ECE 6640
Digital Communications
Dr. Bradley J. Bazuin
Assistant Professor
Department of Electrical and Computer Engineering
College of Engineering and Applied Sciences
Chapter 4
4. Bandpass Modulation and Demodulation/Detection.
1.
2.
3.
4.
5.
6.
7.
8.
9.
ECE 6640
Why Modulate?
Digital Bandpass Modulation Techniques.
Detection of Signals in Gaussian Noise.
Coherent Detection.
Noncoherent Detection.
Complex Envelope.
Error Performance for Binary Systems.
M-ary Signaling and Performance.
Symbol Error Performance for M-ary Systems (M>>2).
2
Sklar’s Communications System
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
3
Signal Processing Functions
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
4
Bandpass Demodulation and Detection
• Focus on Signal of Symbol, Samples, and Detection
• In the presence of Gaussian Noise and Channel Effect
ECE 6640
5
Analog Bandpass Modulation
Includes the RF/IF Frequency
• AM , PM and FM Modulation
st   At   cos2  f 0  t     t 
t


 A  1    m1 t  cos2  f 0  t     p  m2 t   2   f   m3    d 


• The time varying phase components
t   2  f 0  t    t 
t
 2  f 0  t     p  m 2 t   2   f   m 3    d
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6
Phasor Representation
• Taking the positive spectrum complex representation
st   At   Reexpj  2  f 0  t  j    j  t 
• Think in terms of the “analytical signal” representation
– Complex, positive frequencies only
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7
Example: Bandpass Phasor Analysis
of Double Sideband (DSB)
• Given a tone message …
mt   A m  cos2  f m  t 
st   A c  A m  cos2  f m  t   cos2  f c  t 
Ac  Am
s t  
 cos2  f c  f m   t   cos2  f c  f m   t 
2
• A positive frequency phasor can be defined and drawn
– First define the complex signal as (cos  exp)
Ac  Am
s pos  f  t  
 exp j  2  f c  f m   t   exp j  2  f c  f m   t 
4
C
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Phasor Analysis DSB (2)
• A positive frequency phasor can be defined and drawn
Ac  Am
s pos  f  t  
 exp j  2  f c  f m   t   exp j  2  f c  f m   t 
4
C
4
m
A
c
A
A
m
fc
f
m
A
4
c

ECE 6640
f
m

9
Phasor Analysis AM
• Given a tone message …
A m t   1    cos2  f m  t 
st   A c  1    cos2  f m  t   cos2  f c  t 
st   A c  cos2  f c  t  
Ac
A
   cos2  f c  f m   t   c    cos2  f c  f m   t 
2
2
• A positive frequency phasor can be defined and drawn
s t  pos  f  
C
ECE 6640
Ac
A
A
 exp j  2  f c  t   c    exp j  2   f c  f m   t   c    exp j  2   f c  f m   t 
2
4
4
10
Phasor Analysis AM (2)
• A positive frequency phasor can be defined and drawn
s t  pos  f  
C
Ac
A
A
 exp j  2  f c  t   c    exp j  2   f c  f m   t   c    exp j  2   f c  f m   t 
2
4
4
fm
4
c
A


c
2
A

fm
c
A
 4
fc
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11
Narrowband FM & PM Spectrum
s C t   A  expj  2  f c  t  j  t 
• Forming the Quadrature Representation and transforming
the series expanded rig functions
s C t   A  expj  2  f c  t  j  t   A  expj  2  f c  t expj  t 
1


2
s C t   A  expj  2  f c  t 1  j  t     j  t   
2!


• Maintaining the 1st order terms …
s C t   A  expj  2  f c  t 1  j  t 
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Narrowband FM & PM Spectrum (2)
• Taking the Fourier Trasnform of the 1st order
approximation
s C t   A  expj  2  f c  t 1  j  t 
SC f   A  f  f c   f   j   f 
SC f   A  f  f c   j   f  f c 
SC f   A  f  f c    f  f c 
2
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2
2
13
PM and FM Basis
• Based on the previous analysis, we need to determine the
transform of the phase components
t
 PM t    p  m 2 t 
 FM t    f   m 3    d
 PM f    p  M 2 f 
M 3  
 FM t    j   f 
f
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PM Phasor v1
s c t   A  expj  2  f 0  t  j   expj   p  m 2 t 
• The carrier can be removed to describe the baseband signal
as a bounded phase variation about the carrier
p
m
2 t 
Ac
fo
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15
PM Phasor v2
s C t   A  expj  2  f c  t 1  j  t 
• For a cos wave message input
j  p

 j  p
 expj  2  f m  t
 exp j  2  f m  t
s C t   A  expj  2  f c  t 1 
2
2


Ac
fo
ECE 6640
16
FM Phasor
s C t   A  expj  2  f c  t 1  j  t 
• For a cos wave message input
t


s C t   A  expj  2  f c  t 1  j   f   cos2  f m  t   d 


s C t   A  expj  2  f c  t 1  j   f  sin 2  f m  t 

 

