Noise

A voltage signal $v(t)$ applied across a resistor $R_L$ dissipates instantaneous power

(1) $\begin{equation*} P(t) = {v(t)^2 \over R_L} \end{equation*}$

For a periodic signal of period (T), The average power is obtained by integrating over one cycle

(2) $\begin{equation*} P_{av} = \int\limits_{-T/2}^{T/2} {v(t)^2 \over R_L} dt \end{equation*}$

For a random signal, the average power is obtained by integrating over infinitely time

(3) $\begin{equation*} P_{av} = \lim_{T \to \infty} \int\limits_{-T/2}^{T/2} {v(t)^2 \over R_L} dt \end{equation*}$

To simplify calculations, the average power of a random signal ( $x(t)$ ) is expressed as

(4) $\begin{equation*} P_{av} = \lim_{T \to \infty} \int\limits_{-T/2}^{T/2} x(t)^2 dt \quad\quad (V^2) \end{equation*}$

For a random process, the auto-correlation function is defined as

(5) $\begin{equation*} R_x(t,\tau) = E{x(t)x*(t+\tau)} \end{equation*}$

Analog/RF IntgCkts