# How to express FM and PM the most simple way?

I'm trying to understand how to tell phase modulation from frequency modulation by looking at their respective expressions, simplified, particularly frequency modulation. Also, it would be nice for someone to illustrate mathematically the difference between vibrato and FM synthesis.

• Can you please clarify your question as, at the moment, it is a bit too broad. In the meantime, please note that In principle, there is no difference between vibrato and frequency modulation. Particular practical details might differ of course, such as "modulation index". As far as modulation is concerned, this is an excellent starting point. AM happens "outside" of the carrier's argument, FM/PM happens "inside" the carrier's argument. – A_A Aug 2 '16 at 7:46
• @A_A What is the simplified expression for modulating the frequency of a sine wave with another sine wave, so that when the modulators amplitude =1, the carrier frequency doubles up and down. I think that's what I really wanna know. – user23148 Aug 3 '16 at 1:32

An angle modulated signal is defined as: $$x(t) = A_c\cos(\omega_c t + \phi(t))$$ Where $\theta(t) = \omega_c t + \phi(t)$ is the instantaneous angle of the modulated cosine, $\phi(t)$ is the instantaneous phase and $\omega_i = \frac {d\theta(t)}{dt}$ is the instantaneous frequency
Now consider the phase being given by $\phi(t) = K_p m(t)$ , where $m(t)$ is the message signal and $K_p$ is a simple constant for phase modulation. As clearly evident, here the Instantaneous Phase, $\phi(t)$, is directly proportional to the amplitude of the message signal $m(t)$ and hence this type of angle modulation is called as Phase Modulation (PM)
On the other hand, consider a message signal $m(t)$ such that $\phi(t) = K_f \int {m(\tau)d\tau}$, where $K_f$ is the constant for frequency modulation, then the instantaneous frequency is: $$\omega_i = \frac {d\theta(t)}{dt} = \omega_c + K_f m(t)$$ As can bee seen, now instead, the instantaneous frequency in radians is directly proportional to the amplitude of the message signal $m(t)$ and this type of modulation is, therefore, called as Frequency Modulation (FM)
• How would you express "frequency deviation" in terms of the amplitude of the modulator and the carrier (baseline) frequency? Is it $frequency deviation=\omega_\Delta=|(A_m+1)*\omega_c-\omega_c|$ where $A_m$ is the modulator amplitude and $\omega_c$ the carrier frequency? – user23148 Aug 2 '16 at 22:16
• Instanteous frequency deviation is $$\Delta {\omega_i} = \omega_i - \frac{d\theta(t)} {dt} = \frac{d\phi(t)} {dt}$$ you would plug $\phi(t)=K_p m(t)$ for PM and $\phi(t)=\int K_f m(\tau)d\tau$ for FM. – Fat32 Aug 2 '16 at 22:30
• Does $A_m=1$ represent a doubling frequency in the expression $sin(\omega_c t+\frac{\omega_c}{\omega_m} A_m cos(\omega_m t)-\phi)$? – user23148 Aug 3 '16 at 2:03