performance when keeping the value within a reasonable window
|
|
(''slightly larger than one``, as stated in \cite[Sec. 3.2]{proximal_paper}),
|
|
which seems plausible considering its only function is ensuring numerical stability.
|
|
|
|
Summarizing the above considerations, \ldots
|
|
|
|
\begin{itemize}
|
|
\item Conclusion: Number of iterations independent of \ac{SNR}
|
|
\end{itemize}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\begin{axis}[view={75}{30},
|
|
zmode=log,
|
|
xlabel={$E_b / N_0$ (dB)},
|
|
ylabel={$\gamma$},
|
|
zlabel={BER},
|
|
width=\textwidth,
|
|
height=0.75\textwidth,]
|
|
\addplot3[surf,
|
|
mesh/rows=17, mesh/cols=10,
|
|
colormap/viridis] table [col sep=comma,
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_963965.csv};
|
|
\addplot3[RedOrange, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.05},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_963965.csv};
|
|
\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.01},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_963965.csv};
|
|
\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.15},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_963965.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$
|
|
\cite[\text{96.3.965}]{mackay_enc}}
|
|
\end{subfigure}%
|
|
\hfill
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\begin{axis}[view={75}{30},
|
|
zmode=log,
|
|
xlabel={$E_b / N_0$ (dB)},
|
|
ylabel={$\gamma$},
|
|
zlabel={BER},
|
|
width=\textwidth,
|
|
height=0.75\textwidth,]
|
|
\addplot3[surf,
|
|
mesh/rows=17, mesh/cols=10,
|
|
colormap/viridis] table [col sep=comma,
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
|
|
\addplot3[RedOrange, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.05},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
|
|
\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.01},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
|
|
\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.15},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\caption{BCH code with $n=31, k=26$\\[2\baselineskip]}
|
|
\end{subfigure}
|
|
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\begin{axis}[view={75}{30},
|
|
zmode=log,
|
|
xlabel={$E_b/N_0$ (dB)},
|
|
ylabel={$\gamma$},
|
|
zlabel={BER},
|
|
width=\textwidth,
|
|
height=0.75\textwidth,]
|
|
\addplot3[surf,
|
|
mesh/rows=17, mesh/cols=14,
|
|
colormap/viridis] table [col sep=comma,
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot3[RedOrange, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.05},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.01},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.15},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$
|
|
\cite[\text{204.33.484}]{mackay_enc}}
|
|
\end{subfigure}%
|
|
\hfill
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\begin{axis}[view={75}{30},
|
|
zmode=log,
|
|
xlabel={$E_b / N_0$ (dB)},
|
|
ylabel={$\gamma$},
|
|
zlabel={BER},
|
|
width=\textwidth,
|
|
height=0.75\textwidth,]
|
|
\addplot3[surf,
|
|
mesh/rows=17, mesh/cols=10,
|
|
colormap/viridis] table [col sep=comma,
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20455187.csv};
|
|
\addplot3[RedOrange, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.05},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20455187.csv};
|
|
\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.01},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20455187.csv};
|
|
\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.15},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_20455187.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$
|
|
\cite[\text{204.55.187}]{mackay_enc}}
|
|
\end{subfigure}%
|
|
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\begin{axis}[view={75}{30},
|
|
zmode=log,
|
|
xlabel={$E_b / N_0$ (dB)},
|
|
ylabel={$\gamma$},
|
|
zlabel={BER},
|
|
width=\textwidth,
|
|
height=0.75\textwidth,]
|
|
\addplot3[surf,
|
|
mesh/rows=17, mesh/cols=10,
|
|
colormap/viridis] table [col sep=comma,
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_40833844.csv};
|
|
\addplot3[RedOrange, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.05},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_40833844.csv};
|
|
