ECE 6640
Digital Communications
Dr. Bradley J. Bazuin
Assistant Professor
Department of Electrical and Computer Engineering
College of Engineering and Applied Sciences
Chapter 3
Chapter 3
3.1
3.2
3.3
3.4
3.5
ECE 6640
Digital Modulation Schemes
Representation of Digitally Modulated Signals
Memoryless Modulation Methods
Signaling Schemes with Memory
Power Spectrum of Digitally Modulated Signals
Bibliographical Notes and References
Problems
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
95
95
97
114
131
148
148
2
Representation of Signals
• Converting Bits to Symbols: Memoryless or with Memory
• Memoryless: a direct conversion of bits to a symbol
– In a memoryless modulation scheme, the binary sequence is parsed
into subsequences each of length k, and each sequence is mapped
into one of the sm(t), 1 ≤ m ≤ 2k , signals regardless of the
previously transmitted signals.
• With Memory: the time history of bits matters
– the mapping is from the set of the current k bits and the past
(L − 1)k bits to the set of possible M = 2k messages.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
3
Modulation with Memory
(Markov Chain)
•
•
•
The transmitted signal
depends on the current
k bits as well as the most
recent L − 1 blocks of k bits.
This defines a finite-state machine
with 2(L−1)k states.
The mapping that defines the modulation scheme
can be viewed as a mapping from the current state
and the current input of the modulator to the set of
output signals resulting in a new state of the modulator.
The mapping that defines the modulation scheme can be viewed as a mapping
from the current state and the current input of the modulator to the set of output
signals resulting in a new state of the modulator. If at time instant l−1 the
modulator is in state Sl−1 ∈ {1, 2, . . . , 2(L−1)k } and the input sequence is Il ∈ {1,
2, . . . , 2k}, then the modulator transmits the output sml (t) and moves to new state
Sl.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
4
Linear vs. Non-Linear
• Linearity of a modulation method requires that the
principle of superposition apply in the mapping of the
digital sequence into successive waveforms.
• In nonlinear modulation, the superposition principle does
not apply to signals transmitted in successive time
intervals.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
5
Memoryless Modulation Basics
• Binary bits grouped into k-bit symbols.
sm t , 0  m  M  1 for M  2 k
• A new symbol is transmitted every Ts seconds
– Ts is the signal interval and Rs is the signaling rate.
Rs 
1
Ts
– The incoming bit interval, Tb, and bit rate, Rb relate to this as
Tb 
ECE 6640
Ts
Ts

k log 2 M
Rb  Rs  k  Rs  log 2 M
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
6
Symbols Energy and Power
•
The average symbol energy can be computed as
– based on the symbol probability pm as
M
Eavg   pm  Em
m 1
– for equally likely symbols
Eavg
•
1 M

  Em
M m 1
The average energy per bit can be computed as
Ebit  avg 
•
The average signal power
Pavg 
ECE 6640
Eavg
Ts
 Eavg  Rs 
Eavg
k

Eavg
log 2 M
Ebit  avg  log 2 M
Tb  k
 Ebit avg  Rb
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
7
Memoryless Methods
•
•
•
•
Pulse Amplitude Modulation (ASK)
Phase Modulation (PSK)
Quadrature Amplitude Modulation (QAM)
Multidimensional Signaling
–
–
–
–
–
ECE 6640
Orthogonal Signaling – Frequency-Shift Keying (FSK)
Hadamard Signals
Biorthoginal Signaling
Simplex Signaling
Signal Waveforms from Binary Codes
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
8
Pulse Amplitude Modulation
• A signal pulse with various amplitudes.
sm t   Am  pt ,
1  m  M for M  2 k
– pulse duration Ts and amplitude of
Am  2  m  1  M ,
1 m  M
Am  1,3, ,M  1

• Energy
Em 
 Am  pt   dt  Am  E p
2
2
2

Eavg 
Eavg
Ep
M
M
  Am 
m 1
2

Ep
M
M 2
 2   2  m  1  M 
m 1



2  E p M  M 2 1 E p  M 2 1



M
6
3
E p  M 2 1
Ebit  avg 
3  log 2 M

ECE 6640
2

Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
9
PAM Constellations
Note: Gray Code binary representations applied.
see https://en.wikipedia.org/wiki/Gray_code
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
10
Bandpass PAM
• What we described previously is the baseband PAM in
which no carrier modulation is present. In many cases the
PAM signals are carrier-modulated bandpass signals with
lowpass equivalents of the form Amg(t), where Am and g(t)
are real.
sm t   Resml t   exp j  2    f c  t 
sm t   ReAm  g t   exp j  2    f c  t   Am  g t   cos2    f c  t 
– Applying previous definitions, let
pt   g t   cos2    f c  t 
2
A
Em  m  E g
2
ECE 6640
Eavg 


