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Correct Preliminaries

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Andreas Tsouchlos 2024-01-07 21:52:43 +01:00
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letter.tex
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@ -7,6 +7,7 @@
\usepackage{algorithm}
\usepackage{siunitx}
\usepackage{dsfont}
\usepackage{mleftright}
\usepackage{tikz}
\usetikzlibrary{spy, arrows.meta,arrows}
@ -192,31 +193,31 @@ number of parity-checks:
\end{align*}
%
The check nodes $j \in \mathcal{J}:=\left\{1, \ldots, m\right\}$ each correspond
to a parity check, i.e., row of $\boldsymbol{H}$.
The check nodes $j \in \mathcal{J}:=\left\{1, \ldots, m\right\}$ each
correspond to a parity check, i.e., a row of $\boldsymbol{H}$.
The variable nodes $i \in \mathcal{I}:=\left\{1, \ldots, n\right\}$ correspond
to the components of a codeword being subjected to a parity check, i.e., to
columns of $\boldsymbol{H}$.
to the components of a codeword being subjected to a parity check, i.e.,
to the columns of $\boldsymbol{H}$.
The neighborhood of a parity check $j$, i.e., the set of indices of components
relevant for the according parity check, is denoted by
$N_c(j) := \left\{i \mid i \in \mathcal{I}, \boldsymbol{H}_{j,i} = 1 \right\},
$\mathcal{N}_c(j) := \left\{i \in \mathcal{I}: \boldsymbol{H}\negthinspace_{j,i} = 1 \right\},
\hspace{2mm} j \in \mathcal{J}$.
In order to transmit a codeword $\boldsymbol{c} \in \mathbb{F}_2^n$, it is
mapped onto a \textit{binary phase shift keying} (BPSK) symbol via
$\boldsymbol{x} = 1 - 2\boldsymbol{c}$, with
$ \boldsymbol{x} \in \left\{-1, 1\right\}^n$, which is then transmitted over an
$ \boldsymbol{x} \in \left\{\pm 1\right\}^n$, which is then transmitted over an
AWGN channel.
The received vector $\boldsymbol{y} \in \mathbb{R}^n$ is decoded to obtain an
estimate of the transmitted codeword, denoted as
$\hat{\boldsymbol{c}} \in \mathbb{F}_2^n$.
A distinction is made between $\boldsymbol{x} \in \left\{-1, 1\right\}^n$
A distinction is made between $\boldsymbol{x} \in \left\{\pm 1\right\}^n$
and $\tilde{\boldsymbol{x}} \in \mathbb{R}^n$,
the former denoting the BPSK symbol physically transmitted over the channel and
the latter being used as a variable during the optimization process.
The posterior probability of having transmitted $\boldsymbol{x}$ when receiving
$\boldsymbol{y}$ is expressed as a \textit{probability mass function} (PMF)
$p_{\boldsymbol{X}\mid\boldsymbol{Y}}(\boldsymbol{x} \mid \boldsymbol{y})$.
$P_{\boldsymbol{X}\mid\boldsymbol{Y}}(\boldsymbol{x} \mid \boldsymbol{y})$.
Likewise, the likelihood of receiving $\boldsymbol{y}$ upon transmitting
$\boldsymbol{x}$ is expressed as a \textit{probability density function} (PDF)
$f_{\boldsymbol{Y}\mid\boldsymbol{X}}(\boldsymbol{y} \mid \boldsymbol{x})$.
@ -231,25 +232,24 @@ With proximal decoding, the proximal gradient method \cite{proximal_algorithms}
is used to solve a non-convex optimization formulation of the MAP decoding
problem.
When making the equal probability assumption for all codewords, MAP and ML
With the equal prior probability assumption for all codewords, MAP and ML
decoding are equivalent and, specifically for AWGN channels, correspond to a
nearest-neighbor decision.
For this reason, decoding can be done using a figure of merit that describes
the distance from a given vector to a codeword.
For this reason, decoding can be carried out using a figure of merit that
describes the distance from a given vector to a codeword.
