# Confusion In DC power,Average Power,Ac Power and total Power

These are Things I studied.

$$E[X]:-$$ THis gives us Average value or Dc value

$$E[X^2]:-$$Total power

$$\sigma^2=E[X^2]-E[X]^2$$:-Ac power

From Autocorrelation

$$R_x(0)=E[X^2]$$--Total power

And $$\lim_{x\to \infty}{R_x(\tau)}=$$Dc power

Question

The Autocorrelation fuction $$R_x(\tau)$$ of a Wide-sense random process shown in below figure The Average Power is____

So what I did? I just Find Value At $$t=\infty$$ and that is Zero But the official answer is 2.

So they find $$R_x(0)$$.

Here is my confusion. is there any difference between DC power and average power?

I just relate from E[x], which gives the Average value or DC value.

where I am thinking wrong?

What you have studied doesn't seem to include a definition of Average Power. For deterministic signal $$x(t)$$, the instantaneous power delivered at time $$t$$ is $$x^2(t)$$ $${\big (}$$yeh, yeh, nitpickers should choose $$R=1$$ in their beloved formula $$\dfrac{V^2}R\big )$$ which varies from instant to instant. Thus, the average power delivered by $$x(t)$$ can be calculated as $$\text{Average Power} =\lim_{T\to \infty}\frac{1}{2T}\int_{-T}^T x^2(t)\, \mathrm dt.\tag{1}$$ If $$x(t)$$ has a DC value, it is given by $$\displaystyle x_0 = \lim_{T\to \infty}\frac{1}{2T}\int_{-T}^T x(t)\, \mathrm dt$$ and so we can write $$x(t) = x_0 + x_{\text{ac}}(t)$$ (prove for yourself that $$x_{\text{ac}}(t)$$ has DC value $$0$$) so that $$(1)$$ becomes \begin{align}\require{cancel}\text{Average Power} &=\lim_{T\to \infty}\frac{1}{2T}\int_{-T}^T (x_0 + x_{\text{ac}}(t))^2\, \mathrm dt\\ &= \lim_{T\to \infty}\frac{1}{2T}\int_{-T}^T x_0^2 + +2x_0x_{\text{ac}}(t) + x_{\text{ac}}^2(t)\, \mathrm dt\\ &= x_0^2 + 2x_0 \cancelto{0}{\lim_{T\to \infty}\frac{1}{2T}\int_{-T}^T x_{\text{ac}}(t)\, \mathrm dt} + \lim_{T\to \infty}\frac{1}{2T}\int_{-T}^T x_{\text{ac}}^2(t)\, \mathrm dt\\ &= \text{DC Power} + \text{AC Power}\\ &= \text{Total power} \end{align} On the other hand, for wide-sense-stationary random processes, the total power (which is also the same as the average power) is $$R_X(0)$$ by definition, and so for the figure shown, the total power is $$2$$. The total power is the sum of the DC Power and the AC power and since the DC Power is $$0$$ for the $$R_x(t)$$ shown, the average power asked for also has value $$2$$ and ot consists solely of AC power.