Communication SystemsyLecture 9

D I KiDong In KimSchool of Info/Comm Engineering

Sungkyunkwan University

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2

Outline

� 4.1-4.3 Basic definitions and properties

� 4.4 Narrowband FM

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Basic Definitions

�The general form for a modulated signal is

is adjusted in angle modulation.

� Some Definitions

� Instantaneous phase

� Instantaneous frequency

))(2cos()( ttfAtscc

φπ +=

)(tφ

=)(tiθ

=)(tfi

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Basic Definitions

� Frequency deviation : the maximum difference

between the instantaneous frequency and the carrier

frequency.

=∆f

dt

tdf

dt

tdtf

c

i

i

)(

2

1)(

2

1)(

φ

π

θ

π+==

f∆

� Range of instantaneous frequency: 2∆f around fc

� Note: the bandwidth of the output is larger than 2∆f (will

be shown later)

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PM

� Phase Modulation (PM): the phase is proportional to m(t)

))(2cos()( ttfAtscc

φπ +=

))(2cos()( tmktfAtspcc

+= π

:pk

==

dt

tdtf i

i

)(

2

1)(

θ

π

� Instantaneous frequency:

� � The instantaneous freq is determined by

Phase sensitivity factor

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FM ))(2cos()( ttfAtscc

φπ +=

� Frequency Modulation (FM): the freq is proportional to m(t)

),()( tmkftf fci +=

:fk Frequency sensitivity factor.

The phase is the integral of the frequency:

� The frequency deviation becomes:

The amplitude of m(t)

affects the freq deviation.

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Properties of angle modulation

� Constant amplitude � constant transmitted power

� Nonlinearity: does not satisfy superposition principle

� Spectrum analysis and noise analysis are difficult.

� Requires more bandwidth (BW) than AM.

� Better noise performance than AM.

� Can trade off BW and noise performance:

� Increased BW � better noise performance.

�Bandwidth of AM cannot be changed.

))(2cos()( ttfAtscc

φπ +=

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Relationship between PM and FMCarrier

PM output: freq is max when

FM output: freq is max when

)),(2cos()( tmktfAtspcc

+= π

))(22cos()(0∫+=

t

fcc dmktfAts ττππ

PM

FM

.)(

2)(

dt

tdmkftf

p

ciπ

+=

),()( tmkftf fci +=

Message

� Range of instantaneous frequency:

[ ]ffffcc

∆+∆− ,

� But the bandwidth of the output is more than 2∆f , as shown later.

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Relationship between PM and FM

message

PM needs more bandwidth because of the discontinuity.

phase jump2pkπ

=

)),(2cos()( tmktfAtspcc

+= π

))(22cos()(0∫+=

t

fcc dmktfAts ττππ

PM signal has discontinuities,

but FM is continuous.

Example: PM with

FM

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4.3 Relationship between PM and FM

� FM may be viewed as PM of message

)),(2cos()( tmktfAtspcc

+= π ))(22cos()(0∫+=

t

fcc dmktfAts ττππ

∫t

dm0

)( ττ

� PM can be viewed as FM of message dm(t)/dt.

� Therefore we can use PM circuits to generate FM, or

vice versa.

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Outline

� 4.1-4.3 Basic definitions and properties

� 4.4 Narrowband FM

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Spectrum Analysis of Angle

Modulation

� Fourier transform is not as obvious as in AM

� The bandwidth of PM and FM is infinite in theory� But most energy is within a finite range

� Can define the bandwidth as the range the contains certain percentage of total energy.

� Start with special cases:� Single-tone message and narrowband modulation

� Single-tone message and wideband modulation

� Goal: establish an empirical formula to estimate the transmission BW in terms of the message BW and ∆f:� Carson Rule:

WDWfBT

)1(2)(2 +=+∆≈

))(2cos()( ttfAtscc

φπ +=

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FM for Single Tone Modulation

General formula:

Single tone message: )2cos()( tfAtmmmπ=

))(22cos()(0∫+=

t

fcc dmktfAts ττππ

Instantaneous frequency ),()( tmkftf fci +=

� Frequency Deviation:

The max freq deviation is proportional to message amplitude.

)2cos()( tffftfmciπ∆+=

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FM for Single Tone Modulation

� Note: β is defined only for single-tone modulation

� Because it needs fm.

� �Modulated single-tone FM signal:

� Instantaneous phase:

),2cos()( tfAtmmmπ=

� Modulation Index:

β represents the max phase deviation.

mf Akf =∆

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4.4 Narrow-band FM

� Modulated FM signal:

� In narrowband FM, β is very small (β < 0.3).

� In this case,

� The FM signal can be approximated by:

))2sin(2cos()( tftfAtsmcc

πβπ +=

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4.4 Narrow-band FM Implementation

� For general m(t): if ∆f is small:

x

fkπ2

)(tφ

Can be used to generate wideband FM, together with freq multiplier� Armstrong’s indirect method

))(2cos())(22cos()(0

ttfAdmktfAts cc

t

fcc φπττππ +=+= ∫

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4.4 Narrow-band FM

� Consequence of the approximation:

1. Amplitude is no longer constant.

2. Phase contains harmonic distortions (mainly 3rd order)

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Bandwidth of Narrow-band FM

� In the single tone FM example with

� Recall single tone AM with :)2cos()( tfAtmmmπ=

:)2cos()( tfAtmmmπ=