# Amplitude Modulation of a wave

## Homework Statement

I am utterly confused. When I was reading my textbook , I found something unacceptable.

While deriving an expression for a modulated wave,
It's been given that
"A sinusoidal carrier wave can be represented as c (t) = Ac sin
(ωt + Φ)
where c (t) is the signal strength of the carrier wave.
Let m (t) = Am sinωmt represent the message or the modulating signal.
The modulated signal cm (t) can be written as
cm= (Ac+Amsin ωmt) sin ωct

I wonder how's it possible! Shouldn't it be cm (t) = Acsin ωct + Am sinωmt ?

But then I made an adhoc assumption - which was not satisfactory - but I thought it could be justified from a more rigorous application of mathematics. So, I continued reading

On the next page, I found something in contrast to my "assumption".
In the topic "Production of amplitude modulated wave" -
According to my textbook "Here the modulating signal Am
sinωmt is added to the carrier signal Acsinωt to produce the signal x (t). This signal x (t) = Am sinωmt + Ac sin ωct is passed through a square law device."

Now this equation for x (t) is different from the one which was used (in the textbook) earlier.
What even is happening?

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ehild
Homework Helper
If the sum of two signals of different frequencies ω1 and ω2 is the input of a linear network, the output would be again the sum of two signals with the same frequencies.
Using a nonlinear device, one, for example, that produces the square of the input, it will "mix" the frequencies. Expand (Asin(ω1t) + Bsin(ω2t ))2, what do you get? ( there will be components with frequencies of 2ω1, 2ω2, ω12, ω12).
Semiconductor diodes, transistors have nonlinear characteristics, and can be used for "mixing frequencies"

• Prashasti
If the sum of two signals of different frequencies ω1 and ω2 is the input of a linear network, the output would be again the sum of two signals with the same frequencies.
Using a nonlinear device, one, for example, that produces the square of the input, it will "mix" the frequencies. Expand (Asin(ω1t) + Bsin(ω2t ))2, what do you get? ( there will be components with frequencies of 2ω1, 2ω2, ω12, ω12).
Semiconductor diodes, transistors have nonlinear characteristics, and can be used for "mixing frequencies"
Ok.
So, the problem lies in the arrangement of all that in my textbook..

They mentioned the result before deducing the expression for the same.
I got it!
Thanks!!