ece 6640 digital communications - homepages at …bazuinb/ece6640/chap_04.pdf · ece 6640 digital...

ECE 6640Digital Communications

Dr. Bradley J. BazuinAssistant Professor

Department of Electrical and Computer EngineeringCollege of Engineering and Applied Sciences

ECE 6640 2

Chapter 4

4. Bandpass Modulation and Demodulation/Detection.1. Why Modulate? 2. Digital Bandpass Modulation Techniques. 3. Detection of Signals in Gaussian Noise. 4. Coherent Detection. 5. Noncoherent Detection. 6. Complex Envelope. 7. Error Performance for Binary Systems. 8. M-ary Signaling and Performance. 9. Symbol Error Performance for M-ary Systems (M>>2).

ECE 6640 3

Sklars Communications System

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.

Signal Processing Functions

ECE 6640 4Notes 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.

Bandpass Demodulation and Detection

Focus on Signal of Symbol, Samples, and Detection In the presence of Gaussian Noise and Channel Effect

ECE 6640 5

ECE 6640 6

Analog Bandpass ModulationIncludes the RF/IF Frequency

AM , PM and FM Modulation

t

fp dmtmtftmA

ttftAts

3201

0

22cos1

2cos

t

3f2p0

0

dm2tmtf2

ttf2t

The time varying phase components

ECE 6640 7

Phasor Representation

Taking the positive spectrum complex representation

Think in terms of the analytical signal representation Complex, positive frequencies only

tjjtf2jexpRetAts 0

ECE 6640 8

Example: Bandpass Phasor Analysis of Double Sideband (DSB)

Given a tone message tf2cosAtm mm

tf2costf2cosAAts cmmc

tff2costff2cos2AAts mcmcmc

A positive frequency phasor can be defined and drawn First define the complex signal as (cos exp)

tff2jexptff2jexp

4AAts mcmcmc

Cfpos

ECE 6640 9

Phasor Analysis DSB (2)

tff2jexptff2jexp

4AAts mcmcmc

Cfpos

A positive frequency phasor can be defined and drawn

4AA

m

c

4AA

m

c

cfmf

mf

ECE 6640 10

Phasor Analysis AM

Given a tone message

tf2costf2cos1Ats cmc


tf2cos1tA mm

tffjAtffjAtfjAts mccmcccc

Cfpos 2exp4

2exp4

2exp2

tff2cos2

Atff2cos2

Atf2cosAts mccmcccc

ECE 6640 11

Phasor Analysis AM (2)


2A c

4

A c

4

A c

cf

mf

mf

tffjAtffjAtfjAts mccmcccc

Cfpos 2exp4

2exp4

2exp2

ECE 6640 12

Narrowband FM & PM Spectrum

Forming the Quadrature Representation and transforming the series expanded rig functions

tjtf2jexpAts cC

tjexptf2jexpAtjtf2jexpAts ccC

2cC tj!2

1tj1tf2jexpAts

Maintaining the 1st order terms

tj1tf2jexpAts cC

ECE 6640 13

Narrowband FM & PM Spectrum (2)

Taking the Fourier Trasnform of the 1st order approximation

tj1tf2jexpAts cC

fjfffAfS cC

ccC ffjffAfS

2cc22

C ffffAfS

ECE 6640 14

PM and FM Basis

Based on the previous analysis, we need to determine the transform of the phase components

tmt 2pPM

fMf 2pPM

t

3fFM dmt

f

Mjt 3fFM

ECE 6640 15

PM Phasor v1

The carrier can be removed to describe the baseband signal as a bounded phase variation about the carrier

cA

of

tm 2

p

tmjexpjtf2jexpAts 2p0c

ECE 6640 16

PM Phasor v2

For a cos wave message input

cA

of

tj1tf2jexpAts cC

tf2jexp2

jtf2jexp

2j

1tf2jexpAts mp

mp

cC

ECE 6640 17

FM Phasor

For a cos wave message input tj1tf2jexpAts cC

t

mfcC dtf2cosj1tf2jexpAts

tf2sinj1tf2jexpAts mfcC

tf2jexp

2tf2jexp

21tf2jexpAts mfmfcC

See Figure 4.4, p. 173

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 18

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. Sklars 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

20See Figure 4.5 on p. 174

21

Amplitude Shift Keying (ASK)

Digital Symbol Amplitude Modulation On-Off Keying (OOK)

