Angle Modulation
The plot of θ(t ) is tangential to the angle
ω c +θ 0 in interval t 1 <t <t 2 .
The signal ϕ(t )= A cos θ(t ) and
A cos(ω c t +θ0 ) are identical for t 1 <t <t 2 .
Phase modulation
The resulting PM wave is
Frequency modulation
Immunity of Angle Modulation to Nonlinearities
Consider a second-order nonlinear device whose input x(t) and output y(t) are related by
Consider any nonlinear device
A similar nonlinearity in AM not only causes unwanted modulation with carrier
frequencies n ω c but also causes distortion of the desired signal. For instance, if a DSBSC signal m(t )cos ω c t passes through a nonlineaity y (t )=a x (t )+bx 3 (t ) , the output is
Demodulation of FM
The signal ϕ̇ FM (t ) is both amplitude and frequency modulated, the envelope being
A[ω c +k f m(t )] . Because Δ ω=k f m p < ωc , ωc + k f m p >0 for all t, m(t) can be obtained
by envelope detection of ϕ̇ FM (t ) .
Time differentiation can be implemented with LTI system having a frequency response
d
h(t )⇔ j ω H (ω)
dt
In practice, we prefer a frequency-selective network with a transfer function of the form
|H (ω)|=a ω+b
over the FM band, which yield an output proportional to the instantaneous frequency.
Balanced frequency detector
Bandpass Limiter
Consider the incoming angle-modulated signal
(FM)
v 0 (θ)= sign[cos(θ)]
Interference in angle-modulation systems
r (t )= A cos ω c t + I cos(ωc +ω d )t
r (t )= A cos ω c t + I cos ωc t cos ω d t − I sin ωc t sin ω d t
r (t )=[ A+ I cos ω d t ] cos ωc t−[ I sin ωd t ]sin ω c t
r (t )= E (t )cos ψd cos(ωc t )− E (t )sin ψd sin (ω c t ) ,
ψd =tg −1
I sin (ω d t )
A+ I cos(ω d t )
r (t )= E (t )cos(ω c t + ψ d (t )) ,
When the interfering signal is small in comparison to the carrier ( I << A)
I
ψd = sin (ω d t )
A
If the signal r (t )= E (t )cos(ω c t + ψ d (t )) is applied to an ideal demodulator
I
y d (t )= sin ω d t
for PM
A
I ωd
for FM
y d (t )=
cos ωd t
A
The output is inversely proportional to the carrier amplitude A.
Consider an AM signal with an interfering sinusoid I sin(ωc +ω d )t
r (t )=[ A+ m(t )] cos ωc t + I sin(ω c +ω d )t
r (t )=[ A+ m(t )+ I cos ω d t ] cos ω c t − I sin ω d t sin ωc t
The envelope of this signal is
E (t )= √[ A+m(t )+ I cos ω d t ] + I sin ω d t ≈ A+m(t )+ I cos ω d t ,
2
2
2
I≪A
Thus the interference signal at the envelope detector is I cos ω d t which is independent of
the carrier amplitude.
Bandwidth of Angle-Modulation Waves
,
Expanding the exponential e
ωi =ωc + k f m(t )
j k f a (t )
in power series yields
Narrow-Band Angle Modulation
Angle modulation is nonlinear. The principle of superposition does not hold.
If |k f a (t )|≪1
Wide-Band FM
,
,
ωi =ωc +(k f α)cos ω m t
modulation index
e
j βsin ωm t
∞
= ∑ Cne
j nωm t
n=−∞
,
ωm t = x , ,
ω m d t =d x
∞
ϕ FM (t )= A
∑
J n (β )cos(ωc + n ω m )t
n=−∞
For a small modulation index β, only the first sideband corresponding to n = 1 is of
importance. B FM ≈ 2 f m
In general the effective bandwidth of an angle-modulated signal, which contains at least
98% of the signal power, is given by
B FM =2(β +1) f m=2(Δ f + f m )