How to find fundamental frequency of two signals?

Question

I am facing difficulty with finding fundamental frequency of signals

I mean by fundamental frequency $=\frac{1}{\text{Time period}}$

Correct me if I am wrong

consider two continuous time signals with

Time period of signal $~a=T_1~$, frequency $=~f_1 (<f_2) =mfo$

Time period of signal $~b =T_2~$, frequency $=~f_2=nfo~~ : m~$ and $~n~$ are cofactors.

Then, fundamental frequency of

$$a+b \qquad \qquad \qquad ~~~~~~~~~~~~~\text{HCF of two frequencies $f_1$ and $f_2$}$$ $$ \text{multiplication of $~a~$ and $~b~$} \qquad \qquad \qquad \text{Don’t know how to proceed}$$ $$\text{convolution of $~a~$ and $~b~$} \qquad \qquad \text{LCM of two frequencies $~f_1~$ and $~f_2~$}$$

$fo$ is $lcm$ of two frequencies.

Plz also elaborate procedure if any for such equations

Answer 1

Let $y_1 = a_1\sin(b_1x-c_1)+d_1,y_2 = a_2\sin(b_2x-c_2)+d_2$.

Then, you can use trig identities when analyzing the product, for example, $$y_1y_2 = a_1a_2\sin(b_1x-c_1)\sin(b_2x-c_2)+a_1d_2\sin(b_1x-c_1)+a_2d_1\sin(b_2x-c_2)+d_1d_2$$ $$=\frac{a_1a_2}{2}\left(\cos((b_1-b_2)x+(c_2-c_1))-\cos((b_1+b_2)x-(c_1+c_2))\right)+a_1d_2\sin(b_1x-c_1)$$ $$+a_2d_1\sin(b_2x-c_2)+d_1d_2.$$

Now, you have reduced the product of the signals to a sum of signals.