Auto-correlation

Average Power

tfAts

ts

cc 2cos0

1

0

cc f

TAtstsE

tstsE

2cos2

02

11

00

4

0cos0

2210

21

1022

1100

ccOOK

ssssOOK

AT

AP

RPRPP

0tstsE 10

1tp

0tp

1

0

TEAc

2

22

Amplitude Shift Keying (2)

Auto-correlation

Symbol Power Spectral Density

Bandpass Bandwidth Nominally: BT=1/T, first null at Bnull=2/T

c

2c

ss

ss

f2cosT2

AR

0R

11

00

TffTffTAS cccOOK 2222

sincsinc8

TEAc

2

ASK Power Spectrum

From ABC Chapter 11

Baseband or LPF analysis

RF Analysis

23

n

bbbabavv rnfrnPrmfPrfS2222

2

,2

22 AaEAaE nn

trtp b rect

fArf

rAfS

bbvv

4sinc

4

222

bb rf

rfP sinc1

cvvcvvc ffSffSfG 41

24Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Figure 14.1-2

ASK Power Spectrum (2)

fArf

rAfS

bbvv

4sinc

4

222

cvvcvvc ffSffSfG 41 b

b Tr 1

ASK MATLAB Simulation

25

1 2 3 4 5 6 7 8 9 10 11

x 10-5

-1

-0.5

0

0.5

1

Time

Am

plitu

de

Symbol Sequenct in Time

0 1 2 3 4 5 6

x 108

-150

-100

-50

0

Frequency

Mag

nitu

de (d

B)

Symbol Sequence Circular Auto-correlation

3 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4

x 107

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency

Mag

nitu

de (d

B)

Symbol Sequence Circular Auto-correlation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-6

-0.5

0

0.5

1

1.5

2

2.5OOK Demodulation Eye Diagram

Time

Am

plitu

de

26

ASK Transmission Capability

Comparing the ratio of the bit rate to the required signal bandwidth

From the previous slide for the bandwidth

Therefore, the transmission capability is

T

b

BrTP

bT rB

Hzsecondperbit1 T

b

BrTP

M-ary ASK - Noncoherent 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)

27

12

12

1

2222

MmaE

MaEm

ana

na

fMArf

rMAfS

bbvv

4

1sinc12

1 22222

cvvcvvc ffSffSfG

28

M-ary ASK Transmission Capability

Comparing the ratio of the bit rate to the required signal bandwidth For m-ary, the bit rate is

The symbol bandwidth remains


Note that for m-ary ASK, the OOK system has the smallest spectral efficiency

Mrs 2logratebit

sT rB

Hzsecondperbitsloglog 22

MB

MrTPT

s

M-ary ASK - Coherent Use multiple amplitude levels MASK

M/2 positive values and M/2 Negative values: for k=1:M/2 voltage values could be at (k-1/2)/(M/2)

Bipolar levels (with phase reversals)

29

121

121

341

21

61211

411

212

0

22

222

1

22

1

2222

M

nnnnnnnn

aE

kkn

kM

maE

aEm

a

na

n

k

M

kana

na

222

sinc12

1

bb

vv rf

rMAfS

cvvcvvc ffSffSfG Note that AC power

doesnt change, just DC

30

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

(a) transmitter (b) signal constellation : Figure 14.1-3

Binary QAM

Tktpatxk

ki 2 Tktpatx

kkq 12

tftxtftx

Atxcq

cicc 2sin

2cos

222

0

AaE

aEm

na

na

31

Quadrature AM (QAM)

An M-ary Signal 4 complex symbols Quadrature

Auto-correlation, Single Pulse Period

Average Power

tf2cosAitstf2cosA1ts

tf2cosAitstf2cosA1ts

cc3

cc2

cc1

cc0

c

ckk f

AT

tstsE 2cos2

2*

2

A0cos21

T0

AE2

c2cQAM

itp

1tpitp1tp

3

2

1

0

32

QAM

Symbol Cross Correlation

Not that adjacent symbol average correlation is zero for equal probability symbols

TitC

TtC

1,0

0,0 1

TitC

TtC

3,0

2,0 1

02cos1 tfEAitsE cckk

0411

41

411

41

3210

1

1 20201000

TiitstsE

CPCPCPCPtstsE

kk

sssssssskk

33

Quadrature AM Power Spectrum

ss rf

rfP sinc1

trTttp ss

rectrect

cvvcvvc ffSffSfG

22

2 sinc1

ssscvv r

fr

rAfS

Note that the symbol rate is one-half the bit rate.

