APPENDIX
Using the rule of multiplication of probabilities and the main properties of Markov processes, we write expressions for the distribution law of state vector \({{\left[ {{{{\mathbf{z}}}^{T}}\left( k \right),{{{\mathbf{s}}}^{T}}\left( k \right)} \right]}^{T}}\):
$$\begin{gathered} f\left( {{\mathbf{z}}\left( k \right),{\mathbf{s}}\left( k \right)\left| {{\mathbf{D}}_{1}^{k}} \right.} \right) = \frac{1}{{P\left( {{\mathbf{D}}_{1}^{k}} \right)}}f\left( {{\mathbf{z}}\left( k \right),{\mathbf{s}}\left( k \right),{\mathbf{D}}_{1}^{k}} \right) = \frac{1}{{P\left( {{\mathbf{D}}_{1}^{k}} \right)}} \\ \times \,\,\sum\limits_{{\mathbf{s}}\left( {k - 1} \right)} { \cdots \sum\limits_{{\mathbf{s}}\left( 1 \right)} {\int { \cdots \int {f\left( {{\mathbf{z}}\left( k \right),{\mathbf{s}}\left( k \right),{\mathbf{d}}\left( k \right),...,{\mathbf{z}}\left( 1 \right),{\mathbf{s}}\left( 1 \right),{\mathbf{d}}\left( 1 \right)} \right)} } } } \\ \times \,\,\prod\limits_{g = 1}^{k - 1} d {{{\mathbf{z}}}_{g}} = \frac{{P\left( {{\mathbf{D}}_{1}^{{k - 1}}} \right)}}{{P\left( {{\mathbf{D}}_{1}^{k}} \right)}}\sum\limits_{{\mathbf{s}}\left( {k - 1} \right)} {\int {f\left( {{\mathbf{z}}\left( k \right),{\mathbf{s}}\left( k \right),\left. {{\mathbf{d}}\left( k \right)} \right|{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( {k - 1} \right),{\mathbf{d}}\left( {k - 1} \right)} \right)} } \\ \times \,\,f\left( {{\mathbf{z}}\left( {k - 1} \right),\left. {{\mathbf{s}}\left( {k - 1} \right)} \right|{\mathbf{D}}_{1}^{{k - 1}}} \right)d{\mathbf{z}}\left( {k - 1} \right), \\ \end{gathered} $$
(A.1)
where the integration with respect to variable \({\mathbf{z}}\) carried out in area \({{\Re }^{{{{n}_{x}}}}} \times \Omega \) \(\left( {\Omega \subset {{\Omega }_{1}} \times \cdot \cdot \cdot \times {{\Omega }_{{{{n}_{y}}}}}} \right)\).
We transform the first conditional probability density in the integrand:
$$\begin{gathered} f\left( {{\mathbf{z}}\left( k \right),{\mathbf{s}}\left( k \right),\left. {{\mathbf{d}}\left( k \right)} \right|{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( {k - 1} \right),{\mathbf{d}}\left( {k - 1} \right)} \right) \\ = f\left( {{\mathbf{z}}\left( k \right),{\mathbf{s}}\left( k \right),\left. {{\mathbf{d}}\left( k \right)} \right|{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( {k - 1} \right)} \right). \\ \end{gathered} $$
Here, absence \({\mathbf{d}}\left( {k - 1} \right)\) does not matter for given \({\mathbf{z}}\left( {k - 1} \right)\). Using the property of conditional probability densities, we write
$$\begin{gathered} f\left( {{\mathbf{z}}\left( k \right),{\mathbf{s}}\left( k \right),\left. {{\mathbf{d}}\left( k \right)} \right|{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( {k - 1} \right)} \right) \\ = f\left( {\left. {{\mathbf{z}}\left( k \right)} \right|{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( {k - 1} \right),{\mathbf{s}}\left( k \right)} \right) \\ \times \,\,P\left( {\left. {{\mathbf{s}}\left( k \right)} \right|{\mathbf{s}}\left( {k - 1} \right),{\mathbf{z}}\left( {k - 1} \right)} \right) \\ \times \,\,P\left( {\left. {{\mathbf{d}}\left( k \right)} \right|{\mathbf{z}}\left( k \right),{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( k \right),{\mathbf{s}}\left( {k - 1} \right)} \right). \\ \end{gathered} $$
(A.2)
Due to unique transformation \({\mathbf{z}}\left( k \right)\) in \({\mathbf{d}}\left( k \right)\) when quantizing, the conditional probability of a digital signal [17] has form
