41. Shape filter
Response of a LTI to stochastic inputs.
From the response of LTI systems to stochastic input, the auto-correlation of a zero-mean stationary process \(x(t)\) (and thus its auto-covariance) reads
\[\mathbf{R}_{\mathbf{x} \mathbf{x}}(\tau) = \mathbb{E}\left[ \mathbf{x}_t \mathbf{x}^*_{t + \tau} \right] \ .\]
Fourier transform of this auto-correlation is defined as its power spectral density, PSD,
\[\boldsymbol\Phi_{\mathbf{x} \mathbf{x}} (f) := \mathscr{F} \{ \mathbf{R}_{\mathbf{x} \mathbf{x}} (\tau) \} (f) \ .\]
For a linear system whose input-output relation reads
\[\mathbf{y}(s) = \mathbf{H}(s) \mathbf{u}(s) \ ,\]
the PSD of the output can be written as a function of the PSD of the input and the Fourier transform of the transfer function as
(41.1)\[\boldsymbol\Phi_{\mathbf{y}\mathbf{y}} (f) = \overline{\mathbf{H}}(f) \boldsymbol\Phi_{\mathbf{u}\mathbf{u}}(f) \mathbf{H}^T(f) \ .\]
SISO system. The transfer function is a scalar function for a SISO system, and thus
(41.2)\[\Phi_{yy}(f) = \overline{H}(f) \Phi_{uu}(f) H(f) = \left| H(f) \right|^2 \Phi_{uu}(f) \ ,\]
as the product of a complex number by its complex cojugate is equal to the square of its modulus.
White noise.
A white noise \(\xi(t)\) can be defined as a random process with
zero expected value, \(\mathbb{E}[\xi_t] = 0\)
Dirac’s delta auto-correlation, \(\mathbb{E}[\xi_t \xi_{t+\tau}] = \delta(\tau)\).
Fourier transform of a Dirac’s delta is \(1\), and thus the PSD of a white noise is
\[\boldsymbol\Phi_{\xi \xi}(f) = 1 \ .\]
SISO shape filter. A process \(y(t)\) with the desired PSD \(\Phi_{yy}(f)\) is the output of a shape filter - a dynamical system - with TF \(H(f)\) satisfying the relation \(|H(f)|^2 = \Phi_{yy}(f)\), fed by a white noise with unit autocovariance. This can be immediately proved inserting \(\Phi_{uu}(f) = 1\) in the relation (41.2).
41.1. LTI system with non-white noise disturbance
Let the linear system
\[\begin{split}\left\{\begin{aligned}
\dot{\mathbf{x}} & = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} + \mathbf{B}_d \mathbf{d} \\
\mathbf{y} & = \mathbf{C} \mathbf{x} + \mathbf{D} \mathbf{u} + \mathbf{D}_d \mathbf{d} + \mathbf{D}_r \mathbf{r} \\
\end{aligned}\right.\end{split}\]
\[\begin{split}\left\{\begin{aligned}
\dot{\mathbf{x}} & = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} + \mathbf{B}_n \mathbf{n} \\
\mathbf{y} & = \mathbf{C} \mathbf{x} + \mathbf{D} \mathbf{u} + \mathbf{D}_n \mathbf{n} \\
\end{aligned}\right.\end{split}\]
have non-white noise disturbances \(\mathbf{n} = \begin{bmatrix} \ \mathbf{d} & \mathbf{r} \ \end{bmatrix}\), whose PSD is known and that can be modeled as the output of a shape filter fed by a white-noise \(\mathbf{w}\),
\[\begin{split}\left\{\begin{aligned}
\dot{\mathbf{x}}_n & = \mathbf{A}_{n} \mathbf{x}_n + \mathbf{B}_{nn} \mathbf{w}_n \\
\mathbf{n} & = \mathbf{C}_{nn} \mathbf{x}_n \ ,
\end{aligned}\right.\end{split}\]
then, the augmented system becomes
\[\begin{split}\left\{\begin{aligned}
\dot{\mathbf{x}} & = \mathbf{A} \mathbf{x} + \mathbf{B}_n \mathbf{C}_{nn} \mathbf{x}_n + \mathbf{B} \mathbf{u} \\
\dot{\mathbf{x}}_n & = \mathbf{A}_{n} \mathbf{x}_n + \mathbf{B}_{nn} \mathbf{w}_n \\
\mathbf{y} & = \mathbf{C} \mathbf{x} + \mathbf{D}_n \mathbf{C}_{nn} \mathbf{x}_n + \mathbf{D} \mathbf{u} \\
\end{aligned}\right.\end{split}\]
or
\[\begin{split}\left\{\begin{aligned}
\begin{bmatrix} \dot{\mathbf{x}} \\ \dot{\mathbf{x}}_n \end{bmatrix} & =
\begin{bmatrix} \mathbf{A} & \mathbf{B}_n \mathbf{C}_{nn} \\ \cdot & \mathbf{A}_n \end{bmatrix}
\begin{bmatrix} \mathbf{x} \\ \mathbf{x}_n \end{bmatrix}
+ \begin{bmatrix} \mathbf{B} \\ \cdot \end{bmatrix} \mathbf{u} + \begin{bmatrix} \cdot \\ \mathbf{B}_{nn} \end{bmatrix} \mathbf{w}_n \\
\mathbf{y} & = \begin{bmatrix} \mathbf{C} & \mathbf{D}_n \mathbf{C}_{nn} \end{bmatrix} \begin{bmatrix} \mathbf{x} \\ \mathbf{x}_{n} \end{bmatrix} + \mathbf{D} \mathbf{u}
\end{aligned}\right.\end{split}\]
41.2. Example
A 1-dimensional non-white noise process \(d\) has PSD, \(\Phi_{dd}(\omega) = | H(j \omega) |^2\) with
\[H(s) = \frac{s}{(s-p_1)(s-p_2)} = \frac{s}{s^2 + a_1 s + a_0} \ .\]
with \(a_0 = p_1 p_2\), \(a_1 = - ( p_1 + p_2 )\).
Using the controller canonical realization, a state-space representation of the shape filter follows
\[\begin{split}\left\{\begin{aligned}
\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} & = \begin{bmatrix} \cdot & 1 \\ - a_0 & - a_1 \end{bmatrix} \begin{bmatrix} \cdot \\ 1 \end{bmatrix} w \\
y & = \begin{bmatrix} \ 0 & 1 \ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \ .
\end{aligned}\right.\end{split}\]
Lyapunov equation for the coorelation matrix \(\ d\). Correlation matrix of the state can be evaluated as the solution of the Lyapunov equation for the correlation matrix, for systems - like the shape filter - with a white noise input.
Non-strictly proper system. If a LTI fed by a white noise process as an input is not striclty proper, \(\mathbf{D} \ne \mathbf{0}\), …
the output has infinite variance (it can be split as a colored noise + white noise),
PSD doesn’t decay to zero as \(\omega \rightarrow 0\)
Lyapunov equation for variance doesn’t converge to a finite value solution
todo prove these statements
