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$$x(t) = \cos(\pi i t/20+\pi/4) - 2je^{j \,12\pi i t} + 5\sin(2\pi i t/3+\pi/3) $$

I need to find the signal's average power. How can I do that? The $i$ index (represented in red in the below image) just denotes a random integer.

enter image description here

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2 Answers 2

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Solve the integral: $$ P = \lim_{T \to \infty} \frac{1}{T}\int_{-T/2}^{T/2} |x(t)|^2 dt $$ This is usually unwieldy to solve directly; it can be shown that if a signal is periodic, you only need to integrate over the fundamental period $T_0$: $$ P = \frac{1}{T_0}\int_{-T_0/2}^{T_0/2} |x(t)|^2 dt $$

EDIT:

Another "trick" is to note that sinusoidal signals of different frequencies are incoherent, so when integrating over long periods of time, it is only necessary to find the power of the individual components because the cross terms go to zero.

This means that $$ P = P_1 + P_2 + P_3 $$ assuming $x_1$, $x_2$, and $x_3$ are incoherent. Find the individual powers and sum them.

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    $\begingroup$ Waht do I need to do with the exp() component in the absolute value ? $\endgroup$ May 12, 2018 at 13:10
  • $\begingroup$ @ShlomiShabtai think closely, what is $|e^{j\phi}|$ for absolutely arbitrary $\phi$? $\endgroup$ May 12, 2018 at 13:17
  • $\begingroup$ yeah i know but it's an absolute value of a sum and it's not the same as sum of absolute values. Is there another way to break it up that i can't see ? $\endgroup$ May 12, 2018 at 13:20
  • $\begingroup$ yes, read my comment closely; you seem to respond to something I didn't write. $\endgroup$ May 12, 2018 at 13:23
  • $\begingroup$ @MarcusMüller: yes, but you seemed to respond to something he didn't ask $\endgroup$
    – Robert L.
    May 12, 2018 at 13:25
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This is a way you can do that.

As the red $i$ is a fixed integer (its randomness is out of context), and the time terms are of the shape $a_k\pi/b_k$ with $a_k$ and $b_k$ integers, you can find a least common multiple, to ensure that all terms have an integer number of periods inside one interval. Then, you can integrate the power of the signal over that interval.

Then, two options can be used:

  • compute cosine products, linearize them, and compute the integral
  • use orthogonality properties to save a few computations.

If you can show some progress on your computations, you may get additional hints.

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