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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
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
Eeng 360 3
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
Eeng 360 4
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

 
 
 
  
 

Eeng 360 5
Generalized transmitter using the AM–PM generation
technique.
Eeng 360 6
Generalized transmitter using the quadrature
generation technique.
Eeng 360 7
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 

Eeng 360 8
 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

 
 
  
 
Eeng 360 9
 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
Eeng 360 10
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

 
 
Eeng 360 11
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.
Eeng 360 12
 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
Eeng 360 13
Mapping Functions for Various Modulations
Eeng 360 14
Envelope and Phase for Various Modulations
Eeng 360 15
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 

Eeng 360 16
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

Eeng 360 17
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


   

Eeng 360 18
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  
 
  
 
   
  
Eeng 360 19
Example : Amplitude-Modulated Signal
Spectrum of AM signal.
Eeng 360 20
   
   
   
 
 
 
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:
EEE 360 21
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(21000t); 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
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:
EEE 360 23
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:

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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
  • 3. Eeng 360 3 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
  • 4. Eeng 360 4 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             
  • 5. Eeng 360 5 Generalized transmitter using the AM–PM generation technique.
  • 6. Eeng 360 6 Generalized transmitter using the quadrature generation technique.
  • 7. Eeng 360 7 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  
  • 8. Eeng 360 8  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          
  • 9. Eeng 360 9  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
  • 10. Eeng 360 10 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     
  • 11. Eeng 360 11 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.
  • 12. Eeng 360 12  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
  • 13. Eeng 360 13 Mapping Functions for Various Modulations
  • 14. Eeng 360 14 Envelope and Phase for Various Modulations
  • 15. Eeng 360 15 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  
  • 16. Eeng 360 16 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 
  • 17. Eeng 360 17 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       
  • 18. Eeng 360 18 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                
  • 19. Eeng 360 19 Example : Amplitude-Modulated Signal Spectrum of AM signal.
  • 20. Eeng 360 20                   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:
  • 21. EEE 360 21 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(21000t); 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:
  • 23. EEE 360 23 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: