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What i assume here is that LCM of time periods of the two taken signals exist that is signals periods are not like pi/2 and 1 but are rather like 1 and 2 (just an example)

I am given fourier series coefficients of the two signals which have been multiplied.

My approach started like enter image description here

But just as I obtained the first value its frequency was w1+w2 I got confused as as far as I predict (I am not sure about this) the fundamental frequency of multiplication 1/T where T is lcm of timeperiod of first and time period of second.

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The product $x(t)y(t)$ of two periodic signals with fundamental periods $T_x$ and $T_y$ is not a periodic signal unless $T_x$ and $T_y$ are rational multiples of one another; that is, $T_x = aT_y$ where $a$ is a rational number. Thus, except when such a relationship holds, $x(t)y(t)$ does not have a Fourier series.

When $T_x$ is a rational multiple of $T_y$, the corresponding fundamental frequencies $\omega_x$ and $\omega_y$ are also rational multiples of each other and can be expressed as $$\omega_x = m\omega_0, ~~ \omega_y = n\omega_0$$ where $m, n$ are positive integers. The product \begin{align} x(t)y(t) &= \sum_{i=-\infty}^\infty a_i \exp(ji\omega_x t)\sum_{k=-\infty}^\infty b_i \exp(jk\omega_y t)\\ &= \sum_{i=-\infty}^\infty a_i \exp(jim\omega_0 t)\sum_{k=-\infty}^\infty b_i \exp(jkn\omega_0 t)\\ &= \sum_{i=-\infty}^\infty\sum_{k=-\infty}^\infty a_ib_k \exp(j(im+kn)\omega_0 t) \end{align} can be re-arranged into a single sum $\sum_{\ell=-\infty}^\infty c_\ell \exp(j\ell \omega_0 t)$that is recognizable as a Fourier series with fundamental frequency $\omega_0$. Many of the Fourier coefficients $c_\ell$ in this series have value $0$; in particular, $c_1 = 0$.

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