IMPLICATIONS OF THE ABOVE HOLISTIC UNDERSTANDING OF HARMONY ON PROFESSIONAL E...
chap4_lec1.ppt Engineering and technical
1. Eeng 360 1
Chapter4
Bandpass Signalling
Definitions
Complex Envelope Representation
Representation of Modulated Signals
Spectrum of Bandpass Signals
Power of Bandpass Signals
Examples
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
2. Eeng 360 2
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).
c
f
f
Transmission
medium
(channel)
Carrier
circuits
Signal
processing
Carrier
circuits
Signal
processing
Information
m
input m
~
)
(
~ t
g
)
(t
r
)
(t
s
)
(t
g
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 t
j g t
g t x t jy t g t e R t e
Re cos
= cos sin
c
j t
c
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
0
0 0
( ) 2 /
n
jn t
n
n
v t c e T
0
0
1
Re 2 jn t
n
n
v t c c e
0
1
Re ( ) Re 2 c
c c
n
j n t
j t j t
n
n
v t g t e c e e
0
( )
1
( ) 2 c
j n t
n
n
g t c e
0 c
nf f
PROOF: Any physical waveform may be represented by the Complex Fourier Series
*
n n
c 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 1
Re
2 2
THEOREM: Any physical bandpass waveform v(t) can be represented by
where fc is the CARRIER frequency and c=2 fc
Re c
j t
v 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 c
v t x t t y t t
( ) ( )
( ) ( ) Complex Envelope of ( )
j g t j t
g 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 t
t g t
x t
Re cos Envelope and Phase form
c
j t
c
v 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 ( ) 2
c
j t
c c
s t g t e f
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Bandpass Signal Conversion
)
(t
g
)
(t
s
1 1 1
0 0
2
Ac 2
0
Ac 2
n
X
X
Unipolar
Line Coder
cos(ct)
g(t)
Xn
c
A
)
(t
s
On off Keying (Amplitude Modulation) of a unipolar line coded
signal for bandpass conversion.
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Binary Phase Shift keying (Phase Modulation) of a polar line
code for bandpass conversion.
X
Polar
Line Coder
cos(ct)
g(t)
Xn
c
A
)
(t
s
)
(t
g
)
(t
s
1 1 1
0 0
2
Ac 2
2
Ac 2
n
X
Bandpass Signal Conversion
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Spectrum of Bandpass Signals
*
1 1
Re ( ) ( )
2 2
c c c
j t j t j t
v t g t e g t e g t e
t
j
t
j c
c
e
t
g
F
e
t
g
F
t
v
F
f
V
*
2
1
2
1
)
(
f
G
t
g
F
*
*
*
1
( ) - -
2
c c
V f G f f G f f
Theorem: If bandpass waveform is represented by
t
g
F
f
G
f
Pg
Where is PSD of g(t)
Proof:
Thus,
Using and the frequency translation property:
We get,
*
1
Spectrum of Bandpass Signal ( )
2
1
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 ( ) c
j t
v t g t e
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PSD of Bandpass Signals
Re Re c
c j t
j t
v
R v t v t g t e g t e
*
2 1 2 1 2 1
1 1
Re Re Re Re
2 2
c c c c c c
2 ( ) c
j t
c g t e
1
c
j t
c g t e
*
1 1
Re Re
2 2
c c
c c
j t j t
j t j t
v
R g t g t e e g t g t e e
,
2
*
1 1
Re Re
2 2
c c c
j j t j
v
R g t g t e g t g t e e
2
*
1 1
Re Re
2 2
c c c
j j t j
v
R g t g t e g t g t e e
PSD is obtained by first evaluating the autocorrelation for v(t):
Using the identity
where and
- Linear operators
*
g
g t g t R
but
1
Re
2
c
j
v g
R R e
AC reduces to
* *
1
( )
4
v v g c g c g g
P f F R P f f P f f P P f
PSD =>
=>
We get
or g(t)
fc in
s
frequencie
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Evaluation of Power
2
v v
P v t P f df
1 2
j f
v v v
R F P f P f e df
0
v v
R P f df
*
1 1 1
0 Re 0 Re 0 Since Re
2 2 2
c
j
v g v g
R R g t g t R R e
2
1
0 Re
2
v
R g t
2
1
0
2
v
R g t
g t
Theorem: Total average normalized power of a bandpass waveform v(t) is
Proof:
But
So,
or
But is always real
So,
2
2 1
0
2
v v v
P v t P f df R g t
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Example : Amplitude-Modulated Signal
Evaluate the magnitude spectrum for an AM signal:
1
c
g t A m t
Complex envelope of an AM signal:
c c
G f A f A M f
Spectrum of the complex envelope:
1
2
c c c c c
S f A f f M f f f f M f f
AM spectrum:
1 1
, 0
2 2
1 1
, 0
2 2
A f f A M f f f
c c c c
S f
A f f A M f f f
c c c c
Magnitude spectrum:
AM signal waveform:
Re ( ) 1 cos
c
j t
c c
s 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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2 2
2
2 2
2 2
2
Side
2
b
2
a
2
nd
2
2
If DC value of ( ) is zero
1
2
Carrier
1
Powe
1
1
2 2
1
1 2
2
1
1 2
2
1
1
2
1
1
2
Where
1
2
Sideband Po er
r w
s c
c
c
c
c
c m c
m
m
c
P g t A m t
A m t m t
A m t m t
A m t
A A P
P
P m
t
A P
t
P
m
Example : Amplitude-Modulated Signal
Total 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;
2 1000 2 1000
0.8
0.8sin 2 1000
2
j t j t
m t t e e
j
250 100 1000 100 1000
250 100 1000 100 1000
c c c
c c c
S f f f j f f j f f
f f j f f j f f
0.4 1000 0.4 1000
M f j f j f
Fourier transform of m(t):
1
2
c c c c c
S f A f f M f f f f M f f
Spectrum of AM signal:
Substituting the values of Ac and M(f), we have
22. EEE 360 22
0 0
2 2
0 o
cos A=0.8 and 2 1000
2 2
j j
m
A A
R e e
2 2
0 0 1000 1000
4 4
m
A A
P f f f f f f f
2
2
* 1 1
1
g c
c
R g t g t A m t m t
A m t m t m t m t
SA4-2. PSD for an AM signal
Autocorrelation for a sinusoidal signal (A sin w0t )
Autocorrelation for the complex envelope of the AM signal is
Study Examples
2
But 1 1, 0, , 1
m g c m
m t m t m t m t R R A R
2 2
g c c m
P f A f A P f
2
1
g c m
R A R
Thus
1
( )
4
v g c g c
P f P f f P f f
Using
62500 10000 1000 10000 1000
62500 10000 1000 10000 1000
s c c c
c c c
P f f f f f f f
f f f f f f
PSD for an AM signal:
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Study Examples
2
165
s s s
norm rms
P V P f df kW
2 5
1.65 10
3.3 kW
50
s rms
s norm
L
V
P
R
2
2 2
2 2
1 1 0.8
1 500 1 165
2 2 2
s s c m
norm rms rms
P V A V kW
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:
2 5
4.05 10
8.1
50
PEP rms
PEP actual
L
P
P kW
R
SA4-4. PEP for an AM signal
2 2
2 2 2
1 1 1
max 1 max 500 1 0.8 405
2 2 2
PEP c
norm
P g t A m t kW
Normalized PEP:
Actual PEP for this AM voltage signal with a 50 ohm load: