Table 1 Message passing in BP-based channel decoding and state estimation system
From: Performance analysis on joint channel decoding and state estimation in cyber-physical systems
Step | Distribution | Gaussian distribution | Details |
|---|---|---|---|
x(t−1)→x(t) | π x(t−1),x(t)(x(t−1)) | \(\mathcal {N}\left (\mathbf {x}_{t-1}, \mathbf {x}_{\pi _{x},t-1}, \mathbf {P}_{\pi _{x},t-1}\right) \) | * |
x(t)→y(t) | π x(t)(x(t)) | \(\mathcal {N} \left (\mathbf {x}_{t}, \mathbf {x}_{l, t}, \mathbf {P}_{l, t}\right)\) | \(\begin {array}{l}\mathbf {x}_{l, t}= \mathbf {Ax}_{\pi _{x}, t-1}+\mathbf {Bu}_{t-1}; \\ \mathbf {P}_{l, t} = \mathbf {A} \mathbf {P}_{\pi _{x},t-1} \mathbf {A}^{T} +\mathbf {\Sigma }_{n}\end {array}\) |
– | π x(t),y(t)(x(t)) | \(\mathcal {N}\left (\mathbf {x}_{t}, \mathbf {x}_{\pi _{y},t}, \mathbf {P}_{\pi _{y}, t}\right) \) | \(\mathbf {x}_{\pi _{y},t}= \mathbf {x}_{l, t}; \mathbf {P}_{\pi _{y}, t}= \mathbf {P}_{l, t} \) |
y(t)→b(t) | π y(t)(y(t)) | \(\mathcal {N}\left (\mathbf {y}_{t}, \mathbf {y}_{l,t}, \mathbf {S}_{l, t}\right)\) | \( \mathbf {y}_{l,t}= \mathbf {C} \mathbf {x}_{\pi _{y},t} ;\mathbf {S}_{l, t} =\mathbf {C} \mathbf {P}_{\pi _{y}, t} \mathbf {C}^{T} +\mathbf {\Sigma }_{w}\) |
– | π y(t),b(t)(y(t)) | \(\mathcal {N}(\mathbf {y}_{t}, \mathbf {y}_{\pi, t}, \mathbf {S}_{\pi, t})\) | y π,t =y l,t ;S π,t =S l,t |
b(t)→y(t) | λ b(t),y(t)(y(t)) | \(\mathcal {N}(\mathbf {y}_{t}, \mathbf {y}_{\lambda, t}, \mathbf {S}_{\lambda, t})\) | The y λ,t and S λ,t is provided in Section 4. |
y(t)→x(t) | λ y(t),x(t)(x(t)) | \(\mathcal {N}(\mathbf {x}_{t}, \mathbf {x}_{\lambda _{y},t}, \mathbf {P}_{\lambda _{y},t})\) | \(\mathbf {x}_{\lambda _{y},t}= \mathbf {C}^{-1}\times \mathbf {y}_{\lambda, t};{\newline } \mathbf {P}_{\lambda _{y},t} = \mathbf {C}^{-1} (\mathbf {S}_{\lambda, t}+\mathbf {\Sigma }_{w})(\mathbf {C}^{-1})^{T}\) |
x(t)→x(t+1) | π x(t),x(t+1)(x(t)) | \(\mathcal {N}(\mathbf {x}_{t}, \mathbf {x}_{\pi _{x},t}, \mathbf {P}_{\pi _{x},t}) \) | \( \mathbf {P}_{\pi _{x},t}=(\mathbf {P}_{l, t}^{-1}+ \mathbf {P}_{\lambda _{y},t}^{-1})^{-1}; \mathbf {x}_{\pi _{x},t} = \mathbf {P}_{\pi _{x},t}(\mathbf {P}_{l, t}^{-1}\mathbf {x}_{l,t} + \mathbf {P}_{\lambda _{y},t}^{-1}\mathbf {x}_{\lambda _{y},t})\) |