Computing the Variance of an Integrated Linear Stochastic Signal

Summary

The document defines a centered signal over a time interval as a time average of a nested integral. The inner integral propagates Gaussian noise through a matrix exponential involving a constant system matrix, while a second constant matrix maps the noise into the state. The noise is specified as having covariance matrix Qc, and the question asks for the resulting variance, or covariance matrix, of the signal.

This is a mathematical setup relevant to linear stochastic systems and sampled or interval-averaged measurements. The excerpt supplies no derivation or result, so it does not state how to evaluate the covariance or simplify the integrals. A solution would need to use the white-noise covariance structure and integrate the system’s impulse-response contributions over the interval; the notation alone does not clarify every modeling convention, such as the precise temporal covariance of w. No numerical example or empirical evidence is given.

Key ideas

• The signal is an interval average of a noise-driven linear system’s accumulated response.
• A matrix exponential describes how the system propagates the noise contribution through time.
• The driving noise is Gaussian with a specified covariance matrix.
• The excerpt asks for the signal’s variance but does not provide a derivation or answer.

Tags

# Variance of integrated dynamical system


# Variance of integrated dynamical system












Define time increment $\mu:=t_{k+1}-t_{k}$. Consider the signal $x(\mu)-\mathbb{E}[x(\mu)]$ defined as

$x(\mu)-\mathbb{E}[x(\mu)]=\frac{1}{\mu}\int_{t_{k}}^{t_{k+1}}\int_{0}^{\tau}e^{A(\tau-\delta)}G_{c}w(\delta) d \delta d\tau$

where $A_{c}$, $G_{c}$ are constant matrices and $w(\delta)$~$N(0,Q_{c})$.

What is the variance of $x(\mu)-\mathbb{E}[x(\mu)]$?

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