ece 6640 digital communications

ECE 6640Digital Communications

Dr. Bradley J. BazuinAssistant Professor

Department of Electrical and Computer EngineeringCollege of Engineering and Applied Sciences

ECE 6640 2

Chapter 3

3. Baseband Demodulation/Detection.1. Signals and Noise. 2. Detection of Binary Signals in Gaussian Noise. 3. Intersymbol Interference. 4. Equalization..

ECE 6640 3

Sklar’s 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.

Chapter Goals

• Detection of Binary Signals plus Gaussian Noise• The decision process• Define Intersymbol Interference• Error Performance Degradation• Equalization Techniques

ECE 6640 4

Demodulation and Detection

• Focus on Symbols, Samples, and Detection• In the presence of Gaussian Noise and Channel Effects

ECE 6640 5

Demodulation and detection

• Major sources of errors:– Signal Path Loss

• Friis equation relates the received signal power to the transmitted power, antenna gains, and distance that the signal travels in free space

– Thermal noise (AWGN)• disturbs the signal in an additive fashion (Additive)• has flat spectral density for all frequencies of interest (White)• is modeled by Gaussian random process (Gaussian Noise)

– Inter-Symbol Interference (ISI)• Due to the filtering effect of transmitter, channel and receiver,

symbols are “smeared”.• Time spreading effects cause symbols to “overlap”

– Total symbol “length” may easily be 3+ symbol periods!

6

Receiver Block Diagram

• Receive the transmitted symbol plus noise– Symbol filter by channel

• Frequency down-conversion to baseband– Receiver filtering and equalization (if needed) applied

• Symbol filter– Matched filtering with Nyquist shaping for ISI– Optimize the pre-detected signal prior to sampling

• Optimal Time Sampling – Peak filter response timeECE 6640 7

Review Slides from ECE5640

• Chapter 9: Noise• Chapter 10: Noise in analog modulated signals• Chapter 11: Baseband Digital Transmission

ECE 6640 8

9

Noise Approximation• Uniform Noise Spectral Density

– Resistor description (Thevenin Model)

• Available Power from the “noise source”– Source output power into a matched load

TR2fGvv

ssout vR2

Rv

R4

vR1

2v

RvP

2s

2s

2sout

sout

2

N2T

R4TR2

R4fGfG 0vv

ss

2

NR 0ss

10

System Noise• Since the noise power spectrum is uniform, a systems

noise power is the product of the noise power and the integral of the filter power.

20NN

2NN fH

2NfSfHfS

00

0

20

20NN dffHNdffH

2N0R

Noise Equivalent Bandwidth• If we want the total noise power after the filter, we can

integrate the PSD for all frequencies or use the Filtered noise autocorrelation function at zero.– Both of these approaches may be difficult– Could we great a more simple “noise equivalent bandwidth for

filters” that is rectangular?

11

12

EQNEQNPowerDCelrect BHBGaindffHdffH

2_

0

2

mod_0

2 0

EQNPowerDCelrect B

frectGainfH2_mod_

20

2

EQN0H

dffHB

2Power_DC 0HGain

Noise Equivalent Bandwidth

0

2020NN dffH2

2NdffH

2N0R

• When filtering, it is convenient to think of band-limited noise, where the filter is a rect function with bandwidth BEQN

13

Noise Equivalent Bandwidth• Low pass filter

0Hgain_coherent

• For a unity gain filter – assumed when computing receiver input noise power

EQN0EQN0

NNN BNB22

N0RP

2Power_DC 0HGain

20

2

EQN0H

dffHB

0

2EQN dffHB

EQN02

EQN20

NNN BN0HB0H22

N0RP

Model of Received Signal with Noise

© 2010 The McGraw-Hill Companies

15

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

Analog baseband transmission system with noise: Figure 9.4-2

Signal Plus Noise

• Additive Gaussian White Noise

ttf2costAtx c

ttfLtAtx cR 2cos tnttf

LtAts c 2cos

thtnttfLtAteD c

2cosPr

16

Signal-to-Noise Ratio• Comparing the desired signal power to the undesired noise

power.

• To compare signal and noise power, we must assume a filtering operations

tntxty c

TransmittingAntenna

ReceivingAntenna

RF Communication Channel

Noise

Linear Filtering

NonlinearDistortion

Atten-uation

tn

txc

tntxty c

17

Signal-to-Noise Ratio

• Equivalent receiver input signal and noise (ER)

• Equivalent destination signal and noise (D) or pre-demodulation (PreD)

thtntxty RD

tntxty ERERR

tntxty eDeDeD PrPrPr

18

Signal-to-Noise Ratio

• Equivalent receiver input SNR (ER)

EQN

R

EQN

ER

ER

ERR BN

SBN

txEtnEtxESNR

00

2

2

2

• Equivalent destination SNR EQN

D

D

D

eD

eDR BN

SNS

tnEtxESNR

02

Pr

2Pr

can be used to represent receiver noise figure contributions

19

Increase in SNR with filtering• If a filter matched to the input signal is applied, the noise

power would be reduced to the smallest equivalent noise bandwidth that is allowed.– Filter to minimize noise power– Importance of the IF filter in a super-het receiver!