$$\begin{gathered} P\left( {{\mathbf{d}}\left( k \right)|{\mathbf{z}}\left( k \right),{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( k \right),{\mathbf{s}}\left( {k - 1} \right)} \right) \\ = \left\{ {\begin{array}{*{20}{c}} {1,}&{{\mathbf{y}}\left( k \right) \in \Omega } \\ {0,}&{{\mathbf{y}}\left( k \right) \notin \Omega } \end{array}} \right.. \\ \end{gathered} $$
(A.3)
Taking into account the identities
$$P\left( {\left. {{\mathbf{s}}\left( k \right)} \right|{\mathbf{s}}\left( {k - 1} \right),{\mathbf{z}}\left( {k - 1} \right)} \right) \equiv P\left( {\left. {{\mathbf{s}}\left( k \right)} \right|{\mathbf{s}}\left( {k - 1} \right)} \right),$$
$$\begin{gathered} f\left( {\left. {{\mathbf{z}}\left( k \right)} \right|{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( {k - 1} \right),{\mathbf{s}}\left( k \right)} \right) \\ \equiv f\left( {\left. {{\mathbf{z}}\left( k \right)} \right|{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( k \right)} \right), \\ \end{gathered} $$
which are fulfilled according to the conditions of the problem statement, and (A.3) we rewrite (A.2):
$$f\left( {{\mathbf{z}}\left( k \right),{\mathbf{s}}\left( k \right),{\mathbf{d}}\left. {\left( k \right)} \right|{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( {k - 1} \right)} \right) = \left\{ {\begin{array}{*{20}{c}} {f\left( {\left. {{\mathbf{z}}\left( k \right)} \right|{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( k \right)} \right)P\left( {\left. {{\mathbf{s}}\left( k \right)} \right|{\mathbf{s}}\left( {k - 1} \right)} \right),}&{{{{\mathbf{y}}}_{k}} \in \Omega } \\ {0,}&{{{{\mathbf{y}}}_{k}} \notin \Omega } \end{array}} \right..$$
(A.4)
After substituting (A.4) into (A.1), we have
$$\begin{gathered} f\left( {{\mathbf{z}}\left( k \right),{\mathbf{s}}\left( k \right)\left| {{\mathbf{D}}_{1}^{k}} \right.} \right) = \frac{1}{{P\left( {{\mathbf{d}}\left( k \right)\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)}}\,\sum\limits_{{\mathbf{s}}\left( {k - 1} \right)} {\int {f\left( {{\mathbf{z}}\left( k \right)|{\mathbf{z}}\left( {k - 1} \right),{\mathbf{s}}\left( k \right)} \right)} } \\ \times \,\,P\left( {\left. {{\mathbf{s}}\left( k \right)} \right|{\mathbf{s}}\left( {k - 1} \right)} \right)f\left( {{\mathbf{z}}\left( {k - 1} \right),\left. {{\mathbf{s}}\left( {k - 1} \right)} \right|{\mathbf{D}}_{1}^{{k - 1}}} \right)d{\mathbf{z}}\left( {k - 1} \right). \\ \end{gathered} $$
(A.5)
Taking into account the ratios
$${\mathbf{s}}\left( k \right) \triangleq {{\left[ {{{a}_{j}}\left( {{{t}_{k}}} \right),{{b}_{m}}\left( {{{t}_{k}}} \right)} \right]}^{T}};\,\,\,\,P\left( {\left. {{\mathbf{s}}\left( k \right)} \right|{\mathbf{s}}\left( {k - 1} \right)} \right) \triangleq {{\pi }}_{{ij}}^{a}{{\pi }}_{{nm}}^{b}$$
rewrite (A.5):
$$\begin{gathered} f\left( {{\mathbf{z}}\left( k \right),{{a}_{j}},{{b}_{m}}\left| {{\mathbf{D}}_{1}^{k}} \right.} \right) = \hat {f}\left( {{\mathbf{z}}\left( k \right),{{a}_{j}},{{b}_{m}}} \right) = \frac{1}{{P\left( {{\mathbf{d}}\left( k \right)\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)}}\sum\limits_i {\sum\limits_n {{{\pi }}_{{ij}}^{a}{{\pi }}_{{nm}}^{b}} } \\ \times \,\,\int {f\left( {\left. {{\mathbf{z}}\left( k \right)} \right|{\mathbf{z}}\left( {k - 1} \right),{{a}_{j}},{{b}_{m}}} \right)} \hat {f}\left( {{\mathbf{z}}\left( {k - 1} \right),{{a}_{i}}\left( {{{t}_{{k - 1}}}} \right),{{b}_{n}}\left( {{{t}_{{k - 1}}}} \right)} \right)d{\mathbf{z}}\left( {k - 1} \right). \\ \end{gathered} $$
(A.6)
