What is the Frequency Spectrum of a Modulated Carrier?

Can you please explain to me why the frequency spectrum of a carrier wave modulated in amplitude by a signal of one frequency has two additional frequencies ?Thank you so much.
- Suresh (age 63)
Kanpur, India

In amplitude modulation a carrier wave is multiplied by a signal wave.   The simplest example is a sine or cosine carrier multiplied by another sine or cosine.  Let  A = cos( ω₁t)  and  B = cos( ω₂t) where ω₁ is the carrier frequency and ω₂ is the signal frequency.   Then the modulated signal is S = A*B*cos(ω₁) *cos(ω₂)  = A*B*[cos(ω₁+ω₂)  + cos(ω₁-ω₂) ]/2.  So there are two frequencies in the result:   Ω₁ =  ω₁+ω₂   and  Ω₂ = ω₁-ω₂ .

The spectrum of a frequency modulated signal is much more complicated with many more frequencies, all multiples of the modulation frequency. 

For more information I recommend that you take a look at the Wikipedia article on modulation:

https://en.wikipedia.org/wiki/Modulation

LeeH

(published on 11/30/2016)