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This is regarding question 5.1-2 (see picture attached) from Modern digital and analog communications, Lathi & Zhi Ding (2010).

Question

I do not understand what the behavior should be at the discontinuity (sudden jump from 1 to -1). I ran the following code on MATLAB (using simpler values):

a = 0.25;
t = -a:0.0000001:a; % time
wc = 2*pi*1e2; % carrier frequency
m = sawtooth(2*pi*5*t,0.9); % message signal
phi_am = cos(wc*t+(pi/2)*m); % phase modulated signal
plot(t,phi_am); ylim([-2 2]); hold on;
plot(t,m); hold off;

and it showed me a phase shift by $\pi$, and that is what the solution is. But I do not understand how.

Also, if some light on why must $k_p < \pi$ in part (b) then I'd appreciate it.

Abeer
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  • A phase shift by $\pi$ radians is a phase shift of 180 degrees, which is simply a sign change. – Dan Boschen Mar 04 '21 at 22:01
  • I know what it is, but I don't understand how it got there. I mean, in line 4 of the code I slowly changed the last argument of sawtooth() - which denotes how much of the wave is with a positive gradient - from 0.5 to 1.0, and indeed I say a phase shift. But I don't know why. – Abeer Mar 06 '21 at 20:15
  • plot m and then it should be very clear to you (after multiplying by (pi/2) as you do). – Dan Boschen Mar 06 '21 at 20:21

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