Bandpass Signalling & Communication Aspects

Bandpass Signalling & Communication Aspects

1. Chapter 4
Bandpass Signaling

2. In this chapter, we consider the situations where
the information from a source is transmitted at its
non-natural frequency (i.e., shifted frequency).
This process is called the modulation.
1.Representation of modulated signals
2.Spectra
3.Distortions (linear and non-linear)
4.Functional blocks in bandpass
communication systems

3. Basic Model for Bandpass Communication
Source Destination
Source can be analog or digital. The use of channel is restricted around certain
frequency, f
c
(>> 0). For example, a radio station may be given this frequency
range for commercial broadcasting. The goal is to recover the original information,
m, exactly or in the minimum, as closely as possible.
m(t)tm
g(t)tg
tr
ts
tg
tm
ingapproximat signal tedreconstruc :)(
~
ingapproximat signal ddemodulate :)(
~
frequency) (shifted channel in the corrupted signal modulated :)(
frequency) (shifted ed transmittbe tosignal modulated :)(
(baseband) modulated be n toinformatio processed :)(
(baseband) sourceby sent be n toinformatio :)(

4. Definition. A baseband waveform has a spectral
magnitude (and thus its power) concentrated
around f=0 and zero elsewhere.
Definition. A bandpass waveform has a spectral
magnitude concentrated around f=±f
c (f
c >> 0) and
zero elsewhere.
(f
c: carrier frequency)
Definition. Modulation translates the baseband
waveform from a source to a bandpass waveform
with carrier frequency, f
c.
baseband waveform: modulating signal
bandpass waveform: modulated signal

5. Examples of Frequency Spectrum
300 Hz – 20K Hz human voice / sound
50 kHz navigation (ships, submarines, etc)
1 MHzAM radio (20 k Hz channels)
10 MHzCB, short wave
100 MHzFM radio, TV
1 GHzUHF TV, mobile telephony
10 GHzamateur satellite
100 GHzupper microwave
10 T HzInfrared
10
15
Hz Visible light
10
18
HzX-rays

6.  
 
sinusoid. pure a is then constant, a is If
signal) baseband(complex envelopecomplex :)(
signal) baseband (real modulation (PM) phase :)(
signal) baseband positive and (real modulation (AM) amplitude :)(
signal) baseband (real modulation (Q) quadrature :)(
signal) baseband (real modulation (I) phase-in :)(
.)()()()( where
sin)(cos)()(cos)()( ely,Alternativ
.
2
frequency,carrier a tod)heterodyne(or shifted equals I.e.,
.)(Re)(
byrepresent becan ,)( waveform,bandpass physicalAny Theorem.
)(
v(t)g(t)
tg
t
tR
ty
tx
etRtjytxtg
twtytwtxttwtRtv
)
π
w
(fg(t)v(t)
etgtv
tv
tj
ccc
c
c
tjw
c








7.  
 
 
corruption little with recovered becan -
efficentlybandwidth channel theuses -
receivers and rstransmitteimplement easy to -
such that develop toision communicat bandpass of goalmain The
. offunction a is where)(Re)( can write weThus
signal) (bandpass )( signal) (baseband :Modulation
1)( if180
1)( if0
)( and )(1)(
0)( and )(1)(
)(1
(AM), Modulation AmplitudeFor :Example
m(t)
s(t)
)g(
m(t))g(etgts
tsm(t)
tm
tm
ttmAtR
tytmAtx
tmAg(t)
tjw
oc
c
c
c













8.  
   
   .
4
1
Density SpectrumPower
2
1
Spectrum
, and )(Re)( waveform,bandpass aFor Theorem.
*
*
cgcgv
cc
tjw
ffff(f)
ffGffGV(f)
G(f)g(t)etgtv
c



PPP
   
 
 
 
.real is )( and )()( Also
real. is because )()(fact theused wewhere
)()()()(
2
1

)()()()(
2
1
Then, .cos)(1)( Thus
).( where )()()()(1
(AM), Modulation AmplitudeFor :Example
*
**
fff
m(t) fMfM
ffMffffMffA
ffMffffMffAS(f)s(t)
twtmAts
fMm(t)fMfAfGtmAg(t)
ccccc
ccccc
cc
cc











9.  
  