• Front-end filtering goals – a dilemna– Minimize signal power loss (wider bandwidth)– Minimize filter equivalent noise bandwidth

(narrower bandwidths)– A trade-off must be made!

20

Typical Transmission RequirementsSignal Type Freq. Range SNR (dB)Intelligible Voice 500 Hz to 2 kHz 5-10Telephone Quality 200 Hz to 3.2 kHz 25-35AM Broadcast Audio 100 Hz to 5 kHz 40-50High-fidelity Audio 20 Hz to 20 kHz 55-65Video 60 Hz to 4.2 MHz 45-55Spectrum Analyzer 100 kHz-1.8 GHz 65-75

21


Model of a CW communication system with noise: Figure 10.1-1

CW Communication with Noise

ttf2cosLtAtx cc

tnttf2cosLtAtv c

thtnttfLtAtPreD Rc

2cos

ttf2costAtx c

22

Signal and Noise Power• What are the signal and noise powers at the receiver?

• What is the receiver input power

2T

2T

Ts dttxtxT1limP

2n tnEP

2T

2T

ccT

2T

2T

Tv dttntxtntxET1limdttvtvE

T1limP

23

Receiver Signal plus Noise Power• What is the receiver input power

2T

2T

ccT

2T

2T

Tv dttntxtntxET1limdttvtvE

T1limP

2T

2T

2c

2cTv dttntntx2txE

T1limP

2T

2T

2c

2cTv dttnEtnEtx2tx

T1limP

2

2

22

2

2

2

2 021limT

T

T

Tc

T

TcTv dttnEdttxdttx

TP

ns2

2T

2T

2cTv PPtnEdttx

T1limP

24

Signal-to-Noise Ratio (SNR)• The SNR is a measure of the signal power to the noise

power at a point in the receiver.– Typically described in dB– The above computation was performed at the input

• Matlab SNR example: SNR_AM_Example.m– Pre-D “AM” SNR based in filter Beqn

• Effective BEQN RF due to sampling spectrum (B=Fis/2)• AM signal power based on carrier plus signal

n

s

PPSNR

25

Noise Equivalent Bandwidth• Since the noise power spectrum is uniform, a systems

average noise power is the product of the noise power and the integral of the filter power.

20NN

2NN fH

2NfSfHfS

00

0

20

20NN dffHNdffH

2N0R

26

Noise Equivalent Bandwidth

• When filtering, it is convenient to think of band-limited noise, where the filter is a rect function with bandwidth BEQN

0

2020NN dffH2

2NdffH

2N0R

EQN2

EQNPower_DC0

2

elmod_rect0

2 B0HBGaindffHdffH

EQNPowerDCelrect B

frectGainfH2_mod_

20

2

0H

dffHBEQN

2Power_DC 0HGain

27

Noise Equivalent Bandwidth• Low pass filter

0Hgain_coherent

• For a unity gain filter – assumed when computing receiver input noise power

EQN0EQN0

NNN BNB22

N0RP

2Power_DC 0HGain

20

2

EQN0H

dffHB

0

2EQN dffHB

EQNEQNNNN BNHBHNRP 0220 002

20

28

Filtering• What happens if the receiver input is filtered?

• What effect does the filter have on the signal?– None or slight band edge de-emphasis, if and only if the filter is

“wider” than the signal bandwidth– Now you know why a 3dB bandwidth isn’t that useful,

(3dB1/2 power point)!

ththtntxtv 21cf

ththtnththtxtv 2121cf

n

s

PPSNR

29

Filtering• What effect does the filter have on the noise?

– Normally you would expect for two filters

– Assume that the filters follow each other and that the first filter is narrower than the second filter

1_01_

FilterEQN

sFilterPost BN

PSNR

1_01__ FilterEQNFilterPostN BNP

1_0

2_1_02__ ,min

FilterEQN

FilterEQNFilterEQNFilterPostN

BNBBNP

2_02__ FilterEQNfilterpostN BNP 2_0

2_FilterEQN

sFilterPost BN

PSNR

1_02_

FilterEQN

sFilterPost BN

PSNR

30

Filters Provide SNR “Gain”• If filter 2 Beq < filter 1 Beq:

• You expect the IF filter to be smaller than the front-end RF or “pre-filtering” performed

– Think about kTB at different bandwidths and you will derive the same “gain”

– In typical receivers, the IF filter sets the Pre-Demodulation Bandwidth

2Filter_EQN

1Filter_EQN1Filter_Post

2Filter_EQN0

s2Filter_Post B

BSNR

BNPSNR

2Filter_EQN

1Filter_EQNFilter B

BGain

31

Bandpass Noise Processing

• What happens after mixing and lowpass filtering?– assume LPF passes the entire baseband.

cfTc Bf

cfTc Bf

TBTB

20N

tf2cos c Band Pass

FilterLowPass

Filter

Bandpass filter bandwidth may not be centered on fc• an alpha offset

32

Quadrature Noise (1)• Question: Is Quadrature noise a different power than “real”

baseband noise• Noise in a quadrature process

• Noise power is related as

• What about ?