Eg  M 2  1
6
Ebit  avg 


Eg  M 2  1
6  log 2 M
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
11
PAM Dimensionality
• PAM signals are one-dimensional (N = 1) since all are
multiples of the same basic signals. The basis is
pt 
 t  
pt    t   E p
Ep
• Describing the bandpass basis
 t  
ECE 6640
2
 g t   cos2    f c  t 
Eg
 t  
Eg
2
 g t   cos2    f c  t 
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
12
PAM Symbol Representations
• Baseband PAM
sm t   Am 
E p   t 
sm  Am  E p , for Am  1,3, ,M  1
• Bandpass PAM
sm t   Am 
sm  Am 
Eg
2
Eg
2
  t 
, for Am  1,3,,M  1
Figure 3.2-1 Amplitude
Constellations
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
13
PAM Euclidean Distance
• The Euclidean distances
between constellation points can
d  sm  sn
be described as
E
E
d min  sn 1  sn  An 1 
d min  sn 1  sn 
Eg
2
g
2
 An 
 An 1  An 
g
2
Eg
2
2
d min  2  E g  2  E p
• In terms of average bit energy or symbol energy
Ebit  avg 
Eavg 
ECE 6640


Eg  M 2  1
6  log 2 M


Eg  M 2  1
6
d min 
12  log 2 M  Ebit  avg
M 2 1
d min 
12  Eavg
M 2 1
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
14
Bandpass Modulation
• For a purely real signal, this is
similar to DSB.
– Bandpass bandwidth is twice
the baseband bandwidth.
• For a complex signal, this is
similar to SSB.
– Bandpass Bandwidth is
equivalent to positive freq.
baseband bandwidth.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
15
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
16
Binary modulated waveforms
a)
ASK
b)
FSK
c)
PSK
d)
DSB with
baseband pulse
shaping
See Figure 4.5 on p. 174
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
17
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 
Ac
1
1 Ac
 0  
    cos0  
2Notes and
2 figures
2 are T
4 course textbook:
based
 on or taken the
2 E
T
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
18
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
19
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
20
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
21
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
22
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
23
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  
2
 


 a  E an
2
M 1
2
M 2 1
 ma 
12
2
 f  A2  M  12
A2  M 2  1
 sinc  
S vv  f  
  f 
12  rb
4
 rb 
2
Gc  f   S vv  f  f c   S vv  f  f c 
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
24
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
25
Phase Modulation (PSK)
• A signal pulse with various phases.
m 1 

sm t   pt   exp j  2   
1  m  M for M  2 k
,
M 

m 1
 m t   2   
, 1  m  M for M  2 k
M
• Energy

Em 
 pt   dt  E p
2

Eavg 
Ep
M
Ebit  avg 
ECE 6640
M
 1  E p
m 1
Ep
log 2 M
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
26
PSK Constellations
Note: Gray Code binary representations applied.
see https://en.wikipedia.org/wiki/Gray_code
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
27
Bandpass PSK
• The PSK signals are carrier-modulated bandpass signals
with lowpass equivalents of the form g(t), where g(t) is
real.
sm t   Resml t   exp j  2    f c  t 


m 1 

sm t   Re  g t   exp j  2   
  exp j  2    f c  t 
M 



m 1 

 g t   cos 2    f c  t  2   

M


m 1 
m 1 


 g t   cos 2   
  cos2    f c  t   g t   sin 2   
  sin 2    f c  t 
M 
M 


1
Em  Eavg   E g
2
ECE 6640
Ebit  avg 
Eg
2  log 2 M
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
28
PSK Dimensionality
• PSK signals are two-dimensional (N = 2). The basis is
pt 
pt 
 t  
 t   j 
1
Ep
1
Ep
• Describing the bandpass basis
1 t  
2
 g t   cos2    f c  t 
Eg
sm t  
ECE 6640
2 t   
2
 g t   sin 2    f c  t 
Eg
Eg
m 1 
m 1 


 cos 2   
 sin  2   
  1 t  
  2 t 
2
2
M 
M 


Eg
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
29
PSK Symbol Representations
• Baseband PM
m 1 

sm t   E p  cos 2   
  1 t  

M 
m 1 

E p  sin 2   
  2 t 
M 


m 1 
m 1  


sm   E p  cos 2   
, E p  sin 2   
 ,
M 
M 



m  1,2, , M
• Bandpass PM
sm t  
Eg
m 1 
m 1 


 cos 2   
 sin  2   
  1 t  
  2 t 
2
2
M
M




Eg
 Eg
m  1  Eg
m  1  



 cos 2   
 sin s 2   
sm 
,
 ,
 2
M
2
M



 

