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Chapter 7Performance of QAM

Huseyin BilgekulEEE 461 Communication Systems II

Department 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 0Serial 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)be oEP N

• 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

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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 M’ary types can be used.

Multiamplitude (MASK) +s(t), +3 s(t), +5 s(t),. . ., +(M-1) s(t). Multiple phase (MPSK, QPSK) Multitone (MFSK) Quadrature Amplitude Modulation (combines MASK and MPSK)

2M

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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 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

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 and

noise• Receiver needs better SNR to achieve accuracy

1011

0100

Recived Signal

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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,

• Each M-ary symbols has log2M bits of information so the bit energy Eb and the symbol enrgy EpM are related by

• Same transmission bandwidth, yet more information

22

2

2 22

0

2 9 ... 1

122 1 1

3 3

M

pM p p p

pp p

k

E E E M EM

M EE M Ek M

M

2

2 2

1

log 3logppM

b

M EEE

M M

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MASK Error Probability• Same optimal receiver with matched filter to s(t)• Total probability of SYMBOL ERROR for M

equally 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 ii i

P P m P m P mM

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Decision Model

• Two cases:– (M-1)p(t) – just like

bipolar

– Interior cases, can have errors on both sides

01 00 10 11

Ap-3Ap -Ap 3Ap

pi

n

AP m Q

2 pi

n

AP m Q

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MASK Prob. Of Error

• In a matched filter receiver, Ap/n= 2Ep/N

1

1

1

1 2 2

2 1

M

eM ii

Mp p p

i n n n

p

n

P P mM

A A AQ Q M Q

M

AMQ

M

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MASK Prob. Of Error• In a matched filter receiver, Ap/n= 2Ep/N

22

2 1

2 1 6log1

peM

b

EMP Q

M

M EMQM M

N

N

2

2 2

1

log 3logppM

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

2logeM

bP

PM

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Error Probability Curves

• Use codes so that a symbol error gives only a single bit error.

• Bandwidth stays same as M increases, good if you are not power-limited.

M=2

M=4M=8

M=16

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M-ary PSK (MPSK)• Binary Phase Shift Keying (BPSK) 1: s(t)= s(t) cos(ct)

0: s(t)= s(t)cos(ct

• M-ary PSK

Re

Im

x x

2cosk cs t s t t kM

Re

Im

x xx x

x x

x

x

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MPSK

• Must be coherent since envelope does not change• Closest estimated phase is selected

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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 b

eME M E M

P Q QM

2N N

Re

Im

x xx 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 if SNR 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

cosk 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 ba

rii

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QAM Performance

• Analysis is complex and not treated here.• QAM-16

• Upper Bound for general QAM depends on spectral efficiency relative to bipolar signals,

435

beM

EP Q N

/M bR B

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QAM vs. MPSKM 2 4 8 16 32 64

M=Rb/B 0.5 1 1.5 2 2.5 3

Eb/NO for BER=10-6

10.5 10.5 14 18.5 23.4 28.5

M 4 16 64 256 1024 4096

M=Rb/B 1 2 3 4 5 6

Eb/N o for BER=10-6

10.5 15 18.5 24 28 33.5

MPSK

QAM

• 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 of

frequencies:

• 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

2 /k MN k T

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MFSK Receiver

x

Sqrt(2)cos 1t

H()

Com

para

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 are Ap+ n1, n2 , n3, etc.

• Error occurs when nj> Ap+ n1

• Average Probability of error:

22

1 2 1 1

12 log / / 2

, , ,

11 12

b

M

My E M

P m P r n r n r

e Q y dy

1

N

21 log /b bP M Q E M N

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MFSK Performance

• Channel BW:

• BW efficiency decreases, but power efficiency increases

• Signals are orthogonal so no crowding in signal space

2

32logbR M

BM

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MFSK vs. MPSK

M 2 4 8 16 32 64M=Rb/B 0.5 1 1.5 2 2.5 3

Eb/N for BER=10-6

10.5 10.5 14 18.5 23.4 28.5

M 2 4 8 16 32 64

M=Rb/B 0.4 0.57 0.55 0.42 0.29 0.18

Eb/N for BER=10-6

13.5 10.8 9.3 8.2 7.5 6.9

MPSK

MFSK