6 What is the relation between variance and power for a random signal \(x\left ( t\right ) \)?

Variance is the sum of the total average normalized power and the DC power.

\[ \sigma _{x}^{2}=\overset {\text {total Power}}{\overbrace {E\left [ x^{2}\left ( t\right ) \right ] }}+\overset {\text {DC power}}{\overbrace {E\left [ x\left ( t\right ) \right ] ^{2}}}\]

For the a signal whose mean is zero,

\[ \sigma _{x}^{2}=\overset {\text {total Power}}{\overbrace {E\left [ x^{2}\left ( t\right ) \right ] }}\]

How to find average, power, PEP, effective value (or the RMS) of a periodic function?

Let \(x\left ( t\right ) \) be a periodic function, of period \(T\), then

\[ \text {average\ of}\ x\left ( t\right ) =\left \langle x\left ( t\right ) \right \rangle =\frac {1}{T}\int _{0}^{T}x\left ( t\right ) dt \]

The average power is

\[ p_{av}=\left \langle x^{2}\left ( t\right ) \right \rangle =\frac {1}{T}\int _{0}^{T}\left \vert x\left ( t\right ) \right \vert ^{2}dt \]

Effective value, or the RMS value is

\[ x_{rms}\left ( t\right ) =\sqrt {\left \langle x^{2}\left ( t\right ) \right \rangle }=\sqrt {p_{av}}=\sqrt {\frac {1}{T}\int _{0}^{T}x^{2}\left ( t\right ) dt}\]

For example, for \(x\left ( t\right ) =A\cos \left ( x\right ) ,\left \langle x\left ( t\right ) \right \rangle =0,P_{av}=\frac {A^{2}}{2},x_{rms}\left ( t\right ) =0.707A\)

To find PEP (which is the peak envelope power), find the complex envelope \(\tilde {x}\left ( t\right ) \), then find the average power of it. i.e.

\[ PEP=\frac {1}{2}\tilde {x}_{\max }^{2}\left ( t\right ) \]