TELE3113 Analogue and Digital Communications Angle Modulation Wei Zhang [email protected] School of Electrical Engineering and Telecommunications The University of New South Wales
TELE3113 Analogue and DigitalCommunications
Angle Modulation
Wei Zhang
School of Electrical Engineering and Telecommunications
The University of New South Wales
Last two weeks ...
We have studied:
Amplitude Modulation:
s(t) = [1 + kam(t)]c(t).
Simple envelope detection, but low power/BW efficiency.
DSB-SC Modulation:
s(t) = m(t)c(t).
High power efficiency, but low BW efficiency.
SSB Modulation:
s(t) = 1
2Acm(t) cos(2πfct) ∓
1
2Acm̂(t) sin(2πfct).
VSB Modulation: Tailored for transmission of TV signals.
TELE3113 - Angle Modulation. August 18, 2009. – p.1/18
Angle vs Amplitude Modulation
Amplitude modulation: amplitude of a carrier wave varies in
accordance with an information-bearing signal.
Angle modulation: angle of the carrier changes according to
the information-bearing signal.
Angle modulation provides better robustness to noise and
interference than amplitude modulation, but at the cost of
increased transmission BW.
TELE3113 - Angle Modulation. August 18, 2009. – p.2/18
Definitions
Let θi(t) denote the angle of a modulated sinusoidal carrier
at time t.
Assume θi(t) is a function of the information-bearing signal
or message signal m(t).
The angle-modulated wave is
s(t) = Ac cos[θi(t)]
Instantaneous frequency of s(t) is defined as
fi(t) =1
2π
dθi(t)
dt
TELE3113 - Angle Modulation. August 18, 2009. – p.3/18
PM
Two commonly used angle modulation: PM and FM.
Phase modulation (PM): The instantaneous angle is varied
linearly with m(t), as shown by
θi(t) = 2πfct + kpm(t),
where kp denotes the phase-sensitivity factor.
The phase-modulated wave is described by
s(t) = Ac cos[2πfct + kpm(t)].
TELE3113 - Angle Modulation. August 18, 2009. – p.4/18
FM
Frequency modulation (FM): The instantaneous frequency
fi(t) is varied linearly with m(t), as shown by
fi(t) = fc + kfm(t),
where kf denotes the frequency-sensitivity factor.
Integrating fi(t) with time and multiplying 2π, we get
θi(t) = 2π
∫ t
0
fi(τ)dτ = 2πfct + 2πkf
∫ t
0
m(τ)dτ. (1)
The frequency-modulated wave is therefore
s(t) = Ac cos
[
2πfct + 2πkf
∫ t
0
m(τ)dτ
]
.TELE3113 - Angle Modulation. August 18, 2009. – p.5/18
PM versus FM
Phase Modulation Frequency Modulation
θi(t) 2πfct + kpm(t) 2πfct + 2πkf
∫ t
0m(τ)dτ
fi(t) fc +kp
2πddt
m(t) fc + kfm(t)
s(t) Ac cos[2πfct + kpm(t)] Ac cos[
2πfct + 2πkf
∫ t
0m(τ)dτ
]
TELE3113 - Angle Modulation. August 18, 2009. – p.6/18
PM/FM Relationship
Integrator Phase
modulator
Modulating wave FM wave
)2cos( tfA cc π
Differentiator Frequency modulator
Modulating wave PM wave
)2cos( tfA cc π
(b) Scheme for generating a PM wave by using a frequency modulator.
(a) Scheme for generating an FM wave by using a phase modulator.
TELE3113 - Angle Modulation. August 18, 2009. – p.7/18
AM/PM/FM Waves
0 0.5 1 1.5−1
0
1Carrier Wave
0 0.5 1 1.5−1
0
1Message Signal
0 0.5 1 1.5−2
0
2AM Wave
0 0.5 1 1.5−1
0
1PM Wave
0 0.5 1 1.5−1
0
1FM Wave
TELE3113 - Angle Modulation. August 18, 2009. – p.8/18
Properties of Angle Modulation
Property 1 Constancy of transmitted power:The average power of angle-modulated waves is a constant,
as shown by
Pav =1
2A2
c .
Property 2 Nonlinearity of the modulation process:
Let s(t), s1(t), and s2(t) denote the PM waves produced by
m(t),m1(t) and m2(t). If m(t) = m1(t) + m2(t), then
s(t) 6= s1(t) + s2(t).
TELE3113 - Angle Modulation. August 18, 2009. – p.9/18
Properties of Angle Modulation
Property 3 Irregularity of zero-crossings:
A “zero-crossing” is a point where the sign of a function
changes. PM and FM wave no longer have a perfect
regularity in their spacing across the time-scale.