 
22
*
2
2
2
)(
2
1
)(Re
2
1
)0
)0()(Re
2
1
)0(Re
2
1
)0
Then, ).( offunction ation autocorrel
theis where)(Re
2
1
) seen that becan It
).( signal, any timefor trueis 0)( Proof.

).( offunction ation autocorrel
theis where)(
2
1
0)( power,
normalized average total the),( waveform,bandpass aFor Theorem.
tgtg(R
tgtgR(R
tg
)(ReR(τR
tv)(R(f)dftvP
tv
)(Rtg)(R(f)dftvP
tv
v
gv
g
jw
gv
vvv
vvvv
c














P
P

10.  
 
 

).( n,informatioith w
associatedpower :
2
1
powercarrier :
2
1
1
2
1

)(1
2
1
coding, goodand/or signal of
nature todue 0)(With
)()(21
2
1
)()(21
2
1
)(1
2
1
)()()()(
2
1

. spectrum,ngular with triasignal AM :Example
2
2
2
22
222222
tm
PA
A
PA
tmAP
tm
tmtmAtmtmAtmAP
ffMffffMffAS(f)
M(f)
mc
c
mc
cs
cccs
ccccc




 

11.  
2
PEP
PEP
)(max
2
1

by given is , PEP, nomarlized The Theorem.
.peak value
itsat constant held be to were if obtained be that would
power average theis (PEP)power enveloppeak The .Definition
tgP
P
g(t)


12. signals? bandpassfor ion approximat baseband a thereIs -
answer? therate samplingNyquist theIs
n?informatio original theoftion reconstruc
perfectfor sampled be waveformbandpass amust fast How -
power. computing intense requires
0frequency carrier at waveformbandpass a Simulating -
difficult. is waveformsbandpassfor analysis alMathematic -
Questions and nsObservatio
) (f
c


13. Bandpass Signals over Bandpass Channel
Out of Transmitter Into Receiver
Channel
 
c
ffK 
*
2
1
 
c
ffK
2
1
fH
Can we translate this into a baseband model? YES!
 
)(
2
1
)(
2
1
)()(
)(Re)( :response impulse bandpass
*
cc
tjw
ffKffKfHth
etkth
c



14.  
 
 
)( n,informatio
corruptedy potentiall containing envelopecomplex :
)(Re)(
)( input, todue channel theofout waveformbandpass :)(
)( signal, bandpassfor envelopecomplex :
)(Re)(
frequency, bandpass thearound channel for the response impulse :)(
)( n,informatio thecontaining envelopecomplex :
)(Re)(
channel theinto ed transmittbe to waveformbandpass :)(
Notation
2
22
12
1
11
1
tm
(t)g
etgtv
tvtv
thk(t)
etkth
fth
tm(t)g
etgtv
tv
tjw
tjw
c
tjw
c
c
c




15.    
 
 
  )()(
2
1
)()(
2
1
)()(
2
1

thatimplies )()( )( Thus,
)()(
2
1
)(
)()(
2
1
)( )()(
2
1
)(
thatNote .assumptionlinearity from )()()( that trueisIt Proof.
)(
2
1
)(
2
1
)(
2
1

)(
2
1
)(
2
1
)(
2
1

Theorem.
**
11
*
22
12
*
*
111
*
222
12
12
12
cccc
cc
cc
cccc
ffKffKffGffG
ffGffG
fHfVfV
ffKffKfH
ffGffGfVffGffGfV
fHfVfV
fKfGfG
tktgtg









16.  
  