n

aavv rnfrnPrmfPrfS2222

2b

srr

22 sinc1

sscvv r

fr

AfS

22 2sinc4

bb

cvv r

fr

AfS

34

QAM Transmission Capability

Comparing the ratio of the symbol rate to the required signal bandwidth


T

s

BMrTP 2log

Hzsecondperbits2 TP

35

Phase Modulation Methods

Phase shift keying (PSK) is digital PM

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

Multiple discrete frequencies

k

sDkccc TktptfAtx 2cos

k

sDdkccc TktptfatfAtx 22cos

PSK Signal Constellations

36

This is QAM, rotated by pi/4 for 4-PSK

M=44-PSK

M=88-PSK

37

M-PSK

An M-ary Signal M complex symbols Quadrature (2 possible representations)

Auto-correlation, single symbol Period

Average Power, Amplitude to Energy

1Mto0kfor,M

1k2tf2cosAts cck

c

2c

*kk f2cos2

1T

AtstsE

2

0cos210 22 c

cQAMA

TAP

1Mto0kfor,M

1k2sin,M

1k2cosQ,Itp kkk

TEAc

2

38

Binary PSK Signal Symbols

Autocorrelation

Cross Correlation (the definition of antipodal)

tf2cosA1tf2cosAts

tf2cosA0tf2cosAts

cccc1

cccc0

c

2c

*kk f2cos2

1T

AtstsE

c

2c

*10 f2cos2

1T

AtstsE

0010 ssss

RR

39

Binary PSK Signal Symbols

Autocorrelation

Cross Correlation (the definition of antipodal)

tf2cosA1tf2cosAts

tf2cosA0tf2cosAts

cccc1

cccc0

c

2c

*kk f2cos2

1T

AtstsE

c

2c

*10 f2cos2

1T

AtstsE

0010 ssss

RR

BPSK Power Spectrum

From Chapter 11

Baseband or LPF analysis

RF Analysis

40

n

bbbabavv rnfrnPrmfPrfS2222

22,0 AaEaE nn trtp b rect

22

sinc

bbvv r

frAfS

bb rf

rfP sinc1

cvvcvvc ffSffSfG 21

BPSK MATLAB Simulation

41

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10-6

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Am

plitu

de

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 108

-160

-140

-120

-100

-80

-60

-40

-20

Frequency

Mag

nitu

de (d

B)

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

x 107

-120

-100

-80

-60

-40

-20

0

Frequency

Mag

nitu

de (d

B)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-6

-1.5

-1

-0.5

0

0.5

1

1.5BPSK Demodulation Eye Diagram

Time

Am

plitu

de

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)

42

43

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 1 2 3 4 5 6 7 8 9 10

Information m(k) 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

Detect 1 1 0 1 0 1 1 0 0 1

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

Detect 1 1 0 1 0 1 1 0 0 1

44


Figure 14.1-6

Offset-keyed QPSK transmitter (OQPSK)

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!

Digital Frequency Modulation

45

Frequency Shift Keying(FSK)

Continuous Phase FSK(CPFSK)

46

Frequency Shift Keying

Binary FSK

M-ary FSK or MFSK

Desired Condition (makes the time signal continuous at the symbol time boundaries)

tffAtstffAts

dcc

dcc

2cos2cos

1

0

10,2cos MtokfortkffAts stepstartck

intergeranmfor,22 mTf Sstep

47

M-FSK

An M-ary Signal M complex symbols

Desired Condition (normally)

Crosscorrelation

Autocorrelation

10,22cos MtokfortkftfAts stepstartck

intergeranmfor,22 mTfstep

tkfkffET

AtstsE stepstepstartck

22cos

212*

0

kff

TAtstsE stepstartckk 2cos2

12*

Can make expected value zero

48

BFSK

Signal Symbols

Autocorrelation

Cross Correlation

tffAtstffAts

dcc

dcc

2cos2cos

1

0

dcckk ffT

AtstsE 2cos212*

tfftffE

TAtstsE dcdcc 2cos2cos

2*10

dcdc fftfET

AtstsE 222cos212*

10

orthogonal for 2 x2fdxT=2

49

BFSK Quadrature Representation (1)

