1. Representation of system work quality and some statistical concepts

Let the input of a dynamic system be $u (t)$ , the output is $x (t)$ , then the dynamic error is $e (t) = u (t) - x (t)$ . When inputWhen $u$ $($ $t$ $)$ $e (t)$ is the random error.

Generally, under the effect of random input signals, the working quality of the system can be expressed by the random mean square error :
$\overline{e^2} (t) = \lim _{T \rightarrow \infty} \frac{1}{2T} \int_{-T} ^T e^2 (t) {\rm d} t \tag{1}$

Next, some basic concepts of statistics are introduced.
(1) Probability distribution function $F (x)$ . The probability distribution function refers to the random variable $X$ does not exceed a certain value $x$ （即 $X < x$ ):
$F (x) = P (X < x)$ (2)Probability density distribution function $f (x)$ . It can be simply and roughly understood as "random variable $X$ is equal to some value $The probability of x$ ", or mathematically understood as "the probability distribution function $F (x)$ 的导数”即可：
$\frac{ {\rm d} F(x)}{ {\rm d} x} = \lim_{\Delta \rightarrow 0} \frac{ P \left( x \leq X \leq x + \Delta x\right)}{\Delta x} = \lim_{\Delta \rightarrow 0} \frac{F \left( x + \Delta x \right) - F(x)}{\Delta x}$ This leads to
$\int _\infty ^xf(x) {\rm d} x$ random variableThe mathematical expectation of $X$
$\tilde x = M \left[ X \right] = \int _{-\infty} ^\infty xf(x) {\rm d} x$ random variableof $X$ $m$ – 阶矩：
$\tilde x^m = \int _{-\infty} ^\infty x^m f(x) {\rm d} x$ random variableof $X$ $m$ – 阶中心矩：
$\left[ \left( X - \tilde x \right)^m \right] = \int _{-\infty} ^\infty \left( X - \tilde x \right)^m f(x) {\rm d} x$ variance (in fact, second-order central moment):
$\left[ \left( X - \tilde x \right)^2 \right] = \int _{-\infty} ^\infty \left( X - \tilde x \right)^2 f(x) {\rm d} x$ When considering time, the mean of the sample can be equated to the mathematical expectation:
$m_x (t) = \tilde x(t) = M \left[ X(t) \right] = \int _{-\infty} ^\infty xf(x, t) {\rm d} x, \\ D_x (t) = D \left[ X(t ) \right] = \int _{-\infty} ^\infty \left[ X(t) - m_x (t) \right]^2 f(x, t) {\rm d} x$ and for different time $t_1, t_2$ For the random process of , we can also have mathematical expectations:
$\left[ X\left( t_1 \right) X\left( t_2 \right) \right] = \int_{-\infty} ^\infty \ int_{-\infty} ^\infty x_1 x_2 f \left( x_1, t_1, x_2, t_2 \right) {\rm d} x_1 {\rm d} x_2 = R \left( t_1, t_2 \right)$ is calledthe correlation function. The correlation function characterizesthe connection of random variables between different moments.

For a random variable $x$ , its correlation function at different times $R_x \left( t_1, t_2 \right)$ calledAutocorrelation functionof $x$ . And for two different random variables $x, y$ , the correlation function R at different times $R_{xy} \left( t_1, t_2 \right)$ is calledthe cross-correlation function. Obviously, when $t_2 = t_1 + \tau$ , the autocorrelation function can also be expressed as
$(\tau) = M \left[ X\left( t_1 \right) X\left( t_1 + \tau \right) \right] = \int_{-\infty} ^\infty {\rm d} x_1 \int_{-\infty} ^\infty x_1 x_2 f \left( x_1, x_2, \tau \right) {\rm d} x_2$

2. Versatile ergodicity

For a statically determinate process with ergodicity, the following formula holds:
$\tilde x = \bar x, \quad \widetilde{x_1 x_2} = \ overline{x_1 x_2}$ At the same time, due to the ergodicity of various states, the mean value of the random variable will not change due to the sampling period:
$\bar x = \lim _{T \rightarrow \infty} \frac{1}{2T} \int_{-T} ^T x (t) {\rm d} t = \lim _{T \rightarrow \infty} \frac{1}{T} \int_{0} ^T x (t) {\rm d} t$ so the autocorrelation function is
$(\tau) = \ overline{x_1 x_2} = \lim _{T \rightarrow \infty} \frac{1}{T} \int _0 ^T x(t) x(t + \tau) {\rm d} t$

3. Properties of correlation functions

Some properties of the correlation function are given below:
$R_{xy} (\tau) = R_{yx} (-\tau);$ $R_{yx} (\tau) = \lim _{T \rightarrow \infty} \frac{1}{2T} \int _{-T} ^T y(t) x(t + \tau) {\rm d} t;$ $R_{yx} (-\tau) = \lim _{T \rightarrow \infty} \frac{1}{2T} \int _{-T} ^T y(t) x(t - \tau) {\rm d} t$ and if set $t_1 = t - \tau$ 则
$R_{yx} (-\tau) = \lim _{T \rightarrow \infty} \frac{1}{2T} \int _{-T} ^T y(t_1 + \tau) x(t_1) {\rm d} t = R_{xy} (\tau)$ 当 $\tau=0$ 时：
$R_x (0) = M \left[ X^2 (t) \right] = \overline{x^2} = D_x$ 当 $\tau \rightarrow \infty$ 时：
$R_x ( \tau \rightarrow \infty ) = \left( \tilde x \right) ^2 = \left( \bar x \right) ^2$ In particular, the correlation function of white noise is:
$R_x (\tau) = N^2 \delta (\tau)$

4. Experimental approach to determine the correlation function

For a statically determinate ergodic system, its correlation function:
$R_x(\tau) = \overline{X(t) X(t + \tau)} = \lim_{T \rightarrow \infty} \frac{1}{T} \int_0 ^T x(t) x( t + \tau) {\rm d} t$ its estimated value is
$\hat R_x (\tau) = \frac{1}{T} \int_0 ^ T x(t) x(t - \tau) {\rm d} t$ In practice, in order to make the estimated value as accurate as possible, the experimental time $T$ as long as possible.