tftntftntn cqci 2sin2cos

22 2sin2cos tftntftnEtnE cqci

2

0222 NtnEtnEtnE qi

33

Quadrature Noise (2)• Noise in a quadrature process

2cqci

2 tf2sintntf2costnEtnE

tf2sintn

tf2sintf2costntn2tf2costn

EtnE

c22

q

ccqi

c22

i2

222cos

21222cos

21 222 tftntftnEtnE cqci

22

121 0222 NtnEtnEtnE qi

222cos

21222cos

21 222 tfEtnEtfEtnEtnE cqci

Mixing Noise (1)• Think of the two noise bands as

1. The band of interest2. The image band

34

thtftfftntfftn

thtftfftntfftnthtftn

IFLOIFLOqIFLOi

IFLOIFLOqIFLOiIFLO

2cos2sin2cos

2cos2sin2cos2cos

22

11

th

tfftntfftn

tftntftn

thtfftntfftn

tftntftnthtftn

IFIFLOqIFLOi

IFqIFi

IFIFLOqIFLOi

IFqIFiIFLO

22sin22cos

2sin2cos

21

22sin22cos

2sin2cos

212cos

22

22

11

11

thtftntftn

thtftntftnthtftn

IFIFqIFi

IFIFqIFiIFLO

2sin2cos21

2sin2cos212cos

22

11

Mixing Noise (2)• Defining the equivalent IF noise

• But this is the same as quadrature noise

• Mixing doesn’t change the noise power 35

thtftntftn

thtftntftnthtftn

IFIFqIFi

IFIFqIFiIFLO

2sin2cos21

2sin2cos212cos

22

11

thtftntftnthtn IFIFqIFiIF 2sin2cos

th

tftntn

tftntnthtftn IF

IFii

IFii

IFLO

2sin21

21

2cos21

21

2cos

21

21

22

121 02

22

12 NtnEtnEtnE iii

221

21 02

22

12 NtnEtnEtnE qqq

36

Mixing Noise to Baseband• What if we split bandpass noise into two

distinct noise bands, BT/2 above and below the carrier/IF?

• Noise power is related as

• Noise bands get added …

WNBNBNtnE TTcarrier 0002 22

2

thtfftntfftn

thtfftntfftntn

BcqBci

BcqBci

222

111

2sin2cos2sin2cos

WNBNBNtnEtnE TTBelowCAboveC 00

022

21 22

22

WNBNtnEtnEtnE T 002

22

12 2

2TBWfor

37


(a) General case; (b) symmetric-sideband case;

(c) suppressed-sideband case: Figure 10.1-3

Mixing Noise to Baseband

10 or

5.0

5.00

cfTc Bf

cfTc Bf

TBTB

20N

Mix to BasebandW=½ BT

N0BT=2N0W

Mix to IFBT

N0BT

Not Desired

Why did we do these derivations?• The past derivations were all about mixing and filtering.

– Quadrature noise is the noise that gets mixed to the intermediate. The bandwidth and noise power do not change

– Quadrature noise is the noise that gets mixed to baseband. The bandwidth is halved and the noise power is doubled the LPF bandwidth standard noise power.

38

WNBNBNtnE TT 0002 22

2

WNBNtnE T 002 2WBT

2

39

Complex Noise• Noise in a complex process

• Noise power is related as tnjtntn qi

222 tntntnjtntnjtnEtnE qiqqii

2

02 NtnE

Hqiqi tnjtntnjtnEtnE 2

222 tnEtnEtnE qi

tnjtntnjtnEtnE qiqi 2

4

022 NtnEtnE qi

),(: nmrandnnMATLAB 2),(),(: sqrtnmrandninmrandnnMATLAB

40

Noise Envelope and Phase (1)• Noise as a magnitude and phase

ttf2costAtn ncn

nni cosAn nnq sinAn

• The magnitude is a Rayleigh distribution– Mean and moment

nR

2n

R

nnA Au

N2Aexp

NAAp

n

2NAE R

n

R2

n N2AE

41

Noise Envelope and Phase (2)• Probability of An exceeding a value “a”

• Phase Distribution

• Noise Power

Rn N

aaAP 2exp2

2021

nn forp

2212

2cos

02

222

NNNtnE

ttfEtAEtnE

RR

ncn

2

Rn

NAE

Rn NAE 22

42

Noise Characteristics• The noise power does not change based on the

representation, the center frequency, or due to mixing.

• The noise power will change when the bandwidth is further limited in some way!

43


Model of a CW communication system with noise: Figure 10.1-1

CW Communication with Noise

ttf2cosLtAtx cc

tnttf2cosLtAtv c

thtnttfLtAteD Rc

2cosPr

ttf2costAtx c

Chapter 11

• Baseband Transmission of PAM Symbols• PAM Symbol Autocorrelation and Power Spectral Density• Symbol Detection in Noise

ECE 6640 44

45

Digital Pulse-Amplitude Modulation (PAM)

• Also referred to as pulse-code modulation (PCM)• The amplitude of pulse take on discrete number of

waveforms and/or levels within a pulse period T.

