Tele3113 wk5tue

TELE3113 Analogue and DigitalCommunications

Angle Modulation

Wei Zhang

[email protected]

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.


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


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)].


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

]


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.


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


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).


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.


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.



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, · · ·



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, · · ·



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


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).


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


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

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