How to theoretically derive the sideband frequency values?

[bksree]

Hi
If 2 signals of frequencies y1 = A cos $\omega_1$ t and y2 = A cos $\omega_2$ t are combined the resultant signal is given by y = y1 + y2
y = 2A cos (w1-w2)t/2 cos (w1+w2)t/2
If w1~w2 then one can plot the beat phenomenon from this equation.

But if w1 >> w2 as in the case of a carrier wave and a message wave. Then how can one get the sideband frequencies (w1-w2) and (w1+w2) from this ?

TIA

 

[berkeman]

Hi
If 2 signals of frequencies y1 = A cos $\omega_1$ t and y2 = A cos $\omega_2$ t are combined the resultant signal is given by y = y1 + y2
y = 2A cos (w1-w2)t/2 cos (w1+w2)t/2
If w1~w2 then one can plot the beat phenomenon from this equation.

But if w1 >> w2 as in the case of a carrier wave and a message wave. Then how can one get the sideband frequencies (w1-w2) and (w1+w2) from this ?

TIA

There is a difference between adding and multiplying two sinusoids. Which one are you asking about?

http://hyperphysics.phy-astr.gsu.edu/hbase/trid.html

 

[DaveE]

Sidebands are created with modulators and mixers which must be non-linear processes. The dominant operation is multiplication not addition. This is typically the second order term in a Taylor's expansion of the non-linear function, since higher order terms are smaller (i.e. less efficient). Addition does not create sum and difference frequencies.

Edit: Modulators are too broad of a category for this effect . Some modulators won't create sidebands, some will. Really it's mixers that do this.

 

[bksree]

Hi
Thanks for your response.

I want to derive the relation to show the side band frequencies just like that obtained for the beat frequencies. And I want to do that using the simple trig relations you've shown.

I went through some oder posts and saw this (https://www.physicsforums.com/threads/sidebands-in-am-transmission.668449/) by 'sophiecentaur' where it is mentioned

But the easiest way to show how AM produces sidebands is to start with a formula which describes Amplitude Modulating a carrier wave with angular frequency ωc with a cosine modulating signal of frequency ωm does:
A =A0Cos(ωct)(1+Bcos(ωmt))

My waves are y1 = A1 cos $\left( \omega_1 t \right)$ and y2 = A2cos $\left( \omega_2 t \right)$
Now, y1 + y2 = A1 cos $\left( \omega_1 t \right)$ + A2cos $\left( \omega_2 t \right)$
= A1 cos(w1t) ( 1 + A2/A1 * cos(w2t)/cos(w1t) }
Now it looks like I am going off track!

Please help out

TIA

 

[rude man]

Hi
Thanks for your response.

I want to derive the relation to show the side band frequencies just like that obtained for the beat frequencies. And I want to do that using the simple trig relations you've shown.

I went through some oder posts and saw this (https://www.physicsforums.com/threads/sidebands-in-am-transmission.668449/) by 'sophiecentaur' where it is mentionedMy waves are y1 = A1 cos $\left( \omega_1 t \right)$ and y2 = A2cos $\left( \omega_2 t \right)$
Now, y1 + y2 = A1 cos $\left( \omega_1 t \right)$ + A2cos $\left( \omega_2 t \right)$
= A1 cos(w1t) ( 1 + A2/A1 * cos(w2t)/cos(w1t) }
Now it looks like I am going off track!
Please help out
TIA

Your equations are correct, but what are you trying to do?
Summing your two expressions y1 + y2 does NOT produce any sidebands, as you've been advised in previous posts.