s C t   A  expj  2  f c  t 1  f  exp j  2  f m  t   f  exp j  2  f m  t 
2
2


• See Figure 4.4, p. 173
ECE 6640
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Why discuss phasors?
• We are about to describe digital modulation in terms of
one, two, and three dimensional constellation points.
– Amplitude Shift Keying: 1-D array of possible points
– Phase Shift Keying: 2-D circle with points equally spaced on the
circle
– Frequency Shift Keying: N-D space with one point on each of the
N axis
– Quadrature Amplitude Modulation: 2-D 2Mx2M array of points
ECE 6640
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General Notes from ABC
• The following notes are based on Carlson Chapter 14.
• There is a notational difference between Sklar and Carlson
in describing a symbol. Sklar’s more easily lends itself to
defining Eb/No!
ECE 6640
19
Binary modulated waveforms
a)
ASK
b)
FSK
c)
PSK
d)
DSB with
baseband pulse
shaping
See Figure 4.5 on p. 174
20
Amplitude Shift Keying (ASK)
• Digital Symbol Amplitude Modulation
• On-Off Keying (OOK)
p 0 t   0
p1 t   1
s0 t   0
s1 t   Ac  cos2  f c  t   
• Auto-correlation
E s0 t   s0 t     0
 
Ac
E s1 t   s1 t    
    cos2  f c  
2
T 
2
Es 0 t   s1 t     0
• Average Power
POOK  P0   Rs0 s0    P1  Rs1s1  
POOK
2
2
0
Ac
1
1 Ac
 0  
    cos0  
2
2 2
4
T 
Ac 
2 E
T
21
Amplitude Shift Keying (2)
• Auto-correlation
R s0s0    0
2

Ac
R s1s1   
    cos2  f c   
2
T
Ac 
2 E
T
• Symbol Power Spectral Density
2


A
S OOK    c  T 2  sinc 2  f c  f   T   sinc 2  f c  f   T 
8
• Bandpass Bandwidth
– Nominally: BT=1/T, first null at Bnull=2/T
22
ASK Power Spectrum
• From ABC Chapter 11
S vv  f    a  rb  P f   ma  rb  
2
2
2

 Pn  rb    f  n  rb 
n  
• Baseband or LPF analysis
E an  
 
2
pt   rectrb  t 
2
A
A
2
, E an 
2
2
P f  
2
f
1
 sinc 
rb
 rb 
 f  A2
A2
S vv  f  
  f 
 sinc  
4  rb
4
 rb 
• RF Analysis
Gc  f  
1
 S vv  f  f c   S vv  f  f c 
4
23
ASK Power Spectrum (2)
2
 f  A2
A2
 sinc  
S vv  f  
  f 
4  rb
4
 rb 
Gc  f  
1
 S vv  f  f c   S vv  f  f c 
4
rb 
1
Tb
Figure 14.1-2
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
24
ASK MATLAB Simulation
Symbol Sequenct in Time
Symbol Sequence Circular Auto-correlation
0
1
-50
Magnitude (dB)
Amplitude
0.5
0
-0.5
-100
-1
1
2
3
4
5
6
7
8
9
Time
OOK Demodulation Eye Diagram
10
-150
11
0
1
2
-5
x 10
3
Frequency
4
5
6
8
x 10
Symbol Sequence Circular Auto-correlation
2.5
0
-10
2
-20
1.5
Magnitude (dB)
Amplitude
-30
1
0.5
-40
-50
-60
-70
0
-80
-0.5
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
0.7
0.8
0.9
1
-6
x 10
3
3.05
3.1
3.15
3.2
3.25
Frequency
3.3
3.35
25
3.4
7
x 10
ASK Transmission Capability
• Comparing the ratio of the bit rate to the required
signal bandwidth
TP 
rb
BT
– From the previous slide for the bandwidth
BT  rb
– Therefore, the transmission capability is
TP 
rb
 1 bit  per  second Hz
BT
26
M-ary ASK
• Use multiple amplitude levels to represent more than one
bit per symbol
• MASK
– M-1 one states and the off state
– All positive amplitudes (no phase reversals)
ma  E an  
 
 a  E an
2


2
M 1
2
M 2 1
 ma 
12
2
 f  A2  M  12
A  M 1
 sinc  
S vv  f  
  f 
r
12  rb
4
 b
2
2
2
Gc  f   S vv  f  f c   S vv  f  f c 
27
M-ary ASK Transmission Capability
• Comparing the ratio of the bit rate to the required
signal bandwidth
– For m-ary, the bit rate is
bit  rate  rs  log 2 M 
– The symbol bandwidth remains
BT  rs
– Therefore, the transmission capability is
TP 
rs  log 2 M 
 log 2 M  bits  per  second Hz
BT
– Note that for m-ary ASK, the OOK system has the
smallest spectral efficiency
28
Binary QAM
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) transmitter (b) signal constellation : Figure 14.1-3
xi t    a2k  pt  k  T 
 xi t   cos2  f c  t  
xc t   Ac  





x
t
f
t







sin
2
c

 q
k
xq t    a2k 1  pt  k  T 
k
ma  E an   0
 
 a 2  E an 2  A 2
29
Quadrature AM (QAM)
• An M-ary Signal – 4 complex symbols
• Quadrature
s 0 t   1  A c  cos2  f c  t  
p 0 t   1
p1 t   i
s1 t   i  A c  cos2  f c  t  
p 3 t   i
s 3 t   i  A c  cos2  f c  t  
s 2 t   1  A c  cos2  f c  t  
p 2 t   1
• Auto-correlation, Single Pulse Period