 
  
)(
2
1
)(
2
1
)(
2
1
.)()(
4
1
)(
2
1
that truebemust it Thus
)()()()(
4
1
)()()()(
)()()()(
4
1
)()(
2
1
)()(
2
1
)()(
2
1

).(Continued Proof
12
12
**
11
**
1
*
1
*
11
**
11
*
22
fKfGfG
ffKffGffG
ffKffGffKffG
ffKffGffKffG
ffKffGffKffG
ffKffKffGffG
ffGffG
ccc
cccc
cccc
cccc
cccc
cc








17. Equivalent Baseband Model for Bandpass Signals
Out of Transmitter Into Receiver
Channel
Equivalent baseband impulse response
fK
2
1
We can now decouple the complexity of shifted frequency.

18. less.distortiony essentiall is distortionlinear Thus,
. from t reconstruc then to trivialisIt
.input, thefromdelay time
constant has and offactor aby changed amplitude its has channel
thefrom , output, thesatified, are conditions twoabove theIf
delay) group :( 2
)(

.)10Typically change. magnitude(Constant )(
:conditions following the
satisfymust ,)()( function, transfer channel The
DistortionLinear
21
1
2
)(
(t)v(t)v
(t)v
A
(t)v
TT
df
fd
AAfH
efHfH
gg
fj








19. Distortionless Bandpass Channel
 
 
TT
TtwTtAy
TtwTtAx(t)v
twtytwtx(t)v
dg
dcg
dcg
cc
. general,In
sin)(
cos)(
then
,sin)(cos)( If
Example.
2
1





20. . wheresamplest independen2by
interval over time specified completely becan
tolimitedbandwidth with waveformbandpassA Theorem.
baseband.
at thenecessary frequency sampling the twiceis thisy,Essentiall
).( oftion reconstrucfor 4 and 2 Thus, .
and ,
2
Assuming . todbandlimite
signal baseband a is )( wherecos)()( Suppose Example.
. where
2at signal sampled from tedreconstruc becan
tolimitedbandwidth with waveformbandpassA Theorem.
12
21
2
1
12
21
ffB TBN
Tfff
txBfBBBff
BffB
w
fHzB
txtwtxtv
ffB
Bffff
ToT
o
sTc
c
c
c
c
T
Ts









21. Definition. An filter is a linear time-invariant system used for frequency
discrimination. (Filters modify the frequency spectrum of input waveform
to produce output waveform.)
Quality Factor:
o
f
Q
B

: center (or resonant) frequency, : 3 bandwidth
o
f B -dB

22. Types of (Analog) Filter

23. Suited for PCM type signals
Implemented typically in software algorithms (e.g., DFT, DCT)
Practical for inherently digital signals
Advantages
- Flexible
- Ada
Digital Filters
ptive
- Finely tunable (i.e., can be idealized)
A/D
Digital
Filter
D/A
analog x(t) analog y(t)
manipulate digital data

24. 2
1 2
2
1 2
Filter Characterstics
( ) ( ) ... ( )
Transfer function: ( )
( ) ( ) ... ( )
: order of the filter
Design goal: deter
k
o k
n
o n
b b jw b jw b jw
H f
a a jw a jw a jw
n
   

   
mine 's, 's, , and to result in a desired ( ).
There are many different types of filters (according to different classes of ( )).
i i
a b n k H f
H f

26. Signals may be linearly or non-linearly distorted (by the filters, channels, etc).
- Distortion-less transfer function:
- Linear distortion: linear input-output relationship, but no
c d
jw T
Ke

2
1 2
0
0
t in form
(linearity is defined by ( ) () ( ).)
- Non-linear distortion: ...
1
where
!
Fo
c d
i
jw T
o i
n
o o i i n i
n
n
o
n n
i
v
Ke
v t K t v t
v K K v K v K v
d v
K
n dv





    
 
 
 

r non-linear distortion, there exists 0 for 2.

