# How to theoretically derive the sideband frequency values?

• bksree
In summary, if you are trying to obtain sideband frequencies from adding two sinusoids, you will need a nonlinear mixer.
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

bksree said:
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

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.

berkeman
Hi

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!

TIA

bksree said:
Hi

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!
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.

## 1. What is the purpose of theoretically deriving the sideband frequency values?

Theoretical derivation of sideband frequency values is important in understanding the behavior of signals in a communication system. It allows scientists to predict the frequencies at which sidebands will occur, which is crucial in designing and optimizing communication systems.

## 2. How do you theoretically derive the sideband frequency values?

Theoretical derivation of sideband frequency values involves using mathematical equations and principles to calculate the frequencies at which sidebands will occur. This can be done through a process called Fourier analysis, which decomposes a complex signal into its individual frequency components.

## 3. What factors affect the sideband frequency values?

The sideband frequency values are affected by several factors, including the modulation type, carrier frequency, and modulation index. These factors determine the amount of energy that is shifted from the carrier frequency to the sidebands.

## 4. Can the sideband frequency values be calculated for any type of modulation?

Yes, the sideband frequency values can be theoretically derived for any type of modulation, including amplitude modulation, frequency modulation, and phase modulation. However, the equations used to calculate the values may differ depending on the type of modulation.

## 5. How can the derived sideband frequency values be used in practical applications?

The derived sideband frequency values can be used to design and optimize communication systems, such as radio or television broadcasting. They can also be used to analyze and troubleshoot signal interference and distortion in communication systems.

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