bandpass communication and the complex envelope

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Bandpass communicationand the

Complex Envelope

EELE445-14Lecture 25

Complex Envelope

•Review sections 4-1 to 4-4 in the text•look at the complex envelope representations in table 4-1•be able to calculate or find:

•complex envelope of a waveform expression•calculate the signal power•calculate the PEP•define Modulation

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

A bandpass waveform has a spectral magnitude that is nonzero for frequencies in some band concentrated about a frequency f=+/- fc where fc is much greater than zero.

Modulation is the process if imparting the source information onto a bandpass signal with a carrier frequency fc by the introduction of amplitude or phase perturbations or both. The resulting bandpass signal is called the modulated signal s(t), and the baseband source signal is called the modulating signal m(t)

Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0

Figure 4–1 Bandpass Communication system.

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Complex EnvelopeTheorem: Any physical bandpass waveform can be represented by:

{ } { }

)()( )()()()()(

2,)()(

Re)(Re)(

tjtg

ccc

tj

etRetgtjytxtg

fwfrequencycarrierftsofenvelopecomplextg

partrealdenotesetgts c

θ

ω

π

==+=

≡≡≡

•=

∠

Complex Envelope

{ }{ }

{ }tjcc

tjtg

cetgttyttxts

txtytgt

tytxtgtR

ttRtgtyttRtgtx

etRetgtjytxtg

ω

θ

ωω

θ

θθ

)(Re)sin()()cos()()(

)()(tan)()(

)()()()(

))(sin()()(Im)())(cos()()(Re)()()()()()(

1

22

)()(

=−=

⎟⎟⎠

⎞⎜⎜⎝

⎛≡∠=

+≡=

≡=≡=

==+=

−

∠

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Complex Envelopeg(t) = x(t) + jy(t)

x(t) is the “I” or In phase modulation

y(t) is the “Q” or Quadrature modulation

R(t) is >=0 and determines the signal power

[ ][ ] )(

)()(tmforoperatormappingtheisg

tmgtg•

=

g[ ] function determines the bandpass modulation type

Complex Envelope- Vector Modulator

{ }tjcc

cetgttyttxts ωωω )(Re)sin()()cos()()( =−=

)2cos( tfcπ

2π

−

All modulation types may be generated using a vector modulator

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Complex Envelope: time and frequency domain

{ }

[ ]

[ ]

22

*

)(21)0()()(

)()(41)(

)()(21)(

)(Re)(

tgRdffPtvP

ffPffPfP

ffGffGfV

etgtv

vvv

cgcgv

cc

tj c

∫∞

∞−

====

−−+−=

−−+−=

=

:Power

:PSD

:Spectrum

:waveformBandpassω

Complex Envelope, of various modulations

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Complex Envelope, (R θ), of various modulations

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DSB-SCand AM


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DSB-SC – section 5-1 in the text• This is AM without the Carrier• All of the power in the PSD is dependent on the

information m(t)• The modulation efficiency is 100%• Synchronous demodulation is required

)2cos( tfcπ

envelope-pre)()()2cos()()(

tmAtgtftmAts

c

cc

== π

1)(

1)(2 ≤

≤

tm

tm

DSB-SC – Spectrum

[ ])()(2

)(

)()(,1)()()(

)2cos()()(

ccc

c

c

cc

ffMffMAfS

fMAfGtmtmAtg

tftmAts

++−=

=

≤=

=

m(t)) d(normalize envelope-preπ

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DSB-SC – Signals

0 5 .10 4 0.001 0.0015 0.0021

0

1

Carrier VoltageModulation VoltageCarrier VoltageModulation Voltage

Waveforms

time

Vol

ts1

1−

c t( )

m t( )