ECE 6640
m  1,2,, M
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
30
PSK Euclidean Distance
• The Euclidean distances between constellation points can
be described as d  s  s  s  s
m
n
1
m , m 1
2
d min  s0  sm
 Eg
Eg
Eg
m  1   E g
m  1 


 
1 
 cos 2   
0 
 sin  2   
  

M
M
2
2
2
2

 



d min  s0  sm 
d min  s0  sm 
ECE 6640
2

m  1   
m  1 

 1  cos 2   
  sin  2   

M
M
2





 
Eg
2
2
m 1 
m 1 
m 1 



 1  2  cos 2   
  cos 2   
  sin  2   

M
M
M
2






Eg
m 1 

 2  2  cos 2   
d min  s0  sm 

M
2


Eg
m 1 

d min  s0  sm  E g  1  cos 2   

M


Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
2
31
2
PSK Euclidean Distance (cont)
• The Euclidean distances between constellation points can
be described as d  s  s  s  s
m
n
1
m , m 1
m 1 

d min  s0  sm  E g  1  cos 2   

M


– Letting m=2
d min  s0  sm
Em  Eavg
Ebit  avg 
ECE 6640

 2 
 E g  1  cos
 M


 
   2  E g  sin  

M 
2
 
d min  2  E g  sin 
M 
1
 
  Eg
d min  2  Eavg  sin 
2
M 
Eg
2  log 2 M
 
d min  2  Ebbit  avg  log 2 M  sin 
M 
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
32
PSK Euclidean Distance Summary
• Summary
 
d min  2  E g  sin  
M 
 
d min  2  Eavg  sin  
M 
 
d min  2  Ebbit  avg  log 2 M  sin 
M 
– For large M and the sin(x)~x approximation
d min 
d min 
ECE 6640
2 
M
2 
 Eavg
M
Ebbit avg  log 2 M
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
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.
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
34
PSK Signal Constellations
This is QAM,
rotated by pi/4
for 4-PSK
M=4
4-PSK
M=8
8-PSK
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
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 
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
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 
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
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
Quadrature Amplitude Modulation
(QAM)
• A signal pulse with various phases.
sm t   AmI  j  AmQ  pt 
– For square QAM
AnI ,nQ  2  n  1  M , 1  n  M


AnI ,nQ  1,3,, M  1
• Energy
 A

Em 
mI
2



 AmQ  pt   dt  AmI  AmQ  EP
2
2
2
2

Eavg 
Ep
M
M

  AmI  AmQ
m 1
Ebit  avg 
ECE 6640
2
2

Eavg
log 2 M
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
41
QAM Constellations
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
42
Bandpass QAM
• The QAM signals are carrier-modulated bandpass signals
with lowpass equivalents of the form g(t), where g(t) is
real.
sm t   ReAmI  j  AmQ  g t   exp j  2    f c  t 
sm t   AmI  g t   cos2    f c  t   AmQ  g t   sin 2    f c  t 
– letting
rm 
A
mI
2
 AmQ
2

 AmQ 

 AmI 
 m  atan
sm t   rm  g t   cos2    f c  t   m 
– the average energy is based on the signal space defined.
Em 
ECE 6640
Eg
2

2
 AmI  AmQ
2

Eavg 
Eg
2M
M

  AmI  AmQ
m 1
2
2

Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
43
Square QAM Constellations
• Average Energy Computation
Eavg 
Ep
M
M

  AmI  AmQ
m 1
2
2

Eavg 

1 M M
2
2
?? 
  Am  An
M m 1 n 1
?? 
2  M  1
3
2  M  1
 Ep 
3  log 2 M
Eavg  E p 
Ebit  avg
ECE 6640
Eg
2M
M

  AmI  AmQ
m 1
2
2


2  M  M  1
3
Eavg  E g 
M  1
Ebit  avg  E g 
3
M  1
3  log 2 M
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
44
QAM Dimensionality
• QAM signals are two-dimensional (N = 2). The basis is
pt 
pt 
 t  
 t   j 
1
1
Ep
Ep
• Describing the bandpass basis
1 t  
2
 g t   cos2    f c  t 
Eg
sm t  
ECE 6640
Eg
2
2 t   
 AmI  1 t  
Eg
2
2
 g t   sin 2    f c  t 
Eg
 AmQ  2 t 
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
45
QAM Symbol Representations
• Baseband QAM
sm t  
sm 