Property 4 Visualization difficulty of messagewaveform:The message waveform cannot be visualized from PM and
FM waves.
TELE3113 - Angle Modulation. August 18, 2009. – p.10/18
Example of Zero-crossings (1)
Consider a modulating wave m(t) as shown by
m(t) =
at, t ≥ 0
0, t < 0
Determine the zero-crossings of the PM and FM waves produced
by m(t) with carrier frequency fc and carrier amplitude Ac.
TELE3113 - Angle Modulation. August 18, 2009. – p.11/18
Example of Zero-crossings (2)
The PM wave is given by
s(t) =
Ac cos(2πfct + kpat), t ≥ 0
Ac cos(2πfct), t < 0
The PM wave experiences a zero-crossing when the angle is an
odd multiple of π/2, i.e.,
2πfctn + kpatn =π
2+ nπ, n = 0, 1, 2, · · ·
Then, we get
tn =1/2 + n
2fc + kpa/π, n = 0, 1, 2, · · ·
TELE3113 - Angle Modulation. August 18, 2009. – p.12/18
Example of Zero-crossings (3)
The FM wave is given by
s(t) =
Ac cos(2πfct + πkfat2), t ≥ 0
Ac cos(2πfct), t < 0
To find zero-crossings, we may set up
2πfctn + πkfat2n =π
2+ nπ, n = 0, 1, 2, · · ·
The positive root of the above quadratic equation is
tn =1
akf
(
−fc +
√
f2c + akf
(
1
2+ n
)
)
, n = 0, 1, 2, · · ·
TELE3113 - Angle Modulation. August 18, 2009. – p.13/18
Example of Zero-crossings (4)
fc = 0.25, a = 1, kp = π/2 and kf = 1.
−8 −6 −4 −2 0 2 4 6 80
2
4
6
8Message Signal
−8 −6 −4 −2 0 2 4 6 8−1
−0.5
0
0.5
1PM Wave
−8 −6 −4 −2 0 2 4 6 8−1
−0.5
0
0.5
1FM Wave
TELE3113 - Angle Modulation. August 18, 2009. – p.14/18
Narrowband FM (1)
Consider a sinusoidal modulating wave defined by
m(t) = Am cos(2πfmt).
The instantaneous frequency of the FM wave is
fi(t) = fc + kfAm cos(2πfmt) = fc + ∆f cos(2πfmt)
where ∆f = kfAm is called the frequency deviation.
The angle of the FM wave is
θi(t) = 2πfct + β sin(2πfmt)
where β = ∆ffm
is called the modulation index of the FM
wave. TELE3113 - Angle Modulation. August 18, 2009. – p.15/18
Narrowband FM (2)
The FM wave is then given by
s(t) = Ac cos[2πfct + β sin(2πfmt)].
Using cos(x + y) = cos x cos y − sin x sin y, we get
s(t) = Ac cos(2πfct) cos[β sin(2πfmt)]−Ac sin(2πfct) sin[β sin(2πfmt)].
For narrowband FM wave, β << 1. Then, cos[β sin(2πfmt)] ≈ 1
and sin[β sin(2πfmt)] ≈ β sin(2πfmt). Therefore,
s(t) ≈ Ac cos(2πfct) − βAc sin(2πfct) sin(2πfmt).
TELE3113 - Angle Modulation. August 18, 2009. – p.16/18
Generating Narrowband FM
Integrator Product
Modulator
Modulating wave Narrow-
band FM wave
∑
+
__
090− Phase-shifter
Carrier wave
)2cos( tfAcc
π
)2sin( tfA cc π
Narrow-band phase modulator
TELE3113 - Angle Modulation. August 18, 2009. – p.17/18
Narrowband FM vs. AM
For small β, the narrowband FM wave is given by
s(t) ≈ Ac cos(2πfct) − βAc sin(2πfct) sin(2πfmt).
Using sin x sin y = − 1
2cos(x + y) cos(x − y), we get
s(t) ≈ Ac cos(2πfct)+1
2βAc[cos[2π(fc +fm)t]−cos[2π(fc−fm)t]].
Recall the AM of the single-tone message signal is [p.11, Aug-4,
TELE3113]
sAM(t) = Ac cos(2πfct)+1
2µAc[cos[2π(fc+fm)t]+cos[2π(fc−fm)t]].
The only difference between NB-FM and AM is the “sign”.TELE3113 - Angle Modulation. August 18, 2009. – p.18/18