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Suppose that $\omega_1,\omega_2,a,b$ are non-zero real numbers and consider the following sum of sinusoids
$$ x(t)=a\sin(\omega_1t)+b\sin(\omega_2t) $$
The fundamental periods of each of the summands are $T_1=\frac{2\pi}{\omega_1}$ and $T_2=\frac{2\pi}{\omega_2}$, respectively. It is well known that if $$ \frac{T_1}{T_2}=\frac{\omega_2}{\omega_1} $$ is rational, then $x(t)$ is periodic with period $T=qT_1$, where $\frac{p}{q}$ is the rational number $\frac{T_1}{T_2}$. I know that $T$ may not be the fundamental period.

My questions: If $x(t)$ is non-constant function, how we can determine the fundamental period of $x(t)$?

(2) If there a way to determine the fundamental period of $x(t)$, is this procedure can be generalized to a sum of sinusoids with more than two summands, say $$ x(t)=a_1\sin(\omega_1t)+\cdots+a_n\sin(\omega_nt) $$
My attemnt: Note that $x(t+T)=x(t)$ iff $$ a(\sin(\omega_1t+\omega_1T)-\sin(\omega_1t))= b(\sin(\omega_1t+\omega_2T)-\sin(\omega_2t)) $$ that is $$ a\sin(\frac{\omega_1T}{2})\cos(t\omega_1+\frac{\omega_1T}{2})=b\sin(\frac{\omega_2T}{2})\cos(t\omega_2+\frac{\omega_2T}{2}) $$ Can we deduce that $\sin(\frac{\omega_1T}{2})=0$ and $\sin(\frac{\omega_2T}{2})=0$ ?

boaz
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1 Answers1

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When you are looking at the combination of two sinusoids, $qT_1$ is really just an obfuscated expression for the least common multiple of $T_1$ and $T_2.$

For three or more sinusoids, the principle is the same: find the least common multiple of all the periods of the component sinusoids. You can do this only if the ratio between each pair of periods is rational. One irrational ratio give you a non-periodic function.

David K
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  • Thanks David, but why this is the fundamental period? Generally, if $x_1(t),x_2(t)$ are periodic with fundamental periods $T_1,T_2$, respectively, the $lcm(T_1,T_2)$ is not the fundamental period of $(x_1+x_2)(t)$. – boaz Oct 22 '22 at 20:40
  • This is not a proof of the fundamental period. But we can easily show that whatever the phases of the sinusoids are at $t=0,$ they don’t repeat together until $t=\operatorname{lcm}(T_1,\ldots,T_n).$ – David K Oct 22 '22 at 21:11