The sign term for odd bits becomes

tfafAts dkcck 2cos

tfatfAtfatfAts dkccdkcck 2sin2sin2cos2cos

tftfAatftfAts dcckdcck 2sin2sin2cos2cos

2b

drf

trtfAatrtfAts bcckbcck sin2sincos2cos

trtfAatrtfAts bcckkbcck sin2sin1cos2cos

tratrQItbb bkkbkkk sin1,cos,

1ka

50

BFSK Quadrature Representation (2)

tratrQItbb bkkbkkk sin1,cos,

2224

1kb

bbqilp Qr

rfrffGfGfG

The baseband spectrum Glp

2

2

2sinc

2sinc

41

b

bb

b

bk r

rfrrfr

Q

2

222

2

12

cos4

b

b

bk

rf

rf

rQ

2

22

12

cos4224

1

b

b

b

bbqilp

rf

rf

rrfrffGfGfG

51


Figure 14.1-8

Power spectrum of BFSK

222 bd rf

2b

drf

2

22

12

cos4224

1

b

b

b

bbqilp

rf

rf

rrfrffGfGfG

BFSK MATLAB Simulation

52

Not readily observable

The change in frequency is too small

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 108

-150

-100

-50

0

Frequency

Mag

nitu

de (d

B)

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

x 107

-120

-100

-80

-60

-40

-20

0

Frequency

Mag

nitu

de (d

B)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-6

-1.5

-1

-0.5

0

0.5

1

1.5BFSK Demodulation Eye Diagram

Time

Am

plitu

de

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

53

54

Special Versions of FSK Continuous Phase FSK (CPFSK)

t

dccc dxftfAtx0

22cos

TktTkTktaTa

TtTTtaTaTtta

dx

k

k

jj

t

1,

2,0,

1

0

10

0

0

Minimum-Shift Keying (MSK) The binary version of CPFSK Also called fast FSK Capable of using an rb/2 bandwidth

55

CPFSK

Continuous Phase FSK (CPFSK)

The phase is continuous at the transitions between bit. This is most easily accomplished if the phase is or a

multiple of at the start and end of each bit period.

t

dccc dxftfAtx0

22cos

TktTkTktaTa

TtTTtaTaTtta

dx

k

k

jj

t

1,

2,0,

1

0

10

0

0

Binary CPFSK

The binary version of CPFSK is calledMinimum-Shift Keying (MSK) Also called fast FSK Capable of using an rb/2 bandwidth

56

57

MSK Baseband (Sklar Chap 9)

Frequency and phase (history) modulation the previous/current phase determines the next phase point phase x data summation in time

kkk QItbb ,

Tktpcatxk

kkki

cos Tktpcatxk

kkkq

sin

Tktrc bk

2

oddkfork

evenkfork

k

,22

1

,2

58


(a) phase path (b) i and q waveforms: Figure 14.1-11

Illustration of MSK.

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.

59

MSK power spectrum: Figure 14.1-9

Minimum Shift Keying (MSK)

Use 0.25 in BFSK Sim Tfstep2

Sklar Representations

Amplitude Shift Keying

Phase Shift Keying

Frequency Shift Keying

ECE 6640 60

tfTEts cii 2cos

2

tfTEts

ts

c2cos2

0

1

0

10

122cos2

MtokforM

ktfTEts ck

tfTEts

tfTEts

c

c

2cos2

02cos2

1

0

10

2cos2 min

Mtokfor

tfkfTEts stepk

tf

fTEts

tf

fTEts

stepc

stepc

22cos2

22cos2

1

0

Textbook Waveform Energy

Waveform Energy (Symbol Autocorrelation)

Matched Filter

ECE 6640 61

T

ii dttsE0

2

t

dthrthtrtz

tTstuth *

t

dtTsstz0

*

TTT

dsdssdTTssTz0

2

0

*

0

*

Correlation

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.

62bbR T

ES 1

Energy and Power

For

The average power and energy per bit becomes

ECE 6640 63

ETTEdt

TE

dttfETE

dttfETE

dttfTEEE

T

Tc

T

c

T

cb

22

212

222cos

212

2cos2

2cos2

0

0

0

2

0

2

tfTEts c2cos

2

TE

TEAP

22 221

2

64

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)

Eb/No relates the energy per bit to the noise energy(equal to S/N times a time-bandwidth product)

WTNE

WNT

E

NS

b

bbb

1

1

00

WTNS

RW

NS

NE

bb0

b

If you want a higher Eb/No, increase Tb.(Changing W changes the SNR too!)

