Colored Noise

Where white noise is completely independent from time to time, colored noise is dependent. Aka the Autocovariance $\ne 0$.

It can be achieved by passing white noise through a linear plant.

$$y(t)={\int }_{-\infty }^{\infty }H(t-\tau )u(\tau )d\tau$$

Where $u(t)\in \mathbb{R}$ be white noise with zero mean and autocorrelation ${\mathcal{R}}_u(\tau )$. And the causal impulse response is given by

$$H(t)=\Theta (t)[c{e}^{At}b+d\delta (t)]$$

We say that $H$ colors $y$.

We use $w(t)$ for white noise, and $d(t)$ for colored noise.

The Power Density Spectrum (PDS) functions for a colored noise signal is given by

$$S_d(\omega )=H(j\omega )H(-j\omega )$$