• p(t) takes on many different forms, a rect for example

k

k kTtpatx

else0

Tt01tp

T0for,apakTmTpamTx mmk

k

46

Digital Signaling Rate• For symbols of period T,

the symbol rate is 1/T=R

• The rate may be in bits-per-second when bits are sent. A bps rate is usually computed and defined.

• The rate may be in symbols-per-second when symbols are sent. When there are a defined number of bits-per-symbol, the rate may be defined in bits-per-second.– If parity or other non-data bits are sent, the messaging rate and the

signaling rate may differ.

47


(a) Baseband transmission system (b) signal-plus-noise waveform: Figure 11.1-2

Transmission

tnkTttp~atyk

dk

48

Transmission

• The digital signal is time delayed

• The pulse is “filtered” and/or distorted by the channel

• Recovering or Regenerating the signal may not be trivial

– Signal plus inter-symbol interference (ISI) plus noise

tnkTttp~atyk

dk

thtpfntp c~

dt

dmk

kmd tmTnkTmTpaatmTy

~ˆ

49


(a) Unipolar RZ & NRZ

(b) Polar RZ & NRZ

(c) Bipolar NRZ

(d) Split-phase Manchester

(e) Polar quaternary NRZ

ABC Binary PAM formats

50

PAM Power Spectral Density: Polar NRZ

• For a zero mean, polar NRZ of amplitude +/- A and symbol duration Tb

k b

bdk T

TkTtrectatv

bdb

d TTT

Tp 0,1 kjfor,0aaE

aE,0aE

kj

22nn

bbb

vv TTT

tvtvER

,12

bbvv TfTtvtvEwS 22 sinc

222 AaE n

bb

bbvv

rf

rA

TfTAwS

22

22

sinc

sinc

51

PAM Power Spectral Density: Arbitrary Pulse

• Using Poisson’s sum formula

k

dk D

DkTtpatv DT0,D1Tp dd

222, aanan maEmaE

n

avv DfjnRfPD

fS 2exp1 2

0,

0,2

22

nm

nmnR

a

aaa

n

aavv D

nfDnP

DmfP

DfS

222

2

DrDT bb

1,

n

bbbabavv rnfrnPrmfPrfS 2222

rb is symbol rate

52


Figure 11.1-5

Power spectrum of Unipolar, binary RZ signal

bb r

fr

fP2

sinc2

1

2

,2

22 AaEAaE nn

n

bbb

vv rnfnAr

fr

AfS2222

2sinc

162sinc

16

trTttp b

b

2rect

2

rect

0,4

0,2

22

222

nAm

nAmnR

a

aa

a

Power spectrum of Unipolar, binary RZ signal

• For rb=2

53

n

bbb

vv rnfnAr

fr

AfS2222

2sinc

162sinc

16

-8 -6 -4 -2 0 2 4 6 80

0.01

0.02

0.03

0.04

0.05

0.06

0.07Unipolar Binary RZ

freq (f)

Plots from PSD_PCM.m

54

Power spectrum of Unipolar, binary NRZ signal

bb rf

rfP sinc1

2

,2

22 AaEAaE nn

0,4

0,2

22

222

nAm

nAmnR

a

aa

a

nb

bbvv rnfnA

rf

rAfS 2

222

sinc4

sinc4

trTttp bb

rectrect

fArf

rAfS

bbvv

4sinc

4

222

• For rb=2

-8 -6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25Unipolar Binary NRZ

freq (f)

55

Power spectrum of Polar, binary RZ signal (+/- A/2)

4,022 AaEaE nn

0,0

0,42

222

nm

nAmnR

a

aaa

22

2sinc

16

bb

vv rf

rAfS

bb r

fr

fP2

sinc2

1

trTttp b

b

2rect

2

rect

-8 -6 -4 -2 0 2 4 6 80

0.005

0.01

0.015

0.02

0.025

0.03

0.035Polar Binary RZ

freq (f)

• For rb=2

56

Power spectrum of Polar, binary NRZ signal (+/- A/2)

bb rf

rfP sinc1

4,022 AaEaE nn

0,0

0,42

222

nm

nAmnR

a

aaa

22

sinc4

bbvv r

fr

AfS trTttp bb

rectrect

• For rb=2

-8 -6 -4 -2 0 2 4 6 80

0.02

0.04

0.06

0.08

0.1

0.12

0.14Polar Binary NRZ

freq (f)

57

Spectral Attributes of PCMIf Bandwidth W=1/T, then WT=1

Note that WT=0.5 or a bandwidth equal to ½ the symbol rate can be used!