$$\begin{gathered} P\left( {{\mathbf{d}}\left( k \right)\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right) = \sum\limits_j {\sum\limits_m {\sum\limits_i {\sum\limits_n {{{\pi }}_{{ij}}^{a}{{\pi }}_{{nm}}^{b}} \int {\int {f\left( {\left. {{\mathbf{z}}\left( k \right)} \right|{\mathbf{z}}\left( {k - 1} \right),{{a}_{j}},{{b}_{m}}} \right)} } } } } \\ \times \,\,\hat {f}\left( {{\mathbf{z}}\left( {k - 1} \right),{{a}_{i}}\left( {{{t}_{{k - 1}}}} \right),{{b}_{n}}\left( {{{t}_{{k - 1}}}} \right)} \right)d{\mathbf{z}}\left( {k - 1} \right)d{\mathbf{z}}\left( k \right). \\ \end{gathered} $$
(A.7)
To find the probability of one-step prediction of observed digital signal \(P\left( {{\mathbf{d}}\left( k \right)\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)\) integrate (A.7) with respect to variable \({\mathbf{z}}\left( k \right)\) considering that \({\mathbf{z}}(k) = {{\left[ {{{{\mathbf{x}}}^{T}}(k),{{{\mathbf{y}}}^{T}}(k)} \right]}^{T}}\):
$$\begin{gathered} P\left( {{\mathbf{d}}\left( k \right)\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right) = \sum\limits_j {\sum\limits_m {\sum\limits_i {\sum\limits_n {{{\pi }}_{{ij}}^{a}} } } } {{\pi }}_{{nm}}^{b} \\ \times \,\,\int\limits_\Omega {\int\limits_{} {f\left( {{\mathbf{y}}\left( k \right)\left| {{\mathbf{z}}\left( {k - 1} \right),{{a}_{j}},{{b}_{m}}} \right.} \right)} } \\ \times \,\,\hat {f}\left( {{\mathbf{z}}\left( {k - 1} \right),{{a}_{i}},{{b}_{n}}} \right)d{\mathbf{z}}\left( {k - 1} \right)d{\mathbf{y}}\left( k \right). \\ \end{gathered} $$
(A.8)
We represent the posterior probability density of the vector z given that \({{a}_{j}}({{t}_{k}})\), \({{b}_{m}}({{t}_{k}})\)—(A.6) in the form of a system of recurrent equations
$$\begin{gathered} {{{\hat {f}}}_{{jm}}}\left( {{\mathbf{z}}\left( k \right)} \right) = \hat {f}\left( {{\mathbf{z}}\left( k \right),{{a}_{j}},{{b}_{m}}} \right) \\ = \frac{1}{{P\left( {{\mathbf{d}}\left( k \right)\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)}}{{{\tilde {f}}}_{{jm}}}\left( {{\mathbf{z}}\left( k \right)} \right), \\ \end{gathered} $$
(A.9)
where \({{\tilde {f}}_{{jm}}}\left( {{\mathbf{z}}\left( k \right)} \right) = f\left( {{\mathbf{z}}\left( k \right)\left| {{{a}_{j}},{{b}_{m}},{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)\) is the conditional probability density of vector \({{\left[ {{{{\mathbf{z}}}^{T}}(k),{{a}_{j}}({{t}_{k}}),{{b}_{m}}({{t}_{k}})} \right]}^{T}}\):
$${{\tilde {f}}_{{jm}}}\left( {{\mathbf{z}}\left( k \right)} \right) = \sum\limits_i {\sum\limits_n {{{\pi }}_{{ij}}^{a}} } {{\pi }}_{{nm}}^{b}{{\hat {P}}_{{in}}}(k - 1)\int {f\left( {{\mathbf{z}}\left( k \right)\left| {{\mathbf{z}}\left( {k - 1} \right),{{a}_{j}},{{b}_{m}}} \right.} \right)} {{\hat {f}}_{{in}}}\left( {{\mathbf{z}}\left( {k - 1} \right)} \right)d{\mathbf{z}}\left( {k - 1} \right).$$
(A.10)
To find the posterior probability of the discrete component of DCMP \(P\left( {{{a}_{j}},{{b}_{m}}\left| {{\mathbf{D}}_{1}^{k}} \right.} \right)\) integrate (A.6) over variables \({\mathbf{z}}\left( k \right)\), \({\mathbf{z}}\left( {k - 1} \right)\)
$$\begin{gathered} P\left( {{{a}_{j}},{{b}_{m}}\left| {{\mathbf{D}}_{1}^{k}} \right.} \right) = \hat {P}\left( {{{a}_{j}},{{b}_{m}}} \right) = \frac{1}{{P\left( {{\mathbf{d}}\left( k \right)\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)}} \\ \times \,\,\int\limits_\Omega {f\left( {{\mathbf{y}}\left( k \right)|{{a}_{j}},{{b}_{m}},{\mathbf{D}}_{1}^{{k - 1}}} \right)} d{\mathbf{y}}\left( k \right) \\ \times \,\,\sum\limits_i {\sum\limits_n {{{\pi }}_{{ij}}^{a}{{\pi }}_{{nm}}^{b}} \hat {P}\left( {{{a}_{i}}\left( {{{t}_{{k - 1}}}} \right),{{b}_{n}}\left( {{{t}_{{k - 1}}}} \right)} \right)} \\ = \frac{{P\left( {{\mathbf{d}}\left( k \right)|{{a}_{j}},{{b}_{m}},{\mathbf{D}}_{1}^{{k - 1}}} \right)}}{{P\left( {{\mathbf{d}}\left( k \right)\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)}} \\ \times \,\,\sum\limits_i {\sum\limits_n {{{\pi }}_{{ij}}^{a}{{\pi }}_{{nm}}^{b}} \hat {P}\left( {{{a}_{i}}\left( {{{t}_{{k - 1}}}} \right),{{b}_{n}}\left( {{{t}_{{k - 1}}}} \right)} \right)} . \\ \end{gathered} $$
