Señales pasobanda o a radiofrecuencia

1. Eeng 360 1
Chapter4
Bandpass Signalling
 Bandpass Filtering and Linear Distortion
 Bandpass Sampling Theorem
 Bandpass Dimensionality Theorem
 Amplifiers and Nonlinear Distortion
 Total Harmonic Distortion (THD)
 Intermodulation Distortion (IMD)
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University

2. Eeng 360 2
Bandpass Filtering and Linear Distortion
 Equivalent Low-pass filter: Modeling a bandpass filter by using an equivalent low
pass filter (complex impulse response)
]
)
(
Re[
)
( 1
1
t
jwc
e
t
g
t
v  ]
)
(
Re[
)
( 2
2
t
jwc
e
t
g
t
v 
]
)
(
Re[
)
( 1
1
t
jwc
e
t
k
t
h 
*
1 1
( ) ( ) ( )
2 2
c c
H f K f f K f f
   
   
 
c
c f
f
G
f
f
G
f
V 



 *
2
1
)
(
   
t
j c
e
t
g
t
v 
)
(
Re

)
(
1 t
v
)
(
2 t
v
)
(
1 t
h
)
( f
H
Input bandpass waveform
Output bandpass waveform
Impulse response of the bandpass filter
Frequency response of the bandpass filter
H(f) = Y(f)/X(f)
Bandpass filter

3. Eeng 360 3
Bandpass Filtering

4. Eeng 360 4
Bandpass Filtering
     ;
2
1
2
1
2
1
1
2 t
k
t
g
t
g 

     
f
K
f
G
f
G
2
1
2
1
2
1
1
2 
     
f
H
f
V
f
V 1
2 
   
 
c
c f
f
G
f
f
G 


 *
2
2
2
1
   
     
 
c
c
c
c f
f
K
f
f
K
f
f
G
f
f
G 







 *
*
1
1
2
1
2
1
       
       























c
c
c
c
c
c
c
c
f
f
K
f
f
G
f
f
K
f
f
G
f
f
K
f
f
G
f
f
K
f
f
G
*
*
1
*
1
*
1
1
4
1
    ,
0
*
1 


 c
c f
f
K
f
f
G
Theorem:
g1(t) – complex envelope of input
k(t) – complex envelope of impulse response
Also,
Proof: Spectrum of the output is
Spectra of bandpass waveforms are related to that of their complex enveloped
But
    .
0
*
1 


 c
c f
f
K
f
f
G
           


































 c
c
c
c
c
c f
f
K
f
f
G
f
f
K
f
f
G
f
f
G
f
f
G *
*
1
1
*
2
2
2
1
2
1
2
1
2
1
2
1
2
1
The complex envelopes for the input, output, and impulse response of
a bandpass filter are related by

5. Eeng 360 5
Bandpass Filtering
     
f
K
f
G
f
G
2
1
2
1
2
1
1
2 
Taking inverse fourier transform on both sides
     ;
2
1
2
1
2
1
1
2 t
k
t
g
t
g 

Thus, we see that
 Any bandpass filter may be described and analyzed by using an equivalent low-pass
filter.
 Equations for equivalent LPF are usually much less complicated than those for
bandpass filters & so the equivalent LPF system model is very useful.

6. Eeng 360 6
Linear Distortion
  A
f
H 
 
g
T
df
f
d




2
1
  0
2 

 

 g
fT
f
For distortionless transmission of bandpass signals, the channel transfer function
H(f) should satisfy the following requirements:
     
f
j
e
f
H
f
H 

 The amplitude response is constant
A- positive constant
 The derivative of the phase response is constant
Tg – complex envelope delay
  d
fT
f
H
f 
 2
)
( 



  )
( f
H
f 


Integrating the above equation, we get
Are these requirements sufficient for distortionless transmission?
constant
shift
phase
0 


7. Eeng 360 7
Linear Distortion

8. Eeng 360 8
   
  g
g fT
j
j
fT
j
e
Ae
Ae
f
H



 2
2 0
0 




      t
t
y
t
t
x
t
v c
c 
 sin
cos
1 

g
fT
j
e

2

     
     
 
0
0
2 sin
cos 


 






 g
c
g
g
c
g T
t
T
t
Ay
T
t
T
t
Ax
t
v
     
     
 
c
c
g
c
c
g f
t
T
t
Ay
f
t
T
t
Ax
t
v 


 




 sin
cos
2
  d
c
g
c
c T
f
T
f 


 2
0 





     
     
 
d
c
g
d
c
g T
t
T
t
Ay
T
t
T
t
Ax
t
v 




 
 sin
cos
2
    delay
phase
T
shift
phase
carrier
;
2 d 



 c
d
c
c f
T
f
f 


Linear Distortion
The channel transfer function is
     
f
j
e
f
H
f
H 

If the input to the bandpass channel is
Then the output to the channel (considering the delay Tg due to ) is
Using   0
0
2 



 




 g
g T
fT
f
  d
fT
f
H
f 
 2
)
( 



Modulation on the carrier is delayed by Tg & carrier by Td
Bandpass filter
delays input info by
Tg , whereas the
carrier by Td

9. Eeng 360 9
Bandpass Sampling Theorem
T
s B
f 2

      t
t
y
t
t
x
t
v c
c 
 sin
cos 

  2
1
2 f
f
fc 

   
 
 
 
 





































n
n b
b
b
b
c
b
c
b f
n
t
f
f
n
t
f
t
f
n
y
t
f
n
x
t
v




sin
sin
cos
 
b
f
n
x  
b
f
n
y
If a waveform has a non-zero spectrum only over the interval , where
the transmission bandwidth BT is taken to be same as absolute BW, BT=f2-f1, then
the waveform may be reproduced by its sample values if the sampling rate is
2
1 f
f
f 