E p  AmI  1 t   E p  AmQ  2 t 

E p  AmI , E p  AmQ ,
m  1,2, , M
• Bandpass QAM
sm t  
Eg
2
 AmI  1 t  
Eg
2
 AmQ  2 t 
 Eg

Eg

sm 
 AmI ,
 AmQ , m  1,2,, M
 2

2


ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
46
QAM Euclidean Distance
• The Euclidean distances between constellation points can
be described as
d  s s
m
n
2
d min
 Eg
  Eg

Eg
Eg
 sm  sn  
 AmI 
 AnI   
 AmQ 
 AnQ 
2
2
2
2

 

d min  sm  sn 
Eg
2

2
AmI  AnI 2  AmQ  AnQ 2
– for a square constellation


AnI ,nQ  1,3,, M  1
d min  sm  sn 
ECE 6640
Eg
2

22  02
 2  Eg
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
47
QAM Euclidean Distance Summary
• Summary for square constellation
d min  2  E g
d min 
d min 
ECE 6640
6  Eavg
M 1
Eavg  E g 
M  1
Ebit  avg  E g 
3
M  1
3  log 2 M
6  log 2 M
 Ebit  avg
M 1
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
48
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
49
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
50
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


Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
51
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




2
 Pn  r    f  n  r 
2
n  
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
52
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
53
Linear Modulation Summary
• From the discussion of bandpass PAM, PSK, and QAM, it
is clear that all these signaling schemes are of the general
form
sm t   Re Am  g t   exp j  2    f c  t ,
m  1,2,, m
where Am can be real or complex.
• It can beseen that these three signaling schemes belong to
the same family, and PAM and PSK can be considered as
special cases of QAM.
• Also note that in these schemes the dimensionality of the
signal space is rather low (one for PAM and two for PSK
and QAM) and is independent of the constellation size M.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
54
Summary Table
(Text Note: The PAM energy and dimension values are wrong)
Signaling
BP PAM
sm
Am 
Am  g t   cos2    f c  t 
Eavg
Eg
2
m 1 

A  g t   cos 2    f c  t  2   

M 

PSK
QAM
sm t 
E g  M 2  1
 Eg
m  1  Eg



 cos 2   
 sin s 2   
,

 2
M  2
M  



ECE 6640

 Eg
Eg

 AmI ,
 AmQ 

 2
2


12  Eavg
M 2 1
6
A
 Eg
2
m  1  
AmI  g t   cos2    f c  t   AmQ  g t   sin 2    f c  t 
d min
Eg
 
2  Eavg  sin  
M 
M  1

3
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
6  Eavg
M 1
55
Multidimensional Signals
• If we wish to construct signal waveforms corresponding to
higher-dimensional vectors, we may use either the time domain
or the frequency domain or both to increase the number of
dimensions.
– Dimensions in frequency – real (N) or quadrature (N/2)
– Dimensions in “time subintervals” – real (T/N) or quadrature (T/(N/2))
N=12 (or 24)
4 frequencies
3 time slots in Ts
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
56
Orthogonal Signaling
• Signals and Orthonormal basis sets
mn
for 1  m, n  M
mn
 E,
sm t , sn t   
0,
 j t  
s j t 
E
, 1 j  M
mn
for 1  m, n  M
mn
1,
0,
m t , n t   
 E ,0,0,,0
 0, E ,0,,0
s1 
s2
 


sM  0,0,0,, E
ECE 6640

Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
57
Orthogonal Signaling (cont)
• Energy
Eavg  E
Ebit  avg
 E ,0,0,,0
 0, E ,0,,0
s1 
s2
 
E

log 2 M


sM  0,0,0,, E
• Dimensionality
d  sm  sn
d min 
2
2
E  0    E    02  2  E
2
d min  2  log 2 M  Ebit  avg
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
58

Frequency Shift Keying
• The construction of orthogonal signal waveforms that
differ in frequency and are represented as
sm t   Resml t   exp j  2    f c  t 
sml t  
2 E
 exp j  2    m  f  t , 1  m  M and 0  t  T
T
sm t  
2 E
 cos2    f c  t  2    m  f  t 
T
• under the condition
sm t , sn t   0,
for m  n
• Note: this is considered a non-linear modulation scheme
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
59
FSK Condition
Re sml t , snl t    0, for m  n
• For orthogonality to exist
T
sml t , snl t   
0
2 E
2 E
 exp j  2    m  f  t  
 exp j  2    n  f  t   dt
T
T
2 E
sml t , snl t  
exp j  2    m  n   f  t   dt
T 0
T
2  E  e  j2 m n f t  
2  E   1  e  j2 m n f T  


sml t , snl t  

 


