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For the sum to be periodic, $f_1$ and $f_2$ must be commensurable; that is, there most be a number $f_0$ contained in each an integral number of times. Thus, if $f_0$ is the largest such number, $f_1$=$n_1 f_0$ and $f_2=n_2 f_0$ ($f_0$ is the fundamental frequency and $n_1$ and $n_2$ are integers) Which of the signals are periodic:

a) $x_1(t)=2 \cos(2t) + 3\sin(5\pi t)$

b) $x_2(t)=2 \cos(4\pi t) + 5\cos(6\pi t) + 6\sin(22\pi t)$

Find the period of those that are periodic.

How can I find $f_0$? I have an idea but I don't know if it's correct for:

a) We know that $\cos(2t)$ is $\cos wt$, where $w=2\pi f$ so $f_1$ would be $f_1=2/(2\pi)=0.318$ for $3\sin(5\pi t)$, $f_2=2.5$. But how do I find $f_0$ if I don't have $n_1$ or $n_2$??? After that I'm block... what am I missing?

Thanks for your help

wj32
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user43680
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1 Answers1

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DON'T turn the $f_i$ into decimals. You have $$f_1={2\over2\pi},\qquad f_2={5\over2}$$ You want to know whether there are integers $n_1$ and $n_2$ such that $f_1/n_1=f_2/n_2=f_0$. That would make $f_2/f_1=n_2/n_1$, a rational number. So: is $f_2/f_1$ a rational number?

Gerry Myerson
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  • if f2/f1 is a rational number, the sum of the sinusoidals are periodic. Accordingly, with your answer we will obtained the period like this, f2/f1=f0=10pi/4 and the period is To=1/fo=4/10pi rigth??? – user43680 Oct 31 '12 at 20:33
  • Also for the second signal b). How I can employ the same analysis of f1/n1=f2/n2=fo. Thanks for your help! – user43680 Oct 31 '12 at 21:34
  • Is $10\pi/4$ a rational number? For the second signal, can you calculate $x_2(t+1)$, and compare it to $x_2(t)$? – Gerry Myerson Oct 31 '12 at 21:50