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

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

• Isn't that amplitude modulation? Commented Dec 2, 2021 at 3:01
• @DKNguyen no, the multiplication is inside the cos() instead of outside. Commented Dec 2, 2021 at 3:02
• @hobbs Oh, I see. Commented Dec 2, 2021 at 3:03
• (that was my first thought too) Commented Dec 2, 2021 at 3:03