Signal variance and power connection

Question

For a random signal $x(n)$, why is $E(x(n)^2)$ called signal power? Is it really power? Any proof?

Answer 1

The terminology is a bit unfortunate here. "Signal Power" and "Physical Power" are two very different things.

why is E(x(n)2) called signal power?

Because signal power is defined this way.

Is it really power? Any proof?

It's NOT physical power. So there is nothing to proof here. A signal doesn't have power in the physical sense. Physical power only occurs when the signal is associated with some sort of physicals arrangement with an impedance in it. Physical power typical is the product of two complimentary physical field quantities (Voltage/Current, Force/Velocity, etc).

We can relate Signal Power to Physical Power:

1. It never DIRECTLY represents physical power. The units are the square of the original units and for it to be physical power the units of x(n) would have to be $\sqrt W$ which doesn't exist
2. It's often PROPORTIONAL to power (or intensity). That's the case if the impedance across or through which x(n) occurs is real and frequency independent.

Examples:

1. Voltage across an ideal resistor: $<x^2>$ is proportional to the power dissipated in the resistor
2. Sound pressure measured with a pressure microphone sufficiently far away from a sound source: $<x^2>$ is proportional to the intensity of the sound field at the location of the microphone
3. Voltage across a dynamic loudspeaker: $<x^2>$ does not represent anything that can easily be related to physical power.

Answer 2

Let $X$ be a random variable with a pdf (probability density function) $f_X(x)$, mean $\mu_X = E\{X\}$, and variance $\sigma_X^2 = E\{(X-\mu_X)^2\}$. Then the following relation holds :

$$ E\{X^2\} = E\{(X-\mu_x)^2\} + E\{X\}^2 = \sigma_X^2 + \mu_x^2 \tag{1}$$

In Eq.1, the LHS $E\{X^2\}$ is known as the total power of the R.V. $X$, whereas the terms $\sigma_X^2$ and $\mu_X^2$ are known as AC and DC powers respectively. This nomenclature in signal processing stems from their resemblance to the electric circuit variables and their AC/DC power equations.

A discrete-time random process $x[n]=X[n,s)=X_n$ can be viewed as a collection of random variables $X_n$ each indexed by the time $n$. Each random variable has its own associated pdf $f_{X_n}(x)$.

It can be seen that a similar relation to Eq.1 exist between the time-dependent mean and variance of the R.P $X[n,s)$ , and the quantity :

$$E\{x[n]^2\} \tag{4}$$ is referred to as the Average Total Power of the random process $x[n] = X[n,s) = X_n$. This is a useful metric for WSS random processes.

Answer 3

One definition in physics is that power corresponds to some "amount of energy [...] per unit time". If energy is denoted by a square $(\cdot)^2$, and the $E(\cdot)$ symbol for some expected amount (integral averaged for mean probability if you expect ergodicity), then when the quantities exist, you can consider it as a "signal power" but this is more a mundane definition than a definitive proof.