Signals and Systems - Electrical Engineering

164 C H A P T E R 2: Continuous-Time Systems

Let the input to the envelope detector be

x(t)=[p(t)+P] cos( 0 t)

wherePis the minimum ofp(t)scaled. Use MATLAB to solve numerically this problem.

(a) Consider first

p(t)=20[u(t)−u(t− 40 )]− 10 [u(t− 40 )−u(t− 60 )

Let 0 = 2 π,P=1.1|min

(

p(t)|. Generate the signalsp(t)andx(t)for 0 ≤t≤ 100 with an interval of

Ts=0.01.

(b)Consider then the subsystem that computes the absolute value of the inputx(t).

(c)Compute the low-pass filtered signal by using an RC circuit with impulse responseh(t)=e−0.8tu(t).

To implement the convolution use theconvfunction multiplied byTs. Plot together the message signal

p(t), the modulated signalx(t), the absolute valuey(t), and the envelope ofx(t). Does this envelope look

likep(t)?

(d)Consider the message signalp(t)=2 cos(0.2πt), 0 = 10 π, andP=|min

(

p(t)|, and repeat the

process. Scale the signal to get the originalp(t).

2.20. Frequency modulation (FM)—MATLAB

Frequency modulation, or FM, uses a wider bandwidth than amplitude modulation, or AM, but it is not

affected as much by noise as AM is. The output of an FM transmitter is of the form

y(t)=cos(ct+ 2 πν

∫t

0

m(τ)dτ)

wherem(t)is the message andνis a factor in Hz/volt if the units of the message are in volts.

(a) Create as the message a signal

m(t)=cos(t)

Find the FM signaly(t)forν= 10 and then forν= 1. Let the carrier frequencyc= 2 π. Use MATLAB

to generate the different signals for times 0 ≤t≤ 10 at intervals ofTs=0.01. Plotm(t)and the two FM

signals (one forν= 10 and the other forν= 1 ) in the same plot. Is the FM transmitter a linear system?

Explain.

(b)Create a message signal

m 1 (t)=

{

1 whenm(t)≥ 0

− 1 whenm(t) < 0

Find the corresponding FM signal forν= 1.