Signals and Systems - Electrical Engineering

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

0 0.05 0.1 0.15 0.2

− 50

0

50

m

(t)

0 0.05 0.1 0.15 0.2

−0.5

0

0.5

1

x(

t)

t (sec)

− 500 0 500

0

5

10

|M

(Ω

)|

− 500 0 500

0

0.05

0.1

f

|x

(Ω

)|

(a) (b)

500

450

400

350

300

250

200

150

100

50

0

Frequency

0.6/7 0.7/7 0.8/7 0.9/7 1/7 1.1/7 1.2/7 1.3/7 0.2

Time

(c)

FIGURE 6.19

Narrowband frequency modulation: (a) messagem(t)and narrowband FM signalx(t); (b) magnitude spectra of

m(t)andx(t); and (c) spectrogram ofx(t)displaying evolution of its Fourier transform with respect to time.

These instantaneous frequencies are not almost constant as before. The frequency of the carrier

is now continuously changing with time. For instance, for the ramp message the instantaneous

frequency is

IF 2 (t)= 50 π+ 105 tπ

so that for a small time interval [0, 0.1] we get a chirp (sinusoid with time-varying frequency), as

shown in Figure 6.20(b). Figure 6.20 display the messages, the FM signals, and their corresponding

magnitude spectra and their spectrograms. These FM signals are broadband, occupying a band of

frequencies much larger than the messages, and their spectrograms show that their spectra change

with time. n