T  j  2    m  n  f  0
T  j  2    m  n   f 
T
2  E  e  j m n f T   e  j m n f T    j m n f T 

sml t , snl t  
e
T 
j  2    m  n   f

2  E  2  j  sin   m  n   f  T  j m n f T

sml t , snl t  
e
T 
j  2    m  n   f

ECE 6640
 sin   m  n   f  T   j m n f T
sml t , snl t   2  E  
e








m
n
f
T


60
FSK Condition (cont)
• For orthogonality to exist
Re sml t , snl t    0, for m  n
 sin   m  n   f  T   j m n f T
sml t , snl t   2  E  
e
   m  n   f  T 
 sin   m  n   f  T  
Re sml t , snl t    2  E  
  cos  m  n   f  T 
   m  n   f  T 
– using the sincos=1/2 x sin2 identity
 1 sin 2    m  n   f  T  
Re sml t , snl t    2  E   




m

n


f

T
2



Re sml t , snl t    2  E  sinc2    m  n   f  T   0
ECE 6640
61
FSK Condition (cont)
• Therefore orthogonality exists for
Re sml t , snl t    0, for m  n
Re sml t , snl t    2  E  sinc2    m  n   f  T   0
2    m  n   f  T   k  
m  n   f   k
2 T
– The minimum can be found for bandpass real signal as
f 
1
2 T
• This works for the bandpass signal only. If we want the
complex symbols to be orthogonal,
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
62
FSK Condition (cont)
• If we want the complex symbols to be orthogonal, we must
go back to
s t , s t   0,
for m  n
ml
nl
 sin   m  n   f  T   j m n f T
sml t , snl t   2  E  
e
   m  n   f  T 
sml t , snl t   2  E  sinc  m  n   f  T   e j m n f T
  m  n   f  T   k  
– The minimum can be found for bandpass real signal as
f 
1
T
• This works for both baseband and bandpass signals.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
63
“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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
64
Digital Frequency Modulation
Frequency Shift Keying
(FSK)
Continuous Phase FSK
(CPFSK)
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
65
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
66
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


 1
E sk t   sk t     Ac      cos2   f start  f step  k  
Notes and figures are basedTon or2taken the course textbook:
 
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
*
2
67
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  x2f xT=2
Notes and figures are based on or taken the course textbook:
d
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
68
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
69
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 
Notes and figures are based on or taken the course textbook: 
1

4  rb
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
70
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
71
BFSK MATLAB Simulation
0
Not readily observable
Magnitude (dB)
-50
The change in frequency
is too small
-100
-150
BFSK Demodulation Eye Diagram
1.5
0
0.2
0.4
0.6
0.8
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
-120
2.2
2.3
2.4
2.5
2.6
2.7
Notes and figures arex 10based on or taken 2.1
the course
textbook:
Frequency
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
-6
2.8
72
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
73
Time Subinterval Orthogonality
• The Hadamard matrix has orthogonal rows/columns.
• Hadamard matrices Hn are 2n × 2n matrices for n = 1, 2, …
defined by the following recursive relation
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
74
Hadamard Signals
• Based on the n=2 matrix, 4 orthogonal signals can be
defined as
s  E  H2




s1 
E
E
s2
E
 E
E
E
E
 E
s3
s4
E
E
E
 E
 E


E
E
 E

– and
4
sm t    smj   j t ,
1 m  4
j 1
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
75
Hadamard Energy
• The energy calculation is
M 4
4
Eavg  sm t , sm t    E   j t ,  j t  ,1  m  4
j 1
M  2n
2n
Eavg   E   j t ,  j t  ,1  m  2 n
j 1
Eavg  sm t , sm t   4  E
Ebit  avg
ECE 6640
4 E

