Frequency modulation uses the information signal, Vm(t) to vary the carrier frequency within some small range about its original value. Here are the three signals in mathematical form:
- Information: Vm(t)
- Carrier: Vc(t) = Vco sin ( 2 p fc t + f )
- FM: VFM (t) = Vco sin (2 p [fc + (Df/Vmo) Vm (t) ] t + f)
We have replaced the carrier frequency term, with a time-varying frequency. We have also introduced a new term: Df, the peak frequency deviation. In this form, you should be able to see that the carrier frequency term: fc + (Df/Vmo) Vm (t) now varies between the extremes of fc - Df and fc + Df. The interpretation of Df becomes clear: it is the farthest away from the original frequency that the FM signal can be. Sometimes it is referred to as the "swing" in the frequency.
We can also define a modulation index for FM, analogous to AM:
b = Df/fm , where fm is the maximum modulating frequency used.
The simplest interpretation of the modulation index, b, is as a measure of the peak frequency deviation, Df. In other words, b represents a way to express the peak deviation frequency as a multiple of the maximum modulating frequency, fm, i.e. Df = b fm.
Example: suppose in FM radio that the audio signal to be transmitted ranges from 20 to 15,000 Hz (it does). If the FM system used a maximum modulating index, b, of 5.0, then the frequency would "swing" by a maximum of 5 x 15 kHz = 75 kHz above and below the carrier frequency.
The modulation index for fm is
Df = maximum frequency deviation/modulating frequency.
If FM signal is represented as:-
VFM (t) = ac sin(wct + m sin wmt )
The frequency spectrum can be found by rewriting the above expression as a sum of components of constant frequency using the properties of the Bessel Functions. This gives:-
VFM (t) = ac{Jo (Df) sin(wct)
+ J1 (Df)[sin(wc + wm)t - sin( wc - wm)t]
+ J2 (Df)[sin(wc + 2wm)t + sin( wc - 2wm)t]
+ J3 (Df)[sin(wc + 3wm)t - sin( wc - 3wm)t]
+ ...
This expression implies that the FM spectrum consists of a component at wc and an infinite number of lines at wc ± nwm and that the amplitude of the components are given by the Bessel functions.
