909. Find the correlation function of stationary white noise - a stationary random function with constant spectral density $s_{x}(\omega)=s_{0}$.
Solution. We use the formula

$$
k_{x}(\tau)=\int_{-\infty}^{\infty} s_{x}(\omega) \mathrm{e}^{i \tau \omega \omega} \mathrm{d} \omega=s_{0} \int_{-\infty}^{\infty} \mathrm{e}^{l \tau \omega \omega} \mathrm{d} \omega
$$

Taking into account that $\frac{1}{2 \pi} \int_{-\infty}^{\infty} \mathrm{e}^{\tau \tau \omega} \mathrm{d} \omega=\delta(\tau)$, where $\delta(\tau)$ is the delta function. Therefore,

$$
\int_{-\infty}^{\infty} e^{l \tau \omega} d \omega=2 \pi \delta(\tau)
$$

Substituting (**) into (*), we finally obtain the desired correlation function $k_{x}(\tau)=2 \pi s_{0} \delta(\tau)$.

8 7. Transformation of a Stationary Random Function by a Stationary Linear Dynamic System

A stationary linear dynamic system is a device described by a linear differential equation with constant coefficients of the form

$$
\begin{gathered}
{\left[a_{0} Y^{(m)}(t)+a_{1} Y^{(m-1)}(t)+\ldots+a_{n}\right] Y(t)=} \\
{\left[b_{0} X^{(m)}(t)+b_{1} X^{(m-1)}(t)+\cdots+b_{m}\right] X(t)}
\end{gathered}
$$

where $X(t)$ is the input stationary random function (disturbance, excitation), and $Y(t)$ is the output random function (response, reaction).

If the dynamic system is stable, then for sufficiently large values of $t$, i.e., after the transient process has ended, the function $\boldsymbol{V}(t)$ can be considered stationary.

The mathematical expectation of the output function $Y(t)$ is found using the formula $m_{y}=\left(b_{m} / a_{n}\right) m_{x}$, where $m_{x}$ is the mathematical expectation of the input function $X(t)$.

In operator form, equation (*) is written as

$\left.a_{0} p^{n}+a_{1} p^{n-1}+\ldots+a_{n}\right) Y(t)=\left(b_{0} p^{m}+b_{1} p^{m-1}+\ldots+b_{m}\right) X(t)$.

The transfer function of a linear dynamic system is the ratio of the polynomial in $p$ for $X(t)$ to the polynomial in $p$ for $Y(t)$ in the operator equation (**):

$$
\Phi(p)=\left(b_{0} p^{m}+b_{1} p^{m-1}+\ldots+b_{m}\right) /\left(a_{0} p^{n}+a_{1} p^{n-1}+\ldots+a_{n}\right)
$$

The frequency characteristic of a linear dynamic system is the function obtained by replacing the argument $p$ in the transfer function with the argument $i \omega$ (where $\omega$ is a real number): $\Phi(i \omega)=$

$=\left[b_{0}(i \omega)^{m}+b_{1}(i \omega)^{m-1}+\ldots .+b_{m}\right] /\left[a_{0}(i \omega)^{n}+a_{1}(i \omega)^{n-1}+\ldots+a_{n}\right]$.

The spectral densities of the output and input functions are related by the equation $s_{y}(\omega)=s_{x}(\omega) \cdot|\Phi(i \omega)|^{2}$, i.e., to find the spectral density of the output function, one must multiply the spectral density of the input function by the square of the modulus of the frequency characteristic. Knowing the spectral density of the output function, one can find its correlation function

$$
k_{y}(\tau)=\int_{-\infty}^{\infty} s_{y}(\omega) e^{i \tau \omega} \mathrm{d} \omega
$$

and consequently, the variance

$$
D_{y}=k_{y}(0)=\int_{-\infty}^{\infty} s_{y}(\omega) \mathrm{d} \omega
$$