What is the Frequency Modulation. ?

Frequency modulation of a carrier signal (with frequency fC) involves changing the frequency of this carrier based on some input signal s(t). The modulated carrier then contains the information in s(t). This process is useful in radio communications because it is not practical to directly transmit audio frequencies through the air.
 

The modulated carrier is of the form Acos [(t)] , where (t) is the varying phase angle of the signal.
We define the instantaneous frequency of the signal to be

Note that the power of the FM signal is a constant (with value A²).

Figures 1 & 2 show some examples of frequency modulated signals and the corresponding inputs.

To examine the FM signal's spectrum, we will look at the various harmonics comprising the input signal.
If the input signal is s(t) = X cos 2fmt,then the phase angle is

and the FM signal is

 

We define

 

as the modulation index of the FM signal. This gives us a measure of how much the frequency of the FM signal changes, ie this is a measure of the amount of modulation.

Since the FM signal is periodic (for a sine wave input), we can perform a Fourier series expansion of the FM signal to find its spectrum. This analysis yields a Bessel function of the first kind. This tells us that the magnitude of the harmonics is eventually decreasing (ie the function does not monotonicly decrease, but it eventually gets smaller). In fact, the magnitude of the (+1)^th harmonic is negligible, so we assume that there are in fact only b harmonics.

Having done this analysis (skipping the messy mathematical details), we can say that the bandwidth of the FM signal is approximately 2( fm + fm). This equation is known as Carson's Rule and can be written as BW 2(kX + fm). We can thus see that the bandwidth of the FM signal depends on the size of the frequency deviations from the carrier frequency. Also, increasing the frequency of the input signal requires more bandwidth for transmission, which is also intuitive.

For general signals with maximum frequency deviation f (ie f = max{kX}), we can say that BW 2(f + fm).
Note that fm is the maximum frequency of the input signal (ie the input bandwidth is 2fm), but this is not necessarily the maximum frequency deviation, since this depends of the magnitude of the Fourier transform.

Phase Modulation involves changing the phase of the carrier signal based on the input g(t).
In this case, we use (t) = 2 fCt+ kg(t). This is the same as frequency modulation, but with

Thus, we can see that phase modulation and frequency modulation are very similar, so the analysis presented applies to both. In fact, collectively the are described as Angle Modulation . The concept of phase modulation will be useful in later sections.