(A.11)
Taking into account notation
$${{\tilde {P}}_{{jm}}}\left( k \right) = \sum\limits_i {\sum\limits_n {\pi _{{ij}}^{a}\pi _{{nm}}^{b}} \hat {P}\left( {{{a}_{i}}\left( {{{t}_{{k - 1}}}} \right),{{b}_{n}}\left( {{{t}_{{k - 1}}}} \right)} \right)} ,$$
(A.12)
rewrite (A.11) as
$$\begin{gathered} {{{\hat {P}}}_{{jm}}}\left( k \right) = \hat {P}\left( {{{a}_{j}}\left( {{{t}_{k}}} \right),{{b}_{m}}\left( {{{t}_{k}}} \right)} \right) \\ = \frac{{P\left( {{\mathbf{d}}\left( k \right)\left| {{{a}_{j}}\left( {{{t}_{k}}} \right),{{b}_{m}}\left( {{{t}_{k}}} \right),{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)}}{{P\left( {{\mathbf{d}}\left( k \right)\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)}}{{{\tilde {P}}}_{{jm}}}\left( k \right), \\ \end{gathered} $$
(A.13)
where \(P\left( {{\mathbf{d}}\left( k \right),{{a}_{j}},{{b}_{m}}\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)\) is the conditional probability of one-step prediction of the observed digital signal:
$$\begin{gathered} P\left( {{\mathbf{d}}\left( k \right),{{a}_{j}},{{b}_{m}}\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right) \\ = \int\limits_\Omega {f\left( {{\mathbf{y}}\left( k \right)\left| {{{a}_{j}},{{b}_{m}},{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)} d{\mathbf{y}}\left( k \right). \\ \end{gathered} $$
(A.14)
Expression (A.6) corresponds to formula (8); expressions (A.8)–(A.10) correspond to formulas (9)–(11), expressions (A.12)–(A.14) correspond to formulas (13), (12), and (14), respectively.
To derive quasi-optimal algorithms for digital filtering of continuous components of DCMP state vector (19)–(26), we use the assumption of the normality of posterior probability density \({{\hat {f}}_{{in}}}\left( {{\mathbf{z}}\left( {k - 1} \right)} \right)\) at the previous \(k - 1\)th cycle of digital filtering [19]:
$$\begin{gathered} {{{\hat {f}}}_{{in}}}\left( {{\mathbf{z}}\left( {k - 1} \right)} \right) = \frac{1}{{\sqrt {{{{\left( {2\pi } \right)}}^{{{{n}_{x}} + {{n}_{y}}}}}\det {{{{\mathbf{\hat {R}}}}}_{{zz}}}\left( {k - 1,{{a}_{i}},{{b}_{n}}} \right)} }} \\ \times \,\,\exp \left\{ { - \frac{1}{2}{{{\left( {{\mathbf{z}}\left( {k - 1} \right) - {\mathbf{\hat {z}}}\left( {k - 1,{{a}_{i}},{{b}_{n}}} \right)} \right)}}^{T}}} \right. \\ \times \,\,{\mathbf{\hat {R}}}_{{zz}}^{{ - 1}}\left( {k - 1,{{a}_{i}},{{b}_{n}}} \right)\left. {\left( {{\mathbf{z}}\left( {k - 1} \right) - {\mathbf{\hat {z}}}\left( {k - 1,{{a}_{i}},{{b}_{n}}} \right)} \right)} \right\}. \\ \end{gathered} $$
(A.15)
Based on linear transformation (6) of conditionally Gaussian random variable \({\mathbf{z}}\left( {k - 1} \right)\) and taking into account (A.15) we rewrite (A.10):
$$\begin{gathered} {{{\tilde {f}}}_{{jm}}}\left( {{\mathbf{z}}\left( k \right)} \right) = \frac{1}{{\sqrt {{{{\left( {2\pi } \right)}}^{{{{n}_{x}} + {{n}_{y}}}}}} }}\sum\limits_i {\sum\limits_n {{{\pi }}_{{ij}}^{a}{{\pi }}_{{nm}}^{b}} {{{\hat {P}}}_{{in}}}\left( {k - 1} \right)} \frac{1}{{\sqrt {\det {{{{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{R} }}}}_{{zz}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} }} \\ \times \,\,\exp \left\{ { - \frac{1}{2}{{{\left( {{\mathbf{z}}\left( k \right) - {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} }}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)}}^{T}}{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{R} }}_{{zz}}^{{ - 1}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\left( {{\mathbf{z}}\left( k \right) - {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} }}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)} \right\}, \\ \end{gathered} $$