Symbol Detection

Baseband detection and BER defined in the previous chapter.

The following are from ABC Chap. 14

ECE 6640 65

66


(a) parallel matched filters (b) correlation detector: Figure 14.2-3

Optimum binary detection

67


Figure 14.2-2

Conditional PDFs

68


Figure 14.2-1

Bandpass binary receiver

Using superposition of the parallel matched filters, the BPF is the difference of the two filters.

This results in an optimal binary detector

thththBPF 01

69

Binary Receiver

OOK

BPSK

BFSK

thththBPF 01 tTsKth 11 tTsKth 00

tTftTsKthth cBPF 2cos11

tTftTsKthththth cBPF 2cos22 1101

tTsKtTsKthththBPF 0101 tTfftTffth dcdcBPF 2cos2cos

tTftTfth dcBPF 22sin2sin2

70


Figure 14.2-4

Correlation receiver for OOK or BPSK

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.)

71

Optimal Parallel Matched Filter Receiver Error Analysis

Evaluating the expected value

0

0

2012

max

01

22 N

dttstsEzz

T

TTTT

dttsEdttstsEdttsEdttstsE0

20

001

0

21

0

201 2

01010

201 2 EEEdttstsE

T

201 EEEb

0

10

0

102

max

01

222

2 NEE

NEEzz bb

72

Optimal Parallel Matched Filter Receiver Error Analysis

OOK

PSK

FSK

Tb

b dttstsEEEEEE

001

0110

010 E0

2

max

01

2 NEzz b

bEE 1100

2

max

01 22 N

Ezz b

010 E0

2

max

01

2 NEzz b

73

Generalized Probability of Error

Using the optimal BPF filter and sampling for each symbol, the relationship will be based on:

The BER is then based on

Therefore picking arbitrary symbols is possible, but the symbol correlation coefficient will drive the BER performance.

00

102

max

01 12 N

EN

EEzz bb

0

01 12 N

EQzzQP be

74

Generalized FSK

There are multiple orthogonal tone separations. The correlation coefficient can go negative! The minimum occurs at

approximately sinc(1.22) = -0.166

tffAtstffAts

dcc

dcc

2cos2cos

1

0

T

dcdcc dttfftffEAE0

210 2cos2cos

T

dcc dttftfAE

0

2

10 22cos22cos2

T

ddb dttfitfi

TEE

010 22exp22exp2

1

d

d

d

dbb fi

Tfifi

TfiTEE

2222exp

2222exp

21

b

dbdb

d

dbb r

fETfEf

TfTEE 4sinc4sinc

2222sin

Tkff stepd 2

2

MATLAB Coherent Receivers

BASK example code BPSK example code BFSK example code

ECE 6640 75

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!

76

77


Figure 14.3-2

Noncoherent OOK receiver

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

78


Figure 14.3-3

Conditional PDFs for noncoherent OOK

2c

optAV

opteopte VPVP 10

0

0 2exp

NEP be

0

010 2exp

21

21

21

NEPPPP beeee

79


Figure 14.3-5

Noncoherent detection of binary FSK

Noncoherent FSK

Qualitative comments Using envelope detectors on each symbol output, the Rician error

distribution effects the z detection statistic.

80

02

exp21

NEP be

81


(a) coherent BPSK (b) DPSK (c) coherent OOK or FSK (d) noncoherent FSK (e) noncoherent OOK: Figure 14.3-4

Binary error probability curves

0 2 4 6 8 10 12 14 1610-6

10-5

10-4

10-3

10-2

10-1

100BER Simulation for BPSK and BFSK

Eb/No (dB)

BE

R

BPSK simulationBPSK (theoretical)BFSK simulationBFSK (theoretical)

82


(a) coherent BPSK (b) DPSK (c) coherent OOK or FSK (d) noncoherent FSK (e) noncoherent OOK

Binary error probability curves

Figure 14.3-4

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.

83

Quadrature-carrier receiver with correlation detectors

Applicable for: M-QAM M-PSK

84


Figure 14.4-1

85


Figure 14.4-2

Carrier synchronization for quad-carrier receiver

Coherent M-ary PSK receiver

MPSK_Demo.m Fixed N0, varying signal Eb

86


Figure 14.4-3

-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20PreDecode, Es/N0 (dB)=19

Real

Imag

87


Figure 14.4-4

Decision thresholds for M-ary PSK

-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15


Real

Imag

PSK signal constellations

MPSK Symbols are typically Gray-code encoded prior to transmission In the Gray-code, adjacent symbols are only different by 1 bit

value!