\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.01},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_40833844.csv};
|
|
\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.15},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_40833844.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$
|
|
\cite[\text{408.33.844}]{mackay_enc}}
|
|
\end{subfigure}%
|
|
\hfill
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\begin{axis}[view={75}{30},
|
|
zmode=log,
|
|
xlabel={$E_b / N_0$ (dB)},
|
|
ylabel={$\gamma$},
|
|
zlabel={BER},
|
|
width=\textwidth,
|
|
height=0.75\textwidth,]
|
|
\addplot3[surf,
|
|
mesh/rows=17, mesh/cols=10,
|
|
colormap/viridis] table [col sep=comma,
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
\addplot3[RedOrange, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.05},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.01},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
|
|
discard if not={gamma}{0.15},
|
|
x=SNR, y=gamma, z=BER]
|
|
{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\caption{LDPC code (Progressive Edge Growth Construction) with $n=504, k=252$
|
|
\cite[\text{PEGReg252x504}]{mackay_enc}}
|
|
\end{subfigure}%
|
|
|
|
\vspace{1cm}
|
|
|
|
\begin{subfigure}[c]{\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\begin{axis}[hide axis,
|
|
xmin=10, xmax=50,
|
|
ymin=0, ymax=0.4,
|
|
legend style={draw=white!15!black,legend cell align=left}]
|
|
\addlegendimage{surf, colormap/viridis}
|
|
\addlegendentry{$\gamma = \left[ 0\text{ : }0.01\text{ : }0.16 \right] $};
|
|
\addlegendimage{NavyBlue, line width=1.5pt}
|
|
\addlegendentry{$\gamma = 0.01$};
|
|
\addlegendimage{RedOrange, line width=1.5pt}
|
|
\addlegendentry{$\gamma = 0.05$};
|
|
\addlegendimage{ForestGreen, line width=1.5pt}
|
|
\addlegendentry{$\gamma = 0.15$};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\end{subfigure}
|
|
|
|
\caption{BER for $\omega = 0.05, K=100$ (different codes)}
|
|
\label{fig:prox:results_3d_multiple}
|
|
\end{figure}
|
|
|
|
\subsection{Decoding Performance}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[grid=both,
|
|
xlabel={$E_b / N_0$ (dB)}, ylabel={BER},
|
|
ymode=log,
|
|
width=0.48\textwidth,
|
|
height=0.36\textwidth,
|
|
legend style={at={(0.05,0.05)},anchor=south west},
|
|
ymax=1.5, ymin=3e-7,]
|
|
|
|
\addplot [ForestGreen, mark=*]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.15$}
|
|
\addplot [NavyBlue, mark=*]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.01$}
|
|
\addplot [RedOrange, mark=*]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.05$}
|
|
\end{axis}
|
|
\end{tikzpicture}%
|
|
\hfill%
|
|
\begin{tikzpicture}
|
|
\begin{axis}[grid=both,
|
|
xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
|
|
ymode=log,
|
|
width=0.48\textwidth,
|
|
height=0.36\textwidth,
|
|
legend style={at={(0.05,0.05)},anchor=south west},
|
|
ymax=1.5, ymin=3e-7,]
|
|
|
|
\addplot [ForestGreen, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.15$}
|
|
\addplot [NavyBlue, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.01$}
|
|
\addplot [RedOrange, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.05$}
|
|
\end{axis}
|
|
\end{tikzpicture}\\[1em]
|
|
\begin{tikzpicture}
|
|
\begin{axis}[grid=both,
|
|
xlabel={$E_b / N_0$ (dB)}, ylabel={Decoding Failure Rate},
|
|
ymode=log,
|
|
width=0.48\textwidth,
|
|
height=0.36\textwidth,
|
|
legend style={at={(0.05,0.05)},anchor=south west},
|
|
ymax=1.5, ymin=3e-7,]
|
|
|
|
\addplot [ForestGreen, mark=*]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.15$}
|
|
\addplot [NavyBlue, mark=*]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.01$}
|
|
\addplot [RedOrange, mark=*]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.05$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Cmoparison\protect\footnotemark{} of \ac{FER}, \ac{BER} and
|
|
decoding failure rate; $\omega = 0.05, K=100$}
|
|
\label{fig:prox:ber_fer_dfr}
|
|
\end{figure}%
|
|
%
|
|
\footnotetext{(3,6) regular LDPC code with n = 204, k = 102 \cite[\text{204.33.484}]{mackay_enc}}%
|
|
%
|
|
|
|
Until now, only the \ac{BER} has been considered to assess the decoding
|
|
performance.
|
|
The \ac{FER}, however, shows considerably worse behaviour, as can be seen in
|
|
figure \ref{fig:prox:ber_fer_dfr}.
