1. TELE3113 Analogue and Digital
Communications
Angle Modulation
Wei Zhang
w.zhang@unsw.edu.au
School of Electrical Engineering and Telecommunications
The University of New South Wales

2. Last two weeks ...
We have studied:
Amplitude Modulation:
s(t) = [1 + ka m(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 Ac m(t) cos(2πfc t)
2
1
2 Ac m(t) sin(2πfc t).
ˆ
VSB Modulation: Tailored for transmission of TV signals.
TELE3113 - Angle Modulation. August 18, 2009. – p.1/1

3. 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.2/1

4. 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
1 dθi (t)
fi (t) =
2π dt
TELE3113 - Angle Modulation. August 18, 2009. – p.3/1

5. 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) = 2πfc t + kp m(t),
where kp denotes the phase-sensitivity factor.
The phase-modulated wave is described by
s(t) = Ac cos[2πfc t + kp m(t)].
TELE3113 - Angle Modulation. August 18, 2009. – p.4/1

6. FM
Frequency modulation (FM): The instantaneous frequency
fi (t) is varied linearly with m(t), as shown by
fi (t) = fc + kf m(t),
where kf denotes the frequency-sensitivity factor.
Integrating fi (t) with time and multiplying 2π, we get
t t
θi (t) = 2π fi (τ )dτ = 2πfc t + 2πkf m(τ )dτ. (1)
0 0
The frequency-modulated wave is therefore
t
s(t) = Ac cos 2πfc t + 2πkf m(τ )dτ .
0 TELE3113 - Angle Modulation. August 18, 2009. – p.5/1

7. PM versus FM
Phase Modulation Frequency Modulation
t
θi (t) 2πfc t + kp m(t) 2πfc t + 2πkf 0 m(τ )dτ
kp d
fi (t) fc + 2π dt m(t) fc + kf m(t)
t
s(t) Ac cos[2πfc t + kp m(t)] Ac cos 2πfc t + 2πkf 0 m(τ )dτ
TELE3113 - Angle Modulation. August 18, 2009. – p.6/1

8. PM/FM Relationship
Modulating
wave Phase FM wave
Integrator modulator
Ac cos( 2πf c t )
(a) Scheme for generating an FM wave by using a phase modulator.
Modulating
wave Frequency PM wave
Differentiator modulator
Ac cos( 2πf c t )
(b) Scheme for generating a PM wave by using a frequency modulator.
TELE3113 - Angle Modulation. August 18, 2009. – p.7/1

9. AM/PM/FM Waves
Carrier Wave
1
0
−1
0 0.5 Message Signal 1 1.5
1
0
−1
0 0.5 AM Wave 1 1.5
2
0
−2
0 0.5 PM Wave 1 1.5
1
0
−1
0 0.5 FM Wave 1 1.5
1
0
−1
0 0.5 1 1.5
TELE3113 - Angle Modulation. August 18, 2009. – p.8/1

10. Properties of Angle Modulation
Property 1 Constancy of transmitted power:
The average power of angle-modulated waves is a constant,
as shown by
1 2
Pav = Ac .
2
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) = s1 (t) + s2 (t).
TELE3113 - Angle Modulation. August 18, 2009. – p.9/1

11. 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 message
waveform:
The message waveform cannot be visualized from PM and
FM waves.
TELE3113 - Angle Modulation. August 18, 2009. – p.10/1

12. Example of Zero-crossings (1)
Consider a modulating wave m(t) as shown by

 at, t ≥ 0
m(t) =
 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.11/1

13. Example of Zero-crossings (2)
The PM wave is given by

 A cos(2πf t + k at), t ≥ 0
c c p
s(t) =
 Ac cos(2πfc t), t<0
The PM wave experiences a zero-crossing when the angle is an
odd multiple of π/2, i.e.,
π
2πfc tn + kp atn = + nπ, n = 0, 1, 2, · · ·
2
Then, we get
1/2 + n
tn = , n = 0, 1, 2, · · ·
2fc + kp a/π
TELE3113 - Angle Modulation. August 18, 2009. – p.12/1

14. Example of Zero-crossings (3)
The FM wave is given by

 A cos(2πf t + πk at2 ), t ≥ 0
c c f
s(t) =
 Ac cos(2πfc t), t<0
To find zero-crossings, we may set up
π
2πfc tn + πkf at2
n = + nπ, n = 0, 1, 2, · · ·
2
The positive root of the above quadratic equation is
1 2
1
tn = −fc + fc + akf +n , n = 0, 1, 2, · · ·
akf 2
TELE3113 - Angle Modulation. August 18, 2009. – p.13/1

15. Example of Zero-crossings (4)
fc = 0.25, a = 1, kp = π/2 and kf = 1.
Message Signal
8
6
4
2
0
−8 −6 −4 −2 0 2 4 6 8
PM Wave
1
0.5
0
−0.5
−1
−8 −6 −4 −2 0 2 4 6 8
FM Wave
1
0.5
0
−0.5
−1
−8 −6 −4 −2 0 2 4 6 8
TELE3113 - Angle Modulation. August 18, 2009. – p.14/1

16. Narrowband FM (1)
Consider a sinusoidal modulating wave defined by
m(t) = Am cos(2πfm t).
The instantaneous frequency of the FM wave is
fi (t) = fc + kf Am cos(2πfm t) = fc + ∆f cos(2πfm t)
where ∆f = kf Am is called the frequency deviation.
The angle of the FM wave is
θi (t) = 2πfc t + β sin(2πfm t)
∆f
where β = fm is called the modulation index of the FM
wave. TELE3113 - Angle Modulation. August 18, 2009. – p.15/1

17. Narrowband FM (2)
The FM wave is then given by
s(t) = Ac cos[2πfc t + β sin(2πfm t)].
Using cos(x + y) = cos x cos y − sin x sin y, we get
s(t) = Ac cos(2πfc t) cos[β sin(2πfm t)]−Ac sin(2πfc t) sin[β sin(2πfm t)].
For narrowband FM wave, β << 1. Then, cos[β sin(2πfm t)] ≈ 1
and sin[β sin(2πfm t)] ≈ β sin(2πfm t). Therefore,
s(t) ≈ Ac cos(2πfc t) − βAc sin(2πfc t) sin(2πfm t).
TELE3113 - Angle Modulation. August 18, 2009. – p.16/1

18. Generating Narrowband FM
Modulating __
wave Narrow-
Product
Integrator ∑ band
Modulator FM wave
Ac sin( 2πf c t ) +
− 90 0
Phase-shifter
Narrow-band phase
modulator
Carrier wave
Ac cos(2πf c t )
TELE3113 - Angle Modulation. August 18, 2009. – p.17/1

19. Narrowband FM vs. AM
For small β, the narrowband FM wave is given by
s(t) ≈ Ac cos(2πfc t) − βAc sin(2πfc t) sin(2πfm t).
Using sin x sin y = − 1 cos(x + y) cos(x − y), we get
2
1
s(t) ≈ Ac cos(2πfc t) + βAc [cos[2π(fc + fm )t] − cos[2π(fc − fm )t]].
2
Recall the AM of the single-tone message signal is [p.11, Aug-4,
TELE3113]
1
sAM (t) = Ac cos(2πfc t)+ µAc [cos[2π(fc +fm )t]+cos[2π(fc −fm )t]].
2
The only difference between NB-FM and AM is the “sign”.
TELE3113 - Angle Modulation. August 18, 2009. – p.18/1