(A.16)
where
$${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} }}\left( {k,{{a}_{j}},{{b}_{m}}} \right) = {{{\mathbf{\Phi }}}_{{zz}}}({{a}_{j}},{{b}_{m}}){\mathbf{\hat {z}}}\left( {k - 1,{{a}_{i}},{{b}_{n}}} \right),$$
$$\begin{gathered} {{{{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{R} }}}}_{{zz}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right) = {{{\mathbf{\Phi }}}_{{zz}}}({{a}_{j}},{{b}_{m}}){{{{\mathbf{\hat {R}}}}}_{{zz}}}\left( {k - 1,{{a}_{i}},{{b}_{n}}} \right) \\ \times \,\,{\mathbf{\Phi }}_{{zz}}^{{\text{T}}}({{a}_{j}},{{b}_{m}}) + {{{\mathbf{B}}}_{{zz}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right). \\ \end{gathered} $$
As a result of the two-moment Gaussian approximation of the extrapolation probability density (A.16), we write
$$\begin{gathered} {{{\tilde {f}}}_{{jm}}}\left( {{\mathbf{z}}\left( k \right)} \right) = \frac{1}{{\sqrt {{{{\left( {2\pi } \right)}}^{{{{n}_{x}} + {{n}_{y}}}}}\det {{{{\mathbf{\tilde {R}}}}}_{{zz}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} }} \\ \times \,\,\exp \left\{ { - \frac{1}{2}{{{\left( {{\mathbf{z}}\left( k \right) - {\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)}}^{T}}} \right. \\ \times \,\,\left. {{\mathbf{\tilde {R}}}_{{zz}}^{{ - 1}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\left( {{\mathbf{z}}\left( k \right) - {\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)} \right\}, \\ \end{gathered} $$
(A.17)
where
$${\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right) = \frac{1}{{{{{\tilde {P}}}_{{jm}}}\left( k \right)}}\sum\limits_i {\sum\limits_n {{{\pi }}_{{ij}}^{a}{{\pi }}_{{nm}}^{b}} {{{\hat {P}}}_{{in}}}\left( {k - 1} \right)} {{{\mathbf{\Phi }}}_{{zz}}}({{a}_{j}},{{b}_{m}}){\mathbf{\hat {z}}}\left( {k - 1,{{a}_{i}},{{b}_{n}}} \right),$$
(A.18)
$$\begin{gathered} {{{{\mathbf{\tilde {R}}}}}_{{zz}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right) = \frac{1}{{{{{\tilde {P}}}_{{jm}}}\left( k \right)}}\sum\limits_i {\sum\limits_n {{{\pi }}_{{ij}}^{a}{{\pi }}_{{nm}}^{b}} {{{\hat {P}}}_{{in}}}\left( {k - 1} \right)} \left( {{{{\mathbf{\Phi }}}_{{zz}}}({{a}_{j}},{{b}_{m}})} \right.{\mathbf{\hat {R}}}_{{in}}^{{(zz)}}\left( {k - 1} \right){\mathbf{\Phi }}_{{zz}}^{T}({{a}_{j}},{{b}_{m}}) \\ + \,\,{{{\mathbf{B}}}_{{zz}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right) + \left( {{{{\mathbf{\Phi }}}_{{zz}}}({{a}_{j}},{{b}_{m}}){\mathbf{\hat {z}}}\left( {k - 1,{{a}_{i}},{{b}_{n}}} \right){{{{\mathbf{\hat {z}}}}}_{{in}}}\left( {k - 1} \right) - {\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right) \\ \left. { \times \,\,{{{\left( {{{{\mathbf{\Phi }}}_{{zz}}}({{a}_{j}},{{b}_{m}}){\mathbf{\hat {z}}}\left( {k - 1,{{a}_{i}},{{b}_{n}}} \right) - {\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)}}^{T}}} \right). \\ \end{gathered} $$
(A.19)
Taking into consideration
$${\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right) \triangleq {{\left[ {{\mathbf{\tilde {x}}}_{{}}^{T}\left( {k,{{a}_{j}},{{b}_{m}}} \right),{\mathbf{\tilde {y}}}_{{}}^{T}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right]}^{T}}$$
from (A.18) and (A.19) it is easy to obtain expressions (19)–(23).