n
K n 

27. Example of Non-Linear Distortion by Output
Saturation

28. Harmonic Distortion
   
1 1 2 2
This is a non-linear distortion at multiple harmonic frequencies of the input.
Test: Use a single frequency input ( ) sin and measure output ( ).
In general, ( ) cos cos 2
i o o o
o o o o
v t A w t v t
v t V V w t V w t 

      
   
3 3
2
2
1
2
2
2
2
2
2
cos 3 ...
Total Harmonic Distortion (THD%) 100
Example. Second Harmonic Distortion for ( ) sin
The second order term of the output = sin 1 cos2
2
: DC bi
2
o
n
n
i o o
o
o o o
o
V w t
V
V
v t A w t
K A
K A w t w t
K A



  
 

 

2
2
as distortion cos2 : second harmonic distortion
2

o
o
K A
w t

29. 
2
2
Example. Output from filter: ( ) cos
1
Power in each harmonic term =
2
Since the harmonic terms are orthogonal to each other,
1
total power of output signal = .
2
Suppose that the
o n o
n
n
n
n
v t V nw t
V
V


1
2
2 2
1
2
2
2
2
1
4
input and output each has total power of 1 and .
1 1 4
2 1 2 1
2 2
1 4
2 1
2
THD (%) 100 100 = 48.3%
4
n
n
n
n
V
V V
V
V








 
   
        
 
   
 
 
 
  
 
 
 
   



30. Intermodulation Distortion
(IMD)
1 1 2 2
2 1
This is a non-linear distortion caused by the interference among multiple signal
Test: Use input with two sinusoidal frequencies: ( ) sin sin
In general, the second order output term:
s
i
v t A wt A w t
K A
 
   
    
2
2 2 2 2
1 2 2 2 1 1 1 2 1 2 2 2
2 1 2 1 2
2 1 2 1 2 2 1 2 1 2 1 2
in sin sin 2 sin sin sin
2 sin sin : Intermodulation Distortion (IMD)
2 sin sin cos cos
If the bandpass region is narrow eno
wt A w t K A wt AA wt w t A w t
K AA wt w t
K AA wt w t K AA w w t w w t
   
      
   
  
1 2 1 2
1 2 1 2
ugh, both and
components fall outside the bandpass region (since , 0, >>0 ).
Thus the IMD caused by the second order outputs (i.e., the square terms)
is not significant.
w w w w
w w w w
 
 

31.   

3
3 3 2 2
3 1 1 2 2 3 1 1 1 2 1 2
2 2 3 3
1 2 1 2 2 2
3 1
Now consider the third order output term. In general,
sin sin sin 3 sin sin
3 sin sin sin
3
K A wt A w t K A wt A A wt w t
AA wt w t A w t
K A
   

   
   
 
2 2 2
2 1 2 3 1 2 2 1 2 1 2
2 2 2
3 1 2 1 2 3 1 2 1 2 1 2 1
1 2 2
3 1
sin sin sin sin 2 sin 2
2 2
3 1
3 sin sin sin sin 2 sin 2
2 2
The IMD terms correspond to those with the 2 , 2
A wt w t K A A w t w w t w w t
K AA wt w t K AA wt w w t w w t
w w w
 
            
 
 
            
 
   
 
  
 
1 1 2
2 1 1 2 1
2 1 2 2 1
1 2
, 2 , and
2 . Again, if the bandpass region is narrow and , 0, and
0, then, the 2 and 2 frequency componen ts fall outside the
bandpass region. However, 2 and 2
w w w
w w w w w
w w w w w
w w

  
  
  
2 1
1 2
1
IMD 2
3
frequency components are
likely to be within the bandpass region. If , we define,
4
: ratio of desired output to IMD output
3
w w
A A A
K
R
K A

 
 
 
 

32. IMD Analysis for Filter Output

33. 1 3
IMD
1
IMD 10 2
3
1
2
3
1
2
3
2
Example. For a given filter let 1 and 0.01.
Suppose we want 40 .
Find maximum allowed input magnitude.
4
40 20log
3
4
100
3
75
100 75
2
K K
R dB
K
R dB
K A
K
K A
K
K A
A
 

