In real world systems we are affected by disturbances and measurement noise. Since they are unpredictable, we consider them as random processes. We need a way of describing them effectively.

Estimating the Power Spectral Density

E [\frac{1}{T} ∣ F (x_{T} (t)) ∣^{2}] T \to \infty S_{x} (jω)

Terminology

Deterministic process
Completely predictable
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Random process - Uncertain, not completely predictable, and can be characterized using statistical properties

Autocorrelation

Here, it is defined as $R_{x} (t_{1}, t_{2}) = E [X (t_{1}) X (t_{2})]$

Autocovariance
The autocovariance function for a random process $X (t)$ is defined by
$C_{x} (t_{1}, t_{2}) = E [(X (t_{1}) - m_{x} (t_{1})) \cdot (X (t_{2})) - m_{x} (t_{2})]$
The autocovariance of white noise, $u (t) = w (t)$
$A_{u} = C_{x} = δ (t_{1} - t_{2}) Q_{u}, Q_{u} ≻ 0$

Aka $Q_{u}$ is a Positive-Definite Matrix
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