Communication Systems, 5e

Chapter 5: Angle CW Modulatation

A. Bruce CarlsonPaul B. Crilly

© 2010 The McGraw-Hill Companies

Chapter 5: Angle CW Modulation

• Phase and frequency modulation– Signal format and structure – physical layer signal

(time domain & freq. content)

• Transmission bandwidth and distortion– RF spectrum information– Effects from filtering, transmission and reception, and

the RF channel

• Generation and detection of FM and PM • Interference

© 2010 The McGraw-Hill Companies

General Description

• Angle modulation – phase or frequency based

• The instantaneous angle is the argument of the cosine (for complex, the exponential)

3

ttf2cosAts 0

ttf2t 0

tjjtf2jexpReAts 0

dt

tft

ttftfdt

t

2

1221

21

00

Matlab Signal Demo

AM, PM, and FM Spectrum with random noise4

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-150

-100

-50

0

Frequency (Hz)

Pow

er (d

B)

AM, FM, PM Example Power Spectrum

FMPMAM

Phase Modulation (PM)

• The magnitude is constant while the phase changes in time carries the message information

• The instantaneous phase and frequency of a PM system is

5

tmtf2cosAts 2p0 180p 1tm

t

tm2

ftf 2p0

tmtft p 202

Frequency Modulation (FM)

• The magnitude is constant while the frequency changes in time carries the message information

• The instantaneous phase and frequency of a FM system is

6

tmftf 3f0

t

f dmtfAts 30 22cos 1tm

tttf

21

t

f dmtft 30 22

PM and FM Basis

• Based on the previous analysis, we need to determine the transform of the phase components

7

tmt 2pPM

fMf 2pPM

t

3fFM dmt

f

fMjf 3fFM

p is the phase deviation f is the frequency deviation

Demonstrating Tone Modulation

• Using the phase basis for PM and FM

• Define message reference input signals as

• Then

8

tmt 2pPM t

3fFM dm2t

tf2sinAtm mm2 tf2cosAtm mm3

tf2sinAt mpmPM tf2sinf

At mm

fmFM

Tone Modulation (2)

• Defining modulation indexes (using phase and frequency deviation)

• The input signals become

• And we can think in terms of modulation indexes.

9

pmPM A m

fmFM f

A

tft mPMPM 2sin tft mFMFM 2sin

tft m 2sin

Tone Modulation (3)

• Using the 1st order approx. quadrature modulation then becomes (small or very small β, Taylor Series est.)

• And the trig function expansion

10

tf2sintAtf2cosAts cc

tf2sintf2sinAtf2cosAts cmc

tff2cos2

Atff2cos2

Atf2cosAts

mcmc

c

Similar to AM ?!

11

Narrow BandwidthPM/FM with tone modulation

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

(a) Line spectrum; (b) Phasor diagram

tff2cos2

Atff2cos2

Atf2cosAts

mcmc

c

Single Tone Modulation( and Bessel’s Function)

• Establish reference input signals as

• Then

• Using the modulation index (FM and PM same)

12

tf2sinAtm mm2 tf2cosAtm mm3

tf2sinAt mpmPM tf2sinf

At mm

fmFM

pmPM A m

fmFM f

A

tft mPMPM 2sin tft mFMFM 2sin

PM Bounds on

• For PM– abs(m(t)) 1, A = 1, and , therefore the maximum is

– Nominal values

13

pmPM A

PM2 14.357.1 PM

FM Bounds on ?

• For FM– abs(m(t)) 1, and A = 1, therefore the maximum is

– Commercial FM Radio: freq dev 75 kHz, music 15 kHz fmax

– Commercial FM Radio: freq dev 75 kHz, speech 3.4 kHz

– Narrowband FM: referred to when <<1• When only the first two Bessel’s terms matter … 14

m

fFM f

0.51575

kHzkHz

FM

06.224.3

75

kHzkHz

FM

Arbitrary Sinusoidal Input(Bessel’s Function)

• The Quadrature Signal Representation

15

tf2sintsintf2costcosA

ttf2cosAts

cc

c

tf2sintf2sinsintf2costf2sincosAts cmcm

• Mathematical Definition (using Bessel Functions)

0evenn

mn0m tfn2cosJ2Jtf2sincos

0oddn

mnm tfn2sinJ2tf2sinsin

dnsinjexp21Jn

Bessel Functions• Wikipedia definition:

http://en.wikipedia.org/wiki/Bessel_function• In mathematics, Bessel functions, first defined by the

mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation:

• Bessel functions are also known as cylinder functions or cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical coordinates.