  
    
   
 
  
 
 
  
 
 

3
A

34. Cross Modulation (Distortion)
This is a non-linear distortion caused by the modulating signal of one carrier
signal generating interferences to other carrier signals.
Test: Use input with two sinusoidal frequencies and one AM modul
 
   
1 1 2 2
2
2
3 1 2 1 2
ated.
( ) 1 ( ) sin sin .
Again, the second order output terms fall outside the bandpass region.
( ) contains the third order output terms in the form of
3
1 ( ) sin 2
2
i
o
v t A m t wt A w t
v t
K A m t A w w t
  
     
 
 
2
3 1 2 2 1
2
3
1 2 1 2 3 1
3
3 1
3
and 1 ( ) sin 2 .
2
3
If and , the two terms become 1 ( ) sin and
2
3
1 ( ) sin . These are the cross modulation distortion terms.
2
K A m t A w w t
w w A A A K A A m t wt
K A m t wt
    
   
   


35. Limiter
A limiter is a non-linear circuit that compares the input to a certain threshold value.
The output indicates either comparison is true or false (i.e., binary results).
Typically the output is a saturated minimum or maximum value.

36. Mixer
This is a non-linear device whose output is the product of two inputs.
The result is a mathematical multiplication of real time values of the two inputs.
(Different from the audio mixers that combine multiple audio signals into
a single signal.)
input1(t)
input2(t)
output(t)
= input1(t) x input2(t)

37. In communication, mixers are used with bandpass or low-pass filters to shift
the frequency of incoming signal and filter out the undesired components of
the multiplication.

38.  
 
 
 
 
in in in
in
1 in in
Let ( ) be a bandpass signal with ( ) =Reg ( ) .
Thus ( ) is a baseband signal containing the message.
( ) = Re g ( ) Re g ( )
2 2
If the upper frequency is selected (u
c
c o c o
jw t
j w w t j w w to o
v t v t t e
g t
A A
v t t e t e
 
 
 
 
 
 
 
 
2 in
2 in
2 in
2
p converter),
( ) = Re g ( )
2
If the lower frequency is selected (down converter),
( ) = Re g ( ) if
2
or ( ) = Re g ( ) if
2
or
c o
c o
o c
j w w to
j w w to
c o
j w w to
o c
A
v t t e
A
v t t e w w
A
v t t e w w
v





in
( ) = g ( ) if
2
o
c o
A
t t w w 

39. Mixer implementation
- Solid state devices (e.g., FET)
- Non linear devices
- Switching devices
The nonlinear device generates “undesired”
effects of product term between v
in
(t) and
v
LO
(t).

40. Mixer Implementation through Switching

41. Double-Balanced Mixer

42.  
1

Non-linear device that multiplies the input signal frequency by a factor of .
is a positive integer.
( ) ( )cos ( )
( )
Frequency Multiplier
in c
n
n
v t R t wt t
v t
 
   
 
out
( )cos ( ) other terms
( ) ( )cos ( )
n
c
n
c
CR t nwt n t
v t CR t nwt n t


 
  

43. More on Frequency Multiplier
If the message ( ) is contained in ( ), then it is distorted in a non-linear
way, i.e., ( ). (Hence, this is not good for AM modulated signals.)
If the message ( ) is contained in ( ), then it i
n
m t R t
R t
m t t  s NOT distorted, rather it is
amplified by a factor of . This may be a desired property for FM or PM
modulated signal. However, the amplification of the angle makes the required
bandwidth larger.
n