We denote
$$\begin{gathered} \frac{1}{{\sqrt {{{{\left( {2\pi } \right)}}^{{{{n}_{x}} + {{n}_{y}}}}}\det {{{{\mathbf{\tilde {R}}}}}_{{zz}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} }} \\ \times \,\,\exp \left\{ { - \frac{1}{2}{{{\left( {{\mathbf{z}}\left( k \right) - {\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)}}^{T}}} \right. \\ \times \,\,\left. {{\mathbf{\tilde {R}}}_{{zz}}^{{ - 1}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\left( {{\mathbf{z}}\left( k \right) - {\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)} \right\} \\ = f\left( {\left. {{\mathbf{z}}\left( k \right)} \right|{{a}_{j}},{{b}_{m}},{\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right). \\ \end{gathered} $$
Then, instead of (A.9), we have
$${{\hat {f}}_{{jm}}}\left( {{{{\mathbf{z}}}_{k}}} \right) = {{\theta }_{1}}f\left( {\left. {{\mathbf{z}}\left( k \right)} \right|{{a}_{j}},{{b}_{m}},{\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right),$$
(A.20)
where \(\theta _{1}^{{ - 1}} = P\left( {{\mathbf{d}}\left( k \right)\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)\). Considering that \({\mathbf{z}}\mathop = \limits^\Delta {{\left[ {{\mathbf{x}}_{{}}^{T},{\mathbf{y}}_{{}}^{T}} \right]}^{T}}\) and applying the formula for multiplying the probabilities, we write
$$\begin{gathered} f\left( {{\mathbf{x}}\left( k \right),\left. {{\mathbf{y}}\left( k \right)} \right|{{a}_{j}},{{b}_{m}},{\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right) \\ = f{\kern 1pt} \left( {\left. {{\mathbf{y}}\left( k \right)} \right|{{a}_{j}},{{b}_{m}},{\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right) \\ \times \,\,f{\kern 1pt} \left( {\left. {{\mathbf{x}}\left( k \right)} \right|{\mathbf{y}}\left( k \right),{{a}_{j}},{{b}_{m}},\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right). \\ \end{gathered} $$
The first conditional probability density is Gaussian [17]
$$\begin{gathered} f\left( {\left. {{\mathbf{y}}\left( k \right)} \right|{{a}_{j}},{{b}_{m}},{\mathbf{\tilde {z}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right) \\ = \frac{1}{{\sqrt {{{{\left( {2\pi } \right)}}^{{{{n}_{y}}}}}\det {{{{\mathbf{\hat {R}}}}}_{{yy}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} }} \\ \times \,\,\exp \left\{ { - \frac{1}{2}{{{\left( {{\mathbf{y}}\left( k \right) - {\mathbf{\tilde {y}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)}}^{T}}} \right. \\ \times \,\,\left. {{\mathbf{\hat {R}}}_{{yy}}^{{ - 1}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\left( {{\mathbf{y}}\left( k \right) - {\mathbf{\tilde {y}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)} \right\}. \\ \end{gathered} $$
(A.21)
The second conditional probability density has form
$$\begin{gathered} f\left( {\left. {{\mathbf{x}}\left( k \right)} \right|{\mathbf{y}}\left( k \right),{{a}_{j}},{{b}_{m}},\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right) \\ = \frac{1}{{\sqrt {{{{\left( {2\pi } \right)}}^{{{{n}_{x}}}}}\det {{{{\mathbf{\hat {R}}}}}_{{xx}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} }} \\ \times \,\,\exp \left\{ { - \frac{1}{2}{{{\left( {{\mathbf{x}}\left( k \right) - {\mathbf{\hat {x}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)}}^{T}}} \right. \\ \times \,\,\left. {{\mathbf{\hat {R}}}_{{xx}}^{{ - 1}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\left( {{\mathbf{x}}\left( k \right) - {\mathbf{\hat {x}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)} \right\}, \\ \end{gathered} $$
(A.22)
where \({\mathbf{\hat {x}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\), \({{{\mathbf{\hat {R}}}}_{{xx}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\) are conditional estimates of discrete filtering and correlation matrices of their errors, determined from [19]. Using the technique [17], taking into account (A.21), (A.22), we write down conditional estimates of digital filtering and correlation matrices of their errors
