A Message Passing-Assisted Iterative Noise Cancellation Method for Clipped OTFS-BFDM Systems

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. 2022 May 23;22(10):3937. doi: 10.3390/s22103937

Algorithm 1: The procedure of the MP-AIC method.

Require: $β$ , $k_{p}$ , $l_{p}$ , $h_{p}$ , $N_{i t e r}$ , $τ_{max}$ , $σ_{n}^{2}$ , $Δ = 0.6$ , $ε = 0.01$ .
Ensure: ${\hat{d}}_{a} = \underset{a_{j} \in A}{arg max} p_{a} (a_{j})$ and $p_{a b}^{(τ_{max} - 1)}$ .
Initialize $k = 0$ , $j = 0$ , $σ {_{A l l}^{2}}_{(0)} = σ_{A l l}^{2}$ , $p_{a b}^{(0)} = 1 / Q$ , ${\hat{d}}_{(0)} = \hat{d}$ , ${σ_{δ}^{2}}_{(0)} = σ_{δ}^{2}$ .
while $k \leq N_{i t e r} - 1$ do
$σ {_{A l l}^{2}}_{(k)} = β^{- 2} {(σ_{δ}^{2}}_{(k)} + σ_{n}^{2})$ (Equations (20), (24) and (28)).
${\hat{d}}_{(k)} = β^{- 1} {\tilde{s}}_{(k)}$ (Equations (25) and (27)).
while $max_{a, b, a_{j}} |p_{a b}^{(i + 1)} (a_{j}) - p_{a b}^{(i)} (a_{j})| > ε$ or $i \leq τ_{max} - 1$ do
for $a = 1$ to S do
for $b = 1$ to S do
Message from ${\tilde{d}}_{b}$ to $x_{a}$ :
$μ_{b a}^{(i)} = \sum_{c \in S, c \neq a} \sum_{j = 1}^{Q} {p_{c b}^{(i)}}_{(k)} (a_{j}) a_{j} H_{b, c}$
${(σ_{b a}^{(i)})}^{2} = \sum_{c \in S, c \neq a} \sum_{j = 1}^{Q} p {_{c b}^{(i)}}_{(k)} {|a_{j}|}^{2} {|H_{b, a}|}^{2} - \sum_{c \in S, c \neq a} {|\sum_{j = 1}^{Q} p {_{c b}^{(i)}}_{(k)} (a_{j}) a_{j} H_{b, c}|}^{2} + σ {_{a l l}^{2}}_{(k)}$
end for
for end for
for for $a = 1$ to S do
for $b = 1$ to S do
Message from $x_{a}$ to ${\tilde{d}}_{b}$ :
${p_{a b}^{(i)}}_{(k)} \propto \prod_{c \in S, c \neq b} Pr ({\tilde{d}}_{c} | d_{a} = a_{j}, H_{e q})$
${p_{a b}^{(i + 1)}}_{(k)} = Δ {p_{a b}^{(i)}}_{(k)} (a_{j}) + (1 - Δ) Δ {p_{a b}^{(i - 1)}}_{(k)} (a_{j})$
end for
end for
$i = i + 1$
end while
$k = k + 1$
end while