44. Detector Circuits
Source Destinatio
n
Suppose that the sent signal ( ) (which becomes ( ) and ( )) was received
(as ( )) and frequency translated to a baseband signal ( ( )). Now the task
is tp extract ( ) (approximating ) fro
m t g t s t
r t g t
m t m(t)

 m ( ). This task is called the
"detection." I.e., the detector circuits generate ( ), ( ), ( ), and/or ( )
from ( ).
There are different ways to implement the detector circuits. Common ones a
g t
R t t x t y t
g t



re:
Envelop Detector (for AM signals)
Product Detector (for AM and PM signals)
Frequency Modulation Detector (for FM signals)




45. Envelop Detector
 ( ) ( )cos ( )
( ) ( )
The time constant must
be chosen so that the output of
the low pass filter follows the
envelop of the input signal, ( ).
The bandwidth of the low pass
filt
in c
out
in
v t R t w t t
v t KR t
RC
v t


 
 

er must be much smaller than
and much larger than the bandwidth
of ( ).
1

2
c
c
f
m t
B f

  
low pass filter

46.  
 
The envelop detector is suted for AM signals. For AM signals,
( ) 1 ( ) . ( 0 is the strength of the signal. ( ) 1.)
( ) ( ) ( ) 1 ( ) ( )
can be used for automatic gai
c c
out c c c
c
g t A m t A m t
v t KR t K g t KA m t KA KAm t
KA
   
      
n control.
( ) is the detected signal.
1
For AM radios, the cutoff frequency of the low pass filter, 20 ,
2
10 , and 500 1600 .
c
c
KAm t
kHz
RC
B kHz kHz f kHz


  

47. Product Detector
Product detector uses a mixer and manipulates the input signal frequency.

48.    
   
 
 
1
- ( )
( ) ( )cos ( ) cos
1 1
( )cos ( ) ( )cos 2 ( )
2 2
1
After low pass filtering, ( ) ( )cos ( ) .
2
1
If we write ( ) Re ( ) , then ( ) ( ) (
2
o
c o c o
o o o c o
out o o
j j t
out o
v t R t wt t A w t
A R t t A R t w t t
v t A R t t
v t A g t e g t R t e x
 
 
   
 
  
    
 
  
o
) ( ).
With the oscillator signal frequency equalling the input signal frequency,
the detector is "frequency synchronized" (or "tuned") with the incoming signal.
If 0, then the detector output,
o
t jy t
v


o
1
( ) ( ).
2
1
If 90 , then the detector output, ( ) ( ).
2
ut o
o
out o
t A x t
v t A y t

 

49. o
1
If ( ) 0, ( ) ( ). (AM demodulation)
2
1
If ( ) (constant) and 90 , ( ) sin ( ).
2
1
This becomes ( ) ( ) if ( ) is small. (Phase demodulation)
2
Thus, the product det
out o
o
c out o c
out o c
t v t A R t
R t A v t A A t
v t A A t t

 
 
 
  

ector is sensitive to AM and PM.

50. Frequency Modulation Detector
 
 
The input signal, ( ), contains the message, ( ), as part of its
frequency information.
( ) ( )cos ( )
( )
( ) ( )
( )
If the detector is balanced, ( ) .
I
in
in c
out c c
out
v t m t
v t R t wt t
d d t
v t K wt t K w
dt dt
d t
v t K
dt




 
 
    
 
 

mplementation methods
FM-to-AM conversion
Phase-shift or quadrature detection
Zero-crossing detection




51.  
1
Let ( ) ( )cos ( ) , where ( ) ( ) s.
( ) has nothing to do with the message, (). It is a result of something
created during the transmission (e.g., interferences, noise, etc.)
( )
t
in c f
v t A t wt t t K m s d
A t m t
v t
 
 
  


 
 
2
cos ( )
( )
( ) sin ( )
( )
( ) ( )
L c
L c c
out L c L c L f
V wt t
d t
v t V w wt t
dt
d t
v t V w V w V K m t
dt




 
 
   
 
 
 
    
 
 
Slope Detector (FM-to AM Conversion)

52. Slope Detector Circuit

53. Balanced Discriminator

54. Balanced Zero-Crossing Detector

55. A PLL is a device whose output is a periodic signal synchronized in phase
with an input reference (periodic or almost periodic) signal.
Phase Lock Loop (PLL)

