Outline

Outline• Transmitters (Chapters 3 and 4, Source Coding and

Modulation) (week 1 and 2)• Receivers (Chapter 5) (week 3 and 4) • Received Signal Synchronization

(Chapter 6) (week 5)• Channel Capacity (Chapter 7) (week 6)• Error Correction Codes (Chapter 8) (week 7 and 8)• Equalization (Bandwidth Constrained Channels) (Chapter

10) (week 9)• Adaptive Equalization (Chapter 11) (week 10 and 11)• Spread Spectrum (Chapter 13) (week 12)• Fading and multi path (Chapter 14) (week 12)

Channel Capacity (Chapter 7) (week 6)

• Discrete Memoryless Channels• Random Codes• Block Codes• Trellis Codes

Channel Models

• Discrete Memoryless Channel– Discrete-discrete

• Binary channel, M-ary channel– Discrete-continuous

• M-ary channel with soft-decision (analog)– Continuous-continuous

• Modulated waveform channels (QAM)

Discrete Memoryless Channel

• Discrete-discrete– Binary channel, M-ary channel

11

11

........)|(.....)|(

Qq

jiji

p

pxyPxXyYP

P

Probability transition matrix


• Discrete-continuous– M-ary channel with soft-decision (analog)

output x0

x1

x2

.

.

.xq-1

y

)|(

)|( 1

kxXyp

xXyp

P

22 2/)(

21)|(

kxy

k exyp

AWGN


• Continuous-continuous– Modulated waveform channels (QAM)– Assume Band limited waveforms, bandwidth = W

• Sampling at Nyquist = 2W sample/s– Then over interval of N = 2WT samples use an

orthogonal function expansion:

)(

)(

1

tfx

txN

iii

)(

)(

1

tfn

tnN

iii

N

iii tfy

ty

1

)(

)(


• Continuous-continuous– Using orthogonal function expansion:

)(

)(

1

tfx

txN

iii

)(

)(

1

tfn

tnN

iii

N

iiii

N

ii

T

i

N

ii

T

i

N

iii

tfnx

tfdttftntx

tfdttfty

tfy

ty

1

10

*

10

*

1

)(

)()()()(

)()()(

)(

)(


• Continuous-continuous– Using orthogonal function expansion get an

equivalent discrete time channel:

Nx

x

.

.

.

.1

Ny

yy

.

.

.2

1

22 2/)(

21)|( iii xy

iii exyp

111 nxy Gaussian noise

Capacity of binary symmetric channel

• BSC

pppp

11

P

}1,0{ }1,0{ YX

0 0

11

X Yp1

p1

p p


• Average Mutual Information 0 0

11

X Yp1

p1

p p

)1()1()0(1log)1)(1(

)1()0()1(log)1(

)1()1()0(log)0(

)1()0()1(1log)1)(0(

)1()1|1(log)1|1()1(

)0()1|0(log)1|0()1(

)1()0|1(log)0|1()0(

)0()0|0(log)0|0()0();(

XPpXpPppXP

XpPXPpppXP

XPpXpPppXP

XpPXPpppXP

YPXYPXYPXP

YPXYPXYPXP

YPXYPXYPXP

YPXYPXYPXPYXI


• Channel Capacity is Maximum Information– earlier showed:

0 0

11

X Yp1

p1

p p

pppp

XPpXpPppXP

XpPXPpppXP

XPpXpPppXP

XpPXPpppXPYXI

XPXP

2log)1(2log)1(

)1()1()0(1log)1)(1(

)1()0()1(log)1(

)1()1()0(log)0(

)1()0()1(1log)1)(0());(max(C

21)0()1(

21)0()1());(max( XPXPYXI

Capacity of binary symmetric channel• Channel Capacity

– When p=1 bits are inverted but information is perfect if invert them back!

0 0

11

X Yp1

p1

p p

pppp 2log)1(2log)1(C 22

Capacity of binary symmetric channel• Effect of SNR on Capacity

– Binary PAM signal (digital signal amplitude 2A)

0 0

11

X Yp1

p1

p p

)(tg

)(tg)(tg )(tg

)(tgA

A

02

02

/)(

02

/)(

01

1)|(

1)|(

Nr

Nr

b

b

eN

srp

eN

srp

AGWN

Capacity of binary symmetric channel• Effect of SNR


0 0

11

X Yp1

p1

p p

)(tg

)(tg)(tg )(tg

)(tgA

A

)|(2

1)|(

20

0 /)(

01

02

sePN

Q

dreN

seP

b

Nr b

0

1s



0 0

11

X Yp1

p1

p p

0

0

221

121

212

22

2

)|()|(

NAQ

QSNRQ

NQ

sePsePP

bb

b

b



0 0

11

X Yp1

p1

p p

noise rms22

422

212

2 noise rms

21

21

0

0

0

AQAQ

NAQ

NAQP

N

b

Not sure about thisDoes it depend on bandwidth?