58


Figure 11.2-1

Baseband Binary Receiver

• Synchronous Time sampling of maximum filter output

thtnthkTtpaty ink

k

kkk taty n

59


(a) signal plus noise (b) S/H output (c) comparator output: Figure 11.2-2

Regeneration of a unipolar signal

60

Unipolar NRZ Binary Error Probability

• Hypothesis Testing using a voltage threshold– Hypothesis 0

• The conditional probability distribution expected if a 0 was sent

– Hypothesis 1• The conditional probability distribution expected if a 1 was sent

kYkkkYkY tpatapHyp n0|n| 0

kYkkkYkY tpAatapHyp nA|n| 1

kNkY ypHyp 0|

A-| 1 kNkY ypHyp

61


Conditional PDFs Figure 11.2-3

Decision Threshold and Error Probabilities

• Use Hypothesis to establish a decision rule– Use threshold to determine the probability of correctly and

incorrectly detecting the input binary value

V

0Y0e dyH|ypVYPP

V

1Y1e dyH|ypVYPP

kkk taty n

62

Average Error Probability

• Using the two error conditions:– Detect 1 when 0 sent– Detect 0 when 1 sent

• Selecting an Optimal Threshold

• For equally likely binary values

1e10e0error PHPPHPP

21HPHP 10

1e0eerror PP21P

1100 || HVpHPHVpHP optYoptY

10 || HVpHVp optYoptY

63


Figure 11.2-4

Threshold regions for conditional PDFs

2AVopt

21

10 HPHP

64

For AWGN• The pdf is Gaussian

for

for

2

2

2N0Y 2yexp

21ypH|yp

x

2

d2

exp21xQ

2AQVQdyypVYPP

VN0e

2

AQVAQdyAypVYPPV

N1e

2AVopt

21

10 HPHP

21 2 ee PAQVAQP

65

Modification for Polar NRZ Signals (+/- A/2)



– Hypothesis 1• The conditional probability distribution expected if a 1 was

sent

kYkkkYkY tApAatapHyp n

22|n| 0

kYkkkYkY tApAatapHyp n

22|n| 1

2| 0

AypHyp kNkY

2A-| 1 kNkY ypHyp

02A

2AVopt

66

Modification for Polar NRZ Signals (+/- A/2)

• Determining the error probability

• Notice that the error is the same as Unipolar NRZ– The distance between the expected signal values is the

same– The “distance” between the expected values determines

the error …

22

21AQ

VAQdyAypVYPP

V

Ne

22

20AQ

VAQdyAypVYPP

VNe

67

Modification for Bipolar NRZ Signals (+/- A)



– Hypothesis 1• The conditional probability distribution expected if a 1 was

sent

kYkkkYkY tApAatapHyp n|n| 0

kYkkkYkY tApAatapHyp n|n| 1

AypHyp kNkY 0|

A-| 1 kNkY ypHyp

0 AAVopt

68

Modification for Bipolar NRZ Signals

• Determining the error probability

• Notice that the error has been reduced– The distance between the expected signal values may

be twice as large as the unipolar case (using +/- A)

AQVAQdyAypVYPPV

N1e

AQVAQdyAypVYPPV

N0e

69

Relationship to signal power

• Defining the average received signal power– Unipolar NRZ

– Polar NRZ

– Bipolar NRZ

• In terms of SNR

AASR ,0,21 2

2,

2,

41 2 AAASR

Polarfor

NS

UnipolarforNS

21

N4A

2A

R

R

R

22

AAASR ,,2

BipolarforNS

NAA

RR

22

2

2

21limT

TcTR dttx

TES

Probability of error

• The probability of detecting a transmitted symbol correctly is dependent upon the received signal-to-noise ratio …. assuming– Unipolar NRZ (orthogonal)

– Polar NRZ (antipodal)

– Bipolar NRZ (antipodal)

70

Re N

SQAQP21

2

RR NS

NAA

42

22

RR NS

NAA

21

42

22

Re N

SQAQP2

Re N

SQAQP RR N

SNAA

22

21HPHP 10

Power Ratio 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.

71bbR T

ES 1 WNNR 0 ?

R

R

R NS

NS

72

SNR to Eb/No

• 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)

WR

NE

WT1

NE

WNT1E

NS b

0

b

b0

b

0

bb

WTNS

RW

NS

NE

bb0

b

73

Relationship to Eb/No

• Defining the energy per bit to noise power ratiofor a time-bandwidth product of

– Unipolar

– Polar

– Bipolar

0

b

RR

22

NE

NS

21

N4A

2A

0

b

RR

22

NE2

NS

N4A

2A

0

b

RR

22

NE2

NS

NAA

21T

2RTW b

bb

74

Relationship to Bit Error Probability

• Defining the binary bit error probabilityfor a time-bandwidth product, assuming

– Unipolar (orthogonal)

– Polar (antipodal)

– Bipolar (antipodal)

0

berror N

EQ2AQP

0

berror N

E2QAQP

0

berror N

E2Q2AQP

21HPHP 10

75

Bit Error Rate Plot

10-3 10-2 10-1 100 1010

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5B

it E

rror R

ate

Eb/No

Classical Bit Error Rates

OrthogonalAntipodal

EbNo=(0:10000)'/1000;

% Q(x)=0.5*erfc(x/sqrt(2))

