# Why would amplitude modulation result in change of frequency?

Question: (single choice correct; source: JEE Main 2015)

A signal of $5 kHz$ frequency is amplitude modulated on a carrier wave of frequency $2 MHz$. The frequencies of the resultant signal is/are:

1. 2 MHz only
2. 2005 kHz, and 1995 kHz
3. 2005 kHz, 2000 kHz and 1995 kHz
4. 2000 kHz and 1995 kHz

My attempt:

I understand that amplitude modulation is the superimposition of an audio-frequency on a radio frequency carrier wave, in such a manner such that the frequency of the modulated wave is the same as that of the carrier wave, but its amplitude varies in accordance with the instantaneous amplitude of the modulating wave (message signal).

Note the part I highlighted, because, based on that part, I believe that the answer to the quoted question should be (1). However, the official answer was given as (3), and I have no clue why. It might have been correct in frequency modulation, but I am not sure why it's true in amplitude modulation as well.

Note that this is high-school level question, please answer accordingly. Many thanks!

• Have you read this explanation? There you can see that there will be 3 frequency components. The amplitude modulated carrier wave is equivalent to the sum of 3 sinusoids with constant amplitudes. – Matt L. Feb 18 '18 at 11:02
• @MattL. Thanks! That cleared my doubt. It is surprising though, because on that same wikipedia page, the image which is given, shows a constant frequency for the modulated signal in the last row. Either that, or probably the difference between 2.005KHz, 2KHz, and 1.995KHz is too small to be observed on such an approximate graph. – Gaurang Tandon Feb 18 '18 at 11:19
• You don't understand yet. The image is correct and the frequency of the carrier wave is indeed constant. What you need to understand is that a carrier wave with a constant frequency and an amplitude that changes according to a sinusoidal message signal is mathematically equivalent to 3 sinusoids with constant amplitudes. – Matt L. Feb 18 '18 at 11:40
• @MattL. Did you mean the "...frequency of the modulated output wave is indeed constant..." and "that a modulated output wave with a constant frequency and an amplitude..."? Because in my previous comment I did not doubt the carrier waves (2nd row), I had actually meant the "AM signal" (3rd row) – Gaurang Tandon Feb 18 '18 at 11:46
• The frequency of the signal in the 3rd image is constant inside the envelope. However, since its envelope is not constant it cannot consist of a single frequency component (because then its amplitude would need to be constant). – Matt L. Feb 18 '18 at 11:53

$$A[1+\mu\cos(2\pi 5000t)]\sin(2\pi 2000000t)= A\sin(2\pi 2000000t) + \dfrac{A\mu}{2}(\sin(2\pi[2000000-5000]t)+\sin(2\pi[2000000+5000]t))$$