0 0

11

X Yp1

p1

p p

noise rms 2Amplitudeerfc

21

noise rmsAmplitude

21

21QpPb

pppp 2log)1(2log)1(C



0 0

11

X Yp1

p1

p p

SNR Pb1 0.3085382 0.1586553 0.0668074 0.022755 0.006216 0.001357 0.0002338 3.17E-059 3.4E-06

10 2.87E-0711 1.9E-0812 9.87E-1013 4.02E-1114 1.28E-1215 3.19E-1416 6.11E-16

Pb (BER) vs SNR for binary channel

1.40E+01, 1.28E-12

1.20E+01, 9.87E-10

1E-16

1E-15

1E-14

1E-13

1E-12

1E-11

1E-10

1E-09

1E-08

1E-07

1E-06

1E-05

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

SNR = A/rms noise

BER


21

21

bP



0 0

11

X Yp1

p1

p p

pppp 2log)1(2log)1(C 22 SNR Pb C

0 0.5 01 0.308538 0.1085222 0.158655 0.3689173 0.066807 0.6461064 0.02275 0.8433855 0.00621 0.9455446 0.00135 0.9851857 0.000233 0.9968578 3.17E-05 0.9994819 3.4E-06 0.999933

10 2.87E-07 0.99999311 1.9E-08 0.99999912 9.87E-10 113 4.02E-11 114 1.28E-12 115 3.19E-14 116 6.11E-16 1

Capacity © and BER vs SNR for binary channel

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

SNR = A/rms noise

BER


21

21

bP

At capacity SNR = 7, so waste lots of SNR to get low BER!!!

Capacity of binary symmetric channel• Effect of SNRb


0 0

11

X Yp1

p1

p p

pppp 2log)1(2log)1(C 22 SNRb (dB)Pb C

-20 0.443769 0.009143-18 0.429346 0.014452-16 0.411325 0.022809-14 0.388906 0.03591-12 0.361207 0.056319-10 0.32736 0.087793-8 0.286715 0.135561-6 0.239229 0.206245-4 0.186114 0.306729-2 0.130645 0.4407970 0.07865 0.6025972 0.037506 0.7692614 0.012501 0.903056 0.002388 0.9757578 0.000191 0.997366

10 3.87E-06 0.99992512 9.01E-09 114 6.81E-13 1

Capacity C and BER vs SNR for binary channel

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-20 -12 -4 4 12

SNR per bit (dB)

Cap

acity

C a

nd B

ER

b

bb QpP

erfc21

2

Channel Capacity of Discrete Memoryless Channel

• Discrete-discrete– Binary channel, M-ary channel

11

11

........)|(.....)|(

Qq

jiji

p

pxyPxXyYP

P

Probability transition matrix


Average Mutual Information

1

0

1

1

1

0

1

1

)()|(

log)|()(

)()|(

log)|()();(

q

j

Q

i j

jijij

q

j

Q

i j

jijij

yPxyP

xyPxP

yYPxXyYP

xXyYPxXPYXI


Channel Capacity is Maximum InformationOccurs for only if Otherwise must work out max

1)( ,0)(

1

0

1

1)(

1

0

1

1)()(

1

0

)()|(

log)|()(max

)()|(

log)|()(max);(max

q

jjj

j

jj

xPxP

q

j

Q

i j

jijijxP

q

j

Q

i j

jijijxPxP

yPxyP

xyPxP

yYPxXyYP

xXyYPxXPYXIC

jpxP j allfor ,)( symmetric P

)( jxP

Channel Capacity Discrete Memoryless Channel

• Discrete-continuous• Channel Capacity

x0

x1

x2

.

.

.xq-1

y

)|(

)|( 1

kxXyp

xXyp

P

1

0

1

0)()(

)|()()(

where)(

)|(log)|()(max);(max

q

iii

q

i

iiixPxP

xXyYpxXPyYp

dyyYp

xXyYpxXyYpxXPYXICii


• Discrete-continuous• Channel Capacity with AWGN

x0

x1

x2

.

.