Ortho=0.5*erfc(sqrt(EbNo)/sqrt(2));Antipodal=0.5*erfc(sqrt(2*EbNo)/sqrt(2));

semilogx(EbNo,[Ortho Antipodal])ylabel('Bit Error Rate')xlabel('Eb/No')title('Classical Bit Error Rates')legend('Orthogonal','Antipodal')

76

BER Performance, Classical Curveslog-log plot

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1510-7

10-6

10-5

10-4

10-3

10-2

10-1

100B

it E

rror R

ate

Eb/No


OrthogonalAntipodal

77

Antipodal and Orthogonal Signals

• Antipodal– Distance is twice “signal voltage”– Only works for one-dimensional signals

• Orthogonal– Orthogonal symbol set– Works for 2 to N dimensional signals

bE2d

jiforjifor

dttstsE

zT

jiij 111

0

jifor0jifor1

dttstsE1z

T

0jiijbE2d

78

M-ary Signals

• Symbol represents k bits at a time– Symbol selected based on k bits– M waveforms may be transmitted

• Allow for the tradeoff of error probability for bandwidth efficiency

• Orthogonality of k-bit symbols– Number of bits that agree=Number of bits that disagree

k2M

jifor0jifor1

K

bbsumbbsumz

N

1k

jk

ik

K

1k

jk

ik

ij

79

Example 11.2-1

• Unipolar computer network with

– Desired BER is one bit per hour

• Solve for the signal energy

bpsRb610 HzdBHzWN /194/104 20

0

1010336001

be RP

101032

AQPerror

UnipolarforNS

NAA

R

R

R

21

42

22

2.62

A

From p. 790

Rb

R SRNN

22.622.62 0

22

WWSR12620 1054.1105.010444.382

80

Exercise 11.2-1 (1)

• Unipolar system with equally likely digits and SNR = 50

• Calculate the error probabilities when the threshold is set to V=0.4 x A

UnipolarforNSA

R

R

21

2

2

1050212

A

VAQdyAypVYPPV

Ne1

VQdyypVYPPV

Ne0

81

Exercise 11.2-1

• Calculate the error probabilities when the threshold is set to V=0.4 x A

104.00

QVQVYPPe

106.01

QVAQVYPPe

50 105.30.4 QPe

91 105.10.6 QPe

21HPHP 10

1e0eerror PP21P

595 1075.1105.1105.321 errorP

V=0.5 x A

0.5105.010 QQPP ee

710 105.3 erroree PPP

ECE 6640 82-8 -6 -4 -2 0 2 4 6 8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Gaussian PDF and pdf

Gaussian Distribution

The Gaussian probability density function (pdf)

The Gaussian or Normal probability density function is defined as:

xforXxxf X ,

2exp

21

2

2

where X is the mean and is the variance

The Gaussian Probability Distribution Function (PDF)

dvXvxFx

vX

2

2

2exp

21

The PDF can not be represented in a closed form solution!

ECE 6640 83

Gaussian Distribution

The Gaussian Probability Distribution Function is

dvXvxFx

vX

2

2

2exp

21

The PDF can not be represented in a closed form solution!

The PDF is tabulated for a zero mean, unit variance pdf. For these values, it is often described as “normalized” and is defined as

duuxx

u

2

exp21 2

The distribution function is then defined as

XxxFX

When using Appendix D, the negative values in x are derived as

xx 1

84

Q FunctionAnother defined function that is related to the Gaussian (and used) is the Q-function.:

duuxQxu

2

exp21 2

The Q-function is the complement of the normal function, :

xxQ 1

Therefore not that:

xQxQ 1

XxQxFX 1

Q Function Table p. 858

ECE 6640 85

Using MATLAB

Another way to find values for the Gaussian

The error function

duuxerfx

u

0

2exp2

21

21 xerfxQ

22

121

21

211 XxerfXxerfxFX

The error function (Y = ERF(X)) is built-in to MATLAB. .

From MATLAB: ERF Error function. Y = ERF(X) is the error function for each element of X. X must be real. The error function is defined as: erf(x) = 2/sqrt(pi) * integral from 0 to x of exp(-t^2) dt. See also erfc, erfcx, erfinv. Reference page in Help browser doc erf

ECE 6640 86

Using MATLAB (2)

The complementary error function

xerf1xerfc

2

xerfc21xQ

The error function (Y = ERFC(X)) is built-in to MATLAB. .

From MATLAB: ERFC Complementary error function. Y = ERFC(X) is the complementary error function for each element of X. X must be real. The complementary error function is defined as: erfc(x) = 2/sqrt(pi) * integral from x to inf of exp(-t^2) dt. = 1 - erf(x). Class support for input X: float: double, single See also erf, erfcx, erfinv. Reference page in Help browser doc erfc

Qfn and Qfninv• These function are now in the Misc_Matlab zip file on the

web site

function [Qout]=Qfn(x)% Qfn(x) = 0.5 * erfc(x/sqrt(2));Qout = 0.5 * erfc(x/sqrt(2));

function [x]=Qfninv(Pe)% For Qfn(x) = 0.5 * erfc(x/sqrt(2));% The inverse can be found asx=sqrt(2)*erfcinv(2*Pe);

87

88

Properties of Matched Filter• See ECE3800 Notes

– Review from Chapter 9

• Wikipedia– http://en.wikipedia.org/wiki/Matched_filter– “The matched filter is the optimal linear filter for maximizing the

signal to noise ratio (SNR) in the presence of additive stochastic noise.”