 2 E
2
Eavg  2 n  E
Ebit  avg
2n  E
2n  E


n
log 2 2
n
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
76
Biorthogonal Signaling
• A set of M biorthogonal signals can be constructed from ½
M orthogonal signals by simply including the negatives of
the orthogonal signals. Thus, we require N = ½ M
dimensions for the construction of a set of M biorthogonal
signals.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
77
Biorthogonal Signaling
• While we showed that in general orthogonal signal
dimensions where
d min 
2
2
E  02    E    02  2  E
d min  2  log 2 M  Ebit  avg
• Because of the positive and negative considerations and a
reduction in the dimensionality from M to M/2 the
symbols comprise two distance, the previous minimum and
the posite and negative symbols in one dimension
d
ECE 6640
2 E   0
2
2
   02  2  E
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
78
Signaling Schemes with Memory
• Signaling schemes with memory can be best explained in
terms of Markov chains and finite-state machines.
• The state transition and the outputs of the Markov chain
are governed by
ml  f m Sl 1 , I l 
Sl  f s Sl 1 , I l 
– the previous state, S, and the input information vector, I, generate
the next state, S, and the output vector, m.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
79
Differential Encoding
• Each bit uses the previous bit and previous symbol
information to form the symbol to transmit.
bk  ak  bk 1
• A simple for is non-return-to-zero, inverting signaling
– if the bit value is a one, the symbol is inverted
– if the bit value is a zero, the symbol remains the same
NRZI: The “operator”
for computation is
modulo addition
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
80
NRZI as a Markov Process
• Since the information sequence is assumed to be binary,
there are two states in the Markov chain, and the state
transition diagram of the Markov chain is
pn   pn  1  P
1
P 2
 1 2
1 
2
1 
2
Equally probable transitions
2
p 1
1
2

Equally probable states
pn   p0  P n
ECE 6640
Steady state probable states 81
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
Unequal Input Probabilities.
• If the input bits are not equally probable.
Prak  1  1  Prak  0  p
pn   pn  1  P
p 
1  p
P

p
1

p


2
p0  1
1
2

Equally probable iniital states
pn   p0  P n  p0
Steady state probable states
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
82
NRZI Trellis
• Another way to display the memory introduced by the
precoding operation is by means of a trellis diagram.
• The trellis provides exactly the same information
concerning the signal dependence as the state diagram, but
also depicts a time evolution of the state transitions.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
83
Continuous Phase Shift Keying
• FSK Signal generation can cause frequency spectrum
“eruptions” when the frequency is changed.
– this is an undesirable spectral effect!
• Can we define a why to “smoothly” transition?
– what if the frequency is change continuously instead of discretely?
• Constraint: the phase of the carrier is continuous. Because
of this, the symbol must have memory based on the
previous symbol and carrier phase.
ECE 6640
84
CPFSK
• Define a convention PAM based symbol sequence
d t    I n  g t  n  T 
n
I n  Am  1,3,,M  1, for 1  m  M 
• Use the sequence to frequency modulate the baseband
signal.


t

2 E

 exp j  4    f d  Td    d    d   0 
vt  


T

 


– fd is the peak frequency deviation and there is an initial phase
term.
• The bandpass signal is then
 

t

2 E

vt  
 Re exp j  2    f c  t  j  4    f d  Td    d    d   0 

T
 
 


ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
85
CPFSK (cont)
• The phase integration allows discontinuous data steps to
result in smooth phase transitions.
• If we look at the instantaneous phase, it is a phase history
plus a phase ramp based on the new symbol.
n 1
 n  T  t  n  1  T ; I   2    f d  T   I k  4    f d  T  qt  n  T   I n
k  
 n 1  2    f d  T  I n  2  qt  n  T 
n 1
n 1  2    f d  T   I k
k  
ramp t   2    f d  T  I n  2  qt 
ECE 6640
Previous phase
plus phase ramp!
q(t) is a ramp!
0,
t  0, T  t


qt    t
, 0t T
2 T

 1 2 , t  T
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
86
Continuous Phase Modulation (CPM)
• If we let q(t) be some normalized waveform shape, the
phase modulation may
extend for multiple symbol periods.
t
qt    g    d ,
for 0  t  b  T
0
– the symbol transition may be for 1 or more symbol periods
• Possible functions are … rectangular, raised cosine, etc.
CPFSK
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
87
Phase Trajectories
• The continuous phase in time can be plotted to show
possible paths over multiple symbol periods, T.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
88
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
figuresan
are based
or taken the course textbook:
– CapableNotes
of and
using
rb/2onbandwidth
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
89
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.
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
90
Binary CPFSK
• The binary version of CPFSK is called
Minimum-Shift Keying (MSK)
– Also called fast FSK
– Capable of using an rb/2 bandwidth
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
91
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
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
92
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.
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
93
Minimum Shift Keying (MSK)
MSK power spectrum: Figure 14.1-9
2   f step  T   
Use 0.25 in BFSK Sim
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
94
MATLAB Sims
• Looking at Matlab simulations
– MASK
– MPSK
– QAM
ECE 6640
95
Power Spectrum
• For a random sequence of symbols, the transmitted signal
bandwidth is determined by first determining the
autocorrelation of the random sequence and than taking the
Fourier Transform to determine the power spectral density.
• A simpler method to estimate the bandwidth is to use a
deterministic data sequence, a predefined pattern of
alternating symbols, and take the Fourier Transform of the
deterministic signal.
• Finally, both these methods can be applied using a
software simulation tool, such as MATLAB, and the
theoretical PSD can be compared to the simulated PSD.
ECE 6640
96
3.4-2 PSD of Linear Modulation
• We will focus on ASK, PSK and QAM, evaluating the
baseband spectrum.
• The symbol sequence in time can be defined as
vl t  