.xq-1

y

22 2/)(

21)|(

kxy

k exyp

1

01

0

2/)(

2/)(

2/)(

)( 22

22

22

21)(

21

log21)(max

q

iq

i

xyi

xy

xyixP

dyexXP

eexXPC

i

i

i

i


• Binary Symmetric PAM-continuous• Maximum Information when:

x0

x1

x2

.

.

.xq-1

y

21)()( AXPAXP

dyeee

e

dyeee

eC

yAA

A

yAA

A

22

2222

22

22

2222

22

2/2/2/

2/2

2/2/2/

2/2

21

2log

2log21


• Binary Symmetric PAM-continuous• Maximum Information when:

dyeee

e

dyeee

eC

yAA

A

yAA

A

22

2222

22

22

2222

22

2/2/2/

2/2

2/2/2/

2/2

21

2log

2log21


• Binary Symmetric PAM-continuous• Versus Binary Symmetric discrete SNRb (dB)Pb C

-20 0.443769 0.009143-18 0.429346 0.014452-16 0.411325 0.022809-14 0.388906 0.03591-12 0.361207 0.056319-10 0.32736 0.087793-8 0.286715 0.135561-6 0.239229 0.206245-4 0.186114 0.306729-2 0.130645 0.4407970 0.07865 0.6025972 0.037506 0.7692614 0.012501 0.903056 0.002388 0.9757578 0.000191 0.997366

10 3.87E-06 0.99992512 9.01E-09 114 6.81E-13 1

Capacity C and BER vs SNR for binary channel

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-20 -12 -4 4 12

SNR per bit (dB)

Cap

acity

C a

nd B

ER


• Continuous-continuous– Modulated waveform channels (QAM)– Assume Band limited waveforms, bandwidth = W

• Sampling at Nyquist = 2W sample/s– Then over interval of N = 2WT samples use an

orthogonal function expansion:

)(

)(

1

tfx

txN

iii

)(

)(

1

tfn

tnN

iii

N

iii tfy

ty

1

)(

)(


• Continuous-continuous– Using orthogonal function expansion get an

equivalent discrete time channel:

Nx

x

.

.

.

.1

Ny

yy

.

.

.2

1

22 2/)(

21)|( iii xy

iii exyp

111 nxy Gaussian noise


• Continuous-continuous• Capacity is (Shannon)

)(

)(

1

tfx

txM

iii

)(

)(

1

tfn

tnM

iii

M

iii tfy

ty

1

)(

)(

);(1maxlim)(

YXIT

CxpT

ii

N

i i

iijii

NNN

NNNNNNN

dxydyp

xypxpxyp

ddp

pppI

WTN

NN

1 )()|(log)()|(

)()|(log)()|();(

2

XY

yxyxyxxyYX

22 2/)(

21)|( iii xy

iii exyp


• Continuous-continuous• Maximum Information when:

0

2

0

2

1 0

2

21

)(

21log

21log21

21log);(max

NWT

NN

NI

x

x

N

i

xNNxp

YX

22 2/

21)( xix

xi exp

Statistically independent

zero mean Gaussian inputs

then


• Continuous-continuous• Constrain average power in x(t):

22

1

2

0

2

2

)(21

)]([1

xx

N

ii

T

av

WT

N

xE

dttxET

P


• Continuous-continuous• Thus Capacity is:

)(

)(

1

tfx

txM

iii

)(

)(

1

tfn

tnM

iii

M

iii tfy

ty

1

)(

)(

0

0

2

)(

1log

21loglim

);(1maxlim

WNPW

NW

IT

C

av

x

T

NNxpT

YX

22 2/)(

21)|( iii xy

iii exyp


• Continuous-continuous• Thus Normalized Capacity is:

)(

)(

1

tfx

txM

iii

)(

)(

1

tfn

tnM

iii

M

iii tfy

ty

1

)(

)(

WCN

WNC

CPWNP

WC

WCb

b

bavav

/12

1log

but ,1log

/

0

02

02

22 2/)(

21)|( iii xy

iii exyp

etab/No (dB)C/W-1.44036 0.1-1.36402 0.15-1.24869 0.225-1.07386 0.3375-0.8075 0.50625

-0.39875 0.7593750.234937 1.1390631.230848 1.7085942.822545 2.5628915.41099 3.844336

9.669259 5.76650416.65749 8.64975627.92605 12.9746345.69444 19.4619573.22669 29.19293115.4055 43.78939179.5542 65.68408 0.1

1

10

-10 0 10 20 30

Outline

Documents

signal synchronization

channel capacity chapter

modulation week

aat capacity snr

receivers chapter

adaptive equalization

multi path chapter

spread spectrum chapter