# Question about FM and PM and Angle Modulation

I am having a bit of trouble understanding how FM works mathematically. I fully understand the conceptual basics regarding how information is transfered by modulating the frequency, but what I don't get (from the equation below), is how this expression inside the cosine term actually changes the frequency. When I think of trying to change the frequency, it makes more sense to me to 2π(fc)(t) a function of Xm [ex. 2π(fc)(1 + Xm)(t) which comes out to be 2π(fc)(t) + 2π(Xm)(t)]. Why do they have the integral in there? And why is the amplitude of the carrier (Ac) specified, when it shouldn't make a difference to the output signal?

Also, would you be able to clarify the difference between FM and PM and angle modulation (a combination of both), and how/when each should be used. Thanks for the help

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The instantaneous frequency of a signal $A\cos(\phi(t))$ where $\phi(t)$ is an arbitrary function of time is defined as the derivative of $\phi(t)$ if you want to measure frequency in radians per second and as $\frac{1}{2\pi}$ times the derivaive_ of $\phi(t)$ if you want to measure frequency in Hertz. Of course, in the common case of a fixed frequency this corresponds to the familiar $\phi(t) = \omega_c t+\phi_0 = 2\pi f_c t + \phi_0$.

The standard definition of a frequency-modulated signal is one in which the deviation of the instantaneous frequency (at time $t_0$, say), from the carrier frequency is proportional to the value $x_m(t_0)$ of the modulating signal $x(t)$ at time $t_0$. The constant of proportionality is denoted by $f_{\Delta}$ in your notation: a $1$ volt signal creates a deviation of $f_{\Delta}$ Hz. Thus, if $A\cos(\phi(t))$ is the FM signal, then we have that $$\left.\frac{\mathrm d}{\mathrm dt}\phi(t)\right|_{t=t_0} = 2\pi f_c + 2\pi f_{\Delta} x_m(t_0)$$ so that the deviation of the instantaneous frequency $f_c + f_{\Delta} x_m(t_0)$ from the carrier frequency $f_c$ is $f_{\Delta} x_m(t_0)$, just as we want it to be. It then follows from the fundamental theorem of calculus that $$\phi(t_0) = \int_{0}^{t_0}2\pi f_c + 2\pi f_{\Delta} x_m(t_0)\, \mathrm dt = 2\pi f_c t_0 + \int_{0}^{t_0} 2\pi f_{\Delta} x_m(t_0)\, \mathrm dt$$ or, with a slight change in notation, the FM signal can be expressed as $$A\cos\left(2\pi f_c t + \int_{0}^{t} 2\pi f_{\Delta} x_m(\tau)\, \mathrm d\tau\right)$$ the way you have it. Note that $A$ is the amplitude of the FM signal and is fixed; it is the frequency that is varying. Surely we need to distinguish between the FM signal when it is created using a voltage-controlled oscillator with an amplitude of $1$ volt and when it comes out of the power amplifier and goes to the antenna with a power of 10 kW?

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From my standpoint, angle modulation can be FM or PM.

A continuous, progressive, linear angle change to the carrier's phase equates to a fixed frequency displacement of the carrier.

A step angle change is a step phase change of the carrier. After the step change the carrier frequency remains as it was before the change.

As for the maths, I hope someone else can answer. When/how they should be used may be the hardest thing to answer.

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The integral is there because dt is there. It's another way of writing the familiar cos() expression without the integral: it means the same thing. The rewrite is done so as to lead to the analysis of the sidebands in terms of Bessel functions.

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