$$\begin{gathered} {\mathbf{\hat {x}}}(k,{{a}_{j}},{{b}_{m}}) = {\mathbf{\tilde {x}}}(k,{{a}_{j}},{{b}_{m}}) + {{{{\mathbf{\tilde {R}}}}}_{{xy}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right) \\ \times \,\,{\mathbf{\tilde {R}}}_{{yy}}^{{ - 1}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\left( {{\mathbf{\hat {y}}}(k,{{a}_{j}},{{b}_{m}}) - {\mathbf{\tilde {y}}}(k,{{a}_{j}},{{b}_{m}})} \right). \\ \end{gathered} $$
(A.23)
$$\begin{gathered} {{{{\mathbf{\hat {R}}}}}_{{xx}}}(k,{{a}_{j}},{{b}_{m}}) = {{{{\mathbf{\tilde {R}}}}}_{{xx}}}(k,{{a}_{j}},{{b}_{m}}) \\ + \,\,{{{{\mathbf{\tilde {R}}}}}_{{xy}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right){\mathbf{\tilde {R}}}_{{yy}}^{{ - 1}}\left( {k,{{a}_{j}},{{b}_{m}}} \right) \\ \times \,\,\left[ {{\mathbf{I}} - {{{\mathbf{E}}}_{{kb}}}} \right]{\mathbf{\tilde {R}}}_{{xy}}^{T}(k,{{a}_{j}},{{b}_{m}}). \\ \end{gathered} $$
(A.24)
Based on the optimization of the quantization procedure, we write [18]
$$\begin{gathered} {\mathbf{\hat {y}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right) = {\mathbf{\tilde {y}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right) \\ + \,\,{{{\mathbf{T}}}_{{yy}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right){{{\mathbf{Y}}}_{{{\text{opt}}}}}\left( {k,{\mathbf{d}}\left( k \right)} \right). \\ \end{gathered} $$
(A.25)
Substituting (A.25) into (A.23), we finally obtain
$$\begin{gathered} {\mathbf{\hat {x}}}(k,{{a}_{j}},{{b}_{m}}) = {\mathbf{\tilde {x}}}(k,{{a}_{j}},{{b}_{m}}) \\ + \,\,{\mathbf{K}}(k,{{a}_{j}},{{b}_{m}}){{{\mathbf{Y}}}_{{{\text{opt}}}}}\left( {k,{\mathbf{d}}\left( k \right)} \right), \\ \end{gathered} $$
(A.26)
where \({{{\mathbf{Y}}}_{{{\text{opt}}}}}\left( {k,{\mathbf{d}}\left( k \right)} \right) = {{\left[ {y_{{{\text{opt}}}}^{{(1)}},...,y_{{{\text{opt}}}}^{{(u)}},...,y_{{{\text{opt}}}}^{{({{n}_{y}})}}} \right]}^{T}}\), \(u = \overline {1,{{n}_{y}}} \) is the vector of optimal quantization levels. With optimal uniform quantization with step \({{{{\delta }}}_{{{\text{opt}}}}}\) meaning \(y_{{{\text{opt}}}}^{{{\text{(}}u{\text{)}}}}\) and \({{\eta }}_{{{\text{opt}}}}^{{(u)}}\) normalized optimal levels and quantization thresholds are determined by formulas [13]
$$y_{{{\text{opt}}}}^{{(u)}} = \left( {{{l}_{u}} - \frac{{{{L}_{u}} + 1}}{2}} \right){{\delta }}_{{{\text{opt}}}}^{{(u)}},\,\,\,\,{{\eta }}_{{{\text{opt}}}}^{{(u)}} = \left( {{{l}_{u}} - \frac{{{{L}_{u}}}}{2}} \right){{\delta }}_{{{\text{opt}}}}^{{(u)}}.$$
(A.27)
Values \({{\delta }}_{{{\text{opt}}}}^{{(u)}}\) are given in [13, Table 3.2]. There are also values \({{\varepsilon }}_{{{\text{opt}}}}^{{(u)}}\), which are minimum mean squares of quantization errors, which are used to determine the normalized correlation matrix of quantization errors [17]
$${{{\mathbf{E}}}_{{kb}}} = \left[ {{{{{\varepsilon }}}_{{{\text{opt}}}}}{{{{\delta }}}_{{uw}}}} \right],\,\,\,\,u,w = \overline {1,{{n}_{y}}} .$$
Let the condition for the formation of digital signals be constant quantization thresholds, and the difference at the input of the ADC quantizers is quantized \({\mathbf{y}}\left( k \right) - {\mathbf{\tilde {y}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\) where
$$\begin{gathered} {\mathbf{\tilde {y}}}(k,{{a}_{j}},{{b}_{m}}) \\ = \sum\limits_i {\sum\limits_n {\frac{{{{\pi }}_{{ij}}^{a}{{\pi }}_{{nm}}^{b}{{{\hat {P}}}_{{in}}}(k - 1)}}{{{{{\tilde {P}}}_{{jm}}}(k)}}} } {{{\mathbf{\Phi }}}_{{yx}}}({{a}_{j}},{{b}_{m}}){\mathbf{\hat {x}}}(k - 1,{{a}_{i}},{{b}_{n}}), \\ \end{gathered} $$
is the optimal predicted value of vector \({\mathbf{y}}\left( k \right)\). Then, from the conditions we can write
$$\begin{gathered} {{{\mathbf{T}}}_{{yy}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right){{{\mathbf{H}}}_{{{\text{opt}}}}}(k,{{l}_{u}} - 1) < {\mathbf{y}}\left( k \right) \\ - \,\,{\mathbf{\tilde {y}}}(k,{{a}_{j}},{{b}_{m}}) \leqslant {{{\mathbf{T}}}_{{yy}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right){{{\mathbf{H}}}_{{{\text{opt}}}}}(k,{{l}_{u}}), \\ \end{gathered} $$
(A.28)
where
$${{{\mathbf{H}}}_{{{\text{opt}}}}} = {{[{{\eta }}_{{{\text{opt}}}}^{{(1)}},...,{{\eta }}_{{{\text{opt}}}}^{{(u)}},...,{{\eta }}_{{{\text{opt}}}}^{{({{n}_{y}})}}]}^{T}},\,\,\,\,u = \overline {1,{{n}_{y}}} ,$$
is the vector of optimal quantization thresholds determined from (A.27).