88


(a) M=4 (b) M=8Figure 14.5-1

MPSK Eb/N0 Examples

89

-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6


Real

Imag

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8


Real

Imag

-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15


Real

Imag

-20 -10 0 10 20

10-4

10-3

10-2

10-1

100Symbol Error Rate, M=8

Es/No (dB)

SE

R

-20 -10 0 10 20

10-4

10-3

10-2

10-1

100Bit Error Rate, M=8

Eb/No (dB)B

ER

Simulated Performance MPSK MPSK_Ber and MPSK_PP_Plot

90

-5 0 5 10 15 20 2510-7

10-6

10-5

10-4

10-3

10-2

10-1

100MPSK Symbol Error Rate

Es/N0 (dB)

SE

R

M=2 SimM=2 BoundM=4 SimM=4 BoundM=8 SimM=8 BoundM=16 SimM=16 Bound

-5 0 5 10 15 2010-7

10-6

10-5

10-4

10-3

10-2

10-1

100MPSK Bit Error Rate

Eb/N0 (dB)

BE

R


Simulated Performance MFSK MFSK_Ber and MFSK_PP_Plot

91

0 2 4 6 8 10 12 14 1610-7

10-6

10-5

10-4

10-3

10-2

10-1

100MFSK Symbol Error Rate

Es/N0 (dB)

SE

R


-5 0 5 10 1510-7

10-6

10-5

10-4

10-3

10-2

10-1

100MFSK Bit Error Rate

Eb/N0 (dB)

BE

R


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! But the symbol bandwidth (filtering) must be increasing.

92

93


(a) transmitter (b) receiver (c) square signal constellation and thresholds with M=16

Figure 14.4-8

M-ary QAM system

94


410beP

Performance comparisons of M-ary modulation systems

MATLAB Coherent Receivers

MPSK example code MFSK example code QAM example code

ECE 6640 95

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 96

QAM BER Computation

ECE 6640 97

% 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)));endPB3(:,ii) = 4*((1-L^-1)/log2(M))*Qsum;

QAM BER Curves

ECE 6640 98

0 5 10 15 20 25 3010-7

10-6

10-5

10-4

10-3

10-2

10-1

100BER Composite Plot

Bit

Erro

r Rat

e

Eb/No (dB)

4 QMA16 QAM64 QAM256 QAM

QAM BER Curves Detail/Differences

ECE 6640 99

-1 0 1 2 3 4 5 6 7 8 9 1010-2

10-1

100BER Composite Plot

Bit

Erro

r Rat

e

Eb/No (dB)

4 QMA16 QAM64 QAM256 QAM

MPSK Nyquist Filter BERSER vs SNR

ECE 6640 100

0 5 10 15 20 25 30 35 40 45 50 5510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

SNR (dB)

Sym

bol E

rror R

ate

MPSK Simulation: Theory vs. Simulation

T4S4T8S8T16S16T32S32T64S64T128S128T256S256

SklarTheory Plot

MPSK Nyquist Filter BERBER vs Eb/No

ECE 6640 101

-5 0 5 10 15 20 25 30 35 40 4510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

EbNo (dB)

Bit

Erro

r Rat

eMPSK Simulation: Theory vs. Simulation

T4S4T8S8T16S16T32S32T64S64T128S128T256S256

SklarTheory Plot

QAM Nyquist Filter BERSER vs. SNR

ECE 6640 1020 5 10 15 20 25 30 35

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

SNR (dB)

Sym

bol E

rror R

ateQAM Simulation: Theory vs. Simulation

T4S4T16S16T64S64T256S256

SklarTheory Plot

QAM Nyquist Filter BERBER vs Eb/No

ECE 6640 103

SklarTheory Plot

-5 0 5 10 15 20 2510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

EbNo (dB)

Bit

Erro

r Rat

eQAM Simulation: Theory vs. Simulation

T4S4T16S16T64S64T256S256

M-ary Signaling

Does M-ary signaling improve or degrade error performance? This is a loaded question! For coherent orthogonal signaling there appears to be an improvement.

However, the bandwidth must be increasing if the same symbol rate is maintained.

