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Chapter4

Bandpass Signalling Definitions

Complex Envelope Representation

Representation of Modulated Signals

Spectrum of Bandpass Signals

Power of Bandpass Signals

Examples

Huseyin BilgekulEeng360 Communication Systems I

Department of Electrical and Electronic Engineering Eastern Mediterranean University

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Energy spectrum of a bandpass signal is concentrated around the carrier frequency fc.

A time portion of a bandpass signal. Notice the carrier and the baseband envelope.

Bandpass Signals

Bandpass Signal Spectrum

Time Waveform of Bandpass Signal

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DEFINITIONS

Definitions:

The Bandpass communication signal is obtained by modulating a baseband analog or digital signal onto a carrier.

A baseband waveform has a spectral magnitude that is nonzero for frequencies in the vicinity of the origin ( f=0) and negligible elsewhere.

A bandpass waveform has a spectral magnitude that is nonzero for frequencies in some band concentrated about a frequency where fc>>0. fc-Carrier frequency

Modulation is process of imparting the source information onto a bandpass signal with a carrier frequency fc by the introduction of amplitude or phase perturbations or both.

This bandpass signal is called the modulated signal s(t), and the baseband source signal is called the modulating signal m(t).

cff

Transmissionmedium

(channel)

Carrier circuits

Signal processing

Carrier circuits

Signal processing

Information

minput m~

)(~ tg)(tr)(ts)(tg

Communication System

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Complex Envelope Representation The waveforms g(t) , x(t), R(t), and are all baseband waveforms. Additionally all of

them except g(t) are real and g(t) is the Complex Envelope. t

• g(t) is the Complex Envelope of v(t)

• x(t) is said to be the In-phase modulation associated with v(t)

• y(t) is said to be the Quadrature modulation associated with v(t)

• R(t) is said to be the Amplitude modulation (AM) on v(t)

• (t) is said to be the Phase modulation (PM) on v(t)

In communications, frequencies in the baseband signal g(t) are said to be heterodyned up to fc

THEOREM: Any physical bandpass waveform v(t) can be represented as below

where fc is the CARRIER frequency and c=2 fc

( )( ) ( ) ( ) ( ) ( ) j tj g tg t x t jy t g t e R t e

Re cos

= cos sin

cj tc

c c

v t g t e R t t t

x t t y t t

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v(t) – bandpass waveform with non-zero spectrum concentrated near f=fc => cn – non-zero for ‘n’ in the range

The physical waveform is real, and using , Thus we have:

Complex Envelope Representation

00 0( ) 2 /

njn t

nn

v t c e T

00

1

Re 2 jn tn

n

v t c c e

0

1

Re ( ) Re 2 cc c

nj n tj t j t

nn

v t g t e c e e

0( )

1

( ) 2 cj n tn

n

g t c e

0 cnf f

PROOF: Any physical waveform may be represented by the Complex Fourier Series

*n nc c

cn - negligible magnitudes for n in the vicinity of 0 and, in particular, c0=0

Introducing an arbitrary parameter fc , we get

=> g(t) – has a spectrum concentrated near f=0 (i.e., g(t) - baseband waveform)

*1 1Re

2 2

THEOREM: Any physical bandpass waveform v(t) can be represented by

where fc is the CARRIER frequency and c=2 fc

Re cj tv t g t e

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Converting from one form to the other form

Equivalent representations of the Bandpass signals:

Complex Envelope Representation

cos sin Inphase and Quadrature (IQ) form c cv t x t t y t t

( ) ( )( ) ( ) Complex Envelope of ( )j g t j tg t x t jy t g t e R t e v t

Inphase and Quadrature (IQ) Components.

Re ( )cos ( )

Im ( )sin ( )

x t g t R t t

y t g t R t t

Envelope and Phase Components 2 2

1

( ) ( ) ( )

( )( ) ( ) tan ( )

( )

R t g t x t y t

y tt g t

x t

Re cos Envelope and Phase formcj tcv t g t e R t t t

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The complex envelope resulting from x(t) being a computer generated voice signal and y(t) being a sinusoid. The spectrum of the bandpass signal generated from above signal.

Complex Envelope Representation

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Representation of Modulated Signals

• The complex envelope g(t) is a function of the modulating signal m(t) and is given by: g(t)=g[m(t)] where g[• ] performs a mapping operation on m(t).

• The g[m] functions that are easy to implement and that will give desirable spectral properties for different modulations are given by the TABLE 4.1

• At receiver the inverse function m[g] will be implemented to recover the message.Mapping should suppress as much noise as possible during the recovery.

Modulation is the process of encoding the source information m(t) into a bandpass signal s(t). Modulated signal is just a special application of the bandpass representation. The modulated signal is given by:

Re ( ) 2cj tc cs t g t e f

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Bandpass Signal Conversion

)(tg

)(ts

1 1 10 0

2

Ac 2

0

Ac 2

nX

XUnipolarLine Coder

cos(ct)

g(t)XncA

)(ts

On off KeyingOn off Keying (Amplitude Modulation) of a unipolar line coded (Amplitude Modulation) of a unipolar line coded signal for bandpass conversion.signal for bandpass conversion.

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Binary Phase Shift keyingBinary Phase Shift keying (Phase Modulation) of a polar line (Phase Modulation) of a polar line code for bandpass conversion.code for bandpass conversion.

