Signals and Systems - Electrical Engineering

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

c(rad/sec) withm(t), the transmitted signals(t)in angle modulation is of the form

s(t)=Acos(ct+θ(t)) (6.17)

where the angleθ(t)depends on the messagem(t). In the case ofphase modulation, the angle function

is proportional to the messagem(t)—that is,

θ(t)=Kfm(t) (6.18)

whereKf>0 is called themodulation index. If the angle is such that

dθ(t)

dt

=1m(t) (6.19)

this relation definesfrequency modulation. Theinstantaneous frequency, as a function of time, is the

derivative of the argument of the cosine or

IF(t)=

d[ct+θ(t)]

dt

(6.20)

=c+

dθ(t)

dt

(6.21)

=c+1m(t) (6.22)

indicating how the frequency is changing with time. For instance, ifθ(t)is a constant—so that the

carrier is just a sinusoid of frequencycand constant phaseθ—the instantaneous frequency is simply

c. The term1m(t)relates to the spreading of the frequency aboutc. Thus, themodulation paradox

Professor E. Craig proposed in his book [17]:

Inamplitudemodulation the bandwidth depends on thefrequencyof the message, while in

frequencymodulation the bandwidth depends on theamplitudeof the message.

Thus, the modulated signals are

PM: sPM(t)=cos(ct+Kfm(t)) (6.23)

FM: sFM(t)=cos(ct+1

∫t

−∞

m(τ)dτ) (6.24)

Narrowband FM

In this case the angleθ(t)is small, so that cos(θ(t))≈1 and sin(θ(t))≈θ(t), simplifying the spectrum

of the transmitted signal:

S()=F[cos(ct+θ(t))]

=F[cos(ct)cos(θ(t))−sin(ct)sin(θ(t))]

≈F[cos(ct)−sin(ct)θ(t)] (6.25)