I
n  
n
 g t  n  T 
– this represent the information sequence and the “symbol envelope”
• The autocorrelation

Rvl t   , T   E vl t     vl t 
*

 
   *
* 
Rvl t   , T   E    I n  g t    n  T     I m  g t  m  T   

 m  
 n  
ECE 6640
  
*
*
Rvl t   , T   E    I n  I m  g t    n  T   g t  m  T  
n   m  

97
PSD (cont)
• Reforming the summation using n=k+m to an offset from
m for the infinite summation
in n



*
*
Rv t   , T   E    I n  I m  g t    n  T   g t  m  T  
n   m  
l

  
*
*
Rvl t   , T   E    I k  m  I m  g t    k  T  m  T   g t  m  T  
 k   m  

• We must take the time average for a cyclo-stationary
process. Note the symbol values are not functions in time.
1
  
*
*
Rvl t   , T     E    I k  m  I m  g t    k  T  m  T   g t  m  T    dt
T 0  k   m  

T
T

1   
*
*
Rvl t   , T    E    I k  m  I m   g t    k  T  m  T   g t  m  T   dt 
T  k   m  
0

ECE 6640
98
PSD (cont)
• The time function is not an R.V.
T

1   
*
*
Rv t   , T    E    I k  m  I m   g t    k  T  m  T   g t  m  T   dt 
l
T
 k   m  


0
T
1  
*
*
Rvl t   , T      E I k  m  I m   g t    k  T  m  T   g t  m  T   dt
T k   m  
0

• The symbol autocorrelation can be computed

1 
*
Rvl t   , T     RII k     g t    k  T  m  T   g t  m  T   dt
T k  
m   0
T
• The summation of integral segments can be expanded

1 
*
Rvl t   , T     RII k    g t    k  T   g t   dt
T k  

ECE 6640
99
PSD (cont)
• Interpreting the equation

1 
*
Rvl t   , T     RII k    g t    k  T   g t   dt
T k  

• For the symbol autocorrelation, if subsequent symbols are
orthogonal
 R 0, k  0
RII k    II
k 0
0,

1
*
Rvl t   , T    RII 0   g t     g t   dt
T

ECE 6640
100
PSD (cont)
• For orthogonal autocorrelation the Power Spectral Density
becomes
1
2
S v  f    RII 0   G  f 
l
T
• For non-orthogonal autocorrelation the Power Spectral
Density must include the summation and time offset as
1 
2
S vl  f     RII k   G  f   exp j  2    k  f  T 
T k  

1
2
S vl  f    G  f    RII k   exp j  2    k  f  T 
T
k  
ECE 6640
101
Example 3.4-1
• For g(t) = rect(t/T) and a BPSK sequence of +/-1
1,
RII k   
0,
k 0
k 0
t
g t   rect  
T 
G  f   T  sinc f  T 
2
S vl  f  
2
1
2
1  T 2  sinc f  T 
T
S vl  f   T  sinc f  T 
2
ECE 6640
102
An Alternate Derivation
• the following is based on a similar but alternate derivation
to find symbol autocorrelation and the PSD.
• The following defines variations in PAM functions used to
define our previous g(t).
ECE 6640
103
Transmission
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) Baseband transmission system (b) signal-plus-noise waveform: Figure 11.1-2
yt    a k  ~
p t  t d  kT   n t 
k
104
Transmission
y t    a k  ~
p t  t d  kT   n t 
k
• The digital signal is time delayed
td
• The pulse is “filtered” and/or distorted by the channel
~
p t   fn pt   hc t 
• Recovering or Regenerating the signal may not be trivial
– Signal plus inter-symbol interference (ISI) plus noise
yˆ mT  t d   am   ak  ~
p mT  kT   nmT  t d 
k m
105
ABC Binary PAM formats
(a) Unipolar RZ & NRZ
(b) Polar RZ & NRZ
(c) Bipolar NRZ
(d) Split-phase Manchester
(e) Polar quaternary NRZ
106
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
PAM Power Spectral Density:
Polar NRZ
vt  
 t  Td  k  Tb 



a
rect

k
Tb
k  



 
Ea n   0, E a n   2
2
1
pTd   ,
Tb

0  Td  Tb

E a j  a k  0, for j  k



Rvv   E vt   vt      1  ,
 Tb 
2
Tb    Tb
S vv w  E vt   vt     2  Tb  sinc 2  f  Tb 
• For a zero mean, polar NRZ of amplitude +/- A and
symbol duration Tb
S vv w  A2  Tb  sinc 2  f  Tb 
 
