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I am reading about FM.Suppose the message signal $m(t) = \frac{1}{2}cos(t)$ and the frequency of the carrier signal is twice the frequency of the message signal and its amplitude is 1.The resulting modulated signal will be in this form $s(t) = cos((2+\frac{1}{2}cost)t)$.I tried plotting the modulated signal using the Desmos calculator and I get this:

The red line is the message signal $m(t)$ and the blue line is the modulated signal $s(t)$ however I cannot intuitively understand how the data of the message signal is encoded into the modulated signal.Any help?

OverLordGoldDragon
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Volpina
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2 Answers2

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The intuition is that the instantaneous frequency of the modulated signal IS the message. Instantaneous frequency is the time derivative of phase, so the correct expression for the modulated signal will be as follows:

$$s(t) = A\cos\bigg(2\pi f_ct + \beta\int m(t)dt\bigg)$$

Where $\beta$ is a proportionality constant that adjusts the frequency deviation away from the carrier.

The instantaneous phase is given as

$$\phi(t)= 2\pi f_ct + \beta\int m(t)dt$$

If we consider the variation of the phase relative to the carrier (which is the modulation), this is simply the term:

$$\phi_m(t) = \beta\int m(t)dt$$

Thus instantaneous frequency as the time derivative of phase is directly proportional to the message $m(t)$:

$$ \frac{d\phi_m(t)}{dt} = \beta m(t)$$

Here's a corrected version demonstrating frequency modulation where we see that the frequency of the modulated signal is higher when the message amplitude is at its maximum and lower when the message amplitude is at its minimum, and more generally the instantaneous frequency is proportional to the message amplitude, as expected:

FM

Dan Boschen
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  • Clear composition and order of presentation. The last paragraph appears to confuse "carrier" with "modulated signal" though. – OverLordGoldDragon May 20 '23 at 13:26
  • @OverLordGoldDragon Yes, good point, thanks. With FM I had envisioned the carrier itself as changing its frequency but you made me think - that doesn't universally hold for all modulations rather than more appropriately treating the "carrier" as a fixed reference, which I think is your point. – Dan Boschen May 20 '23 at 15:10
  • "Carrier" earlier in the answer refers to the constant $f_c$, and $m(t)$ "relative to the carrier", which is also how OP uses it. Though I'm also unaware of other FM schemes for purposes of message transmission. – OverLordGoldDragon May 20 '23 at 15:29
  • Yes what I "envisioned" wasn't necessarily accurate. I agree, carrier should simply be the constant $f_c$. I updated my answer thanks to your feedback. – Dan Boschen May 20 '23 at 18:58
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A pure sinusoid has the form $a\sin(\omega t+\phi)$ where $a$ is the amplitude and $\phi$ is the phase. Let us denote the modulated signal as $m(t)$, obtained by letting one of the parameters vary.

In amplitude modulation, you can retrieve the amplitude by the ratio

$$\frac{m(t)}{\cos(\omega t+\phi)}=\frac{a(t)\,\cos(\omega t+\phi)}{\cos(\omega t+\phi)}=a(t)$$ (provided the divisions by zero are properly handled).

In frequency modulation, you can retrieve the frequency by

$$\frac{\arccos(m(t))-\phi}t=\frac{\arccos(\cos(\omega(t)\,t+\phi))-\phi}t=\omega(t)$$ (provided the phase wrappings are properly handled).

In phase modulation, you can retrieve the frequency by

$$\arccos(m(t))-\omega t=\arccos(\cos(\omega t+\phi(t)))-\omega t=\phi(t).$$

In all cases, as you know the exact shape of a sinusoid, you can retrieve the perturbation, which encodes the signal.