One such expression, formulated under the assumption of BPSK, is the
\textit{code-constraint polynomial} \cite{proximal_paper}
%
\begin{align*}
h\left( \tilde{\boldsymbol{x}} \right) =
h( \tilde{\boldsymbol{x}} ) =
\underbrace{\sum_{i=1}^{n}
\left( \tilde{x_i}^2-1 \right) ^2}_{\text{Bipolar constraint}}
\left( \tilde{x}_i^2-1 \right) ^2}_{\text{Bipolar constraint}}
+ \underbrace{\sum_{j=1}^{m} \left[
\left( \prod_{i\in N_c \left( j \right) } \tilde{x_i} \right)
\left( \prod_{i\in \mathcal{N}_c \left( j \right) } \tilde{x}_i \right)
-1 \right] ^2}_{\text{Parity constraint}}
.\end{align*}%
%
Its intent is to penalize vectors far from a codeword and favor those close
to one.
Its intent is to penalize vectors far from a codeword.
It comprises two terms: one representing the bipolar constraint
and one representing the parity constraint, incorporating all of the
information regarding the code.
@ -257,18 +257,18 @@ information regarding the code.
The channel model can be considered using the negative log-likelihood
%
\begin{align*}
L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) = -\ln\left(
f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}\left(
\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) \right)
L \mleft( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \mright) = -\ln\mleft(
f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}} \mleft(
\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \mright) \mright)
.\end{align*}
%
The information about the channel and the code are consolidated in the objective
function \cite{proximal_paper}
%
\begin{align*}
g\left( \tilde{\boldsymbol{x}} \right)
= L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
+ \gamma h\left( \tilde{\boldsymbol{x}} \right),
g \mleft( \tilde{\boldsymbol{x}} \mright)
= L \mleft( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \mright)
+ \gamma h\mleft( \tilde{\boldsymbol{x}} \mright),
\hspace{5mm} \gamma > 0%
.\end{align*}
%
@ -280,17 +280,17 @@ introduced, describing the result of each of the two steps:
%
\begin{alignat}{3}
\boldsymbol{r} &\leftarrow \boldsymbol{s}
- \omega \left( \boldsymbol{s} - \boldsymbol{y} \right)
- \omega \mleft( \boldsymbol{s} - \boldsymbol{y} \mright)
\hspace{5mm }&&\omega > 0 \label{eq:r_update}\\
\boldsymbol{s} &\leftarrow \boldsymbol{r}
- \gamma \nabla h\left( \boldsymbol{r} \right),
- \gamma \nabla h\mleft( \boldsymbol{r} \mright),
\hspace{5mm} &&\gamma > 0 \label{eq:s_update}
.\end{alignat}
%
An equation for determining $\nabla h(\boldsymbol{r})$ is given in
\cite{proximal_paper}.
It should be noted that the variables $\boldsymbol{r}$ and $\boldsymbol{s}$
really represent $\tilde{\boldsymbol{x}}$ during different
represent $\tilde{\boldsymbol{x}}$ during different
stages of the decoding process.
As the gradient of the code-constraint polynomial can attain very large values
@ -300,7 +300,7 @@ $\left[-\eta, \eta\right]^n$ by a projection
$\Pi_\eta : \mathbb{R}^n \rightarrow \left[-\eta, \eta\right]^n$, where $\eta$
is a positive constant slightly larger than one, e.g., $\eta = 1.5$.
The resulting decoding process as described in \cite{proximal_paper} is
presented in algorithm \ref{alg:proximal_decoding}.
presented in Algorithm \ref{alg:proximal_decoding}.
\begin{algorithm}
\caption{Proximal decoding algorithm for an AWGN channel.}
@ -395,7 +395,7 @@ decoded bits.%
\end{figure}%
%
An approach for lowering the FER might then be to append an ``ML-in-the-list''
\cite{ml_in_the_list} step to the decoding process shown in algorithm
\cite{ml_in_the_list} step to the decoding process shown in Algorithm
\ref{alg:proximal_decoding}.
This step would consist of determining the $N \in \mathbb{N}$ most probably
wrong bits, finding all variations of the current estimate with those bits
@ -610,7 +610,7 @@ the probability that a given component was decoded incorrectly.%
\label{fig:p_error}
\end{figure}
The complete improved algorithm is depicted in algorithm \ref{alg:improved}.
The complete improved algorithm is depicted in Algorithm \ref{alg:improved}.
First, the proximal decoding algorithm is applied.
If a valid codeword has been reached, i.e., if the algorithm has converged, this
is the solution returned.