Notes on Statistical Dynamics (3) Whole Wave Filter (for personal use)

Whole-wave filters are a class of filters that can integrate statically indeterminate random signals with arbitrary spectral densities. Its input signal is often white noise .

1. Derivation of whole wave filter

From Statistical Dynamics Notes (2) Spectral Density and Dynamic Accuracy of Linear Stochastic System (for self-retention), we can know the system output $x$ importuuThe cross-spectrum density between$u$
$$S_x\left(oh\right)=W(j\omega )W(-j\omega )S_u\left(oh\right)\text{\hspace{1em}( 1 )}$$ When the input is white noise,$S_u\left(oh\right)=S_n\left(oh\right)=1$，则
$$S_x\left(oh\right)=W(j\omega )W(-j\omega )\text{\hspace{1em}( 2 )}$$ In this way, as long as the output terminal$The\; spectral\; density\; of\; x$ is decomposed into twoconjugatethe transfer functionof the system can be obtained. This step is also known asdecomposition of the spectral density.

Example: The spectral density at the output is
$$S_x\left(oh\right)=\frac{4}{4o^2+1}=\frac{2}{2jo+1}\cdot \frac{2}{2(-j\omega )+1}$$Then the transfer function of the system is
$$W(j\omega )=\frac{2}{2jo+1}$$即
$$W(\mathrm{s})=\frac{2}{2\mathrm{s}+1}$$

2. Variance of a random signal at the output of a linear dynamical system

The definition of variance has been given in formula (5) of Statistical Dynamics Notes (2) Spectral Density and Dynamic Accuracy of Linear Stochastic Systems (for self-retention) :
$$D_x=R_x(0)=\frac{1}{2p.m}{\int }_{-\infty }^{\infty }{S}_x\left(\omega \right)d\omega$$代入式(1)
$$D_x=\frac{1}{2p.m}{\int }_{-\infty }^{\infty }W(j\omega )W(-j\omega )S_u\left(\omega \right)d\omega =\frac{1}{2p.m}{\int }_{-\infty }^{\infty }|W(j\omega ){|}^2{S}_u(\omega )d\omega \text{\hspace{1em}( 3 )}$$ The calculation method of formula (3) has the following set of fixed methods, called "$I_n$– Integral法”：
$$I_n=\frac{1}{2p.m}{\int }_{-\infty }^{\infty }\frac{|G(j\omega ){|}^2}{\left|H_n\right(j\omega ){|}^2}\mathrm{d}\omega =\frac{1}{2p.m}{\int }_{-\infty }^{\infty }\frac{G_n(j\omega }{H_n(j\omega )H_n(-j\omega )}\mathrm{d}\omega \text{\hspace{1em}( 4 )}$$ where
$$\begin{gathered} G_n\left(j\omega \right)=b_0(j\omega {)}^{2n-2}+b_1(j\omega {)}^{2n-4}+\cdots +b_{n-1}, \\ {H}_n\left(j\omega \right)=a_0(j\omega {)}^n+a_1(j\omega {)}^{n-1}+\cdots +a_n\text{\hspace{1em}( 5 )} \end{gathered}$$ There are the following points about formula (4)(5):
(1) If the order of the denominator of the integral formula is$n$ , then in the actual system, the order of the numerator will not exceed$2n-2$ .
(2) Integral denominator$H_n(j\omega )H_n(-j\omega )$ isω \omegaEven function of$\omega \; .$
(3) Integral molecule$G_n\left(j\omega \right)$ contains onlyj ω j\omegaEven powersof$j\omega$ . If there is an odd power, it can be ignored directly, because the odd power will be equal to zero after integration. (4) H n ( j ω ) H_n(j \omega)
in the denominator of the integral formula$H_n\left(j\omega \right)$ should be stable.

