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Chapter 7Performance of QAM
Huseyin Bilgekul
EEE 461 Communication Systems IIDepartment of Electrical and Electronic Engineering
Eastern Mediterranean University
Performance of QPSK Comparison of Digital Signaling Systems Symbol and Bit Error Rate for Multilevel
Signaling
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Performance of QPSK Modeled as two BPSK systems in parallel. One using a cosine carrier and the other a
sine carrier
Ts=2 Tb
Re
Im
x x
x
x
0 1 1 1 0 0 1 0
Serial to
Parallel
Converter
x
x
90
cos ct+
0 1 0 1
1 1 0 0
Rb
Rb/2
Rb/2
-BPF
Decision Regions
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Performance of QPSK
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Performance of QPSK Because the upper and lower channels are BPSK receivers the BER is the
same as BPSK.
=Q 2 (Matched Filter Detection)beo
EP
N
Twice as much data can be sent in the same bandwidth comparedto BPSK (QPSK has twice the spectral efficiency with identicalenergy efficiency).
Each symbol is two bits, Es=2Eb
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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 Mary types can be used.
Multiamplitude (MASK) +s(t), +3s(t), +5s(t),. . ., +(M-1)s(t).Multiple phase (MPSK, QPSK)
Multitone (MFSK)
Quadrature Amplitude Modulation (combines MASK and MPSK)
( )2 M
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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 morepower (shorter distances, better SNR) to getgood results
Symbols have different meanings, so what
does the probability of error, PEmean?Bit error probability
Symbol error probability
M-ary Communications
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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 andnoise
Receiver needs better SNR to achieve accuracy
1011
01
00
Recived Signal
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Average Symbol and average Bit Energy
TransmitRm M-ary symbols/sec (Tm=1/ Rm)
Each pulse of form: ks(t)
Assume bit combination equally likely with probability 1/M
The average symbol energy is,
Each M-ary symbols has log2M bits of information so thebit energy Eb and the symbol enrgy EpM are related by
Same transmission bandwidth, yet more information
( )
( ) ( )2
2
2
2 22
0
29 ... 1
122 1 1
3 3
M
pM p p p
pp p
k
E E E M E M
M E E M E k M
M
=
= + + +
= + = ; ?
( )2
2 2
1
log 3log
ppM
b
M EEE
M M
= =
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MASK Error Probability
Same optimal receiver with matched filter to s(t)
Total probability ofSYMBOL ERRORfor Mequally likely signals:
s(T-t)
H(f)
s(t)+n(t) r(t) Threshold
Detector
t=Tp
r(Tp)
+kAp+n(Tp)
( ) ( ) ( )1 1
1M M
eM i i i
i i
P P m P m P m
M
= =
= =
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Decision Model
Two cases:
(M-1)p(t) just like
bipolar
Interior cases, can
have errors on bothsides
01 00 10 11
Ap-3Ap -Ap 3Ap
( ) pin
A P m Q
=
( ) 2 pin
A P m Q
=
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MASK Prob. Of Error
In a matched filter receiver, Ap/ n= 2Ep/N
( )
( )
( )
1
1
1
12 2
2 1
M
eM i
i
M p p p
i n n n
p
n
P P mM
A A AQ Q M Q
M
AM QM
=
=
=
= + +
=
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MASK Prob. Of Error
( )
( )
( )2
2
2 1
2 1 6log
1
p
eM
b
EMP Q
M
M EMQ
M M
=
=
N
N
( )2
2 2
1
log 3log
ppM
b
M EEE
M M
= =
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Bit Error Rate
Need to be able to compare like things
Symbol error has different cost than a bit error
For MASK
2log
eMb
PP
M=
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Error Probability Curves
Use codes so that asymbol error gives only
a single bit error.
Bandwidth stays sameas M increases, good if
you are not power-limited.
M=2
M=4
M=8
M=16
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M-ary PSK (MPSK)
Binary Phase Shift Keying (BPSK) 1: s1(t)= s(t) cos( ct)
0: s0(t)= s(t)cos( ct+ )
M-ary PSK
Re
Im
x x
( ) ( )2
cosk c s t s t t k M
= + Re
Im
x x
x x
x x
x
x
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MPSK
Must be coherent since envelope does not change Closest estimated phase is selected
MPSK P f
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MPSK Performance
Detection error if phase deviates by > /M
Strong signal approximation
( )1 MeMM
P p d
=
2 22 log log2 sin 2b beM E M E M P Q Q
M
2 ; ;
2N N
Re
Im
x x
x x
x x
x
x
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MPSK Waterfall Curve
QPSK gives equivalent performance to BPSK.
MPSK is used in modems to improve performance ifSNR is high enough.
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Quadrature Amplitude Modulation (QAM)
Amplitude-phase shift keying (APK or QAM)
The envelope and phases are,
( ) ( ) ( ) ( )( )
( ) ( )
cos sin
cos
k k c k c
k c k
s t s t a t b t
s t r t
= +
= +
2 2 tan kk k k k k
br a b
a
= + =
ri i
f
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QAM Performance
Analysis is complex and not treated here.
QAM-16
Upper Bound for general QAM depends onspectral efficiency relative to bipolar signals,
43
5
beM
EP Q
N
;
/M bR B =
QAM MPSK
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QAM vs. MPSK
M 2 4 8 16 32 64
M=Rb/B 0.5 1 1.5 2 2.5 3Eb/NO for
BER=10-6
10.5 10.5 14 18.5 23.4 28.5
M4 16 64 256 1024 4096
M=Rb/B 1 2 3 4 5 6
Eb/No for
BER=10-6
10.5 15 18.5 24 28 33.5
M
PS
K
Q
A
M
Very power efficient for high signal configurations, but requires a lot of power
Can give inconsistent results for different bit configurations
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Multitone Signaling (MFSK)
M symbols transmitted by M orthogonal pulses offrequencies:
Receiver:bank of mixers, one at each frequencyBank of matched filters to each pulse
HigherMmeans wider bandwidth needed or tones arecloser together
( )2 /k M N k T = +
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MFSK Receiver
x
Sqrt(2)cos 1t
H( )
C
omp
ara
tor
x
Sqrt(2)cos 2t
H( )
x
Sqrt(2)cos Mt
H( )
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MFSK Performance
When waveform 1 is sent, sampler outputs areAp+ n1, n2, n3, etc.
Error occurs when nj> Ap+ n1
Average Probability of error:
( ) ( )
( )( )
2
2
1 2 1 1
12 log / / 2
, , ,
11 1
2
b
M
M y E M
P m P r n r n r
e Q y dy
= = <