E sk t   sk t   
*
• Average Power
E QAM


 
T
 Ac 2

 2  cos2  f c  

2
0 1
Ac
 A c      cos0  
2
T 2
2
30
QAM
• Symbol Cross Correlation
 
C0, 0 t   1   
T 
 
C0,1 t   i   
T 

C0, 2 t   1   
T 

C0,3 t   i   
T 
• Not that adjacent symbol average correlation is
zero for equal probability symbols …
E sk t   i 
k 1
 Ac  E cos2  f c  t     0
E sk t   sk 1 t   P0  C s0 s0    P1  C s0 s1    P2   C s0 s2    P3  C s0 s2  
1
1
1
1
 
E sk t   sk 1 t    1   i    1    i   
4
4
4
4
 T

0


31
Quadrature AM Power Spectrum
2
 t 
pt   rect   rectrs  t 
 Ts 
f
1
r
P f    sinc 
rs  b
rs
2
 rs 
1
f
2
S vv  f   Ac  rs     sinc 
 rs 
 rs 
f
2 1
S vv  f   Ac   sinc 
rs
 rs 
Note that the symbol rate
is one-half the bit rate.
S vv  f    a  r  P f   ma  r  
2
2
Gc  f   S vv  f  f c   S vv  f  f c 
2
 2 f
Ac  4
S vv  f  
 sinc
rb
 rb
2
2
2


n  



2
Pn  r    f  n  r 
2
32
QAM Transmission Capability
• Comparing the ratio of the symbol rate to the
required signal bandwidth
TP 
rs  log 2 M 
BT
– Therefore, the transmission capability is
TP  2 bits  per  second Hz
33
Phase Modulation Methods
• Phase shift keying (PSK) is digital PM
x t   A   cos2  f  t       p t  k  T 
c
c
c
k
D
s
k
– Points on a unit circle of a constellation plot
– 4-QAM as previously described is using phase to
represent symbols. The magnitude is the same, but
successive symbols differ by 90 degrees in phase.
• Frequency shift keying (FSK) is digital FM
x t   A   cos2  f  t    2  a  f  t   p t  k  T 
c
c
c
k
d
D
s
k
– Multiple discrete frequencies
34
PSK Signal Constellations
This is QAM,
rotated by pi/4
for 4-PSK
M=4
4-PSK
M=8
8-PSK
35
M-PSK
• An M-ary Signal – M complex symbols
• Quadrature (2 possible representations)
2  k  1   ,

s k t   A c  cos 2  f c  t   

M


for k  0 to M  1
  2  k  1   
 2  k  1    
p k t   I k , Q k    cos
, sin 
 ,
M
M



 
for k  0 to M  1
• Auto-correlation, single symbol Period


 1
*
2
E s k t   s k t    A c      cos2  f c  
T 2
• Average Power, Amplitude to Energy
PQAM
2
0 1
Ac
 Ac      cos0  
2
T  2
2
Ac 
2 E
T
36
Binary PSK
• Signal Symbols
s 0 t   A c  cos2  f c  t      0   A c  cos2  f c  t  
s1 t   A c  cos2  f c  t     1  A c  cos2  f c  t  
• Autocorrelation

E s k t   s k t   
*

 1
 A c      cos2  f c  
T 2
2
• Cross Correlation (the definition of antipodal)


 1
*
2
E s 0 t   s1 t     A c      cos2  f c  
T 2
Rs0 s1    Rs0 s0 
37
Binary PSK
• Signal Symbols
s 0 t   A c  cos2  f c  t      0   A c  cos2  f c  t  
s1 t   A c  cos2  f c  t     1  A c  cos2  f c  t  
• Autocorrelation

E s k t   s k t   
*

 1
 A c      cos2  f c  
T 2
2
• Cross Correlation (the definition of antipodal)


 1
*
2
E s 0 t   s1 t     A c      cos2  f c  
T 2
Rs0 s1    Rs0 s0 
38
BPSK Power Spectrum
• From Chapter 11
S vv  f    a  rb  P f   ma  rb  
2
2
2

