Frequency Modulation - Why don't we modulate the carrier by multiplying message signal m(t) directly with f_car?

Question

I am master student of computer science and I have got a problem/question from the domain of communication engineering. So: I'm listening to a lecture that's about Layer 1 stuff, and recently we discussed how exactly FM radio works.

At one point we discussed how a frequency modulated signal s(t) can be obtain a from a message signal m(t). Therefore our lecturer derived the following formula using the instantaneous frequency:

The derivation was done similar to the derivation presented in: FM modulation derivation source - SpringerLink

From my understanding, this formula modulates the frequency by introducing a "changing phase offset"., because our equation introduces another summand, which normally represents a phase.

However, I thought I can point out the obvious and asked my lecture: "why don't we just modulate by directly multiplying our message stream onto the carrier frequency term?" (Which would look like this:)

And my lecturer couldn't really come up with an answer for this, as it looks kind of intuitive to do it like this. So I plotted both approaches using matlab. The resulting plot shows that my intuitive approach doesn't work. However, I am struggling to come up with an explanation, why my approach didn't work. So I thought, maybe someone with a deeper understanding of the mathematics behind the frequency modulation can help me out on this one.

The corresponding m(t) signal was:

Best Regards

Answer 1

I don't know why you think your proposed method of frequency modulation is intuitive. In any case, consider what happens if the modulating signal is 0. In the correct expression for an FM signal, as given in your question, the result reduces to just the carrier as one would expect. However, with your proposed implementation, the result reduces to cos(0) which is a constant value of 1 since m(t) = 0. This is obviously not how FM signals are supposed to work so your expression is incorrect. Note that the correct expression for an FM signal is derived directly from the definition of FM: a signal with a fixed carrier frequency whose instantaneous frequency is shifted from the carrier in proportion to a modulating signal (m(t)). Intuition does not play a role in engineering.