x(k) = Ax(k−1) + B(k)u(k) + v(k)

y(k) = Cx(k) = w(k)

$$\hat{x}\left( k \middle| k - 1 \right) = \hat{x}(k - 1|k - 1)$$

$$\sigma^{2}\left( k \right) = \frac{1}{N - 1}\ \sum_{i - k - N + 1}^{k}\left\lbrack v_{n}\left( i \right) - \overset{\overline{}}{v} \right\rbrack^{2}$$

$$S\left( k \right) = \frac{1}{1 - N}\sum_{i - k - N + 1}^{k}{\lbrack y\left( i \right) - \hat{x}(i - 1|i - 1)\rbrack}^{2}$$

$$K\left( k \right) = \frac{P\left( k - 1 \middle| k - 1 \right) + Q\left( k \right)}{P\left( k - 1 \middle| k - 1 \right) + Q\left( k \right) + R}$$

P(k|k) = (1−K(k))(P(k−1|k−1)+Q(k))

$$\hat{x}\left( k \middle| k \right) = K\left( k \right)y\left( k \right) + \left( 1 - K\left( k \right) \right)\hat{x}(k - 1|k - 1)$$

$$S\left( k \right) = \frac{1}{N - 1}\left\lbrack \sum_{i - k - N + 1}^{k}{(y\left( i \right) - \hat{x}(i - 1|i - 1))}^{2} \right\rbrack = \frac{1}{N - 1}\left\lbrack \sum_{i - k - N + 1}^{k}{(v\left( i \right) + w\left( i \right) + \tilde{x}(i - 1|i - 1))}^{2} \right\rbrack = Q\left( k \right) + R + P\left( k - 1 \middle| k - 1 \right) + \text{Cov}(k,\tilde{x},v,w)$$

$Q\left( k \right) = S - R - P\left( k \middle| k \right) = \frac{1}{N - 1}\left\lbrack \sum_{i - k - N + 1}^{k}\left( y\left( i \right) - \tilde{x}\left( i - 1 \middle| i - 1 \right) \right)^{2} \right\rbrack - R - P\left( k - 1 \middle| k - 1 \right)$

$$Q\left( k \right) = \left( 1 - \alpha \right)Q\left( k - 1 \right) + \alpha\left\lbrack \left( y\left( k \right) - \hat{x}\left( k - 1 \right) \right)^{2} - R - P(k - 1|k - 1) \right\rbrack$$

[x^′,y^′] = f([x,y]);

$$\left\lbrack x,y \right\rbrack = \overrightarrow{r};\ \ \left\lbrack x^{'},y' \right\rbrack = \overrightarrow{r'}$$

h(x,y) = g(x^′,y^′);

$$\left\{ \begin{matrix} x^{'} = \left( x - x_{0} \right)\cos{\varphi + (y - y_{0})\sin{\varphi;}} \\ y^{'} = - \left( x - x_{o} \right)\sin\varphi\left( y - y_{0} \right)\cos\varphi; \\ \end{matrix} \right.\ $$

$$g\left( x^{'},y^{'} \right) = h \bullet \exp\left( - \frac{\left\lbrack {x^{'}}^{2} + k \bullet {y^{'}}^{2} \right\rbrack}{d^{2}} \right);$$

$$A = \text{arctg}\left( \frac{y - y_{0}}{x - x_{0}} \right);A \in < 0,2\pi >$$

φ(A) = φ_ocosA;  φ(A) = φ₀sinA;

$$\varphi\left( A \right) = \varphi_{o}\frac{\left( A - \pi \right)^{2}}{\pi^{2}}\cos\left( A \right)$$

$$F\left( v,k \right) = v \bullet \left\{ 1 - \frac{\exp\left\lbrack - \left( v - v_{1} \right)\left( v - v_{2} \right) - \left( v_{1} - v_{2} \right)^{2} \right\rbrack}{4} - k \bullet \ln\left( \left( v_{1} - v_{2} \right)/2 \right)\rbrack \right\}$$

fg(A) = fg(p(A));

$$p\left( A;A_{1} \right) > \pi = \frac{\left( A - A_{1} \right)\left\lbrack \text{sign}\left( A \right) - \text{sign}\left( A - A_{1} - \pi \right) \right\rbrack}{2} - \frac{\left( A - A_{1} - 2\pi \right)\left\lbrack \text{sign}\left( A - 2\pi \right) - \text{sign}\left( A - A_{1} - \pi \right) \right\rbrack}{2} + \left( 2\pi - A_{1} \right)\left\lbrack 1 - \text{sign}\left( A \right) \right\rbrack$$

$$p\left( A;A_{1} > \pi \right) = \frac{\left( A - A_{1} \right)\left\lbrack \text{sign}\left( A \right) - \text{sign}\left( A - A_{1} + \pi \right) - \text{sign}\left( A - 2\pi \right) \right\rbrack}{2} + \frac{\left( A - A_{1} + 2\pi \right)\left\lbrack \text{sign}\left( A \right) - \text{sign}\left( A - A_{1} + \pi \right) \right\rbrack}{2};$$

z(j+1,k) = exp(v(1,j));

z(j+1,k) = z(j+1,k) − int(z(j+1,k));

z(j,k+1) = exp(v(2,j));

z(j+1,k+1) = z(j+1,k+1) − int(z(j+1,k+1));

z(j,k) = exp(z(j−5, k+1));

z(j,k) = z(j,k) − int(z(j,k));

z(j,k+1) = exp(z(j−5,k));

z(j,k+1) = z(j,k+1) − int(z(j,k+1));

j = 6 : 4000 step 5 and k = 1 : 4000 step 2.

$$\overrightarrow{n_{1}},\overrightarrow{\ n_{2}},\overrightarrow{\ n_{3}}\text{\ and\ }\overrightarrow{n_{4}}$$

$$z = h\left( P \right) = h - \frac{xN_{\text{ij}}^{x} + yN_{\text{ij}}^{y}}{2d}\ ;$$

$\left\{ \begin{matrix} x = k \bullet \frac{D}{2^{n}}; \\ y = 1 \bullet \frac{D}{2^{n}}; \\ \end{matrix} \right.\ $ k, 1 ∈ (−2^n − 1; 2^n − 1);

$\left\{ \begin{matrix} ix = (i - 1) \bullet \ 2^{2} + k; \\ iy = \left( j - 1 \right) \bullet 2^{2} + k; \\ \end{matrix} \right.\ $ i ∈ <1; N > ;j ∈ <1, M > ;k ∈ <1; 2²>

Where $q\left( \overrightarrow{\beta} \right) = \left\{ \begin{matrix} \sqrt{g_{11},} \\ \sqrt{g_{22,}} \\ \end{matrix} \right.\ $ $\frac{\beta^{2} = \left\{ 0,l_{2} \right\},}{\beta^{1} = \left\{ 0,l_{1} \right\}.}$

$\frac{\partial L^{h}}{\partial v_{\beta}^{h}\left( i_{1},\ i_{2} \right)} = 0,$ i_α = 1, 2, …, N_α, α, β = 1, 2