Signal to Noise Ratio Definition

• The filtered response becomes

• The SNR

ECE 6640 89

0

dtntshtnts oo

EQo

o

o

o

Noise

Signalout BN

tsE

tnE

tsEPP

SNR

2

2

2

0

2

2

0

21 dtthN

dtshE

SNR

o

out

Optimized Matched Filter

• The matched filter is

• The resulting SNR

• Or for symbol energy

ECE 6640 90

uTsKh

o

ss

sso

ssT

o

T

out NR

RKN

TtRK

dtTsKN

dtsTsKE

SNR 02

021

21 2

22

0

2

2

0

o

sout N

ESNR 2

dffStsERss220

Autocorrelation versus Integration

• For a matched filter, the integral of the matched filter and the autocorrelation of the signal are approximately equivalent– For signals of limited time duration it can be exact!

• The condition is related to

• But notice that “integrating” and filtering may produce very different results except at t=T.– the integral is monotonically increasing– the filter is not!ECE 6640 91

Ttssss

T

ss

TtRR

dtsTsKTtR

00

Correlation Receivers

• The concept that a matched filter is performing an autocorrelation has resulted in the concept of the Correlation Receiver.

• The symbol of interest is auto-correlated. • All other symbols are cross-correlated.

• It is assumed that the two outputs are different enough to allow symbol detection– Noise power is present based on the “equivalent noise bandwidth”

of the “matched” symbol receivers.

ECE 6640 92

Corr. vs Int. for an RF/IF Envelope

ECE 6640 93

Question?

• Are there some better shapes than others for signals and their matched filters?– Easy to generate.– Frequency band limited. – Finite time duration.– Minimize inter-symbol interference.

• Start with rects or pulses• More advanced use raised cosine or Nyquist filter,

or for matched transmit and receive filters use “square root Nyquist filters”.

ECE 6640 94

95

Defining a Shape or Filter for PulsesSection 3.3

• We want to minimize or zero inter-symbol interference (ISI)

• We want a frequency band limited filter

– Allowable signal rates with as the excess bandwidth

k

dk Tkttpaty

,2,0

01TTt

ttp

fBfP 0

TBBandrwithrBwhere 2

0,2

BrBforBr 2,2

Tr 1

96

Defining a Filter for Pulses• Possible solutions

,2,0

01TTt

ttp

fBfP 0

trtptp sinc

fforfPtp 0

10

dffPp

• Therefore we select

rf

rfPfP rect1

These are considered the Nyquist conditions for the filter

Tr 1

97

Cosine Spectral Shaping

• A candidate filter is (with with as the excess BW)

2rect

42cos

4fffP

From Chap 2Raised cosine

pulse

Raised cosine pulse. (a) Waveform (b) Derivatives(c) Amplitude spectrumFigure 2.5-7

Convolving• Raised Cosine Convolution with Bandlimited Spectrum

98

rf

rfPfP rect1

TBBandrwithrBwhere 2

0,2

20

22242cos1

21

2

rf

rfrrfr

rfr

fP

trt

ttp

sinc412cos

2

• Transforming to the time domain filter

99

Nyquist/Raised Cosine Pulse Shaping

GNU FDL:Oli Filth, Raised Cosine Filter, Impulse Response, en.wikipedia.org, 3 November 2005, Oli Filth

GNU FDL:Oli Filth, Raised Cosine Filter Response , en.wikipedia.org, 3 November 2005, Oli Filth

Tr

rABC

12

Tr

rABC

12

http://en.wikipedia.org/wiki/Raised-cosine_filter

Nyquist Filter (discrete raised cosine)

100

trt

ttp

sinc412cos

2

% function hnyq=nyquistfilt(alpha,M)% or% function hnyq=nyquistfilt(alpha,fsymbol,fsample,k)%% alpha roll-off% fsample rate% fsymbol rate% M = fsample/fsymbol (an integer value)% k is 1/2 the number of symbols in the filter% The filter length is euqal to 2*ceil(k*M)+1%% A discrete time cosine taperd Nyquist filter% Based on frederic harris, Multirate Signal Processing for Communications% Prentice-Hall, PTR, 2004. p. 89

MknMkforMn

MnMn

np

,sinc21

cos

2

r 2

sfnt

10 2

0 r

Mfr s

trtrtrtp

sinc21

cos2

Mntr

MATLAB Raised Cosine Filters (1)• Rcosine (obsolete)

– [NUM, DEN] = RCOSINE(Fd, Fs, ‘fir’, R)– FIR raised cosine filter to filter a digital signal with the digital

transfer sampling frequency Fd. The filter sampling frequency is Fs. Fs/Fd must be a positive integer. R specifies the rolloff factor which is a real number in the range [0, 1].