16

0222

22 yx

dtdyx

dtydx

17

Bessel Function of the 1st Kind

From: Wikipedia®, All text is available under the terms of the GNU Free Documentation License.

dnsinjexp21Jn

Arbitrary Sinusoidal Input (2)

• Substituting and defining constituent parts

18

tf2sintfn2sinJ2A

tf2costfn2cosJ2A

tf2cosJAts

c0oddn

mn

c0evenn

mn

c0

0oddnmcmcn

0evennmcmcn

c0

tfnf2costfnf2cosJ2A

tfnf2costfnf2cosJA

tf2cosJAtsBessel’s Function Form

Bessel’s Function Line Spectrum

• A line spectrum with Bessel magnitudes– Multiple tones would have the sum (superposition) of multiple line spectrum

19

Amplitude

cfmc ff

mc ff

0J 1J

2J

mc f2f

mc f3f

3J

mc f2f

mc f3f

2J

1J

3J

0oddnmcmcn

0evennmcmcn

c0

tfnf2costfnf2cosJ2A

tfnf2costfnf2cosJA

tf2cosJAts“Impulses/Lines” in

Freq. Domain

Freq. Domain Line Spectra

20Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Phasor diagram for arbitrary

21

Tone-modulated line spectra

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

(a) FM or PM with ƒm fixed; (b) FM with Amƒ fixed

14.357.1 PM

22

Rules of Thumb

• The relative amplitudes of the lines will vary with the modulation index

• For <<1 only J0 and J1 are significantfor >>1 there are numerous lines

• Large beta implies a wide bandwidth– Narrowband FM, small <<1– Wideband FM, large

• Table 5.1-2 shows selected values– Keeping scaled terms of 0.01 and higher

Lines above 0.01 in FM based on β

23

Transmission Bandwidth

• From the Bessel’s function discussion, the optimal bandwidth for FM exponential modulation is infinite!– Assuming a “maximum frequency sin/cos wave input”

• A practical bandwidth can be defined based on the magnitude of the spectrum that we wish to keep.– 90% or 99% power for example

24

Transmission Bandwidth (2)

• We must maintain a sufficient number of fmaxlines based on the value of β.– “M” significant sideband pairs or 2M+1 lines– Bandwidth estimate

– The function M() is shown in Fig. 5.2-1 on p. 224 and the next slide

• Approximation: M() = +2, for 2 < < 10

25

1,2 MforfMB m

Transmission Bandwidth (3)

• M vs β

• Approximations– BT=2ꞏ M() ꞏ W, where W ≥ fmax

– M(D) = D+2, for 2 < D < 10– M(D) = D+1, for D < 1 and 10 < D

© 2010 The McGraw-Hill Companies

PM Transmission Bandwidth

• Using the modulation index

• Which is approximated by

27

mpT fB 12

pmPM A

1,2 PMmPM MforfMB

1,2 pmpT MforfMB

1,,12 mmpT AandfWwhereWB

pmPM A

2

Carson’s Rule Estimate for FM

• Carson’s Rule (for >>1 and <<1) (not PM)

• FM Deviation ratio definition

• Note: for the majority of commercial FM, 2 < D < 10 and you want to use (+2)

28

1,,2 mmfT AandfWwhereWB

W

Dwhere,W1D2B fT

W2D2W22B fT

1,,12 mmmT AandfWwherefB

m

fmFM f

A

pmPM A

Commercial FM Bandwidth

• FM Radios– 88 MHz to 108 MHz– 200 kHz channel spacing, centered on odd 100 KHz

steps– Maximum frequency deviation 75 kHz– Message range 30 Hz to 15 kHz

29

51575

kHzkHzD

kHz180kHz15152W1D2BCarson

kHz210kHz15252W2D2B 2C

Commercial FM Bandwidth, Speech

• FM Radios– 88 MHz to 108 MHz– 200 kHz channel spacing, centered on odd 100 KHz

steps– Maximum frequency deviation 75 kHz– Message range 30 Hz to 3.4 kHz

30

06.224.3

75

kHzkHzD

kHzkHzWDBCarson 8.1564.3106.22212

Filter Bandwidths

• You now know the “passband” required for filtering AM, FM and PM waveforms.