|
|
Besides the \ac{BER} and \ac{FER} curves, the figure also shows the
|
|
\textit{decoding failure rate}.
|
|
This is the rate at which the iterative process produces invalid codewords,
|
|
i.e., the stopping criterion (line 6 of algorithm \ref{TODO}) is never
|
|
satisfied and the maximum number of itertations $K$ is reached without
|
|
converging to a valid codeword.
|
|
Three lines are plotted in each case, corresponding to different values of
|
|
the parameter $\gamma$.
|
|
The values chosen are the same as in figure \ref{fig:prox:results}, as they
|
|
seem to adequately describe the behaviour across a wide range of values
|
|
(see figure \ref{fig:prox:results_3d}).
|
|
|
|
It is apparent that the \ac{FER} and the decoding failure rate are extremely
|
|
similar, especially for higher \acp{SNR}.
|
|
This leads to the hypothesis that, at least for higher \acp{SNR}, frame errors
|
|
arise mainly due to the non-convergence of the algorithm instead of
|
|
convergence to the wrong codeword.
|
|
This course of thought will be picked up in section
|
|
\ref{sec:prox:Improved Implementation} to try to improve the algorithm.
|
|
|
|
In summary, the \ac{BER} and \ac{FER} indicate dissimilar decoding
|
|
performance.
|
|
The decoding failure rate closely resembles the \ac{FER}, suggesting that
|
|
the frame errors may largely be attributed to decoding failures.
|
|
|
|
\todo{Maybe reference to the structure of the algorithm (1 part likelihood
|
|
1 part constraints)}
|
|
|
|
|
|
\subsection{Convergence Properties}
|
|
|
|
The previous observation, that the \ac{FER} arises mainly due to the
|
|
non-convergence of the algorithm instead of convergence to the wrong codeword,
|
|
raises the question why the decoding process does not converge so often.
|
|
In figure \ref{fig:prox:convergence}, the iterative process is visualized
|
|
for each iteration.
|
|
In order to be able to simultaneously consider all components of the vectors
|
|
being dealt with, a BCH code with $n=7$ and $k=4$ is chosen.
|
|
Each chart shows one component of the current estimates during a given
|
|
iteration (alternating between $\boldsymbol{r}$ and $\boldsymbol{s}$), as well
|
|
as the gradients of the negative log-likelihood and the code-constraint
|
|
polynomial, which influence the next estimate.
|
|
|
|
\begin{figure}[H]
|
|
\begin{minipage}[c]{0.25\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}[scale = 0.35]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=8cm,
|
|
height=3cm,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_1]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_1]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_1]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_2$}
|
|
\addlegendentry{$\left(\nabla h \right)_2 $}
|
|
\end{axis}
|
|
\end{tikzpicture}\\
|
|
\begin{tikzpicture}[scale = 0.35]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=8cm,
|
|
height=3cm,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_2]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_2]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_2]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_3$}
|
|
\addlegendentry{$\left(\nabla h \right)_3 $}
|
|
\end{axis}
|
|
\end{tikzpicture}\\
|
|
\begin{tikzpicture}[scale = 0.35]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=8cm,
|
|
height=3cm,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_3]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_3]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_3]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_4$}
|
|
\addlegendentry{$\left(\nabla h \right)_4 $}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{minipage}%
|
|
\begin{minipage}[c]{0.5\textwidth}
|
|
\vspace*{-1cm}
|
|
\centering
|
|
|
|
\begin{tikzpicture}[scale = 0.85, spy using outlines={circle, magnification=6,
|
|
connect spies}]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=8cm,
|
|
height=3cm,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_0]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_0]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_0]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_1$}
|
|
\addlegendentry{$\left(\nabla h \right)_1 $}
|
|
|
|
\coordinate (spypoint) at (axis cs:100,0.53);
|
|
\coordinate (magnifyglass) at (axis cs:175,2);
|
|
\end{axis}
|
|
\spy [black, size=2cm] on (spypoint)
|
|
in node[fill=white] at (magnifyglass);
|
|
\end{tikzpicture}
|
|
\end{minipage}%
|
|
\begin{minipage}[c]{0.25\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}[scale = 0.35]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=8cm,
|
|
height=3cm,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_4]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_4]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_4]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_5$}
|
|
\addlegendentry{$\left(\nabla h \right)_5 $}
