Transcript EE422 Lecture 25

Chapter 7
Performance of QAM
 Performance of QPSK
 Comparison of Digital Signaling Systems
 Symbol and Bit Error Rate for Multilevel
Signaling
Huseyin Bilgekul
EEE 461 Communication Systems II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
EEE 461 1
Performance of QPSK
• Modeled as two BPSK systems in parallel. One using a cosine carrier and the
other a sine carrier
Im
• Ts=2 Tb
x
Decision Regions
x
x
Re
x
0101
x
01110010
Rb
Rb/2
Serial to
Parallel
Converter
cos wct
+
90
1100
BPF
-
x
Rb/2
EEE 461 2
Performance of QPSK
EEE 461 3
Performance of QPSK
• Because the upper and lower channels are BPSK receivers the
BER is the same as BPSK.
  Eb


Pe =Q  2 
No  



(Matched Filter Detection)
• Twice as much data can be sent in the same bandwidth compared
to BPSK (QPSK has twice the spectral efficiency with identical
energy efficiency).
• Each symbol is two bits, Es=2Eb
EEE 461 4
EEE 461 5
M-ary Communications
• Send multiple, M, waveforms
• Choose between one of M symbols instead of 1 or 0.
• Waveforms differ by phase, amplitude, and/or frequency
• Advantage: Send more information at a time
• Disadvantage: Harder to tell the signals apart or more bandwidth needed.
• Different M’ary types can be used.
Multiamplitude (MASK) +s(t), +3 s(t), +5 s(t),. . ., +(M-1) s(t).
Multiple phase (MPSK, QPSK)  2 
M
Multitone (MFSK)
Quadrature Amplitude Modulation (combines MASK and MPSK)
EEE 461 6
M-ary Communications
• As M increases, it is harder to make good
decisions, more power is used
• But, more information is packed into a symbol
so data rates can be increased
• Generally, higher data rates require more
power (shorter distances, better SNR) to get
good results
• Symbols have different meanings, so what
does the probability of error, PE mean?
– Bit error probability
– Symbol error probability
EEE 461 7
•
•
Multi-Amplitude Shift Keying (MASK)
Send multiple amplitudes to denote different signals
Typical signal configuration:
– +/- s(t), +/- 3 s(t), ….., +/- (M-1) s(t)
•
•
•
4-ary Amplitude Shift Keying
Each symbol sends 2 bits
Deciding which level is correct gets harder due to fading and
noise
Receiver needs better SNR to achieve accuracy
10
Recived Signal
11
01
00
EEE 461 8
Average Symbol and average Bit Energy
•
•
•
•
Transmit Rm M-ary symbols/sec (Tm=1/ Rm)
Each pulse of form: k s(t)
Assume bit combination equally likely with probability 1/M
The average symbol energy is,
E pM
•
2 
2

E p  9 E p  ...   M  1 E p 

M
M 2
M 2  1 E p M 2 E p
2E p 2

2

 2k  1 

M k 0
3
3
1
Each M-ary symbols has log2M bits of information so the bit
energy Eb and the symbol enrgy EpM are related by
Eb 
•
M
E pM
M


2
 1 E p
log 2 M
3log 2 M
Same transmission bandwidth, yet more information
EEE 461 9
MASK Error Probability
• Same optimal receiver with matched filter to s(t)
• Total probability of SYMBOL ERROR for M
equally likely signals:
M
PeM
s(t)+n(t)
1
  P  mi  P  mi  
M
i 1
s(T-t)
H(f)
r(t)
M
 P  m 
i 1
t=Tp
r(Tp)
i
Threshold
Detector
+kAp+n(Tp)
EEE 461 10
Decision Model
• Two cases:
– (M-1)p(t) – just like
bipolar
 Ap 
P   mi   Q  
n 
– Interior cases, can
have errors on both
sides
 Ap 
P   mi   2Q  
n 
01
-3Ap
00
10
11
-Ap
Ap
3Ap
EEE 461 11
MASK Prob. Of Error
PeM
1

M
M
 P  m 
i 1
i
  Ap 
 Ap 
 Ap  
Q    Q     M  2  2Q   

i 1    n 
n 
  n 
2  M  1  Ap 

Q 
M
n 
1

M
M
• In a matched filter receiver, Ap/n= 2Ep/N
EEE 461 12
MASK Prob. Of Error
• In a matched filter receiver, Ap/n= 2Ep/N
Eb 
PeM
E pM
log 2 M