1.992 10 3−×0 t

DSB-SC – Signal

0 5 .10 4 0.001 0.0015 0.0021

0

1

s(t)m(t)positive envelopenegative envelope

s(t)m(t)positive envelopenegative envelope

DSB-SC and Envelopes

Time

Vol

ts

1

1−

s.dsb tn( )

m tn( )m tn( )

m tn( )−

1.992 10 3−×0 tn

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DSB-SC – Signal

0 5000 1 .104 1.5 .104 2 .1040

0.5

1DSB-SC Voltage Spectrum

Frequency Hz

Vol

ts p

eak

1

0

2 DSBk

20KHz0 1t.max

k⋅

AM – section 5-1

1002

)(min[)](max[

1002

%

1)(1|)(|

)2cos()](1[)()](1[)(

minmax

2

tmtmA

AA

tmtm

tftmAtstmAtg

c

cc

c

−=

−=

≤≤

+=+=

modulation

envelope-preπ

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AM – section 5-1 in the text[ ]

222

2

2222

222

)(21

21

)(

:0)(

)(21)(

21

)(121)(

21)(

tmAA

ts

tm

tmAtmAA

tmAtgts

cc

ccc

c

+=

+=

=

++=

++=

+==

powersidebandpowerCarrier

mean, 0for Sidebandshift CarrierCarrier


Figure 5–1 AM signal waveform.

[ ][ ] powerthedeterminesenvelope

wavformphysical)(1)(

)cos()(1)(tmAtg

ttmAts

c

c

+=+= ω

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Mathcad AM signal waveform, u=50%


Mathcad AM signal waveform μ=0.5

6 8 10 12 140

0.25

0.5

0.75

1

1.25

1.5AM Modulation Voltage Spectrum

Frequency KHz

Vol

ts p

eak

6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5AM Power

Frequency KHz

Wat

ts

.0312

12


Mathcad AM signal waveform μ=1

0 5 .10 4 0.001 0.0015 0.0022

1

0

1

2

s(t)Positive EnvelopeNegative Envelope

s(t)Positive EnvelopeNegative Envelope

AM and Envelopes

Time

Vol

ts

6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5AM Power

Frequency KHz

Wat

ts

.125

6 8 10 12 140

0.25

0.5

0.75

1

1.25

1.5AM Modulation Voltage Spectrum

Frequency KHz

Vol

ts p

eak

AM – Power and modulation efficiency

[ ]

power sidebandcarrier in shift power power Carrier ++=

++=

++=

+==

)(2

)(2

)()(212

)(12

)(21)(

22

22

22

22

22

tmAtmAA

tmtmA

tmAtgts

cc

c

c

c

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AM – section 5-1

[ ]

1002

)(min[)](max[

1002

%

1)(1|)(|

)(1)()()2cos()](1[)(

minmax

2

tmtmA

AA

tmtm

tmAtgtRtftmAts

c

c

cc

−=

−=

≤≤

+==+=

modulation

, :envelope realπ


Figure 4–2 Spectrum of AM signal.

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AM – section 5-1 in the text• use your class notes and review section

5-1 in the text. • also review chapter 4 for AM detection

and the superheterodyne receiver.• AM is used for commercial broadcasting

in order that low cost envelope detection may be used.

Bandpass Digital Signal Examples on board…

EELE445-14

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Asymmetric Sideband SignalsUSSB, LSSB, VSB



Figure 5–4 Spectrum for a USSB signal.

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Figure 5–5 Generation of SSB.


Figure P5–12 Weaver’s method for generating SSB.

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Single Side Band- Phasing Method

)(ˆ tm

)(tm

)(ˆ)()( tmtmtg +=

Single Side Band- Phasing Method

f0

)( fM

1

0

)()()(ˆ fMfHfM h=

f1

-1

)(ˆ)()( fMfMfG +=

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Single Side Band

U(-f)

Single Side Band

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Single Side Band

Single Side Band

20

Single Side Band

Single Side Band

21

Single Side Band

Single Side Band

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Figure 5–5 Generation of SSB.

Single Side Band

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Single Side Band

Single Side Band

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Figure 5–6 VSB transmitter and spectra.

bandpass communication and the complex envelope

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