\$w(t) \longrightarrow \bigg[\frac{\sqrt{2\sigma ^2\beta}}{s+\beta}\bigg]  \longrightarrow \bigg[\frac{1}{s}\bigg] \longrightarrow y\$

$w(t) \longrightarrow \bigg[\frac{\sqrt{2\sigma ^2\beta}}{s+\beta}\bigg]  \longrightarrow \bigg[\frac{1}{s}\bigg] \longrightarrow y$

\usepackage{amsmath}  %可以使用\boldsymbol加粗罗马字符；\mathbf对罗马字符不起作用。

\$\mathbf{x}_{k+1} = \boldsymbol{\phi}_k \mathbf{x}_k + \mathbf{w}_k\$

$\mathbf{x}_{k+1} = \boldsymbol{\phi}_k \mathbf{x}_k + \mathbf{w}_k$

%注意{和}是特殊字符，使用\{和\}

\$\mathbf{Q}_k=E[\mathbf{w}_k\mathbf{w}_k^T]\$

\$=E\big\{   \big[ \int_{t_k}^{t_{k+1}} \boldsymbol{\phi}(t_{k+1}, u) \mathbf{G}(u) \mathbf{w}(u)du \big]  \big[ \int_{t_k}^{t_{k+1}}\boldsymbol{\phi}(t_{k+1},v) \mathbf{G}(v) \mathbf{w}(v)dv \big]^T   \big\}\$

\$=\int_{t_k}^{t_{k+1}} \int_{t_k}^{t_{k+1}} \boldsymbol{\phi}(t_{k+1}, u)\mathbf{G}(u)E[\mathbf{w}(u)\mathbf{w}^T(v)]\mathbf{G}^T(v)\boldsymbol{\phi}^T(t_{k+1},v)dudv\$

$\mathbf{Q}_k=E[\mathbf{w}_k\mathbf{w}_k^T]$

$=E\big\{   \big[ \int_{t_k}^{t_{k+1}} \boldsymbol{\phi}(t_{k+1}, u) \mathbf{G}(u) \mathbf{w}(u)du \big]  \big[ \int_{t_k}^{t_{k+1}}\boldsymbol{\phi}(t_{k+1},v) \mathbf{G}(v) \mathbf{w}(v)dv \big]^T   \big\}$

$=\int_{t_k}^{t_{k+1}} \int_{t_k}^{t_{k+1}} \boldsymbol{\phi}(t_{k+1}, u)\mathbf{G}(u)E[\mathbf{w}(u)\mathbf{w}^T(v)]\mathbf{G}^T(v)\boldsymbol{\phi}^T(t_{k+1},v)dudv$

LaTex与数学公式