Is $x(t)=\sin(5t/2)+\cos(2t/8)+\sin(3t/6)$ periodic or aperiodic? Find the fundamental period and frequency of the signal.

Question

Is $x(t)=\sin(5t/2)+\cos(2t/8)+\sin(3t/6)$ periodic or aperiodic?

$w_1=(5/2)=2.5 \rightarrow T_1 = 2\pi/w_1 = 2\pi/2.5 =2.513$

$w_2=(1/4)=0.25 \rightarrow T_2 = 2\pi/w_2 = 2\pi/0.25=25.13$

$w_3=(1/2)=0.5 \rightarrow T_3 = 2\pi/w_3 = 2\pi/0.5 =12.56$

I think it is aperodic due to the fact that the period for each function is not rational.

Answer 1

$\sin (x) = \sin (x + k2\pi)$ so $\sin(5t/2) = \sin(5(t + p)/2)= \sin(5t/2 + k2\pi)$ if $5p/2 = k2\pi$ i.e if $p = k*4/5*\pi$. $\sin(5t/2)$ has a period of $4/5*\pi$.

Similarly $\cos (2t/8)$ has a period of $8\pi$.

And $\sin(3t/6)$ has a period of $4\pi$.

So $f(t) = \sin(5t/2) + \cos(2t/8) + \sin(3t/6) = f(t + P) = \sin(5(t+P)/2) + \cos(2(t+P)/8) + \sin(3(t+P/6) $ if $5p/2 = n*4/5*\pi$ for some integer n, $2P/8 = 8*m*\pi$ and $3P/6 = 4*k*\pi$ form some integers m, k.

So $P = \operatorname{lcm}(4/5\pi, 8\pi, 4\pi) = 8 \pi$.

Answer 2

Those periodicities all divide $8\pi$ evenly so the period is at most that.