• At RF frequencies the bandpass filters must “pass” both the positive and negative frequency content around the “center” frequency.

• When centered at 0 frequency (a baseband waveform), the lowpass filter bandwidth will be ½ that of the RF bandwidth.

• The filters used in AM, PM and FM examples are based on these bandwidth calculations! 31

Bandpass Filtering in FM Systems(filtering can cause distortion)

• What happens when an FM waveform is filtered?

32

txthty cc

fXfHfY cc

0oddn mcmc

mcmcn

0evenn mcmc

mcmcn

cc0c

fnfffnfffnfffnff

J22A

fnfffnfffnfffnff

J2A

ffffJ2AfX

• For a tone modulated spectrum

Bandpass Filtering in FM Systems (2)

• And the output becomes

33

fXfHfY cc

0

0

0

22

2

2

oddn mcmcmcmc

mcmcmcmcn

evenn mcmcmcmc

mcmcmcmcn

ccccc

fnfffnfHfnfffnfHfnfffnfHfnfffnfH

JA

fnfffnfHfnfffnfHfnfffnfHfnfffnfH

JA

fffHfffHJAfY

• For no distortion, the magnitude of the filter must be equal for all frequencies and the phase should be linear … a perfect filter !?

ABC’s “Linear” Conditions

• Taking baseband signaling

34

𝑥 𝑡𝐴2 ⋅ exp 𝑗 ⋅ 𝜑 𝑡

𝑦 𝑡 Re 𝑥 𝑡 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡

fXffuffHfXfHfY LPccLPLPLP

• Bandpass to Lowpass Filter Consideration(see Example 4.1-1) and Figure 4.1-6

ulBP ffffor,fjexpKfH

cuclccLP ffffffor,ffuffjexpKfH

Y as a basedbandequivalent

Bandpass H shifted to baseband

ABC’s Linear Distortion

• For no distortion:– the gain must remain constant or flat, K, and– the phase should be strictly linear (time delay only).

• But …

• Filters usually have amplitude variation or ripple• Filters usually have phase variation or ripple

– A notable exception is a symmetric digital FIR filter

35

ABC’s Linear Distortion (1)• Assume linear amplitude and phase (distortion) in

the filter, H, and filtered signal, X– Described as the low-pass equivalent output spectrum

36

𝐻 𝑓 𝐾 ⋅ exp 𝑗 ⋅ 𝜃 𝑓 𝑓 ⋅ 𝑢 𝑓 𝑓

𝐾 𝐾 ⋅𝑓𝑓 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑡 ⋅ 𝑓 𝑡 ⋅ 𝑓

𝑌 𝑓 𝐾 𝐾 ⋅𝑓𝑓 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑡 ⋅ 𝑓 𝑡 ⋅ 𝑓 ⋅ 𝑋 𝑓

𝑌 𝑓 𝐾 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑡 ⋅ 𝑓 ⋅ 𝑋 𝑓 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑡 ⋅ 𝑓

𝐾 ⋅1

𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑡 ⋅ 𝑓 ⋅ 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑋 𝑓 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑡 ⋅ 𝑓

ABC’s Linear Distortion (2)• Isolating the changes to the input X (narrowband)

– Transform to time domain

37

dt

ttdxft2jexpf2j

Kttxft2jexpKty

1LPc0

c

1

1LPc00LP

ftjfXfjftjfj

K

ftjfXftjKfY

LPcc

LPcLP

101

100

2exp22exp2

1

2exp2exp

ABC’s Linear Distortion (3)• Taking the derivative of the equivalent lowpass

input , a changing phase in time (PM or FM)

• Results in

38

1101

100

22exp2

2exp

ttxttmjftjfj

KttxftjKty

LPfcc

LPcLP

1LP1f

11

1LP

ttxttm2jdt

ttdttjexp2Aj

dtttdx

tjAtxLP exp

2

1011

0 2exp2

2ttxftj

fjttmKj

Kty LPcc

fLP

ABC’s Linear Distortion (4)• Interpreting the filtered result

• An AM modulated, constant phase shifted baseband waveform!– This is known as FM to AM conversion due to filtering!– In general:

• filters causing AM can result in FM distorted basebands• filters causing FM/PM can result in AM distorted basebands

39

1011

0 2exp2

2ttxftj

fjttmKj

Kty LPcc

fLP

1011

0 2exp ttxftjttmf

KKty LPc

c

fLP