|
|
\end{axis}
|
|
\end{tikzpicture}\\
|
|
\begin{tikzpicture}[scale = 0.35]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=8cm,
|
|
height=3cm,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_5]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_5]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_5]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_6$}
|
|
\addlegendentry{$\left(\nabla h \right)_6 $}
|
|
\end{axis}
|
|
\end{tikzpicture}\\
|
|
\begin{tikzpicture}[scale = 0.35]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=8cm,
|
|
height=3cm,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_6]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_6]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_6]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_7$}
|
|
\addlegendentry{$\left(\nabla h \right)_7 $}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{minipage}
|
|
|
|
\caption{Internal variables of proximal decoder
|
|
as a function of the number of iterations ($n=7$)\protect\footnotemark{}}
|
|
\label{fig:prox:convergence}
|
|
\end{figure}%
|
|
%
|
|
\footnotetext{A single decoding is shown, using the BCH$\left( 7,4 \right) $ code;
|
|
$\gamma = 0.05, \omega = 0.05, E_b / N_0 = \SI{5}{dB}$
|
|
}%
|
|
%
|
|
\noindent It is evident that in all cases, past a certain number of
|
|
iterations, the estimate starts to oscillate around a particular value.
|
|
After a certain point, the two gradients stop further approaching the value
|
|
zero.
|
|
In particular, this leads to the code-constraints polynomial not being
|
|
minimized.
|
|
As such, the constraints are not being satisfied and the estimate is not
|
|
converging towards a valid codeword.
|
|
|
|
While figure \ref{fig:prox:convergence} shows only one instance of a decoding
|
|
task, it is indicative of the general behaviour of the algorithm.
|
|
This can be justified by looking at the gradients themselves.
|
|
In figure \ref{fig:prox:gradients} the gradients of the negative
|
|
log-likelihood and the code-constraint polynomial for a repetition code with
|
|
$n=2$ are shown.
|
|
It is obvious that walking along the gradients in an alternating fashion will
|
|
produce a net movement in a certain direction, as long as the two gradients
|
|
have a common component.
|
|
As soon as this common component is exhausted, they will start pulling the
|
|
estimate in opposing directions, leading to an oscillation as illustrated
|
|
in figure \ref{fig:prox:convergence}.
|
|
Consequently, this oscillation is an intrinsic property of the structure of
|
|
the proximal decoding algorithm, where the two parts of the objective function
|
|
are minimized in an alternating manner using their gradients.
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{subfigure}[c]{0.5\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[xmin = -1.25, xmax=1.25,
|
|
ymin = -1.25, ymax=1.25,
|
|
xlabel={$x_1$}, ylabel={$x_2$},
|
|
width=\textwidth,
|
|
height=0.75\textwidth,
|
|
grid=major, grid style={dotted},
|
|
view={0}{90}]
|
|
|
|
\addplot3[point meta=\thisrow{grad_norm},
|
|
point meta min=1,
|
|
point meta max=3,
|
|
quiver={u=\thisrow{grad_0},
|
|
v=\thisrow{grad_1},
|
|
scale arrows=.05,
|
|
every arrow/.append style={%
|
|
line width=.3+\pgfplotspointmetatransformed/1000,
|
|
-{Latex[length=0pt 5,width=0pt 3]}
|
|
},
|
|
},
|
|
quiver/colored = {mapped color},
|
|
colormap/rocket,
|
|
-stealth,
|
|
]
|
|
table[col sep=comma] {res/proximal/2d_grad_L.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{$\nabla L \left(\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) $
|
|
for a repetition code with $n=2$}
|
|
\end{subfigure}%
|
|
\hfill%
|
|
\begin{subfigure}[c]{0.5\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[xmin = -1.25, xmax=1.25,
|
|
ymin = -1.25, ymax=1.25,
|
|
xlabel={$x_1$}, ylabel={$x_2$},
|
|
grid=major, grid style={dotted},
|
|
width=\textwidth,
|
|
height=0.75\textwidth,
|
|
view={0}{90}]
|
|
\addplot3[point meta=\thisrow{grad_norm},
|
|
point meta min=1,
|
|
point meta max=4,
|
|
quiver={u=\thisrow{grad_0},
|
|
v=\thisrow{grad_1},
|
|
scale arrows=.03,
|
|
every arrow/.append style={%
|
|
line width=.3+\pgfplotspointmetatransformed/1000,
|
|
-{Latex[length=0pt 5,width=0pt 3]}
|
|
},
|
|
},
|
|
quiver/colored = {mapped color},
|
|
colormap/rocket,
|
|
-stealth,
|
|
]
|
|
table[col sep=comma] {res/proximal/2d_grad_h.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{$\nabla h \left( \tilde{\boldsymbol{x}} \right) $
|
|
for a repetition code with $n=2$}
|
|
\end{subfigure}%
|
|
\end{figure}
|
|
|
|
|
|
While the initial net movement is generally directed in the right direction
|
|
owing to the gradient of the negative log-likelihood, the final oscillation
|
|
may well take place in a segment of space not corresponding to a valid
|
|
codeword, leading to the aforementioned non-convergence of the algorithm.
