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I have an angle-modulated signal: $$ s(t)=10\left[\cos\left(10^8\pi · t + 5\sin\left(2\pi\cdot10^3·t\right)\right)\right] $$

How would I go about finding the maximum phase and the maximum frequency deviation?

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  • $\begingroup$ well, you look at the phase of the signal and find the maximum deviation of that? $\endgroup$ Commented Feb 8, 2021 at 16:53
  • $\begingroup$ @MarcusMüller Ok, I guess I should clarify as I'm having trouble finding the phase, as this is different from the general formula of a sin or cos function, how would I go about finding the phase? $\endgroup$
    – B. T.
    Commented Feb 8, 2021 at 18:05
  • $\begingroup$ huh? this doesn't seem any different? Please edit your question to explain explicitly what you consider the "general formula", and how this formula doesn't fit that. $\endgroup$ Commented Feb 8, 2021 at 18:28

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I think this is treating the $10^8\pi\cdot t$ as the carrier frequency and the $5 sin(2\pi\cdot 10^3 t)$ as the modulation. So the maximum phase difference of this signal to the carrier without modulation would be $\pm5$ radians.

To compute the change in frequency you need to find the compute $\frac{d}{dt}$ of the argument of the cos (divided by $2\pi$) and subtract the carrier frequency.

$$\frac{d}{dt} 10^8\pi\cdot t + 5 sin(2\pi\cdot 10^3 t) = 10^8\pi + 2\cdot 5\cdot 10^3\pi cos(2\pi\cdot 10^3 t) $$

Divide by $2\pi$ to get Hz instead of radians.

$$Frequency = 5\cdot 10^7 + 5\cdot 10^3 cos(2\pi \cdot 10^3t)$$

Subtract off the carrier of 50MHz and you have a maximum frequency excursion of $\pm 5$ k Hz

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