5. Properties of Statically Determinate Random Signals Through Linear Dynamical Systems

Suppose a random signal $u (t)$ , its correlation function is $R_u (\tau)$ , then it passes through the pulse signal $K (t)$ system, the output is still a random signal:
$\int_{-\infty} ^\infty K(\lambda) u(t - \lambda) {\rm d} \lambda = K(\lambda) * u( \lambda) \tag{2 }$ is the convolution of the two.
pair output $x$ 求数学期望：
$\left[ x(t) \right] = \int _{-\infty} ^\infty K(\lambda) M \left[ u(t - \lambda) \right] {\rm d} \lambda$ while at $t+\tau$ time:
$\tau) = \int_{-\infty} ^\infty K(\eta ) u(t + \tau - \eta) {\rm d} \eta$ can calculate $x$ correlation function：
$\begin{aligned} R_x (\tau) &= M \left[ x(t) x(t + \tau) \right] \\ &= M \left[ \int_{-\infty} ^\infty K( \lambda) u(t - \lambda) {\rm d} \lambda \int_{-\infty} ^\infty K(\eta) u(t + \tau - \eta) {\rm d} \eta \ right] \\ &= \int_{-\infty} ^\infty K(\lambda) {\rm d} \lambda \int_{-\infty} ^\infty M \left[ u(t - \lambda) u (t + \tau - \eta) \right] K(\eta) {\rm d} \eta \end{aligned}$ 设 $\lambda$ ，则
$\left[ u(t') u(t' + \lambda+ \tau - \eta) \right] = R_u (\tau + \lambda - \eta)$ 代入上式有
$R_x (\tau) = \int_{-\infty} ^\infty K(\lambda) {\rm d} \lambda \int_{-\infty} ^\infty R_u (\tau + \lambda - \eta) K(\eta) {\ rm d} \eta \tag{3}$ In addition, $x$ sum $u$ 的互相关函数为（用到式(2)）：
$\begin{aligned} R_{xu} (\tau) &= M \left[ x(t) u(t - \tau ) \right] \\ &= M \left\{ \left[ \int_{-\infty} ^\infty K(\lambda) u(t - \lambda) {\rm d} \lambda \right] u(t - \tau) \right\} \\ &= \int_{-\infty} ^\infty K(\lambda) M \left[ u(t - \tau) u(t - \lambda) \right] {\rm d} \lambda \end{aligned}$ 任 $\tau = t'$ ,in the case of
$R_{xu} (\tau) = \int_ {-\infty} ^\infty K(\lambda) M \left[ u(t') u(t' + \tau - \lambda) \right] {\rm d} \lambda$ 又由于 $\left[ u(t') u(t' + \tau - \lambda) \right] = R_u (\tau - \lambda)$ ，代入上式有
$R_{xu} (\tau) = \int_{-\infty} ^ \infty K(\lambda) R_u (\tau - \lambda) {\rm d} \lambda \tag{4}$ In the actual situation, when $\lambda < 0$ 时 $K(\lambda) \equiv$ ，故式(3)(4) can also be written as
$R_x (\tau) = \int_0 ^\infty K(\lambda) {\rm d} \lambda \int_0 ^\infty R_u (\tau + \lambda - \eta) K(\eta) {\rm d} \ eta \tag{3}$ $R_{xu} (\tau) = \int_0 ^\infty K(\lambda) R_u (\ tau - \lambda) {\rm d} \lambda \tag{4}$

6. Calculation of the mean square error of the system output

As mentioned earlier, when $\tau=0$ 时：
$R_x (0) = M \left[ X^2 (t) \right] = \overline{x^2} = D_x$ 即语电影方差。上式即（用到式(4)):
$\begin{aligned} R_x (0) &= R_x (\tau) \big\rvert _{\tau = 0} = \int_0 ^\infty K( \lambda) {\rm d} \lambda \int_0 ^\infty R_u (\tau + \lambda - \eta) K(\eta) {\rm d} \eta \Big\rvert _{\tau = 0} \ \ &= \int_0 ^\infty K(\lambda) {\rm d} \lambda \underbrace{\int_0 ^\infty R_u ( \lambda - \eta) K(\eta) {\rm d} \eta}_ {R_{xu} (\lambda)} \\ &= \int_0 ^\infty K(\lambda) R_{xu} (\lambda) {\rm d} \lambda \tag{5} \end{aligned}$ promptD
$D_x = \overline{x^2} = \int_0 ^\infty K(\lambda) R_{ xu} (\lambda) {\rm d} \lambda \tag{6}$

Statistical Dynamics Notes (1) Transformation of Random Signals in Dynamic Systems in the Time Domain (for personal use)

Transformation of Random Signals in Dynamic System in Time Domain