Definition of average power?

Question

There are two kind of average power I encountered in random signal class and textbook:

definition 1: average power =$$E[|x(t)|^2]=R_{xx}(0)=\int^\infty_{-\infty} S_{xx}(f)\,df$$ definition 2: average power =$$\lim_{T\to\infty}\frac{1}{2T}\int^{T}_{-T}|x(t)|^2\,dt $$ It seems that the two definition is different unless the signal is ergodic in $x(t)^2$. By the way, our teacher keep using the term "assuming average power of white noise $n(t)$ is $\sigma_N^2$" in test. But from the above definition, the average power is infinite for continuous white noise. I think in discrete white noise the average power can be well-defined but in continuous one I am wondering our teacher may take the definition wrong.

Answer 1

The first definition works for deterministic as well as for random signals. For random signals we define the autocorrelation by

$$R_x(\tau)=E\{x^*(t)x(t+\tau)\}\tag{1}$$

where $E\{\cdot\}$ is the expectation operator. For deterministic power signals (i.e., signals with finite but non-zero power and infinite energy) we define

$$R_x(\tau)=\lim_{T\to\infty}\frac{1}{2T}\int^{T}_{-T}x^*(t)x(t+\tau)dt\tag{2}$$

From $(2)$ it is clear that for deterministic power signals both definitions of average power are equivalent, because for random signals as well as for deterministic power signals we have

$$S_x(f)=\mathcal{F}\{R_x(\tau)\}\tag{3}$$

Also take a look at this related answer.

You are right about the power of white noise. In continuous time, white noise has infinite power. Only discrete-time white noise has finite power. In sum, teachers and professors are only humans.

Answer 2

The other way of looking at this is to note that for continuous time, the autocorrelation function of white noise is often given as: $$ R_{xx}(t,t+\tau)=\sigma^2 \delta(\tau) $$ where $\delta(t)$ is the Dirac delta function. If you look at this from a probabilistic perspective is says that $P(x(t+\tau)$ is independent from $P(x(t)$ for $\tau > 0$ . From an engineer's perspective, for even very tiny value of $\tau$ , the knowledge of $x(t)$ has no predictive value. By necessity, the bandwidth needs to be infinite.

This is not physically realizable. It is an idealized abstraction.

So if we place an ordinary old fashioned voltmeter across a resistor and measure the rectified voltage due to temperature, the voltmeter does not disappear in a puff of smoke because, thermal noise is typically modeled as being white, and infinite power is after all infinite. The needle will likely have a wiggle related to $\sigma^2$. Essentially white noise is like honesty and virtue, a worthy idealization. White noise is necessarily band limited, but flat over the band we are interested in.

Getting back to the voltmeter which measures continuous time voltage, the measurement itself is an average value over a time aperture, so our $\delta(t)$ is safely inside of an integral.