 Pn  rb    f  n  rb 
2
n  
• Baseband or LPF analysis
pt   rectrb  t 
 
E an   0, E an  A2
2
P f  
f
A
S vv  f  
 sinc 
rb
 rb 
2
2
f
1
 sinc 
rb
 rb 
• RF Analysis
Gc  f  
1
 S vv  f  f c   S vv  f  f c 
2
39
BPSK MATLAB Simulation
-20
1
0.8
-40
0.6
-60
0.2
Magnitude (dB)
Amplitude
0.4
0
-0.2
-80
-100
-0.4
-120
-0.6
-0.8
-140
-1
0.5
1
1.5
2
2.5
3
Time
BPSK Demodulation Eye Diagram
3.5
4
-160
-6
0
1.5
0
1
-20
0.5
-40
0
-80
-1
-100
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
0.7
0.4
0.6
0.8
0.8
0.9
1
-6
x 10
1
1.2
Frequency
1.4
1.6
1.8
2
8
x 10
-60
-0.5
-1.5
0.2
x 10
Magnitude (dB)
Amplitude
0
-120
2.1
2.2
2.3
2.4
2.5
Frequency
2.6
2.7
2.8
40
2.9
7
x 10
Other Forms of PSK
• Differential PSK
– The symbols are the “encoding” of two adjacent bits
– Encoding the bit changes not the bit values
– Typically an exclusive-Or or Exclusive NOR
• QPSK
– Already shown as QAM
• Offset QPSK
– Offset the I and Q bits of QAM by one half the symbol period
– Phase changes at BPSK bit rate, bit absolute phase change is now
always pi/2 (orthogonal)
41
Differential Encoded PSK (DPSK)
• The binary data stream is differentially encoded
– The logical combination of the previous bit sent and the next bit to
be sent. An Exclusive NOR gate can be used.
– Provides an arbitrary start … only phase change by pi is required
to decode the message, not the absolute bit values!
Sample Index
0
Information m(k)
1
2
3
4
5
6
7
8
9
10
1
1
0
1
0
1
1
0
0
1
Diff. Encoding (0)
0
0
0
1
1
0
0
0
1
0
0
DPSK Phase
0
0
0
pi
pi
0
0
0
pi
0
0
1
1
0
1
0
1
1
0
0
1
Detect
Diff. Encoding (1)
1
1
1
0
0
1
1
1
0
1
1
DPSK Phase
pi
pi
pi
0
0
pi
pi
pi
0
pi
pi
1
1
0
1
0
1
1
0
0
1
Detect
42
Offset-keyed QPSK transmitter
(OQPSK)
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.1-6
• Instead of changing I and Q at the same
time, delay the change by T/2.
• Visualize the phase changes … always to
an adjacent symbol!
43
Digital Frequency Modulation
Frequency Shift Keying
(FSK)
Continuous Phase FSK
(CPFSK)
44
Frequency Shift Keying
• Binary FSK
s0 t   Ac  cos2   f c  f d   t  
s1 t   Ac  cos2   f c  f d   t  
• M-ary FSK or MFSK
sk t   Ac  cos2   f start  f step  k  t  ,
for k  0 to M  1
• Desired Condition (makes the time signal
continuous at the symbol time boundaries)
2   f step  TS   m  2 ,
for m an interger
45
M-FSK
• An M-ary Signal – M complex symbols
s t   A  cos2  f  t    2   f  k  t ,
for k  0
k
c
start
step
to M  1
• Desired Condition (normally)
2   f step  T   m  2, for m an interger
Can make expected
value zero
• Crosscorrelation

E s0 t   sk t  
*

 1
 Ac      E cos2   f start  f step  k    2   f step  k  t 
T  2

2

• Autocorrelation

E sk t   sk t   
*

 1
 Ac      cos2   f start  f step  k  
T  2
2
46
BFSK
• Signal Symbols
s0 t   Ac  cos2   f c  f d   t  
s1 t   Ac  cos2   f c  f d   t  
• Autocorrelation


 1
*
2
E sk t   sk t    Ac      cos2   f c  f d   
T  2
• Cross Correlation



*
2
E s0 t   s1 t     Ac     E cos2   f c  f d   t    cos2   f c  f d   t     
T 


 1
*
2
E s0 t   s1 t     Ac      E cos2  2  f d   t  2   f c  f d   
T  2
orthogonal for 2  x2fdxT=2
47
BFSK Quadrature Representation (1)
sk t   Ac  cos2   f c  ak  f d   t  
ak  1
rb
2
fd 
sk t   Ac  cos2  f c  t    cos2  ak  f d  t   Ac  sin 2  f c  t    sin 2  ak  f d  t 
sk t   Ac  cos2  f c  t    cos2  f d  t   ak  Ac  sin 2  f c  t    sin 2  f d  t 
sk t   Ac  cos2  f c  t    cos  rb  t   ak  Ac  sin 2  f c  t    sin   rb  t 
• The sign term for odd bits becomes
sk t   Ac  cos2  f c  t    cos  rb  t    1  ak  Ac  sin 2  f c  t    sin   rb  t 
k


bbk t   I k , Qk   cos  rb  t ,  1  ak  sin   rb  t 
k
48
BFSK Quadrature Representation (2)


bbk t   I k , Qk   cos  rb  t ,  1  ak  sin   rb  t 
k
• The baseband spectrum Glp
Glp  f   Gi  f   Gq  f  
r  
r 
1  
2
  f  b    f  b   rb  Qk 
4  
2 
2 
2



r  
r  
2
Qk 
 sinc  f  b   rb   sinc  f  b   rb 
2 
  2 2  



 f  
cos


rb  
4
2


Qk   2 2  
2
  rb   2  f 


1


 

