# What is the average of $\DeclareMathOperator{\rect}{rect} \rect(\cos(\pi t/2))$?

We have this signal:

$$\operatorname{rect}\left(\cos\left(\frac{\pi t} {2}\right)\right)$$

I must find the average power , how can i get there ?

My solution:

I have seen that
$$-\frac 12 < \operatorname{rect}\left(\cos\left(\frac{\pi t}{2}\right)\right) < 1$$ is for
$$\frac 23+k \pi so in a period it should be rectangular pulse , but how do I calculate the average ?

• Hi! Homework ? Online-Quiz ? Where have you been stuck ? What's that rect() function ? – Fat32 Nov 1 '20 at 21:13
• Hi , rect is the function that is 1 between -1/2 <t < 1/2 – Giovanni Cerciello Nov 1 '20 at 21:18
• Good! So what's the problem ? – Fat32 Nov 1 '20 at 21:19
• and i was doing this exercise and i got 1/3 as average , but i have written that it should be 1/2 , so which one is correct ? – Giovanni Cerciello Nov 1 '20 at 21:19
• You need to show how you arrived at your result. Then we can tell where you went wrong. – Matt L. Nov 2 '20 at 6:46

## 1 Answer

The given signal is clearly periodic. So just follow these steps and find the solution:

1. Find the period $$T$$.
2. Figure out the interval within a period for which the signal equals $$1$$.
3. Compute the average: $$\overline{x(t)}=\frac{1}{T}\int_0^Tx(t)dt$$

If you're convinced that you did everything right, don't worry about a given solution which is different.

• Doing this i find that the period is 4 , and the average is 1/3 is it right ? – Giovanni Cerciello Nov 3 '20 at 8:22
• @GiovanniCerciello: Sounds like a reasonable result. – Matt L. Nov 3 '20 at 12:16