56. 2
VCO: oscillator that produces a period waveform with a frequency that may
be varied around free running frequency, .
VCO output frequency = when ( ) 0.
Phase Detector (PD): o
o
o
f
f v t 
utput is a function of the phase difference between
incoming signal ( ) and ( ).
PLL has two modes:
- Narrowband mode: tracks average frequency of ( ).
- Wideband mode: t
o in
in
v t v t
v t
racks instantaneous frequency of ( ).
Lock: When the PLL tracks the (average or instantaneous) frequency of ( ).
Hold-in range: When the PLL is in lock, the range of frequency of ( ) to remain

in
in
in
v t
v t
v t
in lock. (Also called the lock range.)
Pull-in range: When the PLL is not in lock, the range of frequency of ( ) to
capture a lock. (Also called the ca
in
v t
pture range.)
Maximum locked sweep range: When the PLL is in lock, the maximum rate
of change of the frequency of ( ) to remain in lock.
A PLL can be made in analog (APLL) or d
in
v t
igital (DPLL) circuits.

57. Different Phase Detector Characteristics

58.  
 
 
   
2
1
Let input be ( ) sin ( )
Suppose ( ) cos ( ) where ( ) ( ) .
: VCO gain in rad / volt-second
( ) sin ( ) ( ) sin 2 ( ) ( )
2 2
: multiplier gain,
in i o i
t
o o o o o v
v
m i o m i o
i o o i o
m
v t A w t t
v t A w t t t K v d
K
K AA K AA
v t t t w t t t
K

   
   
 
 
  
     

 

2
no unit
( ) sin ( ) ( )
Phase error: ( ) ( ) ( )
Equivalent PD gain:
2
Impulse response of LPF: ( )
d e
e i o
m i o
d
v t K t f t
t t t
K AA
K
f t

  
  
 


59. The PLL is a negative feedback system such that the error ( ) ( ) ( )
is minimized instantaneously. This is true if is large enough (i.e., sensitive
to the incoming phase ( )).
Note: When
e i o
d
i
e
t t t
K
t
  


 
o
( ) 0, ( ) and ( ) are 90 apart since they are sine and
cosine functions respectively. This is needed to use sin
approximation after the PD.
Then, the overall charatersi
in o
t v t v t
x x


 
tics of the PLL becomes
( ) ( )
( ) ( )
( ) ( )
( ): Transfer function of PLL
( ) 2 ( )
( ) ( ) and ( ) ( )
Note. ( ) is a transfer function for a linear system
e i
d v e
o d v
i d v
o o i i
d t d t
K K t f t
dt dt
f K K F f
H f
f j f K K F f
t f t f
H f
 


 
   

  
 
   
. I.e., the PLL is linearized
based on approximations and assumptions used here.

60. Linearized PLL Model
( ) ( )
( ) 2 ( )
o d v
i d v
f K K F f
f j f K K F f


 

61. Hold-in Range and Pull-in Range
Hysteresis indicates
stored energy
(or inertia)
in the PLL.
Hysteresis is useful against
noises or unexpected interruptions
in received signals. This is
called the “anti ping-pong”
characteristic.

62. 

2
Hold-in range calculation.
( )
( ) sin ( ) ( )
(0)sin ( ) (assuming the input frequency changes slow enough.)
1
(0)
2
Pull-in range is determined by the physical
o
v v d e
v d e
h v d
d t
K v t K K t f t
dt
w K K F t
f K K F




  
  
  
characteristics of
components in the PLL (e.g., PD, VCO).
Note: is always true for PLL.
h p
f f  

63. 2

Let ( ) sin ( )
( ) = ( ) and ( ) ( )
2
Assume that the PLL is in lock (i.e., ).
2
( )
( )
PLL as FM Detector
t
in i c f
t
f
i f i
c o
v
v t A w t D m d
D
t D m d f M f
j f
f f
f
j F f
K
V f
 
  


 
 
 
 
  
  

 
 
 
 


2
2
( )
( ) ( )
2 2
( ) ( )
( ) ( ) (Assuming ( ) 1 for , and .)
2
( ) ( )
f
v
i i
v d v d
f v d
v
D
F f
K
f f
f f
F f j F f j
K K K K
D K K
V f M f F f f B B
K
v t Cm t
 

  
   
    
   
    
 

64. A coherent detector uses an internally generated sinusoidal signal that has
the same frequency and phase with the incoming signal to detect th
PLL as Coherent AM Detector
o o
e messages.
The - 90 phase shifter is needed to eliminate the +90 phase shift between
( ) and ( ) in the basic PLL shown earlier.
o in
v t v t

65. ( ) has the frequency, = where is the frequency of the input
signal, ( ), which is generated by a local oscillator.
PLL as Frequency Synthesizer
o out x x
x
N
v t f f f
M
v t
 
 
 

66. Direct Digital Synthesis (DDS)
One way to generate any waveform is to store the desired waveform in the
memory (e.g., RAM or ROM), and play-back according to the desired timing.
- For arbitrary waveforms, signal is sampled/recorded for a desired length.
- For periodic waveforms, signals can be stored for one period only.
Playback repeats the same pattern over and over.
- For mathematical functions, signal values can be calculated and stored.
With DDS devices, one can eliminate unreliable analog devices
(e.g., oscillator). DDS devices are also useful for sound and music synthesis.

67. Generalized Transmitter (Type 1)
   
Bandpass signal (i.e., signal to be transmitted in the channel)
( ) Re ( ) ( ) ( )cos ( )
Thus the messsage, ( ), is modulated in either AM or PM signal.
c
jw t
c
v t g t e v t R t wt t
m t
   

68. Generalized Transmitter (Type 2)
 ( ) Re ( ) ( ) ( )cos ( )sin
Thus the messsage, ( ), is modulated in quadrature signal.
c
jw t
c c
v t g t e v t x t wt y t wt
m t
   

69. Generalized Receiver
Nearly all receivers are superheterodyne type. Superheterodyne implies
there are two different local oscillator frequencies that the incoming
signals are manipulated. (Heterodyne receivers have only one local operating
frequencies.) Two frequencies are: intermediate frequency (IF) and baseband.
Superheterodyne receivers need two local oscillators: one to bring
the incoming signals to the IF frequency, and another one to bring down
the IF frequency signals to the baseband to recover the messages.

70. The reasons for using the IF filter:
- Making a high quality filter is expensive. Design one good IF filter and use
it to achieve high gain and rejection of unwanted signals.
- Selection of IF filter frequency can be made for better signal processing
capability such as rejection of image frequency, reduction of non-linear
effects, reduction of noise, etc.
Image frequency is those happened to fall in the bandpass region of the filter
after the initial frequency translation.
2 if
For down converters:
2 if
For up converters: 2
c IF LO c
image
c IF LO c
image c IF
f f f f
f
f f f f
f f f
 

 

 

71. Example of Image Frequency

72. Zero IF Receiver
Zero IF receivers has a center frequency of its bandpass region at 0.
Thus it is a lowpass filter. Sometimes it is desired to use a zero IF receiver
to reduce the cost and design a high quality fil
IF
f
ter. For zero IF receivers, there
is no image frequency. When using a zero IF receiver, the system is a
heterodyne system.

73. At the sending transmitter: impurity in sinusoidal carrier
At the receiver: signal overload, non-linearity in RF and IF filters,

Possible Sources of Interference
cross modulation, adjacent signals
In the channel: non-linearity in the transmission medium, signals from adjacent
broadcasting region.

74. Note. If the receivers were made in digital circuit, the incoming signal
must be sampled at the bandpass frequency. It is not easy to do so.
2sin
c
w t