rb 
2


cos   f  

rb 
r  
r 
1  
4


Glp  f   Gi  f   Gq  f     f  b    f  b   2  
2
4  
2 
2    rb   2  f 

1




 
rb 
1

4  rb
49
Power spectrum of BFSK
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.1-8
Glp  f   Gi  f   Gq  f  
r  
1  
  f  b    f
4  
2 


f




cos
rb  
rb 
4 


   2  
2
2    rb   2  f 

1




 
rb 
2
2  2  f d  rb   2
fd 
rb
2
50
BFSK MATLAB Simulation
0
Not readily observable
Magnitude (dB)
-50
The change in frequency
is too small
-100
-150
BFSK Demodulation Eye Diagram
0
0.2
0.4
0.6
0.8
1.5
1
1.2
Frequency
1.4
1.6
1.8
2
8
x 10
0
1
-20
0.5
Magnitude (dB)
Amplitude
-40
0
-0.5
-60
-80
-1
-100
-1.5
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
0.7
0.8
0.9
1
-6
x 10
-120
2.1
2.2
2.3
2.4
2.5
Frequency
2.6
2.7
2.8
51
2.9
7
x 10
Spectrum of M-FSK
• As tones with equal spacing are required, MFSK requires
additional bandwidth for additional symbol tones.
– The bandwidth must grow as a multiple of M,
whereas for M-PSK the bandwidth is based on the symbol period.
– M-FSK is inherently wideband modulation.
– More bits per symbol requires more bandwidth
52
Special Versions of FSK
• Continuous Phase FSK (CPFSK)
t



xc t   Ac  cos 2  f c  t    2  f d   x   d 
0


0t T
 a0  t ,
a  T  a  t  T ,
T  t  2 T
1
 0
t



x
d





0
 k 1

 a j  T   ak  t  k  T ,
k  T  t  k  1  T
 j 0

• Minimum-Shift Keying (MSK)
– The binary version of CPFSK
– Also called fast FSK
– Capable of using an rb/2 bandwidth
53
CPFSK
• Continuous Phase FSK (CPFSK)
t



xc t   Ac  cos 2  f c  t    2  f d   x   d 
0


0t T
 a0  t ,
a  T  a  t  T ,
T  t  2 T
1
 0
t



x
d





0
 k 1

 a j  T   ak  t  k  T ,
k  T  t  k  1  T
 j 0

• The phase is continuous at the transitions between
bit.
– This is most easily accomplished if the phase is π or a
multiple of π at the start and end of each bit period.
54
Binary CPFSK
• The binary version of CPFSK is called
Minimum-Shift Keying (MSK)
– Also called fast FSK
– Capable of using an rb/2 bandwidth
55
MSK Baseband
bbk t   I k , Qk 

xi t    cos k  ak  ck   pt  k  T 
k
ck 
  rb
 t  k  T 
2

xq t    sin  k  ak  ck   pt  k  T 
k
m  ,

k  




n
,

2
for k even
for k odd
• Frequency and phase (history) modulation
56
Illustration of MSK.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) phase path (b) i and q waveforms: Figure 14.1-11
• MSK includes the
phase history with the
frequency slope in time
of the new bit.
• Therefore the phase
plot in time can appear
as shown, with the
corresponding
quadrature
components.
57
Minimum Shift Keying (MSK)
MSK power spectrum: Figure 14.1-9
2   f step  T   
Use 0.25 in BFSK Sim
58
Sklar Representations
• Amplitude Shift Keying
si t  
s0 t   0
2  Ei
 cos2  f c  t   
T
• Phase Shift Keying
2  k  1   
2 E

sk t  
 cos 2  f c  t   

M
T


for k  0 to M  1
• Frequency Shift Keying
sk t  
ECE 6640
2 E
 cos2   f min  k  f step  t   
T
for k  0 to M  1
s1 t  
2 E
 cos2  f c  t   
T
s0 t  
2 E
 cos2  f c  t    0 
T
s1 t  
2 E
 cos2  f c  t     
T
s0 t  


f step 

2 E
  t   
 cos 2   f c 

2 
T



s1 t  


f step 

2 E
  t   
 cos 2   f c 

2 
T



59
Textbook Waveform Energy
• Waveform Energy (Symbol Autocorrelation)
T
Ei   si t   dt
2
0
• Matched Filter
t
z t   r t   ht    r    ht     d

ht   u t   s * T  t 
t
z t    s   s * T  t     d
0
T
T
T
z T    s   s T  T     d   s   s    d   s    d
*
0
ECE 6640
*
0
2
0
Correlation
60
Signal Power vs. Bit Energy
• For continuous time signals, power is a normal
way to describe the signal.
• For a discrete symbol, the “power” is 0 but the
energy is non-zero
– Therefore, we would like to describe symbols in terms
of energy not power
• For digital transmissions how to we go from
power to energy?
– Power is energy per time, but we know the time
duration of a bit. Noise has a bandwidth.
1
S R  Eb 
Tb
61
Energy and Power
• For
2 E
 cos2  f c  t   
T
s t  
• The average power and energy per bit becomes
2

