Ideal and RC filtering of white noise

White noise ($S_W(f)=N_0/2$, $R_W(\tau)=\frac{N_0}{2}\delta(\tau)$) has infinite power and zero correlation time. Passing it through a filter $H(f)$ gives $S_Y(f)=\lvert H(f)\rvert^2\frac{N_0}{2}$ and $R_Y(\tau)=\mathcal{F}^{-1}\{S_Y(f)\}$: the noise becomes band-limited (coloured) and its samples become correlated.

Ideal low-pass filtering

Ideal LPF, gain 1 for $\lvert f\rvert\le B$, zero elsewhere:

$$S_Y(f)=\begin{cases}\frac{N_0}{2}, & \lvert f\rvert\le B\\ 0, & \text{otherwise}\end{cases}$$ $$\begin{aligned} R_Y(\tau) &= \int_{-B}^{B}\frac{N_0}{2}e^{j2\pi f\tau}df=\frac{N_0}{2}\cdot\frac{\sin2\pi B\tau}{\pi\tau} \\ &= N_0B\,\frac{\sin2\pi B\tau}{2\pi B\tau} \end{aligned}$$

• Power $R_Y(0)=N_0B$.
• $R_Y(\tau)$ is a sinc with zeros at $\tau=\pm\frac{k}{2B}$, $k=1,2,\dots$: samples taken at the Nyquist rate $2B$ are uncorrelated (independent if Gaussian).
• It oscillates (takes negative values) and decays slowly as $1/\tau$.

Practical RC low-pass filtering

$H(f)=\frac{1}{1+j2\pi fRC}$, so

$$S_Y(f)=\frac{N_0/2}{1+(2\pi fRC)^2}$$

Using the pair $e^{-a\lvert\tau\rvert}\leftrightarrow\frac{2a}{a^2+(2\pi f)^2}$ with $a=1/RC$:

$$R_Y(\tau)=\frac{N_0}{4RC}\,e^{-\lvert\tau\rvert/RC}$$

• Power $R_Y(0)=\frac{N_0}{4RC}$, i.e. NEB $=\frac{1}{4RC}=\frac\pi2f_{3dB}$.
• $R_Y(\tau)$ is a decaying exponential: always positive, never exactly zero, falls to $1/e$ at $\tau=RC$. Correlation time $\approx RC$.

 R(tau)  ideal LPF: sinc         RC LPF: exponential
   N0B .                     N0/4RC .
      / \                          /|\
  .  /   \  .                     / | \
 -.-/-----\-.-- tau          ----'--+--'---- tau
   -1/2B  1/2B                 -RC  0  RC

Comparison

In both cases a narrower bandwidth gives a wider autocorrelation (longer correlation time) and less noise power.