XPolar

Line Coder

cos(ct)

g(t)XncA

)(ts

)(tg

)(ts

1 1 10 0

2

Ac 2

2

Ac 2

nX

Bandpass Signal Conversion

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Mapping Functions for Various Modulations

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Envelope and Phase for Various Modulations

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Spectrum of Bandpass Signals

*1 1Re ( ) ( )

2 2c c cj t j t j tv t g t e g t e g t e

tjtj cc etgFetgFtvFfV *

2

1

2

1)(

fGtgF **

*1( ) - -

2 c cV f G f f G f f

Theorem: If bandpass waveform is represented by

tgFfG fPgWhere is PSD of g(t)

Proof:

Thus,

Using and the frequency translation property:

We get,

*1Spectrum of Bandpass Signal ( )

21

PSD of Bandpass Signal ( )4

c c

v g c g c

V f G f f G f f

P f P f f P f f

Re ( ) cj tv t g t e

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PSD of Bandpass Signals

tjtjv

cc etgetgtvtvR ReRe

121*212 Re

2

1Re

2

1ReRe cccccc

tj cetgc )(2 tj cetgc1

tjtjtjtjv

cccc eetgtgeetgtgR Re2

1Re

2

1 *

ccc jtjjv eetgtgetgtgR 2* Re

2

1Re

2

1

ccc jtjjv eetgtgetgtgR 2* Re

2

1Re

2

1

PSD is obtained by first evaluating the autocorrelation for v(t):

Using the identity

where and

- Linear operators

gRtgtg *but

cjgv eRR Re

2

1AC reduces to

* *1( )

4v v g c g c g gP f F R P f f P f f P P f PSD =>

=>

We get

or g(t)f c in sfrequencie

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Evaluation of Power

dffPtvP vv2

dfefPfPFR fjvvv

21

dffPR vv 0

0Re2

10Re

2

10 * tgtgRR gv

2Re

2

10 tgRv

2

2

10 tgRv

tg

Theorem: Total average normalized power of a bandpass waveform v(t) is

Proof:

But

So,

cj

gv eRR Re2

1 Since

or

But is always real

So,

22 10

2v v vP v t P f df R g t

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Example : Amplitude-Modulated Signal Evaluate the magnitude spectrum for an AM signal:

1cg t A m t Complex envelope of an AM signal:

c cG f A f A M f Spectrum of the complex envelope:

1

2 c c c c cS f A f f M f f f f M f f AM spectrum:

1 1

, 02 21 1

, 02 2

A f f A M f f fc c c cS f

A f f A M f f fc c c c

Magnitude spectrum:

AM signal waveform: Re ( ) 1 coscj tc cs t g t e A m t t

*

*

*

Because ( ) is real and

do not over

1( ) - -

2

and lap

c c

c c

S f G f f G f f

M f M f f f

G f f G f f

m t

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Example : Amplitude-Modulated Signal

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2 22

2 2

2 2

2Side

2

b2

a2

nd

2

2

If DC value of

1

2

Carrier

1 11

2 21

1

2

Sideband P

( ) i

Powe

1 2 21

1 221

1 2

1 1

2

Wher

s zero

ower r e

s c

c

c

c

mc m c

m

c c

P g t A m t

A m t m t

A m t m t

A m t

A A PP

P m

A P

t

t

P

m

Example : Amplitude-Modulated SignalTotal average power:

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Study Examples

SA4-1.Voltage spectrum of an AM signal

Properties of the AM signal are:g(t)=Ac[1+m(t)]; Ac=500 V; m(t)=0.8sin(21000t); fc=1150 kHz;

tjtj eej

ttm 1000210002

2

8.010002sin8.0

10001001000100250

10001001000100250

ccc

ccc

ffjffjff

ffjffjfffS

10004.010004.0 fjfjfM Fourier transform of m(t):

ccccc ffMffffMffAfS 2

1Spectrum of AM signal:

Substituting the values of Ac and M(f), we have

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00

2cos

2

2

0

2jj

m eeAA

R

1000100044

2

00

2

ffA

ffffA

fPm

tmtmtmtmA

tmtmAtgtgR

c

cg

1

11*2

2

SA4-2. PSD for an AM signal

Autocorrelation for a sinusoidal signal (A sin w0t – ref ex. 2-10)

A=0.8 and 100020

Autocorrelation for the complex envelope of the AM signal is

Study Examples

mRtmtmtmtm ,0 ,11But mcg RAR 12

fPAfAfP mccg22 mcg RAR 12Thus

cgcgv ffPffPfP 4

1)(Using

10001000010001000062500

10001000010001000062500

ccc

cccs

ffffff

fffffffP

PSD for an AM signal:

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Study Examples

kWdffPVP srmssnorms 1652

kW

R

VP

L

rmssnorms 3.3

50

1065.1 52

kWVAVP rmsmcrmssnorms 1652

8.01500

2

11

2

12

2222

SA4-3. Average power for an AM signal

Normalized average power

Alternate method: area under PDF for s(t)

Actual average power dissipated in the 50 ohm load:

kW

R

PP

L

rmsPEPactualPEP 1.8

50

1005.4 52

SA4-4. PEP for an AM signal

kWtmAtgP cnormPEP 4058.015002

1max1

2

1max

2

1 22222 Normalized PEP:

Actual PEP for this AM voltage signal with a 50 ohm load: