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*}

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