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Pb. 4.32 Compute numerically the derivative of the function

2
3
y = x + 2x + 5 in the interval 0 ≤ x ≤ 1
using the difference equations for both d(k) and D(k) for different values of
∆x. Comparing the numerical results with the analytic results, compute the
errors in both methods.

Application
In this application, we make use of the improved differentiator and corre-
sponding integrator (Trapezoid rule) for modeling FM modulation and
demodulation. The goal is to show that we retrieve back a good copy of the
original message, using the first-order iterators, thus validating the use of
these expressions in other communication engineering problems, where reli-
able numerical algorithms for differentiation and integration are needed in
the simulation of different modulation-demodulation schemes.
As pointed out in Pb. 3.35, the FM modulated signal is given by:

 t 
τ
dτ
u t ( ) = A cos2π f t + 2π k ∫ m( )  (4.17)
FM c  c f −∞ 
The following script M-file details the steps in the FM modulation, if the signal
in some normalized unit is given by the expression:

mt() = sinc (10 t) (4.18)

Assuming that in the same units, we have f = k = 25.
c f
The second part of the program follows the demodulation process: the
phase of the modulated signal is unwrapped, and the demodulated signal is
obtained by differentiating this phase, while subtracting the carrier phase,
which is linear in time.

fc=25;kf=25;tlowb=-1;tupb=1;
t=tlowb:0.0001:tupb;
p=length(t);
dt=(tupb-tlowb)/(p-1);

m=sinc(10*t);
subplot(2,2,1)
plot(t,m)
title('Message')

© 2001 by CRC Press LLC