 2 E
Eb   E 
 cos2  f c  t      dt

 T
0


T


2 E

  E cos 2 2  f c  t     dt
T 0
T
2 E
 1 cos2  2 f c  t    

  E 
 dt

2
T 0 2

T
1  2 E
A
  
P
2 2  T
2
2

E
 

T

2 E 1
2 E T

   dt 
 E
T 02
T 2
T
ECE 6640
62
SNR to Eb/No Reminder
• For the Signal to Noise Ratio
– SNR relates the average signal power and average noise
power (Tb is bit period, W is filter bandwidth)
1
Tb  Eb  1
S
 
 
 
 N  N 0  W  N 0  Tb  W
Eb 
– Eb/No relates the energy per bit to the noise energy
(equal to S/N times a time-bandwidth product)
 Eb   S  W  S 

    
    Tb  W 
 N0   N  R b  N 
If you want a higher Eb/No, increase Tb.
(Changing W changes the SNR too!)
63
Symbol Detection
• Baseband detection and BER defined in the previous
chapter.
• The following are from ABC Chap. 14
ECE 6640
64
Optimum binary detection
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) parallel matched filters (b) correlation detector: Figure 14.2-3
65
Conditional PDFs
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.2-2
66
Bandpass binary receiver
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.2-1
• Using superposition of the “parallel matched filters”, the
BPF is the difference of the two filters.
hBPF t   h1 t   h0 t 
• This results in an optimal binary detector
67
Binary Receiver
hBPF t   h1 t   h0 t 
• OOK
h1 t   K  s1 T  t 
h0 t   K  s0 T  t 
hBPF t   h1 t   K  s1 T  t   cos2  f c  T  t 
• BPSK
hBPF t   h1 t   h0 t   2  h1 t   2  K  s1 T  t   cos2  f c  T  t 
• BFSK
hBPF t   h1 t   h0 t   K  s1 T  t   K  s0 T  t 
hBPF t   cos2   f c  f d   T  t   cos2   f c  f d   T  t 
hBPF t   2  sin 2  f c  T  t   sin 2  2  f d  T  t 
68
Correlation receiver for
OOK or BPSK
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.2-4
• Since both optimal filters consist of cosine waveforms,
mix and integrate instead of filter an optimally sample.
– Note that the integrator can be a rectangular window filter that is
optimally sampled. (Provides functionality near synchronization as
well.)
69
Optimal Parallel Matched Filter Receiver
Error Analysis
2
 z1  z0 


2



 max
T

2
E   s1 t   s0 t   dt 

 0
2  N0
• Evaluating the expected value


T

T

T
T
2
2
2
E   s1 t   s0 t   dt   E   s1 t   dt   2  E   s1 t   s0 t   dt   E   s0 t   dt 


0

0

0
0

T
2
E   s1 t   s0 t   dt   E1  2  E10  E0

0
Eb  E1  E0  2
2
2  Eb  2  E10 Eb  E10
 z1  z0 


 
2
2
N
N0




 max
0
70
Optimal Parallel Matched Filter Receiver
Error Analysis
E10    Eb 

T
Eb
 E   s1 t   s0 t   dt 
E1  E0

0
2
• OOK
• PSK
• FSK
E10  0
Eb
 z1  z0 



 2    max N 0
E10   1  Eb
2  Eb
 z1  z0 



N0
 2    max
E10  0
2
2
E
 z1  z0 

  b
 2    max N 0
71
Generalized Probability of Error
• Using the optimal BPF filter and sampling for
each symbol, the relationship will be based on:
Eb  E10 Eb  1  
 z1  z0 




N0
N0
 2    max
2
• The BER is then based on
 E  1   
z  z 
Pe  Q  1 0   Q  b

N
 2 
0


• Therefore picking arbitrary symbols is possible,
but the symbol correlation coefficient will drive
the BER performance.
72
Generalized FSK
s0 t   Ac  cos2   f c  f d   t  
s1 t   Ac  cos2   f c  f d   t  

T
E10  Ac  E   cos2   f c  f d   t  cos2   f c  f d   t   dt 

0
2 T
A
E10  c   cos2  2 f c  t   cos2  2 f d  t  dt
2 0
2
T
E 1
E10  b    expi 2  2 f d  t   exp i 2  2 f d  t  dt
T 2 0
  Eb 
Eb 1  expi 2  2 f d  T  exp i 2  2 f d  T  
 


T 2 
i 2  2 f d
i 2  2 f d

 f 
Eb sin 2  2 f d  T 

 Eb  sinc4  f d  T   Eb  sinc 4  d 
T
2  2 f d
 rb 
k
2 f d  f step 
There are multiple “orthogonal” tone separations.
2T
  Eb 
•
•
The correlation coefficient can go negative! The minimum occurs at
approximately sinc(1.22) = -0.166
73
MATLAB Coherent Receivers
• BASK example code
• BPSK example code
• BFSK example code
ECE 6640
74
Noncoherent Binary Systems
• Synchronous coherent receiver can be very difficult to
design.
• Can noncoherent systems be more easily designed without
giving up significant BER performance?
– For a 1-2 dB Eb/No performance loss, YES!
75
Noncoherent OOK receiver
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.3-2
• Using an envelope detector, the noise pdf for a zero
symbol becomes Rician and is non-longer Gaussian.
• The noise pdf for a one symbol remains Gaussian
76
Conditional PDFs for
noncoherent OOK
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.3-3
Pe 0 Vopt   Pe1 Vopt 