• rcosfir (obsolete)– B = RCOSFIR(R, N_T, RATE, T)– Raised cosine FIR filter. T is the input signal sampling period, in

seconds. RATE is the oversampling rate for the filter (or the number of output samples per input sample). The rolloff factor, R, determines the width of the transition band. N_T is a scalar or a vector of length 2. If N_T is specified as a scalar, then the filter length is 2 * N_T + 1 input samples.

101

MATLAB Raised Cosine Filters (2)• firrcos (also obsolete)

– B=firrcos(N,Fc,DF,Fs)– Returns an order N low pass linear phase FIR filter with a raised

cosine transition band. The filter has cutoff frequency Fc, sampling frequency Fs and transition bandwidth DF (all in Hz).

– The order of the filter, N, must be even.– Fc +/- DF/2 must be in the range [0,Fs/2]– The coefficients of B are normalized so that the nominal passband

gain is always equal to one.– B=firrcos(N,Fc,R,Fs,'rolloff') interprets the third argument, R, as

the rolloff factor instead of as a transition bandwidth.– R must be in the range [0,1]

102

MATLAB Raised Cosine Filters (3)• firrcos

– B = rcosdesign(BETA, SPAN, SPS)– Returns square root raised cosine FIR filter coefficients, B, with a

rolloff factor of BETA. The filter is truncated to SPAN symbols and each symbol is represented by SPS samples. rcosdesigndesigns a symmetric filter. Therefore, the filter order, which is SPS*SPAN, must be even. The filter energy is one.

– Beta [0,1]– SPS number os samples per symbol– SPAN length of filter in number of symbols

103

Textbook Waveform Energy

• Waveform Energy

• Matched Filter

ECE 6640 104

T

ii dttsE0

2

t

dthrthtrtz

tTstuth *

t

dtTsstz0

*

TTT

dsdssdTTssTz0

2

0

*

0

*

Correlation

Optimum binary detection

105


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

Symbols and Matched Filters

2006-01-31 Lecture 3 106

T t T t T t0 2T

)()()( thtsty opti 2A)(tsi )(thopt

T t T t T t0 2T

)()()( thtsty opti 2A)(tsi )(thopt

T/2 3T/2T/2 TT/22

2A

TA

TA

TA

TA

TA

TA

Optimized Error Performance

• Maximize the “distance”, Ed, between the sampled values that are used for detection.– The distance is based on the correlation of the symbols

ECE 6640 107

T

jib dttstsE0

*

TT

ji

T

d dttsdttstsdttsE0

22

0

*

0

21 2

T

ib dttsE0

2

12 bbbbd EEEEE

T

d dttstsE0

221

Optimized Error Performance

• To maximize the distance: = -1– Symbols are said to be antipodal– Examples: +/- 1 Symbols (Bipolar), BPSK– Not always achievable

• Useful performance: = 0– Symbols are said to be orthogonal– Examples: On-Off Keying (ASK), FSK, “independent symbols”

ECE 6640 108

bd EE 2

bd EE

109

Antipodal and Orthogonal Signals

• Antipodal– Distance is twice “signal voltage”– Only works for one-dimensional signals

• Orthogonal– Orthogonal symbol set– Works for 2 to N dimensional signals

bE2d

jiforjifor

dttstsE

zT

jiij 111

0

jifor0jifor1

dttstsE1z

T

0jiijbE2d

110

Relationship to Bit Error Probability

• Defining the binary bit error probabilityfor a time-bandwidth product

– Orthogonal

– Antipodal

0NEQP b

error

0

2N

EQP berror

21HPHP 10

ECE 6640 111

Bit Error Rate Plot-Linear BER

10-3 10-2 10-1 100 1010

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5B

it E

rror R

ate

Eb/No


OrthogonalAntipodal

EbNo=(0:10000)'/1000;

% Q(x)=0.5*erfc(x/sqrt(2))

Ortho=0.5*erfc(sqrt(EbNo)/sqrt(2));Antipodal=0.5*erfc(sqrt(2*EbNo)/sqrt(2));

semilogx(EbNo,[Ortho Antipodal])ylabel('Bit Error Rate')xlabel('Eb/No')title('Classical Bit Error Rates')legend('Orthogonal','Antipodal')

ECE 6640 112

BER Performance Fig. 3.14

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1510-7

10-6

10-5

10-4

10-3

10-2

10-1

100B

it E

rror R

ate

Eb/No


OrthogonalAntipodal

0NEQP b

error

0

2N

EQP berror

Equalization

• An advanced topic that I skip at this time.

• Narrowband communications may go through a magnitude and phase change due to “the channel”

• Wideband communications likely experiences channel effects that may be non-linear across the signal frequency band. To correctly detect the information, an inverse channel filter or “equalizer” is used. – The channel is usually not predictable. Therefore, the equalizer

must “learn” from the transmitted signal how to correct for channel effects. There are “adaptive” algorithms that can do this and are taught in a more advanced course.

ECE 6640 113

ece 6640 digital communications

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