Then for $I_n$– Integral, which is calculated as follows:
$$I_n=(-1{)}^{n+1}\frac{N_n}{2a{\_}_0{D}_n}\text{\hspace{1em}(6)}$$其中
$$D_n=\left|\begin{array}{ccccc}{a}_1& a_0& 0& \cdots & 0\\ a_3& a_2& a_1& \cdots & 0\\ a_5& a_4& a_3& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & a_n\end{array}\right|,\text{\hspace{1em}(7)}$$$$N_n=\left|\begin{array}{ccccc}{b}_0& a_0& 0& \cdots & 0\\ b_1& a_2& a_1& \cdots & 0\\ b_2& a_4& a_3& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ b_{n-1}& 0& 0& \cdots & a_n\end{array}\right|\text{\hspace{1em}(8)}$$$N_n$Just put $D_n$The first column in is replaced by $b_i$。

Example: Let the transfer function of the system be
$$W(\mathrm{s})=\frac{K}{T\mathrm{s}+1}$$The spectral density of the input signal is
$$S_u\left(oh\right)=\frac{D_u}{a^2+{oh}^2}$$Calculate the mean square error of the system.
First get the transfer function of the systematic error:
$${Phi}_e(\mathrm{s})=\frac{1}{1+W\left(s\right)}=\frac{Ts+1}{Ts+1+K}$$Definition(1)Specify the free-flowing function
$$\begin{array}{rl}{S}_e\left(oh\right)& ={|{\Phi }_e\left(j\omega \right)\_}^2{S}_u\left(oh\right)={\left|\frac{T\left(j\omega \right)+1}{T(j\omega )+1+K}\right|}^2\frac{D_u}{a^2+{oh}^2}\\ & =\frac{D_u\left(T^2{o}^2+1\right)}{{|\left(T(j\omega )+1+K\right)\left(j\omega +a\right)|}^2}\\ & =\frac{D_u\left(T^2{o}^2+1\right)}{{T\left(j\omega \right)\_{}^2+\left(\alpha T+1+K\right)j\omega +(1+K)\alpha |}^2}\end{array}$$The mean square error is (analogous to Statistical Dynamics Notes (2) Spectrum Density and Dynamic Accuracy of Linear Stochastic System (for self-retention) - Equation (5)):
$$\overline{e^2}=\frac{1}{2p.m}{\int }_{-\infty }^{\infty }{S}_e\left(\omega \right)d\omega =D_u{I}_2$$Then
$$I_2=\frac{1}{2p.m}{\int }_{-\infty }^{\infty }\frac{\left(T^2{o}^2+1\right)\mathrm{d}\omega }{{T\left(j\omega \right)\_{}^2+\left(\alpha T+1+K\right)j\omega +(1+K)\alpha |}^2}$$We can see
$$G_2\left(j\omega \right)=\underset{b_0}{\underbrace{T^2}}{oh}^2+\underset{b_1}{\underbrace{1}},$$$$H_2\left(j\omega \right)=\underset{a_0}{\underbrace{T}}(j\omega {)}^2+\underset{a_1}{\underbrace{\left(\alpha T+1+K\right)}}j\omega +\underset{a_2}{\underbrace{(1+K)}}$$Determine the equation
$$D_2=\left|\begin{array}{cc}{a}_1& a_0\\ a_3& a_2\end{array}\right|=\left|\begin{array}{cc}\alpha T+1+K& T\\ 0& (1+K)\end{array}\right|=a\left(\alpha T+1+K\right)(1+K),$$$$N_2=\left|\begin{array}{cc}{b}_0& a_0\\ b_1& a_2\end{array}\right|=\left|\begin{array}{cc}{T}^2& T\\ 1& (1+K)\end{array}\right|=\alpha T^2(1+K)-T$$故
$$I_2=(-1{)}^{2+1}\frac{N_2}{2a{\_}_0{D}_2}=-\frac{\alpha T^2(1+K)-T}{2T\alpha \left(\alpha T+1+K\right)(1+K)}$$则
$$\overline{e^2}=D_u{I}_2=\frac{D_u\left[T-\alpha T^2(1+K)\right]}{2T\alpha \left(\alpha T+1+K\right)(1+K)}=\frac{D_u\left[1-\alpha T(1\_+K)\right]}{2a\left(\alpha T+1+K\right)(1+K)}$$Invariant equation
$$\sqrt{\overline{e^2}}=\sqrt{\frac{D_u\left[1-\alpha T(1\_+K)\right]}{2a\left(\alpha T+1+K\right)(1+K)}}$$