2
M
  1 E p
3log 2 M
2  M  1  E p 

Q

 N 
M


2  M  1  6log 2 M  Eb  


Q


2
  M  1  N  
M


EEE 461 13
Bit Error Rate
• Need to be able to compare like things
– Symbol error has different cost than a bit error
• For MASK
PeM
Pb 
log 2 M
EEE 461 14
Error Probability Curves
• Use codes so that a
symbol error gives only
a single bit error.
M=16
M=8
• Bandwidth stays same
as M increases, good if
you are not powerlimited.
M=4
M=2
EEE 461 15
M-ary PSK (MPSK)
• Binary Phase Shift Keying (BPSK)
Im
1: s1(t)= s(t) cos(wct)
• M-ary PSK
2

sk  t   s  t  cos  wct 
M

x Re
x
0: s0(t)= s(t)cos(wct

k

x
x
Im
x
x Re
x
x
x
x
EEE 461 16
MPSK
• Must be coherent since envelope does not change
• Closest estimated phase is selected
EEE 461 17
MPSK Performance
•
Detection error if phase deviates by > /M

PeM  1   M p   d


M
•
x
Im
x
x
x
x
x
x
x
Re
Strong signal approximation
PeM
 2 Eb log 2 M
 
2Q 
sin 


N




 E log M

b
2
2Q 
2N

M





EEE 461 18
MPSK Waterfall Curve
• QPSK gives equivalent performance to BPSK.
• MPSK is used in modems to improve performance if
SNR is high enough.
EEE 461 19
Quadrature Amplitude Modulation (QAM)
• Amplitude-phase shift keying (APK or QAM)
sk  t 
 s  t   ak cos wct   bk sin wct  
 s  t  rk cos wct   k 
• The envelope and phases are,
rk  a  b
2
k
2
k
ri
i
 bk 
k   tan  
 ak 
EEE 461 20
QAM Performance
• Analysis is complex and not treated here.
• QAM-16
PeM
 4 Eb 
3Q 

 5N 
• Upper Bound for general QAM depends on
spectral efficiency relative to bipolar signals,
M  Rb / B
EEE 461 21
QAM vs. MPSK
M
P
S
K
M
M=Rb/B
Eb/NO for
BER=10-6
M
Q
A
M
•
•
M=Rb/B
Eb/N o for
BER=10-6
2
4
8
16
32
64
0.5
1
1.5
2
2.5
3
14
18.5
23.4 28.5
1024 4096
10.5 10.5
4
16
64
256
1
2
3
4
5
6
10.5
15
18.5
24
28
33.5
Very power efficient for high signal configurations, but requires a
lot of power
Can give inconsistent results for different bit configurations
EEE 461 22
Multitone Signaling (MFSK)
• M symbols transmitted by M orthogonal pulses of
frequencies:
wk  2  N  k  / TM
• Receiver:
– bank of mixers, one at each frequency
– Bank of matched filters to each pulse
• Higher M means wider bandwidth needed or tones are
closer together
EEE 461 23
MFSK Receiver
H(w)
Sqrt(2)cos w1t
x
H(w)
Sqrt(2)cos w2t
x
Comparator
x
H(w)
Sqrt(2)cos wMt
EEE 461 24
MFSK Performance
• When waveform 1 is sent, sampler outputs are
•
Ap+ n1, n2 , n3, etc.
Error occurs when nj> Ap+ n1
P   m  1  P  r1  , n2  r1 , , nM
1
 1
2



e

 y  2 Eb log 2 M / N

2
/2
 r1 
1  Q  y  
M 1
dy
• Average Probability of error:
Pb   M  1 Q

Eb log 2 M / N

EEE 461 25
MFSK Performance
• Channel BW:
B
Rb  M  3
2log 2 M
• BW efficiency decreases, but
•
power efficiency increases
Signals are orthogonal so no
crowding in signal space
EEE 461 26
MFSK vs. MPSK
M
M
P M=Rb/B
S
Eb/N for
K
-6
2
4
8
16
32
64
0.5
1
1.5
2
2.5
3
14
18.5
8
16
10.5 10.5
23.4 28.5
BER=10
M
F
S
K
M
M=Rb/B
Eb/N for
BER=10-6
2
0.4
4
0.57 0.55
13.5 10.8
9.3
32
64
0.42
0.29 0.18
8.2
7.5
6.9
EEE 461 27