PHASE MODULATION (PM)
Table of Contents
Intro
Phase Modulation can be created by modifying the phase angle of a radiofrequent carrier 
according to an information signal 
of a lower frequency. The peak deviation 
of a PM signal is responsible for the depth of the modulation and depends on the phase deviation constant and the maximum of the information signal.
according to an information signal of a lower frequency. The peak deviation of a PM signal is responsible for the depth of the modulation and depends on the phase deviation constant and the maximum of the information signal. If the information signal is sinusoidal then is also denoted by the modulation index 
which is defined different from the modulation index 
of the FM. A comparison of FM and PM shows that they are related since both are angle modulation methods:
is also denoted by the modulation index which is defined different from the modulation index of the FM. A comparison of FM and PM shows that they are related since both are angle modulation methods: 
The phase can be obtained by integration of the angular frequency 
If an information signal is integrated before it is being fed into a phase modulator then the output is an FM signal, otherwise if the derivative of an information signal is the input of a frequency modulator then a PM signal is resulted.
Time Domain
On the basis of two sinusoidal signals (representing the information and the carrier) the time dependent function can be calculated:
This is the same formula as modulating to sinusoidal signals by FM.
Frequency Domain
Since FM and PM of sinusoidal signals yield the same results, the spectrum of the PM signal is identical to the FM´s spectrum. But this is just valid for this simple example and can not be generalized. If an information signal consists of more frequency components, the calculation of the spectrum is difficult, because PM as well as FM are nonlinear functions. Therefore you can not combine partial results of individual frequency components applied to the modulator to obtain an overall result.
Comparison (FM - PM)
The different behaviour of FM and PM can be seen by using a linear increasing function as an information signal. The frequency of the FM signal is directly proportional to the information signal, while the frequency of the PM signal is directly proportional to the derivative of the information signal.
%% comparison of PM and FM (time domain)
% variables
init;
syms t
f_c=5e3;u_c=2; % carrier
f_i=1e3; u_s=2; % information
k_PM=1;k_FM=k_PM*2*pi*f_i; % modulator constants
% time spans of functions
t_f1=1e-3; % function 1
t_f2=1e-3; % function 2
t_f3=1e-3; % function 3
t_f4=1e-3; % function 4
t_end=t_f1+t_f2+t_f3+t_f4;
t1=0:1/fs:t_f1;
t2=1/fs:1/fs:t_f2;
t3=1/fs:1/fs:t_f3;
t4=1/fs:1/fs:t_f4;
tt=0:1/fs:t_end;
% partial functions (symbolic and numeric)
f1=@(t) zeros(size(t));s1=f1(t1);
f2=@(t) u_s*(t)/t_f2; s2=f2(t2);
f3=@(t) u_s*ones(size(t));s3=f3(t3);
f4=@(t) u_s*cos(2*pi*f_i*t); s4=f4(t4);
% information signal (numeric)
s=[s1 s2 s3 s4];
% integrated functions (symbolic and numeric)
g1=int(f1,t);g1=double(subs(g1,t,t1));
g2=int(f2,t);g2=double(subs(g2,t,t2))+g1(end);
g3=int(f3,t);g3=double(subs(g3,t,t3))+g2(end);
g4=int(f4,t);g4=double(subs(g4,t,t4))+g3(end);
% integrated signal (numeric)
g=[g1 g2 g3 g4];
y_T=u_c*cos(2*pi*f_c*tt); % carrier
y_PM=u_c*cos(2*pi*f_c*tt+k_PM*s); % PM-signal
y_FM=u_c*cos(2*pi*f_c*tt+k_FM*g); % FM-signal
disp('comparison of FM and PM');
subplot(2,1,1)
hold off
plot(tt,y_T,'LineWidth',1,'LineStyle',':','Color',[0.5,0.5,0.5])
hold on
plot(tt,s,'LineWidth',1,'Color',[0.9,0.3,0])
plot(tt,y_PM,'LineWidth',1,'Color',[0,0.3,0.6])
grid on
legend('carrier','information','PM-signal','location','southeast')
title('PM-signal')
subplot(2,1,2)
hold off
plot(tt,y_T,'LineWidth',1,'LineStyle',':','Color',[0.5,0.5,0.5])
hold on
plot(tt,s,'LineWidth',1,'Color',[0.9,0.3,0])
plot(tt,g,'LineWidth',1,'Color',[0.9,0.3,1])
plot(tt,y_FM,'LineWidth',1,'Color',[0,0.3,0.6])
grid on
legend('carrier','information','information (integrated)','FM-signal','location','southeast')
title('FM-signal')