For coherent phase based signaling there appears to be a degradation. However, for the sample symbol time, the data rate is increasing. It may be decreased when more bits-per-symbol are transmitted.

M-ary signaling provides a method to provide system tradeoffs. Error performance can be traded off with symbol time and bandwidth. Tradeoffs to be discussed in Chap. 9.

ECE 6640 104

Symbol vs Bit Error Rates

SNR and Es/No are related for M-ary symbol transmission. To derive Eb/No, for a known number of symbols

ECE 6640 105

bits

s

symbols

sb k

EMEE

2logsymbolsbits Mk 2log

kWT

NS

RW

NS

NE b

b

b

0Tk

TM

R symbolsb 2log

In every MATLAB Simulation I try to be very careful in defining

SER vs BER and Es/No vs Eb/No.

Notes from Proakis

Notes and figures are based on or taken from materials in the previous course textbook

J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.

ECE 6640 106

4.3 Optimal Detection and Error Probability for Band-Limited Signaling

These are for lower bandwidth, low dimensionality signaling types.

This section is an excellent reference for some of the primary signal types discussed.

Explicit BER vs. Eb/No equations are derived based on the previous material presented. An assumption of equally likely symbols is made for each

derivation. 4.3-1 Derives ASK or PAM 4.3-2 Derives MPSK 4.3-3 Derives QAM

ECE 6640 107Notes 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.

MASK Summary


MASK


MASK Performance


MPSK

The marginal probability density function for a symbol can be defined as

The pdf is a function of the average symbol energy

The higher the number of symbols, the tighter the symbol decision regions must become and more errors can be expected.


MPSK

In general, the integral of p() does not reduce to a simple form and must be evaluated numerically, except for M = 2 and M = 4.

For M=2

For M=4

For other M (M large and SNR large)


MPSK

M=2

M=4

M other


QAM

ECE 6640 114

QAM is dependent upon the symbol constellation selected. Default to square constellations

of 4, 16, 64, & 256 Numerous others are possible

with potentially better system performance

The optimal detector uses 2 basis symbols to resolve the in-phase and quadrature components

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.

Square Constellation QAM

This case appears as two dimensional ASK/PAM


Square Constellation QAM


Comparing QAM and MPSK

Looking at the ratio of the Q(x) arguments

At M=4 the systems are equivalent, but for higher M QAM has better performance.


Demodulation and Detection of Band-Limited Signals

Matched filter involve the basis form of the signals.

ECE 6640 118

Note: Filter are matched to basis, not matched to symbols!

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.

4.4 Optimal Detection and Error Probability for Power-Limited Signaling

These are for wider bandwidth, higher dimensionality signaling types.

BER vs. Eb/No equations are derived based on the previous material presented. An assumption of equally likely symbols is made for each

derivation. 4.4-1 Orthogonal FSK 4.4-2 Biorthogonal 4.4-3 Simplex

ECE 6640 119

Orthogonal Signals - MFSK

For equiprobable, equal-energy orthogonal signals, the optimum detector selects the signal resulting in the largest cross-correlation between the received vector r and each of the M possible transmitted signal vectors {sm}, i.e.,

ECE 6640 120

Ed 2min

Orthogonal Signals (cont)

The probability of correct symbol detection can be described as

assuming Gaussian noise elements where the elements are independent and identically distributed (IID)

with an individual dimension represented as

ECE 6640 121

Orthogonal Signals (cont)

The integral becomes

The error is the complement, therefore

In general, Equation 4.410 cannot be made simpler, and the error probability can be found numerically for different values of the SNR.

To determine bit errors, let us assume that s1 corresponds to a data sequence of length k with a 0 at the first component. The probability of an error at this component is the probability of detecting an sm corresponding to a sequence with a 1 at the first component. Since there are 2k1 such sequences, we have

ECE 6640 122

FSK SignalingAnother Union Bound

The orthogonal case is easier than the previous derivation as all symbols have the minimum distance. Taking the result

For orthogonal signaling

Using M = 2k and Eb = E/k, we have

Note that if

Then Pe 0 as k (not Pe infinite like the text says!!)ECE 6640 123

Ed 2min

Note: a necessary and sufficient condition for reliable comm. that is slightly lower is derived in Chap. 6. It is called the Shannon limit.

Orthogonal Signaling

ECE 6640 124

4.4-2 Biorthogonal Signaling

ECE 6640 125

Ed 2min

Edother 2

Biorthogonal Signaling

ECE 6640 126

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