E 
Pe 0  exp  b 
 2  N0 
Pe 
Vopt 
Ac

E 
1
1
1
 Pe 0  Pe1    Pe 0    exp  b 
2
2
2
 2  N0 
2
77
Noncoherent detection of binary FSK
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.3-5
78
Noncoherent FSK
• Qualitative comments
– Using envelope detectors on each symbol output, the Rician error
distribution effects the z detection statistic.
Pe 

E 
1
 exp  b 
2
 2  N0 
79
Binary error probability curves
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) coherent BPSK (b) DPSK (c) coherent OOK or FSK (d) noncoherent
FSK (e) noncoherent OOK: Figure 14.3-4
10
10
BER
10
10
10
10
10
BER Simulation for BPSK and BFSK
0
-1
-2
-3
-4
BPSK
BPSK
BFSK
BFSK
-5
-6
0
2
simulation
(theoretical)
simulation
(theoretical)
4
6
8
E b/No (dB)
10
12
14
16
80
Binary error probability curves
(a) coherent BPSK (b) DPSK (c) coherent OOK or FSK
(d) noncoherent FSK (e) noncoherent OOK
Figure 14.3-4
81
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Detection for M-ary Systems
• Determine the detection statistic for all symbols
• Select the maximum statistic
• Decode the binary values from the selected symbol
• Notes:
– M-ASK and M-PSK symbols may no longer be orthogonal
– M-FSK symbols may be orthogonal, but the bandwidth W must
increase to contain the symbols.
82
Quadrature-carrier receiver with
correlation detectors
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.4-1
• Applicable for:
– M-QAM
– M-PSK
83
Carrier synchronization
for quad-carrier receiver
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.4-2
84
Coherent M-ary PSK receiver
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.4-3
PreDecode, Es /N0 (dB)=19
20
15
10
Imag
5
• MPSK_Demo.m
– Fixed N0, varying signal Eb
0
-5
-10
-15
-20
-20
-15
-10
-5
0
Real
5
10
15
20
85
Decision thresholds for M-ary PSK
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.4-4
PreDecode, Es /N0 (dB)=19
20
15
10
Imag
5
0
-5
-10
-15
-20
-20
-15
-10
-5
0
Real
5
10
15
20
86
PSK signal constellations
(a) M=4 (b) M=8
Figure 14.5-1
• MPSK Symbols are typically “Gray-code” encoded prior
to transmission
– In the Gray-code, adjacent symbols are only different by 1 bit
value!
87
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
MPSK Eb/N0 Examples
PreDecode, Es /N0 (dB)=1
PreDecode, Es /N0 (dB)=9
8
10
8
6
6
4
4
2
Imag
Imag
2
0
0
-2
-2
-4
-4
-6
-6
-8
-8
-8
-6
-4
20
-2
0
2
PreDecode, Es /N0 (dB)=19
Real
4
6
-10
-10
8
0
-8
-6
-4
-2
Symbol Error Rate, M=8
0
Real 0
10
10
2
4
6
8
Bit Error Rate, M=8
10
-10
20
15
-1
-1
10
10
10
-2
-2
10
0
BER
10
SER
Imag
5
-3
-3
10
-5
-10
10
-4
-4
10
10
-15
-20
-20
-15
-10
-5
0
Real
5
10
15
20
-20
-10
0
10
Es/No (dB)
20
-20
0
10
Eb/No (dB)
88
Simulated Performance MPSK
• MPSK_Ber and MPSK_PP_Plot
MPSK Symbol Error Rate
0
10
-1
-1
10
10
-2
-2
10
10
-3
-3
10
BER
SER
10
-4
M=2 Sim
M=2 Bound
M=4 Sim
M=4 Bound
M=8 Sim
M=8 Bound
M=16 Sim
M=16 Bound
10
-5
10
-6
10
-7
10
MPSK Bit Error Rate
0
10
-5
0
-4
M=2 Sim
M=2 Bound
M=4 Sim
M=4 Bound
M=8 Sim
M=8 Bound
M=16 Sim
M=16 Bound
10
-5
10
-6
10
-7
5
10
Es /N0 (dB)
15
20
25
10
-5
0
5
10
Eb/N0 (dB)
15
20
89
Simulated Performance MFSK
• MFSK_Ber and MFSK_PP_Plot
MFSK Symbol Error Rate
0
10
-1
-1
10
10
-2
-2
10
10
-3
-3
10
BER
SER
10
-4
M=2 Sim
M=2 Bound
M=4 Sim
