Random processes formulas

Holds for any random variable. $E[X^2]$ is the mean-square value (average power for a signal).

Fully described by mean $\mu$ and variance ${\sigma }^2$.

Convention used in these notes: $Q(x)=\frac{1}{2}\mathrm{erfc}(x/\sqrt{2})$. Also $Q(-x)=1-Q(x)$ and $Q(0)=0.5$.

For an ergodic process, $⟨x(t)⟩={\mu }_X$ and $⟨x^2(t)⟩=R_X(0)$.

Depends only on the lag $\tau$ when $X(t)$ is WSS. The process is real-valued here.

$A$ and $f_0$ are constants. A WSS process with zero mean and power $A^2/2$. If $\Theta$ is uniform over only $(0,\pi )$, the mean depends on $t$ and the process is not WSS.

Applies to a WSS process. $S_X(f)$ is two-sided, so it covers negative and positive frequencies.

Integrate over both sides of the spectrum. A nonzero mean contributes an impulse ${\mu }_X^2\delta (f)$ at $f=0$.

$H(0)$ is the DC gain.

Only the magnitude of $H(f)$ matters; phase has no effect on the output PSD.

The Fourier transform of this is $|H(f){|}^2{S}_X(f)$ for real $h(t)$.

Total output mean-square value, including any DC part.

Defined as $R_{YX}(\tau )=E[Y(t+\tau )X(t)]$. For the other order, $R_{XY}(\tau )=h(-\tau )\ast R_X(\tau )$ and $S_{XY}(f)=H^{\ast }(f)S_X(f)$.

Holds when $X(t)$ and $Y(t)$ are jointly WSS and uncorrelated, with at least one of them zero-mean (for example, a signal plus independent zero-mean noise). Otherwise add the cross terms $R_{XY}(\tau )+R_{YX}(\tau )$.

$X(t)$ is WSS and $\Theta \sim U(0,2\pi )$ is independent of $X(t)$. The output power is $P_Y=R_X(0)/2$.

Follows from $S_Y(f)=|H(f){|}^2{S}_X(f)$.

Samples at different instants are uncorrelated (independent if the noise is also Gaussian). Total power is infinite, so ideal white noise is a model that is used only after some filtering.

$H_0$ is the passband gain and $B$ is the one-sided bandwidth in Hz (for a low-pass filter, $|f|\le B$; for a band-pass filter, total width $B$ centred on $\pm f_c$).

Unity gain, passband $|f|\le B$, $\mathrm{sinc}(x)=\frac{\sin \pi x}{\pi x}$. $R_Y(\tau )$ is zero at $\tau =k/(2B)$, $k\ne 0$, so samples taken $1/(2B)$ apart are uncorrelated (independent if the noise is Gaussian).

Width of an ideal filter of the same peak gain that passes the same white-noise power: $P_{\text{out}}=N_0|H{|}_{\max }^2{B}_N$. The peak is at $f=0$ for a low-pass filter and at the centre frequency for a band-pass filter; $f$ in Hz.

Here $B_N=\frac{\pi }{2}{f}_{3\text{dB}}=\frac{1}{4RC}$ with $f_{3\text{dB}}=\frac{1}{2\pi RC}$. The output is coloured noise.

$k=1.38\times 10^{-23}$ J/K, $T$ in kelvin, $B$ in Hz. $\overline{v_n^2}$ is the open-circuit mean-square voltage of a resistor $R$, and $P_{\text{avail}}$ is the power it delivers to a matched load. Thermal noise is Gaussian and is white for frequencies well below $kT/h$ (about $6\times 10^{12}$ Hz at 290 K).

$n(t)$ is zero-mean, WSS and band-pass. $n_c(t)$ and $n_s(t)$ are low-pass, zero-mean and jointly WSS; if $n(t)$ is Gaussian, they are jointly Gaussian.

For white noise $N_0/2$ through an ideal band-pass filter of width $B$ (gain 1): each of the three powers equals $N_{0}B$.

For zero-mean Gaussian narrowband noise with ${\sigma }^2=E[n^2]$, written as $n(t)=r(t)\cos (2\pi f_{c}t+\psi (t))$: the envelope is Rayleigh, and the phase is uniform and independent of the envelope. $P(r>a)=\exp \left(-a^2/2{\sigma }^2\right)$ and $E[r^2]=2{\sigma }^2$. If a sinusoid of amplitude $A$ is added, the envelope becomes Rician: close to Rayleigh for $A\ll \sigma$, and close to Gaussian about $A$ for $A\gg \sigma$.