# Frequency Modulation, signal, spectrum and phasor.

Cuthbert Nyack
In frequency modulation the amplitude is kept constant and the frequency is modulated by the amplitude of the modulating signal. The modulation index for fm is
d = maximum frequency deviation/modulating frequency.
FM signal can be represented as:-
v = 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:-
v = ac{Jo (d) sin(wct)
+ J1 (d)[sin(wc + wm)t - sin( wc - wm)t]
+ J2 (d)[sin(wc + 2wm)t + sin( wc - 2wm)t]
+ J3 (d)[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.

The Applet below shows several features of FM signals which can be seen by changing the parameter Fn which is set by Scrollbar 0.

Fn = 0 Shows the FM spectrum and signal for varying carrier frequency, modulating frequency and modulating index.
Fn = 1 Shows the FM spectrum and signal with the sum of the first and second pair of sidebands.
Fn = 2 Shows the Carrier plus the first pair of sidebands. The result is a signal with varying phase and amplitude.
Fn = 3 Shows the Carrier plus the second pair of sidebands. The result is a signal with varying amplitude but no phase variation. This partly compensates for the amplitude variation introduced by the first pair of sidebands.
Fn = 4 Shows how the sidebands add up to produce a constant amplitude signal with varying phase and frequency.
Fn = 5 Shows a simplified phasor representation of an FM signal. The diagram gets confusing if all individual vectors are shown. Instead the resultant of pairs of sidebands are shown. (For reference the vectors for even n behave similarly to the phasors in an AM signal and for odd n they behave similar to a DSB signal with the carrier shifted by 90°.)
The resultant for the first/second/third/fourth/fifth pair of sidebands are shown in pink/yellow/green/cyan/light magenta.
The overall resultant is shown by the magenta line and as time is changed by Pt(15), the overall resultant moves along the circumference of the Dark Orange circle. This corresponds to a signal with constant amplitude and varying phase and frequency. The amplitude of the phase variation is changed by d.

Fn = 6 shows the signal produced when a carrier is modulated by 2 frequencies wm and wm2 with modulating indices d and d2 respectively.
Fn = 7 Shows the frequency spectrum of an FM signal with 2 modulating frequencies. Unlike AM, FM is a 'nonlinear' form of modulation. The spectrum with 2 modulating frequencies contains components produced by each modulating frequency (wc ± n wm and wc ± m wm2) PLUS mixing or beating components at (wc ± n wm ± m wm2).

When activated, an image of the applet is shown below. Here the applet is used to show the spectrum of an FM signal with 2 modulating frequencies. 