Signals and Systems - Electrical Engineering

384 C H A P T E R 6: Application to Control and Communications

m 1 (t)

m 2 (t)

m 2 (t)

s(t) r(t)

cos(Ωct)

×

+

LPF

LPF

π

2 shift

π

2 shift

̃

m ̃ 1 (t)

Transmitter Receiver

cos(Ωct)

×

× ×

FIGURE 6.17

QAM transmitter and receiver:s(t)is the transmitted signal andr(t)is the received signal.

To simplify the computation of the spectrum ofs(t), let us consider the messagem(t)=m 1 (t)−jm 2 (t)

(i.e., a complex message) with spectrumM()=M 1 ()−jM 2 ()so that

s(t)=Re[m(t)ejct]

=0.5[m(t)ejct+m∗(t)e−jct]

where∗stands for complex conjugate. The spectrum ofs(t)is then given by

S()=0.5[M(−c)+M∗(+c)]

=0.5[M 1 (−c)−jM 2 (−c)+M∗ 1 (+c)+jM∗ 2 (+c)]

where the superposition of the spectra of the two messages is clearly seen. At the receiver, if we

multiply the received signal (for simplicity assume it to bes(t)) by cos(ct), we get

r 1 (t)=s(t)cos(ct)

=0.25[m(t)+m∗(t)]+0.25[m(t)ej^2 ct+m∗(t)e−j^2 ct]

which when passed through a low-pass filter, with the appropriate bandwidth, gives

0.25[m(t)+m∗(t)]=0.25[m 1 (t)−jm 2 (t)+m 1 (t)+jm 2 (t)]

=0.5m 1 (t)

Likewise, to get the second message we multiplys(t)by sin(ct)and pass the resulting signal through

a low-pass filter.

Frequency-Division Multiplexing

Frequency-division multiplexing (FDM) implements sharing of the spectrum by several users by allo-

cating a specific frequency band to each. One could, for instance, think of the commercial AM or FM