PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 443

Example 4.22

Modulated signals are often encountered in telecommunications and consist of mixing

(multiplying) two analog signals referred to as the message signal m(t), consisting of low

frequencies; and the carrier signal c(t), consisting of a high-frequency sinusoidal.

Let the modulating and carrier signals be given by m(t) = 3 cos(5t) and c(t) = cos(50t),

respectively.

Create the MATLAB script fi le telecom_signals that returns

1. Expressions of
   a. C(w) = ℑ[c(t)]
   b. M(w) = ℑ[m(t)]
   c. M(w) + C(w) = ℑ[m(t)] + ℑ[c(t)]
   d. AM_DSB_SC(w) = ℑ[m(t)  c(t)] = 1/2π(ℑ[M(w) ⊗ C(w)]*
   e. AM_DSB_WC(w) = ℑ[(m(t) + 5)  c(t)]
   f. FM(w) = ℑ[fm(t)] = ℑ[cos(50 t + m(t))]


2. The following time and frequency plots:
   a. m(t) versus t ↔ M(w) versus w
   b. c(t) versus t ↔ C(w) versus w
   c. [m(t) + c(t)] versus t ↔ [M(w) + C(w)] versus w
   d. m(t)  c(t) versus t ↔ ℑ [m(t)  c(t)]
   e. [m(t)+5}  c(t)] versus t ↔ ℑ [(m(t) + 5)  c(t)] versus w
   f. cos[50 t + m(t)] versus t ↔ ℑ{cos[(50t) + m(t)]} versus w

ANALYTICAL Solution

An AM signal is a signal in which (m(t) cos(w 0 t)), the amplitude of the carrier, is modu-

lated by the message wave m(t); hence the information content of m(t) is carried by the

amplitude variations of the carrier.

If the angle of the carrier varies in some manner with respect to the modulation

signal m(t), then the modulation technique is referred to as angle modulation given by

the following equation:

cos(w 0 t) + f(m(t))

The following two methods are used in the generation of angle modulation:

1. PM


2. FM
   PM and FM are defi ned by the following equations:


3. pm(t) = A cos[w 0 t + kam(t)]


4. fm(t) = A cos[w 0 t + kb ∫m(t) dt(m(t)]
   where A, ka, and kb are constants.
   It i s e a sy to appr e c i ate t h at a lt houg h PM a nd F M a r e d i f fe r e nt for m s of a ng le mo du l at ion,
   they are essentially very similar, specially if the message signal is a sinusoidal. Therefore,
   it is unnecessary to discuss the two modulation methods, and one is suffi cient to illustrate
   the process. For simplicity, fm(t) will be referred as an angular-modulated signal.
   AM spectrum expressions can directly be obtained by using the MATLAB function
   fourier.

*^ AM stands for amplitude modulation, DSB for double side band, SC for suppress carrier, WC for with carrier,

and FM for frequency modulation.