# How do I obtain the fourier series coefficients for a signal obtained by multiplication of two signals of different frequency?

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

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.

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, $$c_1 = 0$$.