# 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

## 3 Answers

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?

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.

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.