Taking into account condition (3) \({\mathbf{y}}\left( k \right) = \int {{\mathbf{v}}\left( {{\tau }} \right)} d{{\tau }}\), \({{\tau }} \in \left[ {{{t}_{{k - 1}}},{{t}_{k}}} \right]\), expression (A.28) has form
$$\begin{gathered} {{{\mathbf{T}}}_{{yy}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right){{{\mathbf{H}}}_{{{\text{opt}}}}}(k,{{l}_{u}} - 1) \\ < \int\limits_{{{t}_{{k - 1}}}}^{{{t}_{k}}} {{\mathbf{v}}(\tau )d\tau - {\mathbf{\tilde {y}}}(k,{{a}_{j}},{{b}_{m}})} \leqslant {{{\mathbf{T}}}_{{yy}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right){{{\mathbf{H}}}_{{{\text{opt}}}}}(k,{{l}_{u}}). \\ \end{gathered} $$
To derive the conditional probability of one-step prediction of observations (14) from (A.13), we define the probability of digital signal
$$\begin{gathered} P\left( {{\mathbf{d}}\left( k \right)\left| {{{a}_{j}},{{b}_{m}},{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right) \\ = \int\limits_\Omega {f\left( {{\mathbf{y}}\left( k \right)\left| {{{a}_{j}},{{b}_{m}},{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right)} d{\mathbf{y}}\left( k \right). \\ \end{gathered} $$
(A.29)
Considering expressions (A.13), (A.20), and (A.21), we can conclude that the conditional integrand in (A.29) is Gaussian:
$$\begin{gathered} f\left( {{\mathbf{y}}\left( k \right)\left| {{{a}_{j}},{{b}_{m}},{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right) \\ = \frac{1}{{\sqrt {{{{\left( {2\pi } \right)}}^{{{{n}_{y}}}}}\det {{{{\mathbf{\tilde {R}}}}}_{{yy}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} }} \\ \times \,\,\exp \left\{ { - \frac{1}{2}\left( {{\mathbf{y}}\left( k \right) - {\mathbf{\tilde {y}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)} \right. \\ \times \,\,\left. {{\mathbf{\tilde {R}}}_{{yy}}^{{ - 1}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\left( {{\mathbf{y}}\left( k \right) - {\mathbf{\tilde {y}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)} \right)} \right\}. \\ \end{gathered} $$
(A.30)
Taking this into account, we rewrite (A.30) in form
$$\begin{gathered} P\left( {{\mathbf{d}}\left( k \right),{{a}_{j}},{{b}_{m}}\left| {{\mathbf{D}}_{1}^{{k - 1}}} \right.} \right) \\ = \prod\limits_{u = 1}^{{{n}_{y}}} {\left\{ {F\left\{ {\frac{{{{\eta }}_{{{\text{opt}}}}^{{(u)}}({{l}_{u}}) - {{{\tilde {y}}}_{u}}(k,{{a}_{j}},{{b}_{m}})}}{{{{{{\tau }}}_{u}}(k)}}} \right\}} \right.} \\ \left. { - \,\,F\left\{ {\frac{{{{\eta }}_{{{\text{opt}}}}^{{(u)}}({{l}_{u}} - 1) - {{{\tilde {y}}}_{u}}(k,{{a}_{j}},{{b}_{m}})}}{{{{{{\tau }}}_{u}}(k)}}} \right\}} \right\}, \\ \end{gathered} $$
where \({{\tilde {y}}_{u}}(k,{{a}_{j}},{{b}_{m}})\) is ith element vector \({\mathbf{\tilde {y}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\); \({{{{\tau }}}_{u}}\) is the diagonal element of lower triangular matrix \({{{\mathbf{T}}}_{{yy}}}\left( {k,{{a}_{j}},{{b}_{m}}} \right)\); and \(F\left\{ \cdot \right\}\) is the integral of the probability.