M=4 Bound
M=8 Sim
M=8 Bound
M=16 Sim
M=16 Bound
10
-5
10
-6
10
-7
10
MFSK Bit Error Rate
0
10
0
2
4
-4
M=2 Sim
M=2 Bound
M=4 Sim
M=4 Bound
M=8 Sim
M=8 Bound
M=16 Sim
M=16 Bound
10
-5
10
-6
10
-7
6
8
Es /N0 (dB)
10
12
14
16
10
-5
0
5
Eb/N0 (dB)
10
15
90
Comparing MPSK and MFSK
• MPSK
– More Eb/N0 required for higher M for symbol error rate
– 2- and 4-PSK have the same BER
• Otherwise higher BER for higher M
• MFSK
– More Eb/N0 required for higher M for symbol error rate,
BUT it does not increase as fast as MPSK
– Less Eb/N0 required for higher M for BER!
– How could this be?
• The symbols are all orthogonal!
91
M-ary QAM system
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) transmitter (b) receiver (c) square signal
constellation and thresholds with M=16
Figure 14.4-8
92
Performance comparisons of M-ary
modulation systems
Pbe  104
93
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
MATLAB Coherent Receivers
• MPSK example code
• MFSK example code
• QAM example code
ECE 6640
94
Notes on BER
• For MPSK and QAM
– Sklar
• QAM p. 565
• MPSK p. 229-230
– J.G. Proakis & M. Salehi, Digital Communications, 5th ed.
• QAM p. 196-200
• MPSK p. 190-195
– Jianhua Lu; Letaief, K.B.; Chuang, J. C-I; Liou, M.-L., "M-PSK
and M-QAM BER computation using signal-space concepts,"
Communications, IEEE Transactions on , vol.47, no.2, pp.181,184,
Feb 1999.
ECE 6640
95
QAM BER Computation
% Sklar (bit error rate)
PB1(:,ii) = 2*((1-L^-1)/log2(L))*Q_fn(sqrt(3*log2(L)*2*Es_No/((M-1)*bitpersym)) );
% Proakis (symbol error rate)
PB2(:,ii) = 2*(1-L^-1)*Q_fn(sqrt(3*log2(M)*Es_No/((M-1)*bitpersym)));
PB2(:,ii) = 2*PB2(:,ii).*(1-0.5*PB2(:,ii));
PB2(:,ii) = PB2(:,ii)/bitpersym;
% Lu, Lataief, Chuang, and Liou (bit error rate)
Qsum = 0;
for jj=1:L/2
Qsum=Qsum+Q_fn((2*jj-1)*sqrt(3*log2(M)*Es_No/((M-1)*bitpersym)));
end
PB3(:,ii) = 4*((1-L^-1)/log2(M))*Qsum;
ECE 6640
96
QAM BER Curves
BER Composite Plot
0
10
4 QMA
16 QAM
64 QAM
256 QAM
-1
10
-2
Bit Error Rate
10
-3
10
-4
10
-5
10
-6
10
-7
10
ECE 6640
0
5
10
15
Eb/No (dB)
20
25
30
97
QAM BER Curves Detail/Differences
BER Composite Plot
0
10
Bit Error Rate
4 QMA
16 QAM
64 QAM
256 QAM
-1
10
-2
10
ECE 6640
-1
0
1
2
3
4
5
Eb/No (dB)
6
7
8
9
10
98
MPSK Nyquist Filter BER
SER vs SNR
Sklar
Theory
Plot
MPSK Simulation: Theory vs. Simulation
1
10
0
10
-1
10
-2
Symbol Error Rate
10
-3
10
T4
S4
T8
S8
T16
S16
T32
S32
T64
S64
T128
S128
T256
S256
-4
10
-5
10
-6
10
-7
10
ECE 6640
0
5
10
15
20
25
30
SNR (dB)
35
40
45
50
55
99
MPSK Nyquist Filter BER
BER vs Eb/No
Sklar
Theory
Plot
MPSK Simulation: Theory vs. Simulation
0
10
-1
10
-2
Bit Error Rate
10
-3
T4
S4
T8
S8
T16
S16
T32
S32
T64
S64
T128
S128
T256
S256
10
-4
10
-5
10
-6
10
-7
10
ECE 6640
-5
0
5
10
15
20
EbNo (dB)
25
30
35
40
45
100
QAM Nyquist Filter BER
SER vs. SNR
Sklar
Theory
Plot
QAM Simulation: Theory vs. Simulation
1
10
0
10
-1
Symbol Error Rate
10
-2
10
-3
10
T4
S4
T16
S16
T64
S64
T256
S256
-4
10
-5
10
-6
10
-7
10
ECE 6640
0
5
10
15
20
SNR (dB)
25
30
35
101
QAM Nyquist Filter BER
BER vs Eb/No
Sklar
Theory
Plot
QAM Simulation: Theory vs. Simulation
0
10
-1
10
-2
Bit Error Rate
10
-3
10
-4
T4
S4
T16
S16
T64
S64
T256
S256
10
-5
10
-6
10
-7
10
ECE 6640
-5
0
5
10
EbNo (dB)
15
20
25
102