Theorem:
Quadrature bandpass representation
Let fc be center of the bandpass:
x(t) and y(t) are absolutely bandlimited to B=BT/2
The sampling rate required to represent the baseband signal is T
b B
B
f 
2
Quadrature bandpass representation now becomes
Where and samples are independent , two sample values
are obtained for each value of n
Overall sampling rate for v(t): T
b
s B
f
f 2
2 


10. Eeng 360 10
Bandpass Dimensionality Theorem
0
2 T
N B T

 Assume that a bandpass waveform has a nonzero spectrum only over a frequency
interval , where the transmission bandwidth BT is taken to be the absolute
bandwidth given by BT=f2-f1 and BT<<f1.
The waveform may be completely specified over a T0-second interval by N Independent
pieces of information. N is said to be the number of dimensions required to specify the
information.
2
1 f
f
f 


11. Eeng 360 11
Received Signal Pulse
   
 
t
j c
e
t
g
t
s 
Re

       
t
n
t
h
t
s
t
r 


     
 
 
 
t
n
e
T
t
Ag
t
r c
c f
t
j
g 

 

Re
   
   
t
n
e
t
g
t
r t
j c

 
Re
The signal out of the transmitter
Transmission
medium
(Channel)
Carrier
circuits
Signal
processing
Carrier
circuits
Signal
processing
Information
m
input m
~
)
(
~ t
g
)
(t
r
)
(t
s
)
(t
g
g(t) – Complex envelope of v(t)
If the channel is LTI , then received signal + noise
n(t) – Noise at the receiver input
Signal + noise at the receiver input
Signal + noise at the receiver input
- carrier phase shift caused by the channel, Tg – channel group delay.
)
( c
f

A – gain of the channel

12. Eeng 360 12
Amplifiers
Non-linear Linear
Circuits with memory and circuits with no
memory
Memory - Present output value ~ function of present input + previous input values
- contain L & C
No memory - Present output values ~ function only of its present input values.
Circuits : linear + no memory – resistive ciruits
- linear + memory – RLC ciruits (Transfer function)
Nonlinear Distortion

13. Eeng 360 13
Nonlinear Distortion
Assume no memory  Present output as a function of present input in ‘t’ domain
   
t
Kv
t
v i

0
K- voltage gain of the amplifier
• If the amplifier is linear
• In practice, amplifier output becomes saturated as the amplitude of
the input signal is increased.











0
2
2
1
0
0
n
n
i
n
i
i v
K
v
K
v
K
K
v
output-to-input characteristic (Taylor’s expansion):
Where
0
0
!
1










i
v
n
i
n
n
dv
v
d
n
K
0
K
i
v
K1
2
2 i
v
K
- output dc offset level
- 1st
order (linear) term
- 2nd
order (square law) term

14. Eeng 360 14
  t
A
t
vi 0
0 sin

   
t
A
K
t
A
K 0
2
0
2
2
0
0
2 2
cos
1
2
sin 
 

      









 3
0
3
2
0
2
1
0
1
0 3
cos
2
cos
)
cos( 




 t
V
t
V
t
V
V
t
vout
100
V
THD(%)
1
2
n
2




 n
V
Nonlinear Distortion
Let the input test tone be represented by
Harmonic Distortion associated with the amplifier output:
Then the second-order output term is
In general, for a single-tone input, the output will be
Vn – peak value of the output at the frequency nf0
2
2 i
v
K =
To the amplifier input
The Percentage Total Harmonic Distortion (THD) of an amplifier is defined
by
2nd
Harmonic
Distortion with 2
2
0
2 A
K

15. Eeng 360 15
Nonlinear Distortion
Intermodulation distortion (IMD) of the amplifier:
If the input (tone) signals are   t
A
t
A
t
vi 2
2
1
1 sin
sin 
 

Then the second-order output term is
   
t
A
K
t
t
A
A
K
t
A
K
t
A
t
t
A
A
t
A
K
t
sn
A
t
A
K
2
2
2
2
2
2
1
2
1
2
1
2
2
1
2
2
2
2
2
2
1
2
1
1
2
2
1
2
2
2
2
1
1
2
sin
sin
sin
2
sin
sin
sin
sin
2
sin
sin

















IMD
Harmonic distortion at 2f1 & 2f2
Second-order IMD is:
 
   
 
 
2
1
2
1
2
1
2
2
1
2
1
2 cos
cos
sin
sin
2 




 


 t
A
A
K
t
t
A
A
K

16. Eeng 360 16
Nonlinear Distortion
Third order term is  
)
sin
sin
sin
3
sin
sin
3
sin
(
sin
sin
2
3
3
2
2
2
1
2
2
1
2
1
2
2
2
1
1
3
3
1
3
3
2
2
1
1
3
3
3
t
A
t
t
A
A
t
t
A
A
t
A
K
t
A
t
A
K
v
K i














 
t
t
A
A
K
t
t
A
A
K 1
2
2
2
1
3
2
1
2
2
2
1
3 2
cos
1
sin
2
3
sin
sin
3 


 

   
 










 t
t
t
A
A
K 2
1
2
1
2
2
2
1
3 2
sin
2
sin
2
1
sin
2
3





   
 










 t
t
t
A
A
K
t
t
A
A
K
1
2
1
2
1
2
2
1
3
2
2
1
2
2
1
3
2
sin
2
sin
2
1
sin
2
3
sin
sin
3







The third term is
The second term (cross-product) is
Intermodulation terms at nonharmonic frequencies
For bandpass amplifiers, where f1 & f2 are within the pasband, f1 close to f2,
the distortion products at 2f1+f2 and 2f2+f1 ~ outside the passband
Main Distortion Products