2
E an   2  A 2
f 
A2

 sinc 2  
rb
 rb 
107
PAM Power Spectral Density:
Arbitrary Pulse (pulse width D)
vt  
 t  Td  k  D 
a
p




k
D


k  

pTd  
 
E an   ma , E an   a  ma
2
2
1
,
D
0  Td  D
2

1
2
S vv  f    P f    Ra n   exp j  2  f  D 
D
n  
 a 2  ma 2 ,
Ra n    2
ma ,
n0
Tb  D, rb 
n0
rb is symbol rate
• Using Poisson’s sum formula
2
2
2
1
D
a
n
2
 ma    n 

S vv  f  
 P f       P    f  
D
D

 D  n    D 
S vv  f    a  rb  P f   ma  rb  
2
2
2

 Pn  rb    f  n  rb 
n  
2
108
Power spectrum of
Unipolar, binary RZ signal
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 11.1-5
2
2
 f  A2 
A2
n
 
 sinc
  sinc    f  n  rb 
S vv  f  
16  rb
 2
 2  rb  16 n  

 t
pt   rect
 Tb
 2
P f  


  rect2  rb  t 


 f 
1

 sinc

2  rb
2
r
 b
 
A
A2
2
E an   , E an 
2
2
 2
A2
2
,n  0
 a  ma 
2
Ra n   
2
m 2  A ,
n0
 a
4
109
Power spectrum of
Unipolar, binary RZ signal
2
2
 f  A2 
A2
n
 
S vv  f  
  sinc    f  n  rb 
 sinc
16  rb
 2
 2  rb  16 n  
Unipolar Binary RZ
0.07
• For rb=2
0.06
0.05
0.04
0.03
0.02
0.01
0
-8
Plots from PSD_PCM.m
-6
-4
-2
0
freq (f)
2
4
6
8
110
Power spectrum of
Unipolar, binary NRZ signal
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
2
 f  A2 
A2
2
S vv  f  
  sincn    f  n  rb 
 sinc  
4  rb
4 n  
 rb 
 t 
pt   rect   rectrb  t 
 Tb 
2
 f  A2
A
S vv  f  
  f 
 sinc  
r
4  rb
4
 b
2
Unipolar Binary NRZ
P f  
f
1
 sinc 
rb
 rb 
 
0.25
• For rb=2
0.2
0.15
0.1
A
A2
2
E an   , E an 
2
2
 2
A2
2
,n  0
 a  ma 
2
Ra n   
2
m 2  A ,
n0
 a
4
0.05
0
-8
-6
-4
-2
0
freq (f)
2
4
6
8
111
Power spectrum of
Polar, binary RZ signal (+/- A/2)
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
 f 
A2

 sinc
S vv  f  
16  rb
 2  rb 

 t
pt   rect
 Tb
 2
2
• For rb=2
Polar Binary RZ
0.035
P f  


  rect2  rb  t 


 f 
1

 sinc
2  rb
 2  rb 
 
0.03
2
2
E an   0, E an  A
0.025
4
0.02
 2  m 2  A2 , n  0
a
a
4
Ra n   
ma2  0,
n0
0.015
0.01
0.005
0
-8
-6
-4
-2
0
freq (f)
2
4
6
8
112
Power spectrum of
Polar, binary NRZ signal (+/- A/2)
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
f 
A
 sinc 
S vv  f  
4  rb
 rb 
2
 t 
pt   rect   rectrb  t 
 Tb 
2
• For rb=2
Polar Binary NRZ
0.14
P f  
f
1
 sinc 
rb
 rb 
 
2
2
E an   0, E an  A
0.12
0.1
4
0.08
 2  m 2  A2 , n  0
a
a
4
Ra n   
ma2  0,
n0
0.06
0.04
0.02
0
-8
-6
-4
-2
0
freq (f)
2
4
6
8
113
Spectral Attributes of PCM
If Bandwidth W=1/T,
then WT=1
Note that WT=0.5 or a
bandwidth equal to ½ the
symbol rate can be used!
114