|
|
This also partly explains the difference in decoding performance when looking
|
|
at the \ac{BER} and \ac{FER}, as it would lower the amount of bit errors while
|
|
still yielding an invalid codeword.
|
|
|
|
When considering codes with larger $n$, the behaviour generally stays the
|
|
same, with some minor differences.
|
|
In figure \ref{fig:prox:convergence_large_n} the decoding process is
|
|
visualized for one component of a code with $n=204$, for a single decoding.
|
|
The two gradients still start to fight each other and the estimate still
|
|
starts to oscillate, the same as illustrated on the basis of figure
|
|
\ref{fig:prox:convergence} for a code with $n=7$.
|
|
However, in this case, the gradient of the code-constraint polynomial iself
|
|
starts to oscillate, its average value being such that the effect of the
|
|
gradient of the negative log-likelihood is counteracted.
|
|
|
|
In conclusion, as a general rule, the proximal decoding algorithm reaches
|
|
an oscillatory state which it cannot escape as a consequence of its structure.
|
|
In this state, the constraints may not be satisfied, leading to the algorithm
|
|
returning an invalid codeword.
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=0.6\textwidth,
|
|
height=0.45\textwidth,
|
|
scale only axis,
|
|
xtick={0, 100, ..., 400},
|
|
xticklabels={0, 50, ..., 200},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_0]
|
|
{res/proximal/extreme_components_20433484_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_0]
|
|
{res/proximal/extreme_components_20433484_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_0]
|
|
{res/proximal/extreme_components_20433484_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L\right)_1$}
|
|
\addlegendentry{$\left(\nabla h\right)_1$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Internal variables of proximal decoder as a function of the iteration ($n=204$)}
|
|
\label{fig:prox:convergence_large_n}
|
|
\end{figure}%
|
|
|
|
|
|
\subsection{Computational Performance}
|
|
|
|
\begin{itemize}
|
|
\item Theoretical analysis
|
|
\item Simulation results to substantiate theoretical analysis
|
|
\item Conclusion: $\mathcal{O}\left( n \right)$ time complexity, implementation heavily
|
|
optimizable
|
|
\end{itemize}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Improved Implementation}%
|
|
\label{sec:prox:Improved Implementation}
|
|
|
|
As mentioned earlier, frame errors seem to mainly stem from decoding failures.
|
|
This, coupled with the fact that the \ac{BER} indicates so much better
|
|
performance than the \ac{FER}, leads to the assumption that only a small
|
|
number of components of the estimated vector may be responsible for an invalid
|
|
result.
|
|
If it was possible to limit the number of possibly wrong components of the
|
|
estimate to a small subset, an \ac{ML}-decoding step could be performed on
|
|
a limited number of possible results (``ML-in-the-List'' as it will
|
|
subsequently be called) to improve the decoding performance.
|
|
This concept is pursued in this section.
|
|
|
|
\begin{itemize}
|
|
\item Decoding performance and comparison with standard proximal decoding
|
|
\item Computational performance and comparison with standard proximal decoding
|
|
\item Conclusion
|
|
\begin{itemize}
|
|
\item Summary
|
|
\item Up to $\SI{1}{dB}$ gain possible
|
|
\end{itemize}
|
|
\end{itemize}
|
|
|