# Frequency and Phase Modulation

Cuthbert Nyack
A carrier can have its phase and frequency modulated. If the carrier signal is sin(w ct) and the phase modulating signal is cos(w mt), the FM and PM signals can be described by the following equations:-
For FM the frequency deviation is independent of frequency and the phase deviation decreases with increasing frequency. For PM the phase deviation is independent of frequency and the frequency deviation increases with increasing frequency.

For Fn = 1 the applet above shows frequency modulated(red) and phase modulated(cyan) signals.
Fn = 2 shows the FM and PM with the maximum frequency deviation of the FM = 10rad/s and the maximum phase deviation of the PM is 10rad.
Because the modulating frequency = 1, then in this case the phase deviation of the FM is also 10rad and the frequency deviation of the PM is 10rad/s. At any given modulating frequency, PM and FM signals can look alike, the difference being that the maximum frequency deviation of the FM occurs at the maximum of the modulating signal and the maximum frequency deviation for PM occurs at the maximum gradient of the modulating signl. Fn = 3 shows a case where the carrier is at 40rad/s and the frequency deviation is from 30 to 50rad/s. Here the Horiz scale is chosen so that a signal with 1 period across the screen has a frequency of 50rad/s. ie the max frequency which occurs at the max of the mod signal.
Fn = 4 shows that at the min of the mod signal the frequency of the FM is 30rad/s.

Fn = 5 and 6 show that the same occurs for the PM.

Fn = 7 shows the phase variation of the PM. The phase is equal to that of the carrier at the center of the screen. One can easily verify that between the left and the middle of the screen, the total phase of the PM is 10rad less than that of the carrier. Between the middle and right of the screen the phase of the PM is 10rad larger than that of the carrier. At the zero of the modulating signal, the phase of the carrier is delayed by its maximum amount and vice versa.

Fn = 8 shows that the same thing occurs for the FM.
Fn = 9 to 12 show what happens to the FM and PM when the modulating signal is changed. As the modulating signal is reduced, the phase deviation of the FM increases to keep the frequency deviation constant while for the PM the frequency deviation decreases to keep the phase deviation constant.

Fn = 13 shows the spectra of the FM and PM.
Fn = 14 to 19 show what happens to the spectra as the modulating frequency is reduced from 2.5rad/s to 0.25rad/s. When the frequency of the modulating signal is changed, the bandwidth of the FM signal remains constant as lower frequencies produce higher modulation indexes. With PM the bandwidth reduces with decreasing modulating frequency. A PM signal can only use the maximum allowed bandwidth at the highest modulating frequency. With FM the full bandwidth is used at all modulating frequencies and a more efficient transmission system results.
For binary transmission however, PSK is better than FSK.

As for AM FM and PM can be expressed as different combinations of sines and cosines. 4 possibilities are shown below.

(1) Vfm(t) = sin(w ct + Mf/wm sin (w mt));
Vpm(t) = sin(w ct + Mp cos (w mt));

(2) Vfm(t) = sin(w ct - Mf/wm cos (w mt));
Vpm(t) = sin(w ct + Mp sin (w mt));

(3) Vfm(t) = cos(w ct + Mf/wm sin (w mt));
Vpm(t) = cos(w ct + Mp cos (w mt));

(4) Vfm(t) = sin(w ct - Mf/wm cos (w mt));
Vpm(t) = sin(w ct + Mp sin (w mt));

Fn = 20 to 27 shows these 4 possibilities. The middle plot shows the phase of the carrier and modulating signal. Besides phase changes, the FM and PM look the same in all cases. For these plots (Blue Magenta) shows the frequency of the FM and (Pink Magenta) shows the phase of the PM. The frequency deviation of the FM is shown by Mw and the phase deviation is shown by the blue text at the top.
The phase deviation of the PM is shown by Mf and the frequency deviation is shown by the blue text at the top.

Fn = 20 shows case 1, Fn = 21 shows a specific case of Fn = 20.
Fn = 22 shows case 2, Fn = 23 shows a specific case of Fn = 22.
Fn = 24 shows case 3, Fn = 25 shows a specific case of Fn = 24.
Fn = 26 shows case 4, Fn = 27 shows a specific case of Fn = 26.

With sinusoidal modulation FM and PM signals look very similar. However with nonsinusoidal modulation the difference between them can be much clearer.

Fn = 28 shows FM and PM with triangular wave modulation, Fn = 29 shows a specific case of Fn = 28.
Here the FM frequency changes linearly from high to low and back and the phase variation is quadratic.
The PM phase changes linearly and when differentiated it produces a constant frequency so the PM looks like a FSK signal.
Fn = 30 shows FM and PM with square wave modulation, Fn = 31 shows a specific case of Fn = 30.
Here the FM frequency changes between 2 values(FSK) and the phase variation is linear.
The PM phase changes between 2 values(PSK) and when differentiated it produces a frequency variation containing impulses at the phase changeover points.