Power content of a signal:

For a deterministic periodic with a period of $N$ samples its power content is:

$$P_{x} = \frac{1}{N} \sum_{k=n}^{n+N-1}\vert x[k]\vert^2 = \frac{1}{N}\underline{\textbf{x}}_n^h \underline{\textbf{x}}_n$$

With a runtime vector of $N$ samples.

As we may recall from stochastic processes theory we have:

• Mean: $m_{x}[n] = E\left\lbrace x[n]\right\rbrace$
• Correlation: $r_{xx}[k,m] = E\left\lbrace x[k]^* x[m] \right\rbrace$
• Power: $\mathcal{P}_x = r_x[0] = E\left\lbrace x[n]^* x[n] \right\rbrace = E\left\lbrace \vert x[n]\vert^2 \right\rbrace$
• If the process is stationary in wide sense (wss) $m_{x}[n] = m_{x}$ the statistics of the process doesn't depend on the time instant choosen⁴ so we can define:
  • Mean: $m_{x} = E\left\lbrace x[n]\right\rbrace$
  • Correlation: $r_{xx}[n] = E\left\lbrace x[k]^* x[n+k] \right\rbrace$
  • Covariance: $C_{xx}[k,m] = E\left\lbrace (x[k]-m_x)^* (x[m]-m_x) \right\rbrace = r_{xx}[k,m] - m_x^2$
  • Variance: $\sigma_x^2 = E\left\lbrace \vert x[k]-m_x\vert^2 \right\rbrace$
  • Standard deviation: $\sigma_x$

For a process, the power content is defined as:

$$P_x = E\left\lbrace \mathcal{P}_x\right\rbrace = E\left\lbra... ...ace = \lim_{N\to\infty}\frac{1}{N}\sum_{k=0}^{N-1} r_{xx}[k,k]$$

Where we have used that the expected value is a linear operator and can enter inside the integral.